Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Healing of defects in random antiferromagnetic spin chains
(USC Thesis Other)
Healing of defects in random antiferromagnetic spin chains
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
HEALING OF DEFECTS IN RANDOM ANTIFERROMAGNETIC SPIN CHAINS by Arash Roshani 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) December 2019 Copyright 2019 Arash Roshani Dedication To my Grandfather, Samad... ii Contents Dedication ii 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Entanglement Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Quantum Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.1 Criticality in non-disordered systems . . . . . . . . . . . . . 4 1.3.2 Criticality in disordered systems . . . . . . . . . . . . . . . . 5 1.3.3 Infinite randomness fixed point (IRFP) . . . . . . . . . . . . 6 1.4 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Formalism 8 2.1 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Random singlet phase . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Renormalization group rules . . . . . . . . . . . . . . . . . . 10 2.1.3 Renormalization group flow . . . . . . . . . . . . . . . . . . 13 2.2 Exact Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Numerical Subtleties . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Single-Impurities 22 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Healing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Some Numerical Checks . . . . . . . . . . . . . . . . . . . . . . . . 27 3.4 Direct Diagonalization versus Renormalization Group, Box versus Infinite Randomness Fixed Point Distribution . . . . . . 30 4 Quantum-dot Impurities 33 4.1 Kondo physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.2 Minimum resistivity . . . . . . . . . . . . . . . . . . . . . . 34 4.1.3 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 iii 4.2.1 Clean chains - case b and c . . . . . . . . . . . . . . . . . . . 37 4.2.2 Disordered chain - Symmetric q-dots - Case b and c . . . . . 39 4.2.3 Disordered chain - Asymmetric q-dots - Case b and c . . . . 41 4.2.4 Disordered chain - Asymmetric q-dots - Case a . . . . . . . 42 4.3 Conclusion for observations . . . . . . . . . . . . . . . . . . . . . . 50 Reference List 51 A Codes 53 A.1 Direct Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . 53 A.2 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . 59 A.3 Singlet Formation Rate . . . . . . . . . . . . . . . . . . . . . . . . . 65 B Pit and Bump 71 C Broken Q-dots 75 iv Chapter 1 Introduction 1.1 Background The effect of quantum impurities in many body quantum systems has been an area of interest for condensed matter physicists even before celebrated J. Kondo’s paper in 1967 [19] which explained the baffling problem of existence of a resistance minimum in dilute magnetic alloys in low enough temperatures [6]. In the past decade there has been a regain of interest in this field and people have attempted to study these impurity problems by studying different parameters such as entan- glement entropy and Loschmidt echo. In particular, entanglement entropy is well recognized to reflect important features of the behavior of many body systems and particularly the ones with these impurities. Also, the dynamics of these systems can be studied through quantum quenches which are usually sudden changes in the Hamiltonian, such as a sudden change in the on-site potential of a q-dot in a fermionic chain. There are different approaches one can adopt to explore such sys- tems. Conformal field theory (CFT) and renormalization group (RG) are amongst the most used ones to study pure systems and systems with disorder respectively. BecauseofthelackoftranslationalinvarianceindisorderedsystemsusuallyRGhas to be used and even further, we are going to see that the RG scheme is especially convenient for studying entanglement entropy. 1 1.2 Entanglement Entropy Entanglement entropy falls into the intersection of a few different fields of physics and different physics communities use it to gain insight towards the behav- ior of the system. In specific, focusing on the connections of quantum informa- tion and many-body physics, people from both communities have realized that entanglement entropy is a great parameter to study the ground states of different Hamiltonians to monitor criticality and topological order. There are different measures and definitions for entanglement entropy and the most common one and the one that we are going to use throughout this thesis is the von Neumann entropy. If we divide our system into two subsystems M and ¯ M, for a pure quantum state with the density matrix ρ the von Neumann entropy is defined as: S =− trρ M log 2 ρ M =− trρ ¯ M log 2 ρ ¯ M in whichρ M is the reduced density matrix (Tracing outρ over ¯ M givesρ M ). Notice that the entanglement is symmetric with respect to M and ¯ M which is what one would expect since regardless of if we choose subsystem M or its compliment ¯ M, the entanglement is supposed to measure how connected the two parts M and ¯ M are. Along the same line, another feature of entanglement entropy is that if the state of the system is not a tensor product of pure states from subsystems M and ¯ M then the entanglement is not going to be zero which is again something that we expect. Now one might ask how useful entanglement entropy actually is. A simple and fastanswertothisquestionwouldbethatitisaparameter/toolthattellsusabout the state of our many-body condensed matter system when the more conventional 2 parameters such as correlation functions do not help much. For example, when the system gets closer to the critical points the correlation length diverges and at criticality it becomes infinity and we really cannot use it anymore. Though, we still have access to entanglement entropy and that would give us insight towards our system. Entanglement of the ground state is a very important parameter that can be used to understand quantum phase transitions [33]. There are other measures available which satisfy ones basic expectations of entropy. A famous one is “Renyi entropy" which is defined as the fallowing: S α = 1 1−α log 2 trρ α M In the limit ofα→ 0 Renyi entropy becomes simply the logarithm of the size ofM. For α→∞ it can be determined by the biggest eigenvalues. And, interestingly forα→ 1 we can revive the von Neumann entropy from Renyi entropy. So, Renyi entropy is the generalization of von Neumann entropy in a sense. In this thesis we are going to use von Neumann entropy as our main tool to gain insight about the behavior of our systems. Also, as mentioned RG provides us with a powerful tool to calculate this entanglement. 1.3 Quantum Criticality We are familiar with phase transitions caused by thermal fluctuations when the temperature is finite but big enough. On the other hand, when the temperature is small enough (and ideally zero) that the quantum fluctuations take over thermal fluctuations a new class of critical systems emerge. At temperature zero, phase transition is still possible by changing the coupling constants or parameters other 3 than temperature such as the external field. In that case phase transitions would be the transitions in the characteristics of the ground state of the many-body system. Quantum critical systems are the systems at the transition between these quantum phases and they are associated with the divergence of correlation length. At criticality the system becomes scale-less and this fact gives rise to interesting behaviors in systems at criticality such as self-similarity [15, 4, 29]. For systems far from criticality it has been shown that the bipartite entangle- ment grows proportional to the area of the segment for which we are calculating the entanglement for, which is know by the area law [31, 9]. In one dimensional systems it can be shown that the entanglement of any finite subsystem in an infi- nite chain scales as S∝A logξ in which ξ is the correlation length andA is the number of boundary points. An oversimplified intuition towards this fact is that since the bipartite entanglement should be the same for a segment of a system and its compliment therefore it should be a function of a parameter that is com- mon between the two and that resembles area. At the same time, given that far from criticality the correlation length is not infinite anymore it makes sense that the entanglement scales proportional to the surrounding area of the segment since the number of connections between the segment and it’s compliment should grow proportional to the surface area of the segment. 1.3.1 Criticality in non-disordered systems Innon-disorderedmany-bodychainsthecontinuumofthesystemisconformally invariant and CFT methods can be used. In some examples one can map the one dimensional quantum critical system to a 1+1 quantum field theory (CFT). Using this methodology one can derive that in the limit ofL→∞ (in whichL is the size 4 of the segment for which we want to calculate the entanglement) the entanglement scales as: S∼ c + ¯ c 6 log 2 L in which c and ¯ c are the holomorphic and anti-holomorphic central charges of the CFT theory [16, 33, 2, 28]. What we observe here addresses a very important and rather fundamental fact that in the scaling limit (system size andL→∞) there exists a universal behavior for critical systems. Regardless of the microscopic details of the original model the scaling falls into a universality class to which the system belongs [33]. 