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Entanglement parity effects in quantum spin chains

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Entanglement parity effects in quantum spin chains

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Content ENTANGLEMENT PARITY EFFECTS IN QUANTUM SPIN CHAINS by Chunyu Tan A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) May 2024 Copyright 2024 Chunyu Tan Acknowledgements The Ph.D. journey is one of the most important journeys in my life. This journey would not have been possible without the support, guidance, and encouragement from many people, to whom I owe my deepest gratitude. Firstly, I want express profound gratitude to my advisor, Professor Stephan Haas. Stephan is a super kind and warm-hearted person. He has been a constant source of support, not just in my research but in my personal life and career development as well. His kindness, generosity, and insightful guidance have been pivotal to my growth as a researcher. I also want to give special thanks to my co-advisor Professor Hubert Saleur for his invaluable help to my research. His exceptional expertise and keen insights have significantly shaped my work, offering me the clarity and direction needed to finish my research projects. I want to thank my collaborators Henning Schloemer and Yuxiao Hang for their wonderful collaborations. Working with such talented and dedicated individuals is a very happy experience for me. Their help and suggestions are very valuable to me. Gratitude is also extended to the committee members of my defense and qualifying exam: Stephan Haas, Hubert Saleur, Moh El-Naggar, Aiichiro Nakano, Vitaly Kresin and Paolo Zanardi. Their insightful comments and suggestions help me a lot. Furthermore, I am grateful to all the teachers who have taught and guided me. I have learned a lot from them. My friends also deserve my heartfelt thanks for their unwavering support. ii Last but certainly not least, I must express my deepest gratitude to my parents. Their endless love, encouragement, and support have been my foundation throughout this journey. iii Table of Contents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2: Parity effects and universal terms of O(1) in the entanglement near a boundary . . . . 5 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Single impurity link: numerical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 Charge fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Single impurity link: analytical insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1 Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.2 Charge fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Extended impurities (towards the SSH model) . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.2 SSH model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Chapter 3: Entanglement parity effects in the Kane Fisher problem . . . . . . . . . . . . . . . . . . 38 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Physics around the split fixed-point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.2 The relevant and irrelevant case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Physics around the homogeneous fixed-point . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.2 Perturbative calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.1 Symmetries between µ and −µ, λ and −λ . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.2 Symmetries between λ and 1 λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Chapter 4: Entanglement parity effects in the disordered system . . . . . . . . . . . . . . . . . . . . 58 4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 Single Impurity and Quantum Dot Impurity . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Chapter 5: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 iv References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 v List of Figures 2.1 Entanglement entropy in a tight binding system with a single impurity link and open boundaries. ⃝a Illustration of the model. The bond placed between sites at ℓ and ℓ + a (corresponding to site indices j0 and j0 + 1, respectively) is modified, and has amplitude λJ. The total system size is set to be L = Zℓ. ⃝b Entanglement entropy of subsystem A (boxed region in ⃝a ) with B (non-circled region in ⃝a ) for λ = 1 (left) and λ = 0.8 (right) as a function of the total system size L = Zℓ, with Z fixed to Z = 10. For λ < 1, non-decaying parity effects of O(1) are observed, illustrated by the shaded region. The inset shows the predicted central charges ceff(λ) compared to the numerically extracted leading order behavior for varying impurity strength. ⃝c Even component c˜ ′e 1 (s) (upper panel, blue circles) and odd component c˜ ′o 1 (s) (upper panel, red circles) of the entanglement, the corresponding difference c˜ ′e 1 (s) − c˜ ′o 1 (s) (lower panel, black circles) and the isolated boundary parts c˜ ′e/o 1 (s) − c ′ 1 (s)/2 (lower panel, purple diamonds and green squares). c˜ ′e 1 (s), c˜ ′o 1 (s) and c ′ 1 (s)/2 are obtained by numerically fitting the data to Eq. (2.14) for open and Eq. (2.13) for periodic boundaries. . . . . . . . . . . . . . . . . . . . 9 2.2 Entanglement entropy difference δS between subsystem size even and odd when λ = 0.8. To simulate bulk effects, we apply periodic boundary conditions, and in particular choose Z = 10. The entanglement differences are seen to scale as ∼ 1/L, as depicted in the double logarithmic plot in the inset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Charge fluctuations in a tight binding system with a single impurity link and open boundaries. ⃝a Illustration of the model. An impurity is placed between sites at ℓ and ℓ + a (correpsonding to site indices j0 and j0 + 1, respectively), with impurity hopping λJ. The total system size is set to be L = Zℓ, where we choose Z = 10. ⃝b Fluctuations in subsystem A (circled region in ⃝a ) for λ = 1 (left) and λ = 0.8 (right). For λ < 1, non-decaying parity effects of O(1) are observed, illustrated by the gray background. The inset shows the predicted values of C(λ), cf. Eq. (2.20), compared to the numerically extracted leading order behavior for varying impurity strength. ⃝c Even component C˜′e 1 (upper panel, blue circles) and odd component C˜′o 1 (upper panel, red circles) of the charge fluctuations, the corresponding difference C˜′e 1 − C˜′o 1 (lower panel, black circles) and the isolated boundary parts C˜′e/o − C ′ 1 /2 (lower panel, purple diamonds and green squares) for varying λ. C˜′e 1 (λ), C˜′o 1 (λ) and C ′ 1 (λ)/2 are obtained by numerically fitting the data to Eq. (2.25) for open and Eq. (2.28) for periodic boundary conditions. . . . . . . . . . . . . . . 16 vi 2.4 ⃝a The slope of the entanglement entropy differences δS(λ) fitted in the window λ = (1 − ϵ). . . 1 and extrapolated to the limit ℓ, L → ∞ for ℓ = L/2 (black circles) and ℓ = L/3 (red squares). In the limit ϵ → 0, the slope is given by the analytically predicted value Eq. (2.52), here shown by horizontal dashed lines. ⃝b The same for fluctuation differences δF(λ), where the slope approaches the limit Eq. (2.75). . . . . . . . . . . . . . . 23 2.5 Parity effects of ⃝a the entanglement entropy and ⃝b the fluctuations when transitioning from the single impurity link to an “SSH-like” impurity with a collection of alternating bonds J, λJ. Upon increasing the number of impurities Nimp = 1, 3, 5, 7 with all λ ⟨2n,2n+1⟩ = λ, the differences quickly saturate to ± ln 2 (upper panel ⃝a ) and ±1/4 (upper panel ⃝b ). This can be understood in terms of an effective, stronger single impurity with strength λ Nimp . When rescaling δS and δF accordingly, the curves become equivalent (lower panels). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 Comparing two different ways of defining the fluctuation differences between the parities. (i) We both change the subsystem’s parity as well as the topological phase, i.e., in the language of impurities, we “move” the impurity together with the subsystem’s border (see the blue curve). Though locally the systems look the same at the subsystem’s border, the edge modes appearing in the topologically non-trivial phase lead to a 1/4 contribution for all λ. (ii) While changing the parity of the subsystem, the geometry of the underlying model is fixed. Here, only local differences are seen, leading to a functional dependence different from the step function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.7 Modes |Ψ(x)| 2 in vicinity of zero energy for various tight-binding models. Values of |Ψ(x)| 2 are marked in an alternating fashion by red and blue dots for A and B sublattices. Each model is illustrated in the center of the left panel, with strong (weak) bonds marked by black (red) lines. Right panels show the energy spectra close to zero energy, with the two symmetric modes closest to ϵ = 0 marked in blue. ⃝a The pure SSH chain features localized zero energy modes, having support only on sublattice A (B) for the left (right) edge mode. Here, λ = 0.8. ⃝b An SSH impurity of length 20 attached to two leads of size 30 on either side, again with λ = 0.8. The long SSH impurity polarizes the wave functions close to zero energy, having support only on the A (B) sublattice on the left (right) to the extended impurity. ⃝c A single impurity bond with 40 metallic sites to each left and right side for λ = 0.8. ⃝d The same as ⃝c but with λ = 0.1. . . . . . . . . . . . . . . . . . . . . . 35 3.1 The different types of flows near the split fixed-point . . . . . . . . . . . . . . . . . . . . . 41 3.2 Crossover in the bulk entanglement entropy for Hamiltonian 3.4. . . . . . . . . . . . . . . 42 3.3 Healing flow of entanglement difference for Jz = −0.5. Here Z = 2 and the Hamiltonian is (3.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 Healing flow of entanglement difference for Jz = −0.5. Here Z = 2 and the Hamiltonians (3.4), (3.9) are compared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5 Flow of entanglement difference for Jz = 0.5. Here Z = 2 and the Hamiltonian is (3.9). The split fixed-point is recovered in the IR, while the healed fixed-point is not reached in the UV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 vii 3.6 The different types of flows near the homogeneous fixed-point . . . . . . . . . . . . . . . . 46 3.7 Flow for Jz = 0.5; Here, Z = 2. The weakly perturbed chain flow to the split fixed-point in the IR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.8 (a) Exponent in (3.28) (b) Exponent 1 − g −1 in (3.29) . . . . . . . . . . . . . . . . . . . . . . 53 3.9 (a) Example of fit. (b)Study of the slope of δS near the homogeneous fixed point for Hamiltonian(3.10). (c)Study of the slope of δS near the homogeneous fixed point for Hamiltonian (3.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.10 Jz=0.5 δS(µ) and -δS(−µ) comparison (a) Comparison in large range of LΘB. (b) Detailed comparison in the scaling limit where µ is very small. . . . . . . . . . . . . . . . 54 3.11 Jz=-0.5 δS(λ) and δS(−λ) comparison (a) Comparison in large range of LTB. (b) Detail comparison in the scaling limit where λ is very small. . . . . . . . . . . . . . . . . . . . . . 54 3.12 Jz=-0.5, δS(λ) and -δS( 1 λ ) comparison, -δS( 1 λ ) curve shifted according to 1 1+∆ . (a) Comparison in large range of LTB. (b) Detail comparison in the scaling limit where λ is very small. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.13 Jz=-0.5, δS(λ) and -δS( 1 λ ) comparison, with Hamiltonian 3.9(a) Comparison in large range of LTB. (b) Detail comparison in the scaling limit where λ is very small. . . . . . . . 57 4.1 Illustration of one RG step, we find the largest bond Jn and assume that a singlet form between the associated spins and we take that term out of the Hamiltonian and introduce an effective new bond instead. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Numerical RG calculation of the scaling function of entanglement entropy and theoretical formula. (a) The periodic case. (b) The open boundary odd case. (c) The open boundary even case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3 entanglement difference between the even and odd . . . . . . . . . . . . . . . . . . . . . . 65 4.4 Numerical RG calculation of the scaling function of entanglement entropy and theoretical formula. (a) The periodic case. (b) The open boundary odd case. (c) The open boundary even case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.1 Comparison of the large ℓ expansion Eq. (.11) with exact numerical evaluation of Eq. (.8). The inset shows that errors are of order O ℓ −2 . . . . . . . . . . . . . . . . . . . . . . . . 75 vii Abstract In the presence of boundaries, the entanglement entropy in lattice models is known to exhibit oscillations with the (parity of the) length of the subsystem, which however decay to zero with increasing distance from the edge. We point out in this dissertation that, when the subsystem starts at the boundary and ends at an impurity, oscillations of the entanglement (as well as of charge fluctuations) appear which do not decay with distance, and which exhibit universal features.We conduct a detailed study of parity effects across a range of spin chains, investigating both systems that exhibit a renormalization group (RG) flow and those that do not. We start with the XX chain with one modified link (a conformal defect), a case where the system does not exhibit RG flow, we study parity effects both numerically and analytically. We then generalize our analysis to the case of extended (conformal) impurities, which we interpret as SSH models coupled to metallic leads. In this context, the parity effects can be interpreted in terms of the existence of non-trivial topological phases. Subsequently, we investigate parity effects on entanglement in systems exhibiting a RG flow. Specifically, we focus on the XXZ chain with a single impurity, a scenario known as the Kane-Fisher problem. We find that the difference of entanglement depend on ℓTB, where TB is a characteristic energy scale akin to the Kondo temperature in Kondo problem. Finally we study the parity effects when strong disorder exists. ix Chapter 1 Introduction Quantum impurity problems have been an interest area in condensed matter physics for many years, a wide range of systems with quantum impurities has been studied. The existing of quantum impurities can arise a lot of fascinating physical phenomena. A notable example is the Kondo effect, due to magnetic impurities, normal metals have an unusual minimum in electrical resistivity with temperature. In Kondo problem, there is a crossover energy scale named Kondo temperature TK, many physical properties are governed by Kondo temperature TK and could be expressed as universal functions of T /TK [24, 19]. The exploration of the Kondo effect has significantly propelled the development of the renormalization group method, a cornerstone in contemporary theoretical physics. Similar crossover scales, serving roles akin to the Kondo temperature, are found in other quantum impurity systems. For instance, the Kane-Fisher problem, which concerns a weak link in a Luttinger liquid. When the interactions are repulsive, electrons are completely reflected by even the smallest scatterer as energy is swept across the crossover scale TB, leading to two separated parts. On the other hand when the interactions are attractive, electrons could perfectly transmit through the strongest scatterer in the scaling limit[21]. The tunneling in the fractional quantum Hall effect is another example of quantum impurity problems, some experiments utilized a Quantum Point Contact method where two leads are attached to a quantum impurity, they directly observed the existence of a fractional charge[31, 37]. 1 Quantum entanglement is at the heart of quantum mechanics, its diverse theoretical interest covering topics from quantum information science [28] to condensed matter [3, 7] and high energy physics [6, 40]. A important measure of bipartite entanglement is the von Neumann entanglement entropy. Let ρ be the density matrix of the total system and the system can be divided into two regions A and B. The reduced density matrix of A could be obtained by tracing out region B, ρA = TrB ρ. The corresponding von Neumann entanglement entropy is, SA = TrAρA ln ρA (1.1) Similar we can have SB and when the total system is in a pure quantum state, SA = SB. Another measure, the Rényi entropy, is defined as, S (n) A = 1 1 − n ln Tr ρ n A (1.2) Rényi entropy S (n) A is a generalization of von Neumann entropy SA with SA = limn→1 S (n) A . In many systems the von Neumann entanglement entropy S would depend only on the surface of separation between the two regions A and B. For a d-dimensional system S ∼ ℓ d−1 , where ℓ is of the order of the size of one of the block. This is known as the area law, while in some critical systems there are logarithmic corrections to the area law[41, 16]. The von Neumann entanglement entropy could be used to detect topological order in the system. Specifically, consider a disk in the plane, with a smooth boundary of length L, the entanglement entropy would scale as S = αℓ − γ + O ℓ −1 . The coefficient α is non universal and ultraviolet divergent, but the O(1) term γ is a universal additive constant characterizing a global topological properties of the system. This constant O(1) term γ = log D is called topological entanglement entropy, where D is the total quantum dimension[22, 26]. For 1 + 1 dimensional conformal field theories (CFTs), it is possible to derive universal predictions for the entanglement entropy (see e.g. [5] for a review) SA = c 3 ln ℓ a + c ′ 1 , (1.3) where c is the central charge of the CFT, A is a single interval of length ℓ, a is a UV cutoff, and we dropped terms which vanish as a → 0. While the logarithmically divergent term is governed only by the central charge c of the CFT, the constant term, c ′ 1 , is non-universal, and can be changed by a redefinition of the cutoff. Boundaries or impurities may alter in a significant way the the entanglement entropy. In the presence of a boundary, however, the constant term becomes more relevant, as the dependence on the UV cutoff can be eliminated. If A denotes again a single interval of length ℓ, now starting at the boundary of a semi-infinite system, one finds [5] SA,bdr = c 6 ln 2ℓ a + ˜c ′ 1 , (1.4) with the same cut-off a as in Eq. (1.3), along with c˜ ′ 1 − c ′ 1 2 = ln g, (1.5) where ln g is the Affleck-Ludwig boundary entropy [2]. Note the factor of 2 in the argument of the logarithm in Eq. (1.4), first pointed out in [47]. Many results related to the impurity and boundary contribution to the entanglement are reviewed in [1]. Terms of O(1) are also physically meaningful for CFTs with topological defects [30, 17, 35, 4, 8]. The case where such defects are inside the interval is well understood, and closely related to the boundary case 3 (via folding). The case where the topological defect is at the border of the interval1 is more controversial [34, 33]. Here, we report on yet other situations where universal terms of O(1) do in fact appear, with interesting physical interpretations. In this dissertation, we study entanglement entropy when combining quantum impurities and boundaries. We find that the O(1) term in entanglement entropy exhibits strong parity effects, it differs between the even subsystem size case and odd subsystem size case. The corresponding difference of O(1) term δS is universal and shows many interesting properties. The dissertation is organised as following, in chapter 2, we start with the non-interacting XX spin chain, we study the parity effects in entanglement entropy and charge fluctuations for the single impurity case. We also discuss extended impurities and the connection with SSH model and topological phase. In chapter 3, we study the entanglement entropy of the XXZ chain with impurities, where the anisotropic interaction is considered. When interaction exists the system is similar to Kane-Fisher problem, it has a RG flow towards to a uniform chain or to two separate chains depends on the sign of the interaction. In chapter 4, we study the XX spin chain with strong disorder, the disorder could induce a RG flow and drive the system to the random singlet phase. We find parity effects in entanglement still exists within strong disorder. Finally we give our conclusions in chapter 5 1To avoid confusion, we use the words boundary and edge to refer to physical boundaries, and the word border to refer to the extremity or extremities of the sub-systems. 4 Chapter 2 Parity effects and universal terms of O(1) in the entanglement near a boundary 2.1 Background We start with the non-interacting XX spin chain, which could be mapped to a free fermion chain by using Jordan–Wigner transformation. When combining a conformal defect with a boundary1 , we find universal terms of O(1) in entanglement entropy with very interesting properties. Specifically, we consider an XX chain with a modified bond (sometimes also referred to as impurity bond) of value J ′ (the value of the bulk bonds is J). In the bulk, the entanglement of a region of length ℓ ending on the modified bond is known to take the form [11] SA,imp ≈ (ceff(λ) + c) 6 ln ℓ a + (c ′ 1 (λ) + c ′ 1 ) 2 , (2.1) where we have set λ ≡ J ′ J . (2.2) In Eq. (2.1), ceff(λ) is an “effective central charge” whose expression is known analytically, interpolating between ceff(0) = 0 and ceff(1) = c. The term of O(1) also depends on λ, with c ′ 1 (λ = 1) = c ′ 1 . There are no parity effects at large distances. 1The case of topological defects in the presence of boundaries can be handled by conformal methods [15]. 5 In this article, we combine a conformal defect with a boundary by considering the XX chain with free boundary conditions, and choosing the interval of length ℓ so that it starts at the boundary and ends on the modified bond. As discussed below, we find that the entanglement entropy now exhibits strong parity effects 2 , and that one can write3 S e A,imp+bdr = ceff(λ) 6 ln 2ℓ a + ˜c ′e 1 (λ), S o A,imp+bdr = ceff(λ) 6 ln 2ℓ a + ˜c ′o 1 (λ). (2.3) where superscripts e and o denote the cases where the size of subsystem A is even and odd, respectively. The surprising observation is that the parity effects in Eq. (2.3) do not decay with ℓ but remain of order one. In particular, We claim that c˜ ′e 1 (λ) − c˜ ′o 1 (λ) ≡ δS(λ) is a universal, ℓ independent term. Here, by universal we mean that this term does not depend on the cutoff (in contrast with e.g. c ′ 1 ), and can be calculated using field theory with only a few renormalized couplings - here, in fact, the phase shift at the Fermi surface. While parity effects for the entanglement entropy are well known to occur in the presence of a boundary4 [25, 1], these effects, without an impurity, decay with large ℓ (like ℓ −1 for the XX chain5 [12]). 2A different but somewhat related observation was made earlier in [10]. 3Here - as in all similar statements below - our results apply to the limits ℓ, L → ∞ (with ℓ L fixed), that is, in the scaling limit. Most results would not hold in finite size. 4 Parity effects are known to occur in the bulk as well for general Rényi entropies. 5The entropy for the XXZ chain exhibits strong, oscillating corrections to the conformal result, decaying with an amplitude ℓ −K, where K is the Luttinger liquid exponent (the dimension of the most relevant operator in the bulk theory). The XX case has been studied in considerable detail [12], with the result that SA,bdr = 1 12 ln ℓ + O(1) + f1 sin[(2ℓ + 1)kF ]ℓ −1 + o(ℓ −1 ) Here, f1 is a non-universal constant, kF is the Fermi momentum, so for kF = π 2 (no magnetic field) we get the usual oscillating term (−1)ℓ . 6 It has often been observed in the past that charge fluctuations exhibit features qualitatively similar to those of the entanglement [39, 23], while being easier to handle technically and more realistic to measure in an experiment. As we will see below, this is still the case in our problem, where the qualitative physics of universal parity effects of O(1) is the same as for the entanglement entropy. This chapter is organized as follows. In section 2.2, we analyze the XX chain with a single modified bond, and establish strong evidence for the aforementioned conjectures. In particular, we study the vonNeumann entropy in 2.2.1 and the charge fluctuations in 2.2.2. In Section 2.3, we present analytical insights into the problem in the continuum limit and obtain perturbative results in the regime λ ≲ 1. Finally, we briefly study extended conformal impurities with several alternating modified bonds in Sec. 2.4, yielding interesting physical insights into the single impurity model in terms of topological phases of the SSH model. 2.2 Single impurity link: numerical study We start by analyzing non-decaying parity effects for both the entanglement entropy and charge fluctuations in models with a single conformal defect located at the border of the subsystem, as introduced in Sec. 2.1. 2.2.1 Entanglement entropy To our knowledge, the function ceff(λ) in Eq. (2.1) has been determined analytically only in the bulk case of an XX chain with impurity bond, with the result [11, 35, 4] ceff = − 6 π 2 n [(1 + s) ln(1 + s) + (1 − s) ln(1 − s)] ln s + (1 + s)Li2(−s) + (1 − s)Li2(s) o . (2.4) 7 Here, Li2(z) = − Z z 0 dxln(1 − x) x , (2.5) and we have introduced the variable s = sin(2 arctanλ) = 2JJ′ J 2 + (J ′) 2 . (2.6) We recall that J ′ is the modified link, and λ = J ′ J . With the identity Li2(s) = π 2 6 − Li2(1 − s) − ln s ln(1 − s), (2.7) we can rewrite Eq. (2.4) as ceff = s − 1 − 6 π 2 n [(1 + s) ln(1 + s) ln s + (1 + s)Li2(−s) + (s − 1)Li2(1 − s) o , (2.8) an expression which is also often found in the literature. We note that ceff is rather well approximated by ceff ≈ s 2 . Furthermore, from Eq. (2.6), it is evident that the variable s obeys the symmetry s(λ) = s 1 λ , (2.9) and therefore, ceff being analytical in s, we have ceff(λ) = ceff 1 λ . (2.10) 8 < 1 =1 =0.8 t t t t ... ... 1 ` c e ↵ leading order Fluctuations summarizing plot theory numerics L = Z` L S ( L ) ⇠ ce↵() 6 ln(L) S J J J J J constant terms 1/ < 1 s = sin(2 arctan()) - c - - 0 1/2 c0 c˜ 1/2 0e 1 c˜ 0o 1 c˜ 0e 1 c˜ 0o 1 c˜ 0o 1 c˜ 0e 1 ` + a Figure 2.1: Entanglement entropy in a tight binding system with a single impurity link and open boundaries. ⃝a Illustration of the model. The bond placed between sites at ℓ and ℓ + a (corresponding to site indices j0 and j0 + 1, respectively) is modified, and has amplitude λJ. The total system size is set to be L = Zℓ. ⃝b Entanglement entropy of subsystem A (boxed region in ⃝a ) with B (non-circled region in ⃝a ) for λ = 1 (left) and λ = 0.8 (right) as a function of the total system size L = Zℓ, with Z fixed to Z = 10. For λ < 1, non-decaying parity effects of O(1) are observed, illustrated by the shaded region. The inset shows the predicted central charges ceff(λ) compared to the numerically extracted leading order behavior for varying impurity strength. ⃝c Even component c˜ ′e 1 (s) (upper panel, blue circles) and odd component c˜ ′o 1 (s) (upper panel, red circles) of the entanglement, the corresponding difference c˜ ′e 1 (s) − c˜ ′o 1 (s) (lower panel, black circles) and the isolated boundary parts c˜ ′e/o 1 (s) − c ′ 1 (s)/2 (lower panel, purple diamonds and green squares). c˜ ′e 1 (s), c˜ ′o 1 (s) and c ′ 1 (s)/2 are obtained by numerically fitting the data to Eq. (2.14) for open and Eq. (2.13) for periodic boundaries. We also note (see Sec. 2.3.2 for details) that the variable s has a physical meaning: s = cos ξ, where ξ is the phase shift at the Fermi surface6 , |ξ(λ)| ≡ π 2 − 2 arctanλ, λ ∈ [0, 1]. (2.11) It is known that ceff depends only on this phase shift, and not on the microscopic mechanism (such as the exact realization of the “impurity”) giving rise to it, see e.g. [46]. In the following, we consider a non-interacting, one-dimensional chain of hopping electrons. Fig. 2.1 ⃝a for an illustration. Specifically, we consider a system of size L (with N = L/a sites and a the lattice spacing), which is fixed by the size of subsystem A, i.e., ℓ, via L = Zℓ, where Z ∈ N. We introduce a 6The sign of ξ is discussed below. 9 single impurity link of strength J ′ = λJ at the border of subsystem A at site j0 = ℓ/a. The corresponding Hamiltonian is given by H = −J X N j=1 c † j+1cj + c † j cj+1 − J(λ − 1) c † j0+1cj0 + c † j0 cj0+1 . (2.12) By first using periodic boundary conditions, we check that the entanglement is given by the natural generalization of Eq. (2.1) for finite sizes, i.e., SA,imp ≈ (ceff(λ) + c) 6 ln L πa sin πℓ L + (c ′ 1 (λ) + c ′ 1 ) 2 , (2.13) where we evaluate the entanglement entropy via the correlation matrix of the tight-binding model [29]. Moreover, we verify that the parity effects decay with increasing size, as expected in a system with closed boundaries. This is true more precisely when ℓ, L become large, with fixed ratio Z = L/ℓ (including the case ℓ ≪ L). In the following, we fix Z = 10, i.e., the size of subsystem A is 10% of the total system size, with the modified bond located at the right border. An example of the decay of oscillatory behavior is illustrated in Fig. 2.2. The difference of terms of O(1) between the even and odd cases, denoted by δS, is found to vanish as ℓ −1 (see inset of Fig. 2.2), in agreement with standard predictions for the XX chain. Fluctuations summarizing plot L L =0.8 S ⇠ 1/L | S| Figure 2.2: Entanglement entropy difference δS between subsystem size even and odd when λ = 0.8. To simulate bulk effects, we apply periodic boundary conditions, and in particular choose Z = 10. The entanglement differences are seen to scale as ∼ 1/L, as depicted in the double logarithmic plot in the inset. 