1.3.2 Criticality in disordered systems Introducing randomness to quantum critical systems usually modifies the char- acteristics of those systems and these modifications are more intense when the dimension of the system is lower. Though, parallel to the non-disordered chains for which the correlation length was diverging at criticality, in the case of disor- dered chains a similar situation happens. For disordered chains singlets form over arbitrary long distances. In other words, parts of the system that are arbitrarily far from each other become correlated. Strictly speaking, as Daniel Fisher showed for a spin-1/2 Heisenberg chain any amount of disorder (even small) causes the spin chain to factorize into singlets that are forming over arbitrary long distances which is called random-singlet phase [12]. This can be observed in the average correla- tions between a site at the origin and a site at L, < ˆ S z (0) ˆ S z (L)>. The sites at 0 andL being locked in a singlet is a rare event to happen with the probability which falls as 1/L 2 . Though, these events dominate the average < ˆ S z (0) ˆ S z (L)> and we 5 get< ˆ S z (0) ˆ S z (L)>∼ 1/L 2 as well [12, 13, 14, 26]. The infinite-randomness fixed point therefore is the parallel of pure CFT’s in disordered systems. In the absence of conformal invariance which is the case for our disordered systems, we cannot rely on CFT methods anymore and we need new machinery to analyze the ground-state of our chain. Renormalization group (RG) is the remedy of this situation and the perfect tool to study the infinite randomness fixed point phase (IRFP). We are going to discuss IRFP after we learn how the RG scheme changes the distribution of the bonds in section 2.1.3, and we are going to see that IRFP is actually the attractive fixed point of the RG scheme which exactly means that with any small amount of disorder the system flows to infinite disorder. Though, in favor of illuminating criticality in disordered systems and to give the reader an overview we briefly discuss the RG scheme and the IRFP here. 1.3.3 Infinite randomness fixed point (IRFP) As mentioned in the last section, our disordered chains are factorized into singlets that are forming over arbitrary long distances. Using RG, one can find these factorized singlets. At each step of RG one would find the largest bond of the chain and assume that the singlets in that bond are locked together and it is safe to exclude that term from the Hamiltonian. Though, in addition to that, an effective bond should be replaced. This process is called decimation (Fig 1.1). The newly introduced bond J eff is rather small in comparison to all of the bonds that we are excluding from the Hamiltonian in one step of decimation. This is the cause of the increase in the amount of disorder as we go forward through RG. 6 Decimation 1 2 3 4 1 2 4 3 Figure 1.1: In each RG step we find the largest bond 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. Here J 2 is the largest bond and we replace J 1 , J 2 , and J 3 by J eff . 1.4 Thesis Structure The structure of this thesis is as the fallowing: In chapter 2 we develop and study the formalism that is needed to understand RG,healingandtocalculateentanglemententropy. Welearnaboutrandomsinglet phase which might be one of the main ideas at the core of this thesis. In chapter 3 we explain the concept of healing and explore single-impurity defects as well as the limitations of our numerical machinery. In chapter 4 we start by explaining the Kondo physics and how it relates to this research. We then study quantum dot (q-dot) defects and we explore some characteristics that are common between systems with single-impurities and q-dot impurities as well as new emerging phenomena 1 . 1 Such as the emerging bump that we are going to talk about more 7 Chapter 2 Formalism Inthischapterweelaborateonthemethodsweusetocalculateentanglemententropy and the numerical subtleties that one has to deal with using these methods. Knowing that Jordan-Wigner transformation introduces a mapping between spin chains and fermionic chains, we are going to study them interchangeably and depending on our method we will choose the more suitable one. 2.1 Renormalization Group Renormalization group (RG) is at the core of the approaches we use to under- stand the behavior of our 1D chains with quenched randomness both theoretically and numerically. As we are going to see, RG provides us with a very convenient scheme to calculate entanglement entropy in such systems. The ground state of these chains are almost 1 factorized into singlet pairs and RG provides us with the necessary tool to keep track of the formation of these singlets which ultimately helps us understand the universal features of our systems. Before we delve deep into the RG scheme we need to gain a good understanding of the ground state of the chains that we are studying. In this section first we are going to start by learning more about the concept of random singlet phase which illuminates the structure of the ground state of our chains. Next we are going to derive the RG rules for an example and give the generic recipe for other situations for which RG 1 We use the term “almost" not to undermine the fact that this picture is an approximation of the ground state of the hamiltonian. 8 Figure 2.1: Random Singlet Phase - The ground state of the system is factorized into singlets that are forming over arbitrary long distances. can be used. In the section that comes afterwards we are going to use the RG rules that we have derived to explore how the distribution of the initial couplings of our chains evolve throughout the RG process. That would allow us to take a more mature look at the infinite randomness fixed point concept again. 