10 Near a boundary, the slope of the entanglement is divided in half, just like for ordinary, homogeneous systems. We note that there doesn’t seem to be a simple derivation of this result in the literature, in particular when finite size effects are taken into account. We have checked numerically that Eq. (2.3) gets modified as announced in the introduction, i.e., S e A,imp+bdr = ceff(λ) 6 ln 2L πa sin πℓ L + Ae (λ) ℓ + ˜c ′e 1 (λ) S o A,imp+bdr = ceff(λ) 6 ln 2L πa sin πℓ L + Ao (λ) ℓ + ˜c ′o 1 (λ), (2.14) where we explicitly take into account the leading order (λ dependent) decaying terms ∼ ℓ −1 for a better quality of the numerical fits. Numerical results for ceff are presented in the inset of Fig. 2.1 ⃝b , in excellent agreement with the predicted value, Eq. (2.4). The crucial point is now that the parity effects do not decay with L. This is illustrated in detail in Fig. 2.1 ⃝b , where the entanglement entropy is shown when increasing L for fixed Z = 10. In particular, the left panel shows the case λ = 1, in which case oscillations decay as ∼ L −1 . However, when λ ̸= 1, finite oscillations persist and saturate towards the value δS(λ) for L → ∞, illustrated by the grey shaded region in Fig. 2.1 ⃝b . We numerically evaluate the entanglement entropies for varying impurity strengths λ (where we allow both λ < 1 and λ > 1), and extract the values of c˜ ′e/o 1 (λ) by fitting the data to Eq. (2.14). The upper panel of Fig. 2.1 ⃝c shows the even and odd contributions c˜ ′e/o 1 (λ) as a function of s = sin(2 arctan(λ)), featuring strong and weak dependencies on the tunable impurity strength in the even and odd case, respectively. Remarkably, the difference between these two cases, δS(λ) = ˜c ′e 1 (λ) − c˜ ′o 1 (λ) (see the black curve in the lower panel of Fig. 2.1 ⃝c ), is a seemingly simple function of s, exhibiting the symmetry δS(s) = −δS(−s), (2.15) 11 or, equivalently, δS(λ) = −δS 1 λ . (2.16) Like the effective central charge ceff, we expect δS to be a function of the phase shift at the Fermi surface only. Our curves δS(s) should thus be the same for every similar problem once s = cos ξ is being used. Examples of this will be given below, in particular when we extend to impurities that include several modified bonds in Sec. 2.4. We further note that δS(s)seems to be well approximated by a simple ellipse with semi-major (-minor) axis 1 (ln 2), δS ≈ ± ln 2p 1 − s 2, (2.17) which, however, turns out to be not exact - a fact that can be proven analytically, as discussed in Sec. 2.3.1. In analogy to Eq. (1.5), we further isolate the boundary contribution of the entanglement entropy, c˜ ′e/o bdry(λ) = ˜c ′e/o(λ) − c ′ 1 (λ) 2 , (2.18) where we compute c ′ 1 (λ) by fitting the numerical outcome of an impurity system with periodic boundary conditions to Eq. (2.13). The boundary contributions c˜ ′e/o bdry(λ) are observed to lie on the same curve, see the lower panel of Fig. 2.1 ⃝c - with the subtle distinction that c˜ ′p bdry(λ) = ˜c ′p¯ bdry(1/λ), (2.19) where o¯ = e, e¯ = o. Indeed, from Eq. (2.19), the (anti)-symmetry of δS(λ) in Eq. (2.16) immediately follows. The limiting cases of strong and weak modified bonds can be intuitively interpreted in a simplified valence bond picture [1], the different scenarios in the even and odd cases depicted in Tab. 2.1. For λ ≪ 1, 12 Table 2.1: Simplified valence bond picture for strong (weak) impurity bonds λ ≪ 1 (1/λ ≪ 1) for both even and odd scenarios. The bonds separating the regions of the bipartition are illustrated by grey lines, and inter- (intra-) subsystems valence bonds are depicted by black (dark blue) curved lines. Each blue line corresponds to an entanglement entropy of ln 2. ⌧ 1 1/ ⌧ 1 even odd it is unfavorable to form a valence bond over the impurity. This, in turn, leads to an additional contribution to the entanglement entropy between the two subsystems on the left and right of the impurity of ln 2 in the odd case compared to the even case. For a strong impurity bond, 1/λ ≪ 1, on the other hand, it is highly favorable to form a valence bond over the impurity. Focusing first on the even case, this leaves an odd number of sites on both sides, that have to be “assigned” a valence bond partner, leading to an extra valence bond between the two subsystems - and hence an additional ln 2 entanglement entropy compared to the odd case. Note that the valence bond picture does not represent the exact ground state of the impurity chain, however it provides an intuitive understanding of the observed entanglement entropy differences in the limits of weak and strong impurities, cf. Fig. 2.1 ⃝c . 2.2.2 Charge fluctuations As mentioned in the introduction, fluctuations usually behave in a way that is qualitatively similar to entanglement [39]. Although there is, to our knowledge, no precise, universal version of this statement, many examples have been found, especially in the non-interacting case. As we will show, our system is no exception. We thus consider the same model as before, but now instead of the entanglement of the subsystem situated between the boundary and the impurity, we consider the fluctuations of the number of electrons it contains. 13 First, with no impurity and in the bulk, it is well known that the charge fluctuations in the noninteracting case - up to decaying, non-universal parity terms - take the form ⟨(Q − ⟨Q⟩) 2 ⟩A = FA(ℓ) = C π 2 ln ℓ a + C ′ 1 , (2.20) where the coefficient C takes the value C = 1, while C ′ 1 is non-universal, and changes as usual under cutoff re-definitions7 . We emphasize that C is not related with the central charge: with interactions of Luttinger type for instance, C would vary while c = 1 is a constant. In the presence of an impurity and in the bulk, one can also show (see Sec. 2.3.2) that FA,imp = Ceff(λ) + 1 2π 2 ln ℓ a + (C ′ 1 (λ) + C ′ 1 ) 2 , (2.21) with Ceff = cos2 ξ = s 2 . (2.22) Finally, near a boundary but without impurity, one has [39] FA,bdr = 1 2π 2 ln 2ℓ a + C ′ 1 2 . (2.23) Note the similarity of all these results with those for the entanglement entropy. This carries over to finite size effects as well - for instance, equation (2.23) when the system has total length L becomes FA,bdr = 1 2π 2 ln 2L πa sin πℓ L + C ′ 1 2 . (2.24) 7 For the XX chain, and the cutoff given by the lattice spacing, one can show that C ′ 1 = 1+γ+ln 2 π2 , where γ is Euler’s constant [39]. 14 Parity effects are also known to occur for the fluctuations. Like in the case of the entanglement, they decay with ℓ large8 . We shall see now that, in the presence of both the impurity and the boundary, we have9 F e A,imp+bdr = Ceff(λ) 2π 2 ln 2L πa sin πℓ L + Ae (λ) ℓ + B e (λ) ln ℓ ℓ + C˜′e 1 (λ) F o A,imp+bdr = Ceff(λ) 2π 2 ln 2L πa sin πℓ L + Ao (λ) ℓ + B o (λ) ln ℓ ℓ + C˜′o 1 (λ). (2.25) For the single impurity link model, we numerically evaluate the correlator Gij = ⟨c † i cj ⟩, from which the density-density correlations can be computed, ⟨c † i ci c † j cj ⟩ = ⟨c † i ci⟩ ⟨c † j cj ⟩ + ⟨c † i cj ⟩ ⟨ci c † j ⟩ = ⟨c † i ci ⟩ ⟨c † j cj ⟩ + ⟨c † i cj ⟩ δij − ⟨c † j ci ⟩ . (2.26) The charge fluctuations in a given interval of length ℓ are then evaluated via F(ℓ) = X ℓ i,j=1 ⟨ninj ⟩ − ⟨ni⟩ ⟨nj ⟩ = X ℓ i,j=1 ⟨c † i cj ⟩ δij − ⟨c † j ci ⟩ . (2.27) Note that the open boundary, non-interacting model with an impurity can also be solved semi-analytically using a plane-wave ansatz that scatters at the impurity. The plane wave amplitudes as well as the condition for the momentum quantization can then be solved numerically using the scattering ansatz together with the boundary conditions, from which the fluctuations can be computed. In our numerical results however, we stick to the same strategy as used for evaluating the entanglement entropy, that is we use numerical diagonalization of the tight-binding matrix, compute the fluctuations and fit the data to Eq. (2.25). In 8 For the XX model for instance, the leading decaying terms for a homogeneous system with a boundary read FA,bdr = 1 2π2 ln 2 ˜ℓ a + C ′ 1 2 + 1 2π2(2˜ℓ) − (−1)ℓ π2(2˜ℓ) h ln 2˜ℓ + γ + ln 2i + O(ℓ −2 ), where ˜ℓ = (L/aπ) sin πℓ/L. Note the similarity with the result for the entanglement entropy, however with the additional oscillating contribution ∝ (−1)ℓ ln ˜ℓ/˜ℓ. This additional oscillating term has previously been analyzed for Luttinger liquids (LL), where amplification (suppression) of the decaying parity effects have been observed for LL parameters K < 1 (K ≥ 1) [39]. 9The coefficients A o , Ae are not the same as those appearing in the entanglement. We kept the same notation for simplicity. F ( ` ) < 1 1/ < 1 Fluctuations summarizing plot theory numerics F leading order t t t t ... ... 1 ` L = Z` J J J J J =1 =0.8 constant terms - s = sin(2 arctan()) L -C0 1/2 -C0 1/2 C˜0o 1 C˜0e 1 C˜0e 1 C˜0o 1 C˜0o 1 C˜0e 1 ` + a ⇠ Ce↵() 2⇡2 lnL Ce ↵ ( ) Figure 2.3: Charge fluctuations in a tight binding system with a single impurity link and open boundaries. ⃝a Illustration of the model. An impurity is placed between sites at ℓ and ℓ+a (correpsonding to site indices j0 and j0 + 1, respectively), with impurity hopping λJ. The total system size is set to be L = Zℓ, where we choose Z = 10. ⃝b Fluctuations in subsystem A (circled region in ⃝a ) for λ = 1 (left) and λ = 0.8 (right). For λ < 1, non-decaying parity effects of O(1) are observed, illustrated by the gray background. The inset shows the predicted values of C(λ), cf. Eq. (2.20), compared to the numerically extracted leading order behavior for varying impurity strength. ⃝c Even component C˜′e 1 (upper panel, blue circles) and odd component C˜′o 1 (upper panel, red circles) of the charge fluctuations, the corresponding difference C˜′e 1 −C˜′o 1 (lower panel, black circles) and the isolated boundary parts C˜′e/o − C ′ 1 /2 (lower panel, purple diamonds and green squares) for varying λ. C˜′e 1 (λ), C˜′o 1 (λ) and C ′ 1 (λ)/2 are obtained by numerically fitting the data to Eq. (2.25) for open and Eq. (2.28) for periodic boundary conditions. Sec. 2.4, we use the plane-wave ansatz to exemplify that the physics is governed solely by the properties of the system at the Fermi level. Our numerics are presented in Fig. 2.3. Akin to the entanglement entropy, we observe strong parity effects of O(1) for non-trivial impurity bonds λ ̸= 1, see i.p. Fig. 2.3 ⃝b . The parameter Ceff(λ), which plays the role of the effective central charge and which we calculate from a generalization of Eq. (2.25) similar to what has been done for the entanglement entropy, is further verified to coincide with analytical predictions (see Sec 2.3.2 for details). Furthermore, the fitted even (odd) components C˜′e (s) (C˜′o (s)) of the O(1) contributions to the fluctuations vary strongly (slightly) with the impurity strength λ, but collapse onto a symmetric curve when considering their difference δF(λ) = C˜′e (λ) − C˜′o (λ). Note that for the charge fluctuations, the relevant scale is given by 1/4 (in comparison to ln 2 for the 16 entanglement), which can be understood by considering a mapping from the non-interacting tight binding chain to the XX spin model, where ⟨S z i S z i ⟩ = 1/4 at half-filling, i.e., without a magnetic field. We further extract the constant term for a system with periodic boundaries, where we again explicitly take into account the leading order parity terms, i.e., we fit the impurity system with periodic boundaries to FA,imp = Ceff(λ) + 1 2π 2 ln 2L πa sin πℓ L + A(λ) ℓ + B(λ) ln ℓ ℓ + (C ′ 1 (λ) + C ′ 1 ) 2 . (2.28) Computing C ′ 1 (λ), we can calculate the isolated boundary contributions to the fluctuations, C˜ ′e/o bdry(λ) = C˜′e/o(λ) − C ′ 1 (λ)/2. Numerically, we again observe that C˜′p bdry(λ) = C˜′p¯ bdry(1/λ), (2.29) see the lower panel of Fig. 2.3 ⃝c . From Eq. (2.29) immediately follows the symmetry of δF(λ), δF(λ) = −δF(1/λ). (2.30) 2.3 Single impurity link: analytical insights After this detailed, phenomenological study, we now turn to some analytical considerations. In particular, we present a general analysis of the scaling limit together with perturbative calculations, both for the entanglement entropy in Sec. 2.3.1 and fluctuations in Sec. 2.3.210 . 10Unfortunately, we have not been able to derive the exact form of the curves δS(λ) or δF(λ). 17 2.3.1 Entanglement entropy We start by recalling the formulas for the decomposition of lattice fermions into continuous fields, cj 7→ e ikF jψR + e −ikF jψL. (2.31) At half-filling, kF = π 2 . For a chain starting at j = 1, to take into account the chain termination we have formally cj=0 = 0, so at that extremity the boundary conditions read ψL = −ψR. We use the bosonization formulas (see e.g. [9]) ψR = 1 √ 2π ηRe i √ 4πϕR ; ψL = 1 √ 2π ηLe −i √ 4πϕL , (2.32) where ηR,L are anti-commuting cocycles (Klein factors) necessary to ensure anti-commutation of the fermion fields11. We see that the boundary conditions correspond to Dirichlet boundary conditions, i.e., ϕ(x = 0, t) ≡ ϕR + ϕL = n + 1 2 √ π. (2.33) If we consider a homogeneous chain starting at j = 1 with a single modified link at j0 with Hamiltonian12 H = −J X∞ j=1 c † j+1cj + c † j cj+1 − J(λ − 1) c † j0+1cj0 + c † j0 cj0+1 , (2.34) the bosonized form of the perturbation reads13 c † j0+1cj0 + c † j0 cj0+1 7→ (−1)j0 2π iηRηLe −i √ 4πϕ(j0a) − iηLηRe i √ 4πϕ(j0a) + h.c. = (−1)j0 π 2iηRηL cos √ 4πϕ(j0a). (2.35) 11We use the convention that ϕR, ϕL commute; we also use non normal-ordered exponentials, so there is no need for the usual 1/ √ a factor 12We assume that the Hamiltonian is properly normalized so that its continuum limit is relativistically invariant, J = 1 2 . 13We have only retained the leading contribution, which is exactly marginal in the RG sense. All higher order terms are irrelevant, and should not contribute in the limit ℓ ≫ 1. 18 We have used that η † R,L = ηR,L and η 2 R,L = 1. Note that (iηRηL) 2 = −ηRηLηRηL = 1; in what follows we will treat the term iηRηL as a number (say 1) 14. The crucial point is that the perturbation in the field theory has a component whose sign differs between the cases j0 even and j0 odd. We can now attempt to carry out a perturbative analysis of the entanglement in the limit of weak impurities 1−λ ≪ 1. In the continuum limit, the Hamiltonian reads (we take J = 1 2 to ensure a relativistic dispersion relation of low energy excitations with velocity unity) H = 1 2 Z ∞ 0 Π 2 + (∂xϕ) 2 − 1 π (λ − 1)(−1)j0 cos √ 4πϕ(x = ℓ). (2.36) Since the boson sees Dirichlet boundary conditions, we have the non-trivial one-point function in the half-plane, ⟨e iβϕ(x) ⟩HP = 1 (2x) β2 4π . (2.37) Note that the physical dimension of the r.h.s. is compatible with the values of the conformal weights h = h¯ = β 2 8π . We now consider the Rényi-entropy of the interval (0, ℓ) using the replica approach of [5]. This involves a p-sheeted Riemann surface Rp (with p an integer) with a cut extending from the origin to the point of coordinates (x = ℓ, τ = 0). We expand the partition function, and thus need for this the one point function of e iβϕ on the corresponding surface. We obtain it by uniformizing via two mappings: If w is the coordinate on Rp, the map u = − w − iℓ w + iℓ1/p (2.38) 14A more sophisticated approach would be to realize the cocycles using σ matrices σx, σy, and diagonalize the product σxσy = iσz. 19 maps onto the disk |u| ≤ 1, while the map z = −i u − 1 u + 1 (2.39) maps the disk onto the ordinary half-plane. Using Eq. (2.37), we obtain ⟨e iβϕ(w,w¯) ⟩Rp = 2ℓ p 2h [(w − iℓ)(w + iℓ)( ¯w − iℓ)( ¯w + iℓ)]h( 1 p −1) (w + iℓ) 1/p( ¯w − iℓ) 1/p − (w − iℓ) 1/p( ¯w + iℓ) 1/p2h . (2.40) Here, we have used the fact that vertex operators are primary fields, and thus transform without any anomalous terms. Since two transformations are in fact required to map Rp onto the half-plane w 7→ u 7→ z, we have refrained from writing the intermediate steps, and only give the final result. The perturbative calculation of the entanglement proceeds [36] (in the Euclidian version) by considering the ratio (we set 2 J ′−J π = J ′−J πJ = λ−1 π ≡ µ and take j0 even for the time being), Rp(µ) ≡ Zp (Z1) p = R twist[Dϕ1] . . . [Dϕp] exp − Pp i=1 A[ϕ]i + µ R dτi cos βϕi(ℓ, τi) R [Dϕ] exp{− A[ϕ] + µ R dτ cos βϕ(ℓ, τ ) } p , (2.41) and the integral in the numerator is taken with the sewing conditions ϕi(0 ≤ x ≤ ℓ, τ = 0+) = ϕi+1(0 ≤ x ≤ ℓ, τ = 0−). (2.42) As usual, we trade this problem of p copies of the field for a single field on Rp [5]. With no modified bond, we get the known result, Rp(µ = 0) ∝ a ℓ 2hp , hp = c 24 p − 1 p , (2.43) so S = − d dp Rp p=1 = 1 6 ln ℓ a . (2.44) This is the usual entanglement entropy near a boundary - and we have discarded terms of order 1 which are independent of µ. We now proceed to calculate the ratios Rp(µ)/Rp(µ = 0). To leading order in |µ| ∝ 1 − λ ≪ 1, one finds Rp(µ) Rp(µ = 0) = 1 + µ X p i=1 Z dτi⟨cos βϕ(ℓ, τi)⟩Rp − p Z dτ ⟨cos βϕ(ℓ, τ )⟩HP! . (2.45) Here, τi parametrizes the p copies (on the p sheets of Rp) of the line along which the perturbation is applied (in the Euclidian formulation). It is now time to set h = 1 2 for our perturbation, cf. Eqs. (2.35)&(2.37). We observe that we move from one sheet to the next by (w −iℓ) → e 2iπ(w −iℓ), an operation which does not change the one-point function. We also observe that ⟨cos √ 4πϕ(w, w¯)⟩Rp = 2ℓ p τ 2 (τ 2 + 4ℓ 2 ) 1 2 1 p −1 (τ 2 + 4ℓ 2) 1/p − τ 2/p , (2.46) with the asymptotics ⟨cos √ 4πϕ⟩ ≈ (2ℓ) −1 , τ ≫ ℓ, and ⟨cos √ 4πϕ⟩ ≈ 1 p (2ℓ) −1/pτ 1 p −1 , τ ≪ ℓ. Because of the subtraction coming from the numerator, we see that the integral in Eq. (2.45) converges at large τ . It also converges at small τ , and can in fact be written elegantly using τ = 2ℓ tan θ, i.e., Rp(µ) Rp(µ = 0) = 1 + 2µ Z π 2 0 dθ cos2 θ " (cos θ) 1 p −1 cos2 θ 1 − (cos θ) 2/p − p # ≡ 1 + 2µIp. (2.47) We see from Eq. (2.44) that we get a correction of O(1) to the entanglement, given by S = 1 6 ln ℓ a + 2µ d dpIp p=1 . (2.48) Using the integral Z 1 0 dx (1 − x 2) 3/2 1 + 1 + x 2 1 − x 2 ln x = − π 6 , (2.49) 21 we find d dp Ip p=1 = π 6 and thus S = 1 6 ln ℓ a + π 3 µ = 1 6 ln ℓ a + 1 3 (λ − 1), (2.50) where we used that J = 1 2 . Recall now that this was done for j0 even. The opposite term of O(1) is obtained for j0 odd, leading to δS = 2 3 J ′ − J J = 2 3 (λ − 1). (2.51) We see now that Eq. (2.17) would lead to δS ≈ ln 2(λ − 1), instead of the exact result δS = 2 3 (λ − 1), hence establishing that the ellipse approximation cannot be exact. We note that finite size effects can be studied in perturbation theory as well. Following again [5], we find that the leading term is obtained upon using the more general integral d dp Z ∞ 0 dx 1 q 1 + sin2 πℓ L x 2   x 2 (1 + x 2 ) 1 2 ( 1 p −1) (1 + x 2) 1 p − x 2 p − p   , (2.52) which is equal to π 6 for ℓ ≪ L, and ≈ 0.500125 for ℓ = L 2 . We then find δS = 0.636779(λ − 1) (2.53) instead of the 2 3 slope for an infinite system. When instead fixing ℓ = L 3 , the slope turns out to be given by 0.642286. We can check these results by ‘ab-initio’ measurements of the slopes close to λ ≲ 1. Using a varying fitting window of λ = (1 − ϵ). . . 1 and extrapolating the slopes to infinite system size, we see how the slope close to λ ≲ 1 approaches the analytically predicted value Eq. (2.52) for vanishing window size ϵ → 0, cf. Fig. 2.4 ⃝a . The growing discrepancies for larger fitting values away from ϵ ≳ 1 suggests that there are other terms, possibly of the form λ log λ, which we have not considered in our lowest order perturbative calculation. 2 Fluctuations summarizing plot ✏ 0.271377 0.273147 ` = L/2 ` = L/3 slope for `,L !1 ✏ Entanglement entropy Fluctuations 0.636779 0.642286 Figure 2.4: ⃝a The slope of the entanglement entropy differences δS(λ) fitted in the window λ = (1 − ϵ). . . 1 and extrapolated to the limit ℓ, L → ∞ for ℓ = L/2 (black circles) and ℓ = L/3 (red squares). In the limit ϵ → 0, the slope is given by the analytically predicted value Eq. (2.52), here shown by horizontal dashed lines. ⃝b The same for fluctuation differences δF(λ), where the slope approaches the limit Eq. (2.75). To the order we have considered, the fact that the entanglement gets finite terms of O(1) arises technically from the facts (i) that the term coupled to λ − 1 acquires a finite expectation value in the presence of a boundary, and (ii) that, while this expectation value decays with ℓ, the integral over imaginary time leads to a finite contribution. While calculating entanglement perturbatively at vanishing temperature leads usually to difficulties [36], we note that no divergence is encountered at leading order. At this order, the (anti-)symmetry under λ → 1 λ also appears clearly. While our calculation started out with a specific model (the XX chain with one modified bond), any other system in the universality class of free fermions and with a set of modified bonds should be equivalent to the field theory action of a free boson perturbed by the term Eq. (2.35) up to irrelevant terms. The same calculation should then apply, and we expect therefore our term of O(1) to be a universal function of the effective amplitude in the Hamiltonian Eq. (2.36). 23 2.3.2 Charge fluctuations We first recall briefly what happens without an impurity or a boundary. Using the same factorization Eq. (2.31) as before, we have for the charge density c † j cj 7→ ψ † RψR + ψ † LψL + e 2iKF jψ † RψL + e −2iKF jψ † LψR, (2.54) where, at half-filling, KF = π 2 . One usually considers (but see below) that the oscillating terms can be neglected when considering the fluctuations of the charge Q = P j c † j cj in a given interval. Going over to a continuous description ρ(x = ja) = c † j cj , we obtain ⟨ρ(x)ρ(x ′ )⟩c ≈ ⟨: ψ † RψR : (x) : ψ † RψR : (x ′ )⟩ + R ↔ L + rapidly oscillating terms. (2.55) We now recall that ⟨ψ † R (x)ψR(x ′ )⟩ = i 2π(x − x ′) ⟨ψ † L (x)ψL(x ′ )⟩ = − i 2π(x − x ′) , (2.56) so we find, with Q = R ℓ 0 ρ(x)dx, ⟨(Q − ⟨Q⟩) 2 ⟩ = 2 × − 1 4π 2 Z ℓ 0 dxdx′ 1 (x − x ′) 2 = 1 π 2 ln ℓ a , (2.57) 24 where we regularized UV divergences with the cut-off a 15. Note that this result is pretty much the bosonic propagator since, upon bosonizing and by using ϕ = ϕR + ϕL, we obtain ⟨ρ(x)ρ(x ′ )⟩c ≈ 1 π ⟨∂xϕR(x)∂xϕR(x ′ )⟩ + ⟨∂xϕL(x)∂xϕL(x ′ )⟩ + . . . · · · + rapidly oscillating terms ≈ 1 π ⟨∂xϕ(x)∂xϕ(x ′ )⟩, (2.58) and therefore ⟨(Q − ⟨Q⟩) 2 ⟩ ≡ FA ≈ 1 π ⟨[ϕ(ℓ) − ϕ(0)]2 ⟩. (2.59) Using the free boson propagator on the infinite line, ⟨ϕ(x)ϕ(x ′ )⟩c = − 1 2π ln |x − x ′ | a , (2.60) we recover Eq. (2.28). We now turn to the continuous description of the impurity in the fermionic language. We have, to leading order, c † j cj+1 + h.c. 7→ 2i(−1)j (ψ † LψR − ψ † RψL). (2.61) Solving the Schrödinger equation in the continuum limit gives, for a modified bond at position j0 and setting ℓ ≡ j0a,   ψR(ℓ+) ψL(ℓ−)   =   cos ξ sin ξ − sin ξ cos ξ     ψR(ℓ−) ψL(ℓ+)   . (2.62) 15Although we use the same notation, this cutoff is only proportional to the cutoff discussed in the numerical calculations. 2 This defines the phase shift ξ at the Fermi-surface. It is, in general, a non-universal quantity which must be calculated starting from the lattice model. In our case, we find ξ = (−1)j0 π 2 − 2 arctan J ′ J , (2.63) where the oscillating factor (−1)j0 originates from the alternating term in Eq. (2.61). We see that shifting the weak bond by one site (i.e., exchanging the odd and even cases) amounts to changing the sign of ξ, or equivalently changing λ = J ′ J → 1 λ = J J′ . In the presence of a weak link but no boundary, we can now see how this result is modified. We can for instance decompose the fermion fields into modes. Assuming the weak link is at x = ℓ, this gives ψR(x, t) = R k>0 dk 2π e ik(x−t)α(k) + e −ik(x−t)β † (k) ψL(x, t) = R k>0 dk 2π e −ik(x+t)α¯(k) + e ik(x+t)β¯† (k)    for x < ℓ, (2.64) and similarly ψR(x, t) = R k>0 dk 2π e ik(x−t)a(k) + e −ik(x−t) b † (k) ψL(x, t) = R k>0 dk 2π e −ik(x+t)a¯(k) + e ik(x+t)¯b † (k)    for x > ℓ, (2.65) with matching conditions a = cos ξ α + e −2ikL sin ξ a¯ b † = cos ξ β† + e 2ikL sin ξ ¯b † α¯ = cos ξ a¯ − e 2ikL sin ξ α β¯† = cos ξ ¯b † − e −2ikL sin ξ β† . (2.66) 26 We must now consider the modes α, β† , a, ¯ ¯b † (and their conjugates) as independent, while α, ¯ β¯† , a, b† (and their conjugates) follow from the matching relations. While the RR and LL correlators are not modified, we now have a non-zero RL correlator, ⟨ψ † R (x)ψL(x ′ )⟩ = − sin ξ (2iπ)(x + x ′ − 2ℓ) . (2.67) This leads to fluctuations (for an interval in the bulk, with an impurity on one of its borders) of FA,imp ≈ 1 2π 2 (1 + cos2 ξ) ln ℓ a . (2.68) We see that the slope interpolates between 1 π2 when ξ = 0 (no impurity) to 1 2π2 when the chain is cut in half. The ‘impurity part’ has a slope proportional to cos2 ξ = s 2 = 4(JJ′ ) 2 (J2+J ′2) 2 , where we now recover the result Eq. (2.22) and see how C plays a role similar to the effective central charge. The case with a boundary and no impurity is slightly more convenient to handle via bosonization. In this case, the left and right components of the bosonic field are not independent any longer due to the Dirichlet boundary conditions, ⟨ϕ(x, τ )ϕ(x ′ , τ ′ )⟩ = − 1 4π ln |z − z ′ | 2 − ln |z − z¯ ′ | 2 , (2.69) where z ≡ −τ + ix. It follows that (using ϕ(0) = 0), ⟨[ϕ(0) − ϕ(ℓ)]2 ⟩ = ⟨[ϕ(ℓ)]2 ⟩ = 1 2π ln ℓ a + 1 2π ln 2, (2.70) and thus FA,bdr = 1 2π 2 ln 2ℓ a . (2.71) 2 While the field theory does not control the terms of order one, it does control their difference, as seen by comparing Eqs. (2.57),(2.71) with Eqs. (2.20),(2.25). Finally, when both the boundary and the impurity are present, we can do perturbation theory, once again in the bosonized language. What matters then are the connected terms in the correlator, ⟨[ϕ(ℓ, τ = 0)]2 cos βϕ(ℓ, τ )⟩c = 1 4π ln 1 + 4ℓ 2 τ 2 2 × 1 2L . (2.72) Hence, the leading correction to the charge fluctuations is given by ⟨(Q − ⟨Q⟩) 2 ⟩ (1) = µ × 1 π × 1 4π × 1 2ℓ Z ∞ −∞ dτh ln 1 + 4ℓ 2 τ 2 i2 = µ 2π 2 Z ∞ 0 dxh ln 1 + 1 x 2 i2 , (2.73) where the integral is tabulated and yields 4π ln 2. Subtracting the opposite term for the odd case, it follows that ⟨(Q − ⟨Q⟩) 2 ⟩ (1) e − ⟨(Q − ⟨Q⟩) 2 ⟩ (1) o = 4µ π ln 2 = 4 π 2 J ′ − J J ln 2 = 4 ln 2 π 2 (λ − 1), (2.74) where we recall that J ′ J = λ. With 4 ln 2 π2 ≈ 0.280922 ̸= 1 4 we see that the result is, again, not supported by the naive ellipse shape mentioned previously. We note that the leading term ∝ ln ℓ does not get corrections to first order in µ, which is in agreement with the formula for the slope in Eq. (2.68). The foregoing calculation holds when the interval from the edge to the impurity is part of a halfinfinite system. We can otherwise expect finite-size corrections, like for the entanglement. It is possible 28 to calculate the effect of these corrections on the first order term by evaluating the propagators on a strip instead of in the half-plane, using a conformal map, which yields ⟨(Q − ⟨Q⟩) 2 ⟩ (1) = µ 2π 2 Z ∞ 0 dx q 1 + sin2 πℓ L x 2 ln 1 + 1 x 2 2 . (2.75) We find that 4 ln 2 π2 ≈ 0.280922 is replaced by 0.271377 when ℓ = L 2 , or by 0.273147 when ℓ = L 3 - these are small but non negligible effects. We again check these results by ‘ab-initio’ measurements of the slopes close to λ ≲ 1, obtained by diagonalizing the tight binding Hamiltonian. As for the entanglement entropy, we observe that the slope is given by Eq. (2.75) for vanishing fitting window, cf. Fig. 2.4 ⃝b . Finally, we note that the behavior of the slope of δF close to λ = 1 can also be derived from first order perturbation theory around the exact plane wave solution of the homogeneous chain with open boundaries, yielding identical results. 2.4 Extended impurities (towards the SSH model) In this section, we would like to illustrate the fact that the fluctuation oscillations only depend on the phase shift at the Fermi level. Let us for this purpose consider a system with Nimp impurity bonds arranged in an alternating fashion, i.e., H = −J X m c † m+1cm+h.c. − J (λ ⟨0,1⟩ − 1)c † j0 cj0+1 + (λ ⟨2,3⟩ − 1)c † j0+2cj0+3 + . . . · · · + (λ ⟨2Nimp,2Nimp+1⟩ − 1)c † j0+2Nimp cj0+2Nimp+1 + h.c. , (2.76) where λ ⟨i,j⟩ is the impurity strength between sites j0 + i, j0 + j. 2 2.4.1 General results To solve the system, we make a plane wave ansatz, |k⟩ = j X0−1 j=−∞ Ake ikj + Bke −ikj ) |j⟩ + j0+2 X Nimp−1 j=j0 c j k |j⟩ + X∞ j=j0+2Nimp Cke ikj + Dke −ikj |j⟩, (2.77) with H |k⟩ = −2J cos(k)|k⟩. (2.78) By comparing coefficients after applying the Hamiltonian to Eq. (2.77), we find for c 0 k and c 1 k c 0 k = Ak + Bk λ ⟨0,1⟩ c 1 k = Ake ik + Bke −ik . (2.79) The coefficients c 3 k ...c 2Nimp−1 k are then constructed with the recursion relation c n k =    2 cos(k)c n−1 k − λ ⟨n−2,n−1⟩ c n−2 k for n even 1 λ⟨n−1,n⟩ (2 cos(k)c n−1 k − c n−2 k ) for n odd. (2.80) This now expresses all tight-binding coefficients lying inside the impurity with the plane wave coefficients Ak, Bk. We further get two additional equations relating c2Nimp−1 and c2Nimp−2 with Ck and Dk, given by 2 cos(k)c 2Nimp−1 k = λ ⟨2Nimp−2,2Nimp−1⟩ c 2Nimp−2 k + Cke 2ikNimp + Dke −2ikNimp , c 2Nimp−1 k = Cke ik(2Nimp−1) + Dke −ik(2Nimp−1) . (2.81) 30 s() s(Nimp) < 1 1/ < 1 < 1 1/ < 1 Nimp F s() s(Nimp) < 1 1/ < 1 < 1 1/ < 1 Nimp =1 Nimp =3 Nimp =5 Nimp =7 S Figure 2.5: Parity effects of ⃝a the entanglement entropy and ⃝b the fluctuations when transitioning from the single impurity link to an “SSH-like” impurity with a collection of alternating bonds J, λJ. Upon increasing the number of impurities Nimp = 1, 3, 5, 7 with all λ ⟨2n,2n+1⟩ = λ, the differences quickly saturate to ± ln 2 (upper panel ⃝a ) and ±1/4 (upper panel ⃝b ). This can be understood in terms of an effective, stronger single impurity with strength λ Nimp . When rescaling δS and δF accordingly, the curves become equivalent (lower panels). Eqs. (2.82),(2.80) fully solve the Nimp alternating impurity system. In case of half-filling, these equations simplify considerably, such that c n k =    (−1)n/2 (Ak + Bk) Qn/2−1 i=0 λ ⟨2i,2i+1⟩ for n even (−1)(n+1)/2 i(Ak − Bk)/ Q(n−1)/2 i=0 λ ⟨2i,2i+1⟩ for n odd. (2.82) Now, using Eq. (2.81), we find (Ak + Bk) Nimp Y−1 i=0 λ ⟨2i,2i+1⟩ = Ck + Dk , Ak − Bk = (Ck − Dk) Nimp Y−1 i=0 λ ⟨2i,2i+1⟩ , (2.83) where i = 0..Nimp − 1 now runs through all impurity bonds. 31 Finally, with Q i λ ⟨2i,2i+1⟩ = Λ, this results in   B C   =   2Λ2 1+Λ2 1−Λ2 1+Λ2 − 1−Λ2 1+Λ2 2Λ2 1+Λ2     D A   . (2.84) We thus see that the collection of impurities behaves, in fact, like a single one of effective strength Λ. Comparing with Eqs. (2.62),(2.63), we see that the phase shift at the Fermi surface16 is now given by ξ = (−1)j0 " π 2 − 2 arctan Y i λ ⟨2i,2i+1⟩ !# . (2.85) On the other hand, we can study numerically the fluctuations or the entanglement for these extended impurities as well. In analogy with the case of the single modified bond (Nimp = 1), we define the entanglement and fluctuation differences as the difference between even and odd parities of the part of the system situated between the boundary and the border of the extended impurity. In practice, this corresponds to comparing two systems where the cut defining the subsystem as well as the impurity are both shifted by one site. Restricting to the simplest case where all impurity strengths in the chain are identical, Λ = λ Nimp , we find for instance that the curves for the Nimp-impurity system coincide with those for a single impurity upon the rescaling λ → λ Nimp , see Fig. 2.5. Here, we always choose an odd number of impurities and cut the system through the central impurity bond. The observation that the fluctuations become identical upon rescaling λ → λ Nimp is a clear indication that the parity effects in the scaling limit are governed by the physics at the Fermi level, ultimately supporting our conjecture that the differences are universal. Note that these results are indeed independent on how one defines the subsystem’s border. For instance, when defining the cut to go through one of the outermost impurity bonds instead of the central 16The phase shift depends on k in general. However, it becomes independent of k (and equal to ξ) for low-energy excitations, that is excitations whose energy is much smaller than the band-width, and this irrespective of the modified couplings. 32 one, we numerically checked that the curves are identical17- which is expected since the phase shift at the Fermi surface is independent of the definition of the subsystem. When again focusing on the upper curves in Fig. 2.5, we see that δS(λ) (δF(λ)) approaches a step function taking values ± ln 2 (±1/4) for λ ≷ 1 in the limit of an infinitely large Nimp ≫ 1 impurity. 2.4.2 SSH model We now observe that entanglement entropy differences in the SSH model [43] look somewhat similar to large alternating impurities. It is interesting to see what becomes of our observations in the latter case. First, we note that the way we define the entanglement entropy and the fluctuation differences both for the single impurity bond as well as the extended (alternating) impurities corresponds to comparing the two distinct topological phases in the limit of a pure SSH model (all weak and strong bonds are exchanged and the border of the subsystem is moved by one unit). In this case, the local environment around the border of the subsystem looks identical for both parities, and its influence gets cancelled when taking the difference, see the upper part of the inset of Fig. 2.6. However, the boundary of the system features localized modes in the topologically non-trivial regime, which remain when we subtract the entanglement from the trivial regime (where the edge modes don’t exist). This, in turn, results in an entanglement entropy (fluctuation) difference of exactly ln 2 (1/4) - in fact for all values of λ, cf. the blue curve in Fig. 2.6. Had we, on the other hand, fixed the geometry of the SSH chain and focused on entanglement and fluctuation differences arising from merely shifting the subsystem’s border from even ↔ odd, we would only observe local differences around the impurity (cf. the lower part of the inset of Fig. 2.6). This results in ln 2 (1/4) differences of the entanglement (fluctuations) only in the limit λ = 0, in which case we can 17In this case both definitions have the same Nimp limit and both scale as described above, making the curves indeed identical. 33 ... ... ... - ... ... ... ... - ... s() F Figure 2.6: Comparing two different ways of defining the fluctuation differences between the parities. (i) We both change the subsystem’s parity as well as the topological phase, i.e., in the language of impurities, we “move” the impurity together with the subsystem’s border (see the blue curve). Though locally the systems look the same at the subsystem’s border, the edge modes appearing in the topologically nontrivial phase lead to a 1/4 contribution for all λ. (ii) While changing the parity of the subsystem, the geometry of the underlying model is fixed. Here, only local differences are seen, leading to a functional dependence different from the step function. again rely on the valence bond picture: local valence bonds between dimers lead to differences of ln 2 (1/4) when changing the parity of the subsystem’s border, see the red curve in Fig. 2.6. It is not clear a priori to what extent the limit Nimp ≫ 1 can be considered as identical with a pure SSH model since, in all our calculations, we have always considered an impurity embedded in a gapless bulk (or metallic) system. We observe however that the corresponding curves for the entanglement or fluctuation differences are the same step functions. This means that the two leads coupling to the impurity become effectively decoupled in the limiting case of spatially large SSH-type impurities, ultimately also leading to a contribution of ln 2 (1/4) to the entanglement (fluctuation) differences for all impurity strengths. It seems therefore that the phenomenology of the SSH model and the SSH-type impurity are similar: in the former case the differences arise from the edge modes appearing in the non-trivial phase, and in the latter they arise from the leads being effectively decoupled18 . This decoupling of the leads is further confirmed by an analysis of the single particle energies and wave functions of the SSH model when we add leads of increasing length to the edges starting with either 18In the language of the SSH model, this corresponds to having negligible overlap between the two exponentially decaying edge modes. 34 site | ( x )| 2 =0.8 =0.8 =0.8 =0.1 ✏ n Figure 2.7: Modes |Ψ(x)| 2 in vicinity of zero energy for various tight-binding models. Values of |Ψ(x)| 2 are marked in an alternating fashion by red and blue dots for A and B sublattices. Each model is illustrated in the center of the left panel, with strong (weak) bonds marked by black (red) lines. Right panels show the energy spectra close to zero energy, with the two symmetric modes closest to ϵ = 0 marked in blue. ⃝a The pure SSH chain features localized zero energy modes, having support only on sublattice A (B) for the left (right) edge mode. Here, λ = 0.8. ⃝b An SSH impurity of length 20 attached to two leads of size 30 on either side, again with λ = 0.8. The long SSH impurity polarizes the wave functions close to zero energy, having support only on the A (B) sublattice on the left (right) to the extended impurity. ⃝c A single impurity bond with 40 metallic sites to each left and right side for λ = 0.8. ⃝d The same as ⃝c but with λ = 0.1. of the two topological phases. We observe that when we add leads to the edges of a non-trivial SSH chain, the zero energy modes remain- though the gap around them closes. Nevertheless, the central SSH chain fully polarizes the system in the same sense it does for the conventional (full) SSH chain, i.e., the wave function of the zero modes localized on the right (left) edge only have support on the A (B) sublattice. This is illustrated in Fig. 2.7. The left panel in ⃝a shows a linear combination of the zero modes to have weight on both boundaries in the non-trivial SSH chain, localized at the edges of the system on sublattice A (red dots) on the left and B (blue dots) on the right. The single particle spectrum around zero energy is shown in the right panel of Fig. 2.7 ⃝a , whereby edge (bulk) modes are marked by blue (black) dots. When 35 coupling metallic leads to an SSH chain of size 20, we see that - up to a small split due to the shorter SSH part- zero modes are still present, and the corresponding wave function is polarized on sublattice A (B) to the left (right) of the chain, see Fig. 2.7 ⃝b . When now keeping the same system size but further shrink the SSH impurity part, we notice how the polarization around the impurity weakens but qualitatively remains intact even in the extreme limit of a single impurity bond, shown for λ = 0.8 in Fig. 2.7 ⃝c . By decreasing the ratio λ, the polarization is, as expected, enhanced, shown in Fig. 2.7 ⃝d for λ = 0.1. Indeed, we find that for λ < 1 the energy split of the two modes around zero energy scales as ∼ e ln(λ)Nimp = λ Nimp , akin to the scaling law we found for the fluctuation and entanglement differences19 . In the language of the earlier impurity discussion, the above examples shown in Fig. 2.7 correspond to the “even case”. If we instead consider the trivial phase of the SSH chain (“odd case”) and couple leads to it, there are no wave functions that are strongly polarized on one sublattice only. Hence, when comparing the two opposing geometries (i.e. “even” and “odd”, which corresponds to non-trivial and trivial in the full SSH limit), we can now get an intuition for the resemblance of the SSH and single impurity bond model: both systems feature low energy modes that polarize the system on a scale that grows exponentially with the length of the (central) SSH chain ∝ λ Nimp . This, in turn, leads to non-quantized values of entanglement/fluctuation differences for small impurity lengths and strengths. Note that the problem we considered in this paper is fundamentally different from the SSH model, in particular because the theory in the bulk is conformal instead of being gapped. However, it is tempting to think of the oscillations we have exhibited as some sort of measure of “topological phase precursor”, even though the meaning of the (non-topological) numbers δS (δF) other than ln 2 (1/4) remains unclear. Lastly, note that the above scaling results would break down if the modified bonds did not obey the alternating pattern where unmodified bonds sit between two modified ones. An interesting example with 19In both cases, a consequence of the scaling of the phase shift at the Fermi energy. 36 such a different behavior is the case where two successive bonds are modified, leading to the so-called resonant level model. 37 Chapter 3 Entanglement parity effects in the Kane Fisher problem 3.1 Background We further study the case when the impurity induces an RG flow. Specifically, we consider the case of an XXZ model with a single modified bond. As discussed in chapter 2, for the single modified link in XX chain, its effect corresponds to a marginal perturbation, there is no RG flow in this case. Unlike the non-interacting case, the anisotropic interaction will change the system significantly, a XXZ chain with a single modified bond also exhibits RG flow which is similar to the Kane-Fisher problem[21, 44]. Depends on the sign of interaction term Jz, the system will flow to a homogeneous chain or a split chain. The effective central charge in this system would vary as a function of ℓTB where TB is a characteristic energy scale akin to the Kondo temperature in Kondo problem. We find that the difference of O(1) term between even and odd also depend on ℓTB For an interval in the bulk that ends on the impurity link, the entanglement obeys a scaling relation of the type [44] dSA,imp d ln ℓ = G(ℓTB), (3.1) where G is a sort of running effective central charge. Note that in Eq. (3.1) we considered only the derivative of the entanglement with respect to ℓ, not S itself, since the latter quantity involves several non-universal 38 terms of O(1) because of the additional TB dependency [44]. Once again, we can ask what happens in the presence of such an impurity combined with a boundary as before. We will then see that dSe A,imp+bdr d ln ℓ = F(ℓTB) + f e (ℓTB) dSo A,imp+bdr d ln ℓ = F(ℓTB) + f o (ℓTB), (3.2) where F, f o , f e are non-trivial, universal functions1 . The difference, f e (ℓTB) − f o (ℓTB) ≡ δ dS(ℓTB) d ln ℓ , (3.3) turns out to exhibit a most interesting crossover between the UV (ℓTB ≪ 1) and the IR (ℓTB ≫ 1) regimes. 3.2 Physics around the split fixed-point 3.2.1 Generalities We consider first hamiltonians near the "split fixed-point" HA = X ℓ j=0 σ x j σ x j+1 + σ y j σ y j+1 + Jzσ z j σ z j+1 + λ σ x ℓ σ x ℓ+1 + σ y ℓ σ y ℓ+1 + Jzσ z ℓ σ z ℓ+1 + X∞ j=ℓ+1 σ x j σ x j+1 + σ y j σ y j+1 + Jzσ z j σ z j+1 (3.4) Set g = 2 − 2 π arccosJz (3.5) 1 It is not so clear how F is related to G in the bulk because of potential interactions between the two borders of the interval. 3 so g > 1 for Jz > 0 and g < 1 for Jz < 0. Near the split fixed point λ = 0 , a small nearest-neighbor interaction as in (3.4) is an operator of dimension (length) −g . This means that near the split f ixed point λ is relevant if Jz < 0 near the split f ixed point λ is irrelevant if Jz > 0 (3.6) Relevant means that for a fixed λ the chain at large distance appears healed. This is clearly seen if we consider the entanglement of the region of size ℓ to the left of the cut with the rest of the system: the bulk behavior should interpolate from S = 0 to S ≈ 1 6 ln ℓ as ℓ is increased at fixed λ. This is illustrated on figure 3.2 where we have plotted the derivative of the entanglement entropy for the system in the particular case L = 2ℓ. Recall that, from finite size scaling results, the entanglement in the healed case always has the leading behavior c 6 lnL with c = 1 here. This is seen on figure 3.2 as the the two curves go to 6dS/d lnL = 1 in the IR. Note that the results look quite different for odd and even lengths ℓ (represented by S e and S o respectively. It is this difference δS ≡ S e − S o we shall be interested in. In this context, relevant means δS will evolve from something smaller than ln 2 to 0 as L increases for a fixed λ (that is, healing occurs) while irrelevant means δS evolves from something small than ln 2 to exactly ln 2 as L increases for a fixed λ. The RG flows are thus as on the figure 3.1 . Writing the perturbation as λO this product must have dimension (length) −1 and thus, if O has dimension ( length ) −g , this means dim[λ] = ( length ) g−1 (3.7) Hence we can manufacture a quantity of dimension ( length ) −1 (a temperature) by considering TB ∝ λ 1/(1−g) (3.8) 40 Figure 3.1: The different types of flows near the split fixed-point 3.2.2 The relevant and irrelevant case We expect that in the limit of small λ and large L, and for the relevant case (g<1), results should have a universal dependency on the product LTB. Since results depend only on LTB, increasing L at fixed λ in the scaling limit is like increasing λ at fixed L: in other words, at fixed λ the long distance physics corresponds to healing, behaviours like a homogeneous chain. While the phenomenology is well understood in general, we focus here on aspects of entanglement in the presence of a boundary that have not been studied before except in the special free fermion case Jz = 0 [38]. Results confirming the qualitative RG picture are given below 2 . We plot the difference of entanglements with the subsystem starting at the boundary and ending in the middle of the modified bond (i.e. containing the spins j = 1, . . . , j = ℓ) for the cases ℓ even and ℓ odd, δS ≡ S e − S o . The total system size is taken to be L = Zℓ, with Z a factor taken to Z = 2 unless otherwise specified. 2Our numerical results are obtained by using the density matrix renormalization group (DMRG) algorithmand the Tenpy package [18]. 41 Figure 3.2: Crossover in the bulk entanglement entropy for Hamiltonian 3.4. We note that there are in fact possible variants of the problem: the other Hamiltonians HB = X ℓ j=0 σ x j σ x j+1 + σ y j σ y j+1 + Jzσ z j σ z j+1 + λ σ x ℓ σ x ℓ+1 + σ y ℓ σ y ℓ+1 + X∞ j=ℓ+1 σ x j σ x j+1 + σ y j σ y j+1 + Jzσ z j σ z j+1 (3.9) Since theσ z ℓ σ z ℓ+1 term is irrelevant near the split fixed point, it does not affect the low-energy physics, although it it affects corrections to scaling, it simply disappears in the limit λ → 0, ℓ → ∞. We first give results for Hamiltonian 3.4 in figure 3.3. Totally identical results are obtained in the scaling limit for the Hamiltonian 3.9 as shown in figure 3.4. In particular, the value of TB is the same for the two curves. And we also find that varying Z does not change results much. Then we consider the The irrelevant case, we start from a small tunneling term but are driven at low-energy to the split situation again. This can be seen in the fact that LTB increases at fixed λ when increasing L but increases at fixed L when decreasing λ. Hence, large L behaves like small λ, and the splitfixed point is reached at large distances. Going to small LTB is formally equivalent to increasing λ and 4 Figure 3.3: Healing flow of entanglement difference for Jz = −0.5. Here Z = 2 and the Hamiltonian is (3.4) thus, one would hope, to getting closer to the healed fixed-point. However, in this limit, other irrelevant operators will start playing a role, and there is not chance to reach this fixed-point without fine tuning. In practice, this simply means that the left-hand side of the curves plotting δS as a function of LTB are not fully universal. See figure 3.5 for some illustrations. 43 Figure 3.4: Healing flow of entanglement difference for Jz = −0.5. Here Z = 2 and the Hamiltonians (3.4), (3.9) are compared 44 Figure 3.5: Flow of entanglement difference for Jz = 0.5. Here Z = 2 and the Hamiltonian is (3.9). The split fixed-point is recovered in the IR, while the healed fixed-point is not reached in the UV. 3.3 Physics around the homogeneous fixed-point 3.3.1 Generalities We can also consider the vicinity of the homogeneous (uniform) fixed point. In this case the Hamiltonian is HA = X ℓ j=0 σ x j σ x j+1 + σ y j σ y j+1 + Jzσ z j σ z j+1 + (1 − µ) σ x ℓ σ x ℓ+1 + σ y ℓ σ y ℓ+1 + Jzσ z ℓ σ z ℓ+1 + X∞ j=ℓ+1 σ x j σ x j+1 + σ y j σ y j+1 + Jzσ z j σ z j+1 (3.10) 4 Figure 3.6: The different types of flows near the homogeneous fixed-point And HB = X ℓ j=0 σ x j σ x j+1 + σ y j σ y j+1 + Jzσ z j σ z j+1 + (1 − µ) σ x ℓ σ x ℓ+1 + σ y ℓ σ y ℓ+1 + Jzσ z ℓ σ z ℓ+1 + X∞ j=ℓ+1 σ x j σ x j+1 + σ y j σ y j+1 + Jzσ z j σ z j+1 (3.11) Perturbing the coupling near the uniform fixed point corresponds in the continuum limit to an operator of dimension (length) −g−1 .We see that the regions of relevance and irrelevance are switched with respect to the previous section near the homogeneous f ixed point µ is irrelevant if Jz < 0 near the homogeneous f ixed point µ is relevant if Jz > 0 (3.12) The corresponding flows are sketched in figure3.6. We see now that dim µ = ( length ) g−1−1 (3.13) 4 Figure 3.7: Flow for Jz = 0.5; Here, Z = 2. The weakly perturbed chain flow to the split fixed-point in the IR. Using the same kind of scaling argument as for the case of an almost split chain, we now expect the properties to have a universal dependency on LΘB with ΘB ∝ µ 1/(1−g−1 ) (3.14) In the relevant case, we now flow from the homogeneous to the split fixed point, this is illustrated in figure 3.7. 47 3.3.2 Perturbative calculations Like in our previous paper we shall consider mostly perturbation near the homogeneous fixed point. The steps are standard: we first represent the σj s in terms of fermionic creation and annihilation operators using the Jordan-Wigner transformation. For a homogeneous chain for instance this gives H = X j σ x j σ x j+1 + σ y j σ y j+1 + Jzσ z j σ z j+1 = X j c † j cj+1 + c † j+1cj + Jz c † j+1cj+1 − 1 2 c † j cj − 1 2 (3.15) Left and right moving fermion fields are then introduced and bosonized, and the quadratic interaction term eliminated by a suitable rotation. In the end, we get the continuum theory with bulk Hamiltonian: H = v 2 Z dx h Π 2 + (∂xΦ)2 i (3.16) The compactification radius of the boson is R ≡ r g 4π (3.17) the the sound velocity is given by v = π 2 p 1 − J 2 z arccos Jz (3.18) one finds the crucial bosonization formula σ x j σ x j+1 + σ y j σ y j+1 + Jzσ z j σ z j+1 = (−1)j 2c ± 1 + c z 1 cos Φ R (x = j) (3.19) 4 where c ± 1 is a constant equal to 1 π in the non-interacting case, but otherwise not known exactly (see below). Note we only represented the leading term, of dimension 1 4πR2 = g −1 . The next term would be proportional to ∂Φ ∂x 2 , of dimension 2. It thus becomes the most relevant one when g −1 = 2, that is Jz < − √ 2 2 . We will not study this region for now. In the non-interacting case there is no σz term, g = 1, R = √ 4π and we recover the results used in [38]. The Hamiltonian ( 3.10) corresponding to is then H = v 2 Z ∞ 0 dx h Π 2 + (∂xΦ)2 i − µR π (−1)ℓ cos Φ(x = ℓ) R (3.20) Note that in 3.20 we have introduced a renormalized coupling constant µR. Indeed, while in the noninteracting case µR = µ as defined with the lattice Hamiltonian , in the presence of interactions, renormalization effects lead to µR = Zµµ. The constant Zµ is not universal. Its value for the XXZ spin chain is not known exactly, but has been determined numerically to a great accuracy in [20] (for earlier work see [42]). We will use the values deduced from [20], after the correspondence Zµ = √ 8π 2 q B ± 1 + Jz 2 p Bz 1 , while Jz = ∆ ( so that Zµ = 1 for Jz = 0). For the Hamiltonian 3.11, we don’t need to consider this term σ z j σ z j+1 = c z 1 (−1)j cos Φ(x=j) R , to leading order as well, for Jz > − √ 2 2 . In this region, the Hamiltonian in the continuum limit if still 3.20, but with different value of the renormalization constant Zµ = q 8π 2B ± 1 = √ 4π 2c ± 1 . We note that, when flowing from the split fixed point, the infra-red, healed fixed point is approached along a combination of the stress-energy tensor (of dimension 2) and the dual tunneling operator of dimension ( length ) −g−1 . Hence the same Hamiltonian can be taken to describe the vicinity of the homogeneous fixed point in cases where it is the IR and the UV fixed point of the flow. 4 We start with Jz ≤ 0, 1 g > 1, where the problem corresponds to perturbing the homogeneous fixed point by an irrelevant perturbation. Since the results depend only on ℓΘB, this means, assuming δS is perturbative and has a contribution at first order (see below), that we expect the entropy in the IR to go as δS = −α(g) (ℓΘB) 1−g−1 (3.21) We can actually check this and determine α(g) using the same kind of calculation carried out in [38]. What is called (λ − 1) in eq. 43 ( in this section, equation numbers from reference [38] will be indicated by eq. ) is called µ here, while in eq. 44 we have β = 1 R , so β 2 4π = g −1 and h = 1 2 g −1 . Everything up until eq. 52 works as well in our case, although now we have cos Φ R (w, w¯) Rp = 2ℓ p 2h v 2 τ 2 v 2 τ 2 + 4ℓ 2 h 1 p −1 h (v 2τ 2 + 4ℓ 2) 1/p − (vτ ) 2/pi2h (3.22) The asymptotic behaviors at large distance are D cos ϕ(w,w¯) R E Rp ≈ (2ℓ) −2h , τ >> ℓ and D cos ϕ(w,w¯) R E Rp ≈ h 1 p (2ℓ) −1/pτ 1 p −1 i2h , τ << ℓ. Like for when h = 1 2 , the resulting integral is still convergent - in fact, thanks to the subtraction coming from the denominator, it turns out to be always convergent. Setting τ = 2ℓ tan θ we have, replacing eq. 54 Rp(µR) Rp(µR = 0) = 1 + 2µR πv (2ℓ) 1−2h Z π/2 0 dθ cos2 θ " p 1−2h (sin θ) 2h( 1 p −1) cos4h θ [1 − (sin θ) 2/p] 2h − p # ≡ 1 + 2µR π (2ℓ) 1−2h Ip (3.23) Like in [38] we get the correction to the entanglement by S = 1 6 ln ℓ a + 2µR πv (2ℓ) 1−2h d dpIp p=1 (3.24) 50 Remarkably, the resulting integral differs from the one when h = 1 2 by a simple factor: d dp Ip p=1 (h) = 2h d dp Ip p=1 (h = 1 2 and we find in the end S = 1 6 ln ℓ a + 1 3 g −1 µR v (2ℓ) 1−g−1 (3.25) where we used that 2h = g −1 and d dp Ip p=1 h = 1 2 = π 6 . As in [38] the result only holds for L = ∞. When the ratio ℓ L is finite, finite size effects have to be taken into account. Comparing odd and even cases amounts to changing the sign of µ as discussed in [38]. This leads immediately to δS = 2 3 g −1 µR v (2ℓ) 1−g−1 (3.26) In the non-interacting case Jz = 0 we have g = 1 and we find δS = 2 3 µ like in eq.58 of that reference. When the system has finite size L = 2ℓ (so Z = 2 ), we find, generalizing eq. 60 δSZ=2 = .636779g −1 µR v 4ℓ π 1−g−1 = .636779g −1Zµ v µ 4ℓ π 1−g−1 (3.27) As mentioned above we now observe that the integrals encountered in this calculation are always convergent, irrespective of the relevance of the perturbation. It follows that ( 3.27) should hold as well when the perturbation is relevant, i.e. when Jz > 0. The numerics indeed do not see anything happening when Jz = 0 is crossed. On the other hand, result ( 3.27) only makes sense when the hopping term is the leading (ir)relevant operator. As Jz crosses the value − √ 2 2 , the term of (Jz independent) dimension 2 dominates, and thus ( 3.27) ceases to be valid. 5 The numerical analysis is a little tricky because, even in the absence of a local perturbation, the entanglement is known to already exhibit an alternating dependency upon ℓ, leading to δSZ=2(µ = 0) = a(g)l −g−1 (3.28) This correction is well identified in the literature, and the exponent usually written as K, the Luttinger constant, with K = π 2(π−arcos Jz) = 1 g . It is due to the leading irrelevant bulk oscillating term in the chain. We have first checked the result ( 3.28), as illustrated on figure 3.8 (a). To leading order, we expect the correction ( 3.28) and the correction induced by the µ ̸= 0 perturbation to simply add up, so we should have: δSZ=2(µ) − δSZ=2(µ = 0) = .636779g −1Zµµ 4ℓ π 1−g−1 (3.29) We have therefore studied in what follows the quantity δSZ=2(µ)−δSZ=2(µ = 0). Measures of the exponent are obtained by plotting ln [δSZ=2(µ) − δSZ=2(µ = 0)] vs ln ℓ for small values of µ. It give excellent µ independent results as illustrated in figure 3.8 (b). To obtain results for the slope .636779g −1 Zµ v , we fit δSZ=2(µ) − δSZ=2(µ = 0) vs µ 4ℓ π 1−g−1 , an example of such fit is given in figure 3.9 (a). The resulting slopes are then compared with theoretical values in figure 3.9 (b), (c), for the two possible Hamiltonians. Note the excellent agreement both in the relevant and irrelevant case as long as Jz is not too close to ±1. 3.4 Symmetries 3.4.1 Symmetries between µ and −µ, λ and −λ The entanglement entropy is expected to possess several interesting symmetries in the scaling limit. The first such symmetry can be seen from the point of view of the perturbed homogeneous chain, where we 5 Figure 3.8: (a) Exponent in (3.28) (b) Exponent 1 − g −1 in (3.29) Figure 3.9: (a) Example of fit. (b)Study of the slope of δS near the homogeneous fixed point for Hamiltonian(3.10). (c)Study of the slope of δS near the homogeneous fixed point for Hamiltonian (3.11). have seen in section 3.3.2 that in the field theory Hamiltonian (eq. (3.20)) translation of the cut by one site amounts to µR → −µR. Of course this is true only to first order in µR, but since the results in the scaling limit are valid in the limit µR → 0, ℓ → ∞ with µℓ1−g−1 finite, it is only this order that matters. Hence we conclude: δS(µ) = −δS(−µ) (3.30) The second symmetry is δS(λ) = δS(−λ) (3.31) 53 It follows from the discussion of the perturbation expansion around the split fixed point. The relationship (3.30) is illustrated in figure 3.10 while the relationship (3.31) is illustrated in figure 3.11 . Note that, as emphasized above, the relationships are only expected to hold in the scaling limit, µ → 0 (resp. λ → 0) and L → ∞ with the appropriate combinations ΘB (resp. TB) finite. (a) (b) Figure 3.10: Jz=0.5 δS(µ) and -δS(−µ) comparison (a) Comparison in large range of LΘB. (b) Detailed comparison in the scaling limit where µ is very small. (a) (b) Figure 3.11: Jz=-0.5 δS(λ) and δS(−λ) comparison (a) Comparison in large range of LTB. (b) Detail comparison in the scaling limit where λ is very small. 54 3.4.2 Symmetries between λ and 1 λ To see the third symmetry, imagine we consider a chain with λ >> 1 i.e. with a coupling between sites ℓ and ℓ + 1 greatly enhanced. To facilitate the discussion we introduce a slightly more general Hamiltonian Hℓ = σ x ℓ−1σ x ℓ + σ y ℓ−1 σ y ℓ + Jzσ z ℓ−1σ z ℓ + λ σ x ℓ σ x ℓ+1 + σ y ℓ σ y ℓ+1 + ∆σ z ℓ σ z ℓ+1 + σ x ℓ+1σ x ℓ+2 + σ y ℓ+1σ y ℓ+2 + Jzσ z ℓ+1σ z ℓ+2 (3.32) where we have allowed for the coupling with amplitude λ to have a different anisotropy ∆. In the limit λ >> 1, the spins ⃗σℓ and ⃗σℓ+1 are almost paired into a singlet.The Hamiltonian can then be replaced in this limit , by its first-order perturbation theory approximation Hℓ 7→ −Es + X ti |⟨s|Hℓ |ti⟩|2 Es − Eti (3.33) where the energies of the term coupling spins ℓ and ℓ + 1 are Es, Eti respectively. For the singlet we have Es = −λ 1 2 + ∆ 4 while the “triplet” now splits into states (for spins ℓ, ℓ + 1) | + +⟩ and | − −⟩ with energies Et1 = Et3 = λ∆ 4 and |+−⟩−|−+⟩ √ 2 with Et2 = λ 1 2 − ∆ 4 . A straightforward calculation then gives, up to an irrelevant additional constant Hℓ 7→ 1 λ σ + ℓ−1 σ − ℓ+2 + σ − ℓ−1 σ + ℓ+2 1 + ∆ + ∆2σ z ℓ−1σ z ℓ+2! (3.34) Observe that, while initially the modified bond was between sites ℓ, ℓ + 1, after this renormalization it is now between sites ℓ − 1 and ℓ + 2 which, after a relabelling starting as usual from the left, becomes between sites ℓ − 1 and ℓ. Hence we have exchanged the odd and even impurity problems. Notice also that the anisotropy of the Hamiltonian is not preserved in general. This only occurs in the XXX case when ∆ = 1, for which we recover an XXX Hamiltonian, and the coupling has gone from λ to 1 2λ and in the XX case when ∆ = 0 for which we recover an XX Hamiltonian but the coupling has gone from λ to 1 λ . The duality is best seen for Hamiltonian HB (3.9) which corresponds to ∆ = 0. In this case we expect, in the scaling limit δS(λ) = −δS 1 λ (3.35) In general, since we have argued and checked that dependency of the δS curve on the exact form of the modified Hamiltonian can entirely be absorbed into a redefinition of TB, we expect the results for the problem and its dual to be identical (up to the exchange of odd and even) in the scaling limit. Moreover, in the case of Hamiltonians HA and HB, the redefinition of TB can be obtained simply by the substitution λ → 1 λ(1+∆) . This relationship is illustrated in 3.12, while the equation3.35 is illustrated in 3.13. (a) (b) Figure 3.12: Jz=-0.5, δS(λ) and -δS( 1 λ ) comparison, -δS( 1 λ ) curve shifted according to 1 1+∆ . (a) Comparison in large range of LTB. (b) Detail comparison in the scaling limit where λ is very small. 56 (a) (b) Figure 3.13: Jz=-0.5, δS(λ) and -δS( 1 λ ) comparison, with Hamiltonian 3.9(a) Comparison in large range of LTB. (b) Detail comparison in the scaling limit where λ is very small. 57 Chapter 4 Entanglement parity effects in the disordered system 4.1 Background In this chapter, we discuss the effects of strong disorder. We specifically consider an XX spin chain, described by the Hamiltonian: H = X ℓ i=0 Ji S x i S x i+1 + S y i S y i+1 + λJℓ S x ℓ S x ℓ+1 + S y ℓ S y ℓ+1 + X∞ i=ℓ+1 Ji S x i S x i+1 + S y i S y i+1 (4.1) Here Ji are random variables drawn from a probability distribution. The presence of disorder can significantly change the behavior of physical systems. In the absence of impurity (λ = 1), this disodered XX chain has been extensively studied. It can be described by the random-singlet phase using the real space renormalization group approach [13, 27]. In this random-singlet phase, singlets can form over all length scales. The typical correlation between two spins is weak, decaying exponentially with distance, whereas the average correlation decays according to a power law. In the random singlet phase, the system is quantum critical with the mean entanglement entropy scales logarithmically as S ≈ ln 2 3 ln ℓ [32]. The introduction of impurity has been shown to induce healing in the disordered XX chain, even in the absence of interaction [45]. 4.2 Methods In this section we review the real space renormalization group (RSRG) approach, a powerful method to deal with disordered systems. The procedure of RSRG is iteratively finding the strongest bond in the system and treating the corresponding spins as a singlet.Following this, it connects the neighboring spins on either side of the strongest bond with a new effective bond. Consider the strongest bond to be Jn. In cases of strong disorder, the nearest neighbors Jn−1 and Jn+1 are typically much smaller than Jn, so we can deal with it perturbatively. The local unperturbed Hamiltonian is: H0 = Jn(S x nS x n+1 + S y nS y n+1) (4.2) The eigenstates of this Hamiltonian are: |s⟩ = 1 √ 2 (| ↑↓⟩ − | ↓↑⟩) |t1⟩ = | ↑↑⟩ |t0⟩ = 1 √ 2 (| ↑↓⟩ + | ↓↑⟩) |t−1⟩ = | ↓↓⟩ (4.3) The corresponding energies are Es = − 1 2 Jn, Et1 = Et2 = 0 and E0 = 1 2 Jn, so the ground state is a singlet with energy Es = − 1 2 Jn. Then the interactions with the neighbors are then considered as a perturbation, this perturbation Hamiltonian is: V = Jn−1(S x n−1S x n + S y n−1S y n ) + Jn+1(S x n+1S x n+2 + S y n+1S y n+2) (4.4) 59 Figure 4.1: Illustration of one RG step, we find the largest bond Jn and assume that a singlet form between the associated spins and we take that term out of the Hamiltonian and introduce an effective new bond instead. Applying perturbation theory, the correction to the ground state energy is calculated as: X i |⟨s|V |ti⟩|2 1 Es − Et = −J 2 n−1 + J 2 n+1 4Jn + Jn−1Jn+1 Jn S x n−1S x n+2 + S y n−1S y n+2 (4.5) Then we write the perturbed ground state energy as: Es → E ′ s + J ′ S x n−1S x n+1 + S y n−1S y n+2 (4.6) where E′ s = − 1 2 Jn − 1 4Jn J 2 n−1 + J 2 n+1 and J ′ = Jn−1Jn+1 Jn . So during one RG step, we eliminate the spins Sn and Sn + 1 that connected by the strongest bond Jn from the original Hamiltonian. Then we add the energy E′ s and an effective bond J ′ between spins Sn−1 and Sn+2. Figure 4.1 shows the illustration of this process. The new Hamiltonian has the same form of the original Hamiltonian but with fewer degrees of freedom and a reduced energy scale. During each RG step, as we delete and add bonds, the distribution of bonds P(J) evolves. Understanding how this bond distribution changes during the RG process is of particular interest. Given that the new effective bond strength is Jef f = Jn−1Jn+1 Jn , we define some new variables to get linear equations which are much more easily to solve. Firstly we define: Γ = ln Ω0 Ω (4.7) where Ω0 is the initial largest bond and Ω is the strongest bond in the current Hamiltonian, Γ is referred to as the RG time parameter. Next, we define another variable: β = ln Ω J (4.8) Consequently, Jef f = Jn−1Jn+1 Jn transforms into βeff = βn−1+βn+1. Utilizing these logarithmic variables, we denote PΓ(β) as the distribution of the bonds β at RG time Γ. The evolution of this distribution as the RG time Γ goes from Γ to Γ + dΓ has two primary sources. One source is the change in the definition of β due to the change of Γ, as the strongest bond is removed and the new strongest bond is defined, requiring each J to be adjusted by new Ω yield the updated β. The second source is the addition of a new renormalized bond upon the removal of the strongest bond. Combining these two factors, we get the renormalization group flow equation: ∂PΓ(β) ∂Γ = ∂PΓ(β) ∂β + PΓ(0) Z dβ1dβ2PΓ (β1) PΓ (β2) δ (β1 + β2 − β) (4.9) The fix point distribution of this flow equation is: P(β) = 1 Γ e −β/Γ (4.10) This distribution is known as the random singlet fixed point distribution and it is a stable fixed point of the RG process [27, 13]. As the RG time Γ increases, the bond distribution broadens, indicating an increase 61 in disorder throughout the RG process, makes RG asymptotically exact. In the random singlet phase, the ground state is made by a brunch of singlets, so the entanglement of a segment of a discorded XX chain is the number of singlets that connect the segment with the rest of the chain. 4.3 Single Impurity and Quantum Dot Impurity In this section we discuss the entanglement entropy in disordered XX chain with a single impurity and a dot impurity. As previously discussed in chapter 2 there is no RG flow in clean XX chain with a single impurity. However the presence of disorder induces the healing, which restores homogeneity at low energy. This healing behaviour could also be observed in a clean XXZ chain with Jz < 0, as explored in chapter 3. This phenomenon of healing effect in the presence of disorder was extensively studied in [45]. In the model under consideration, the impurity link is situated in the middle of the chain, dividing it into two equal segments. Our focus is on the bipartite entanglement entropy between these two segments. To compute the entanglement entropy, we need to determine the number of singlets forming across the left and right boundaries of the segments. In the absence of an impurity link (λ = 1), the average number of decimations over a specific bond N¯, scales with ln Γ. The average duration between two consecutive singlet formations on a specific bond is 3, leading to the entanglement entropy SL = 1 3 · 2 ln Γ + k = ln 2 3 log2 L + k [32]. When impurity link exists, the initial probability distribution for the weak link is : QΓ0 (β) = 1 Γ0 e − β+ln λ Γ0 θ(β + ln λ) (4.11) Given that the strength of the weak link β = ln Ω0 λJ0 > ln λ −1 , it can not be decimated until the RG time Γ exceeds Γ ∗=ln λ. It has been shown that for a periodic chain, when the RG time Γ reaches Γ ∗ , the original weak link would have the same strength as the other bonds, as a result the average duration between 62 decimations remains 3. The effect of the weak link is only to shift the origin of the RG time. Therefore, the the bipartite entanglement entropy with periodic boundary conditions is given by: S = Z ∞ Γ0 dΓ Γ g Γ √ L ln 2 Z Γ Γ0 dΓ ′/Γ ′ 3 + Z Γ Γ0 dΓ ′/Γ ′ 3 θ Γ ′ + ln λ (4.12) Here g is related to the distribution strength of the last bond being decimated with dP = dΓ Γ g(Γ/ √ L) [14]. In equation 4.12, the first term represents the entanglement contribution from the normal left boundary and the second term accounts for the entanglement contribution of the weak link. Taking the scaling limit Γ → ∞, λ → 0 with x = Γ − ln λ fixed, we can get the universal scaling function of entanglement entropy with crossover scale L ⋆ ∼ (ln λ) 2 [45]: ∂S ∂ lnL = ln 2 3 1 − 2 π X∞ n=0 (−1)n 2n + 1 e −π 2 (2n+1)2L/4L⋆ ! (4.13) However we find the formula in equation 4.13 only works well in periodic boundary condition. The parity effects which we discussed in previous chapters also manifest in the disordered XX chain with open boundary condition. This results in an actual scaling function that is different from equation 4.13. As illustrated in figure 4.2, the theoretical formula 4.13 agrees well with numerical calculations for the periodic case. In contrast, for open boundary odd case, equation 4.13 only matches numerical results when L/L∗ is either very large or very small. Intriguingly, in open boundary even case, numerical scaling functions exhibit a noticeable "bump" that cannot be explained by equation 4.13. This "bump" in the open boundary even case presents a fascinating and challenging problem to rigorous solve analytically. We propose a hypothesis that might may explain this phenomenon. Similar to the periodic boundary case, the weak link cannot be decimated until the RG time Γ exceeds Γ ∗ . Upon the RG time reaching Γ ∗ , the strength of the weak bond becomes stronger than the other bonds due to the open boundary condition. This causes it to be decimated at a faster rate, with an estimated average duration of 6 (a) (b) (c) Figure 4.2: Numerical RG calculation of the scaling function of entanglement entropy and theoretical formula. (a) The periodic case. (b) The open boundary odd case. (c) The open boundary even case. 2 instead of 3. After the first decimation of the weak bond, it returns to the original decimation rate akin to other bonds. Based on this hypothesis, we propose the following expression for entanglement entropy: S (2) = ln 2 Z Γ˜ Γ∗ dΓ Γ g Γ √ L Z Γ Γ∗ (dΓ ′/Γ ′ ) 2 + ln 2 Z ∞ Γ˜ dΓ Γ g Γ √ L Z Γ Γ∗ (dΓ ′/Γ ′ ) 2 + Z Γ Γ˜ (dΓ ′/Γ) 3 (4.14) We take the scaling limit and consider the derivative with respect to lnL, this yields a new formula: ∂S ∂lnL = ln 2 6 " 1 − 1 π X∞ n=0 (−1)n 2n + 1 3e −(2n+1)2π 2L/(4L∗) − e −(2n+1)2π 2L/(4e 4L∗ ) # (4.15) As shown in figure 4.2c, this hypothesis regarding the accelerated rate of the first decimation explains the observed "bump" in the entanglement entropy scaling function quite well. In the disordered XX chain, the entanglement difference between the even and odd case δS still exhibits a universal dependency on the product L/L∗, it goes from ln2 to 0 as the healing occurs. Notably, the behavior of δS in the disordered XX chain is similar to that observed in the pure XXZ chain with Jz<0. The entanglement difference δS and its derivative with respect to L/L∗ are illustrated in figure 4.3. We also investigated the disordered XX chain with dot impurity. The results are shown in figure 4.4. It is very interesting that in the case of a periodic boundary condition, the the scaling function for single 64 (a) (b) Figure 4.3: entanglement difference between the even and odd impurity case 4.13 also applies to the dot impurity case, except with a rescaled L ∗ . In the open boundary odd case, the scaling functions for both dot and single impurity cases exhibit similar characteristics. However in the open boundary even case, the "bump" observed in the dot impurity case is significantly more pronounced than in the single impurity case, suggesting a stronger boundary-induced parity effect. 65 (a) (b) (c) Figure 4.4: Numerical RG calculation of the scaling function of entanglement entropy and theoretical formula. (a) The periodic case. (b) The open boundary odd case. (c) The open boundary even case. 66 Chapter 5 Conclusion In conclusion, we have conducted research on parity effects in entanglement across various types of spin chain systems, studying both systems that exhibit a RG flow and those that do not. In an XX chain with a single impurity, there is no RG flow as the effect of impurity is marginal perturbation in the RG sense, we find that the parity effects in entanglement and fluctuations that do not decay with distance when the boundary is combined with an impurity. These parity effects contain interesting physical information: they can be construed as indicating the presence of a topological phase in the SSH model for impurities preserving the symmetry of reflection through a link (conformal defects). The systems within a RG flow are also studied, the XXZ chain with a impurity will flow to the fully split chain fixed point or the healed chain fixed point. The parity effects in entanglement are studied near both of the two fixed points, we find the difference of entanglement δS depends on a universal function of ℓTB. For the XX chain with strong disorder exists, the properties of entanglement in a single impurity case and a dot impurity case are similar in the periodic boundary condition. When combine with open boundary, the parity effects in entanglement still exists and behave similar to the pure XXZ chain with Jz < 0 case, while a dot impurity case have stronger parity effects than a single impurity case. From a technical point of view, we note that it would be very interesting to come up with a derivation of closed-form analytical expressions for the functions δS(s) and δF(s)for XX chain. This would presumably 67 require an extension of the derivation [35] of ceff in the presence of a boundary. Besides, we get the analytical expressions for universal functions of δS(ℓTB) in XXZ chain. In the disordered XX chain, we come up a hypothesis that could explain the "bump" in the scaling function. 68 References [1] Ian Affleck, Nicolas Laflorencie, and Erik S Sørensen. “Entanglement entropy in quantum impurity systems and systems with boundaries”. In: Journal of Physics A: Mathematical and Theoretical 42.50 (Dec. 2009), p. 504009. doi: 10.1088/1751-8113/42/50/504009. [2] Ian Affleck and Andreas W. W. Ludwig. “Universal noninteger “ground-state degeneracy” in critical quantum systems”. In: Phys. Rev. Lett. 67 (2 July 1991), pp. 161–164. doi: 10.1103/PhysRevLett.67.161. [3] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. “Entanglement in many-body systems”. In: Rev. Mod. Phys. 80 (2 May 2008), pp. 517–576. doi: 10.1103/RevModPhys.80.517. [4] E. Brehm and I. Brunner. “Entanglement entropy through conformal interfaces in the 2D Ising model”. In: Journal of High Energy Physics 2015.9 (2015), p. 80. doi: 10.1007/JHEP09(2015)080. 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[47] Huan-Qiang Zhou, Thomas Barthel, John Ove Fjærestad, and Ulrich Schollwöck. “Entanglement and boundary critical phenomena”. In: Phys. Rev. A 74 (5 Nov. 2006), p. 050305. doi: 10.1103/PhysRevA. 74.050305. 72 Appendices A: Fluctuations in the homogeneous, periodic chain Here, we show how the leading order of the charge fluctuations in a homogeneous, periodic chain can be derived from a simple plane wave analysis. Let the system be of total length L. The Hamiltonian reads H = −J X m c † m+1cm + h.c., (.1) where periodic boundary conditions are imposed. Fourier transforming the creation and annihilation operators leads to c † k = 1 √ L X L n=1 e −ikamc † m, (.2) which diagonalizes the Hamiltonian. Eigenstates c † k |0⟩ have eigenenergies ϵ(k) = −2J cos(ka), and k = 2πl/L, −L/2a < l ≤ L/2a, such that k ∈ (−π/a, π/a]. The many-body ground state is then given by |GS⟩ = Y |k|<kF c † k |0⟩. (.3) 73 Let us look at the correlator Gij = ⟨c † i cj ⟩, G(i, j) = ⟨0|    Y |k|<kF ck      1 √ L X k1 e ik1aic † k1     1 √ L X k2 e ik2ajck2      Y |k ′ |<kF c † k ′    |0⟩ = 1 L e ik1ai−ik2aj ⟨0|    Y |k|<kF ck    c † k1 ck2    Y |k ′ |<kF c † k ′    |0⟩ | {z } →|k1|≤kF | {z } →δk1k2 . (.4) Therefore, we have G(i, j) = G(i − j) = 1 L X |k|<kF e ika(i−j) = a 2π Z kF −kF e ika(i−j) = akF π sinc[akF (i − j)]. (.5) Now, we are interested in the density-density correlations, ⟨c † i cic † j cj ⟩ = ⟨c † i ci⟩ ⟨c † j cj ⟩ + ⟨c † i cj ⟩ ⟨cic † j ⟩ = ⟨c † i ci⟩ ⟨c † j cj ⟩ + ⟨c † i cj ⟩ δij − ⟨c † j ci⟩ . (.6) Using G(0) = 1/2 at half filling kF = π/2 and a = 1 for simplicity, this results in the density fluctuations F(i − j) = ⟨ninj ⟩ − ⟨ni⟩ ⟨nj ⟩ =    1/4 for i = j − 1 4 sinc2{(i − j)π/2} for i ̸= j. (.7) The charge fluctuations in a given interval of length ℓ are hence F(ℓ) = X ℓ i,j=1 F(i − j) = 1 4 ℓ − X ℓ i,j=1 i̸=j sinc2 {(i − j)π/2} . (.8) 7 ✏ / `2 Figure 5.1: Comparison of the large ℓ expansion Eq. (.11) with exact numerical evaluation of Eq. (.8). The inset shows that errors are of order O ℓ −2 . The double sum can be simplified into a single sum due to the dependence on solely the distances i − j, F(ℓ) = ℓF(0) +X l d=1 (ℓ − d)[F(d) + F(−d)] = 1 4 ℓ − 2 X ℓ d=1 (ℓ − d)sinc2 {dπ/2} , (.9) where we used that F(d) = F(−d). The finite sum Eq. (.9) is tabulated, yielding 4π 2F(ℓ) = π 2 ℓ + (−1)ℓ ℓψ(1)[(ℓ + 1)/2] + (−1)ℓ+1ℓψ(1)[(ℓ + 2)/2] − 4(−1)ℓΦ(−1, 1, ℓ + 1) −π 2 ℓ + 4ℓψ(1)[ℓ + 1] + 4ψ (0)[ℓ + 1] + 4γ + 4 log(2). Here, the first term on the RHS stems from ℓ/4 in Eq. (.9), while the rest stems from the analytical expression of the sum. γ = 0.57721 . . . is Euler’s constant, ψ (n) is the n’th derivative of the digamma function, and Φ(x, s, a) is the Lerch transcendent, Φ(z, s, a) = X∞ k=0 z k (a + k) s . (.10) 7 For large ℓ ≫ 1, the first derivative of the digamma function behaves as ψ (1)(ℓ) = ℓ −1 + O(ℓ −2 ), such that (−1)ℓ ℓ ψ (1)[(ℓ + 1)/2] − ψ (1)[(ℓ + 2)/2] = 2(−1)ℓ 1 1 + 1/ℓ − 1 1 + 2/ℓ + O ℓ −1 = O ℓ −1 and ℓψ(1)[ℓ + 1] = 1 + O ℓ −1 . For the Lerch transcendent, we have limℓ→∞ Φ(−1, 1, ℓ) = 0. Lastly, ψ (0)(ℓ) = log(ℓ) + O l −1 . Ultimately, we find for ℓ ≫ 1, F(ℓ) = 1 π 2 [log(ℓ) + 1 + γ + log(2)] + O ℓ −1 . (.11) The term ∝ 1 + γ + ln(2) is the known term of O(1) for the fluctuations in the homogeneous, onedimensional tight-binding chain, see e.g. [39]. This is underlined in Fig. 5.1, where the large ℓ limit Eq. (.11) is compared to the exact evaluation of Eq. (.9). B: Standard perturbation theory for λ ≈ 1 To further study the behavior of the slope of δF close to λ = 1 presented in Sec 2.3.2, we can perturb the exact plane wave solution of the homogeneous chain with open boundaries, given by |k (0)⟩ = C X j sin(kj)|j⟩, (.12 with |j⟩ the tight-binding basis and C = p 2/(L + 1). The corresponding energies read E (0) k = −2t cos(k); k = πl L + 1 with l = 1, ..., L, (.13) where we see that -in contrast to periodic boundary conditions- the states are non-degenerate. Hence, when introducing a perturbing potential V˜ (λ), V˜ (λ) = (1 − λ)t |ℓ + 1⟩ ⟨ℓ| + H.c. | {z } =V , (.14) we can perform the standard first order perturbation theory for systems with non-degenerate spectra. In first order, the corrected energies then read Ek = E (0) k + (1 − λ)E (1) k = −2t cos(k) + (1 − λ)⟨k (0)|V |k (0)⟩ = −2t cos(k) + (1 − λ) sin[k(ℓ + 1)] sin[kℓ]. (.15) Perturbation of the standing waves leads to |k⟩ = |k (0)⟩ + (1 − λ)|k (1)⟩ = |k (0)⟩ + (1 − λ) X k1̸=k ⟨k (0) 1 |V |k (0)⟩ E0 k − E (0) k1 |k (0) 1 ⟩ (.16) For the correlator Gij = ⟨c † i cj ⟩ = P k<kF ⟨i|k⟩ ⟨k|j⟩ then follows in linear order (1 − λ), Gij = X k<kF ⟨i|k (0)⟩ ⟨k (0)|j⟩ | {z } =G (0) ij +(1 − λ) X k<kF ⟨i|k (1)⟩ ⟨k (0)|j⟩ + ⟨i|k (0)⟩ ⟨k (1)|j⟩ | {z } G (1) ij . (.17) 77 Here, ⟨i|k (0)⟩ = C sin(ki), ⟨i|k (1)⟩ = C 3 X k1̸=k sin[k1(ℓ + 1)] sin[kℓ] + sin[k(Nimp + 1)] sin[k1ℓ] 2[cos(k1) − cos(k)] sin(k1i). (.18) For computing the difference of the fluctuations between even/odd scenarios, we will consider the two perturbing potentials V e , V o , resulting in different perturbed states and correlation functions G e/o ij . The fluctuations at half filling are given by F(ℓ) = ℓ 2 − X ℓ i,j=1 i̸=j G 2 ij . (.19) The first order correction in (1 − λ) to ∆F = δF (0) + (1 − λ)δF (1) hence reads δF (1) = 2    ℓ Xo i,j=1 i̸=j G (0) ij G (1) ij,o − ℓ Xe i,j=1 i̸=j G (0) ij G (1) ij,e    . (.20) Using Eqs. (.17) & (.18), Eq. (.20) can be evaluated numerically. To compare with the results presented above, we compute the slopes ∆F (1) for different chain lengths and different impurity bond positions (i.e. at 1/2 and 1/3 of the chain), and extrapolate the values for L → ∞. We find that Impurity at L/2 : 0.27138 Impurity at L/3 : 0.27315, (.21) which is in correspondence with the predictions presented in the main text, Sec. 2.3.2. 78

Abstract (if available)

Abstract In the presence of boundaries, the entanglement entropy in lattice models is known to exhibit oscillations with the (parity of the) length of the subsystem, which however decay to zero with increasing distance from the edge. We point out in this dissertation that, when the subsystem starts at the boundary and ends at an impurity, oscillations of the entanglement (as well as of charge fluctuations) appear which do not decay with distance, and which exhibit universal features.We conduct a detailed study of parity effects across a range of spin chains, investigating both systems that exhibit a renormalization group (RG) flow and those that do not. We start with the XX chain with one modified link (a conformal defect), a case where the system does not exhibit RG flow, we study parity effects both numerically and analytically. We then generalize our analysis to the case of extended (conformal) impurities, which we interpret as SSH models coupled to metallic leads. In this context, the parity effects can be interpreted in terms of the existence of non-trivial topological phases. Subsequently, we investigate parity effects on entanglement in systems exhibiting a RG flow. Specifically, we focus on the XXZ chain with a single impurity, a scenario known as the Kane-Fisher problem. We find that the difference of entanglement depend on $\ell T_B$, where $T_B$ is a characteristic energy scale akin to the Kondo temperature in Kondo problem. Finally we study the parity effects when strong disorder exists.

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Creator Tan, Chunyu (author)

Core Title Entanglement parity effects in quantum spin chains

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Degree Conferral Date 2024-05

Publication Date 03/04/2024

Defense Date 02/14/2024

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag entanglement,impurities,SPiN

Format theses (aat)

Language English

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Advisor Haas, Stephan (committee chair), El-Naggar, Moh (committee member), Kresin, Vitaly (committee member), Nakano, Aiichiro (committee member), Saleur, Hubert (committee member)

Creator Email chunyuta@usc.edu,tcy2017@gmail.com

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Type texts

Source 20240304-usctheses-batch-1127 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA

Repository Email cisadmin@lib.usc.edu