2.1.1 Random singlet phase In the random singlet phase (RSP), the ground state of the system is factorized into pairs of singlets that are forming over arbitrary long distances[FIG 2.1]. We are studying random antiferromagnetic spin chains for which the ground state can be assumed to be made out of spin couples that are locked with each other in separate singlets that are forming over arbitrary long distances. The fact that these singlets can form between spins far away is very intriguing and it might come strange at the beginning since we only have the nearest neighbor couplings in our chain. Though, the idea is that the singlets with shorter length scales are mediating between the spins far away and that is how these long distance singlet form. One should notice that this picture is an approximation of the exact ground state and not exactly the ground state itself. Though, the beauty of RG is that 9 Figure 2.2: In this picture we have shown the segment for which we want to calculate the entanglement for (M) with the red box. For example here, 3 singlets crosstherightboundaryofM and 2crosstheleftone. Therefore, it’sentanglement with respect to the rest of the system would be S M = 5 ln(2). when it comes to the universal features of a model, the mistakes that one might make by undermining the details of the model at the initial steps of RG are going to vanish as the RG time goes forward. In other words, assuming that the ground state of our chain is factorized into singlets is not going to be a problem if we look at RG asymptotically [12]. Now that we are familiar with the structure of the ground state, knowing that the entanglement between one spin 1 2 and the other spin 1 2 in an entangled pair is ln 2 to calculate the entanglement between a segment of the system and its compliment one can simply count the number of singlets connecting the two subsystems and multiply that by ln 2 [FIG 2.2]. At this point the RG scheme comes very handy for counting the singlets. The code which calculates the entanglement entropy by counting the singlets can be found in A.2. 2.1.2 Renormalization group rules The RG scheme is basically as the following: First, we find the strongest bond and we treat the corresponding spins with that bond in between as a singlet. The 10 Decimation 1 2 3 4 1 2 4 3 Figure 2.3: In each RG step we find the largest bond 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. Here J 2 is the largest bond and we replace J 1 , J 2 , and J 3 by J eff . neighboring bonds on the right and left of the strongest bond are most probably very small in comparison to that bond in the middle and by using second order perturbation theory one can find an effective interaction between the spins of those neighboring bonds and replace the strongest bond alongside with its neighboring bonds with the new effective bond. This well known calculation was developed by Ma and Dasgupta [21, 5]. This process is called decimation and schematically it can be shown as (...,J i−1 ,J i ,J i+1 ,...) N → (..., J i−1 J i+1 2J i ,...) N−2 in which N is the length of the chain before decimation and J i is the strongest bond. From perturbation theory one can find that the effective bond which replaces J i−1 , J i , andJ i+1 isJ eff = J i−1 J i+1 2J i [FIG 2.3]. To derive this we use the notation of figure 2.3. The biggest bond is J 2 its neighboring bonds are J 1 and J 3 which we can safely assume that are small in comparison to J 2 . The unperturbed Hamiltonian would be H 0 =J 2 S 2 ·S 3 in which S i s are spin vector operators. The eigenvectors of H 0 are one singlet|si = 1 √ 2 (|↑↓i−|↓↑i) and three triplets|ti =|↑↑i, 1 √ 2 (|↑↓i +|↓↑i), |↓↓i. The corresponding energies are E s =− 3 4 J 2 and E t = 1 4 J 2 . Since we are studying the system in very low temperatures as explained before we are going 11 to assume that a singlet forms between spins 2 and 3. We treat the rest of the chain as perturbation. The only terms that are going to be involved with our perturbation calculation are the terms that come from the couplings between spins 1 and 2 and also spins 3 and 4 and we have V = J 1 S 1 ·S 2 +J 3 S 3 ·S 4 to be our perturbative part. Our goal is to find the corrections to the ground state energy of the hamiltonian H 0 after adding perturbation. First thing to notice is that the first order of perturbation is simply zero since E (1) s =hs|V|si = 0. Therefore, the first correction toE s either comes from the second order of perturbation or higher. We are going to calculate the second order correction which would be P t |hs|V|ti| 2 Es−Et in which we are summing over the triples 2 : V|↑↑i = J 1 2 S 1 · (|↓↑i ˆ x +i|↓↑i ˆ y +|↑↑i ˆ z) + J 3 2 (|↑↓i ˆ x +i|↑↓i ˆ y +|↑↑i ˆ z)·S 4 V|↓↓i = J 1 2 S 1 · (|↑↓i ˆ x−i|↑↓i ˆ y−|↓↓i ˆ z) + J 3 2 (|↓↑i ˆ x−i|↓↑i ˆ y−|↓↓i ˆ z)·S 4 V|↑↓i = J 1 2 S 1 · (|↓↓i ˆ x +i|↓↓i ˆ y +|↑↓i ˆ z) + J 3 2 (|↑↑i ˆ x−i|↑↑i ˆ y−|↑↓i ˆ z)·S 4 V|↓↑i = J 1 2 S 1 · (|↑↑i ˆ x−i|↑↑i ˆ y−|↓↑i ˆ z) + J 3 2 (|↓↓i ˆ x +i|↓↓i ˆ y +|↓↑i ˆ z)·S 4 Therefore: hs|V|↑↑i = 1 √ 2 [ 1 2 J 3 (ˆ x +iˆ y)·S 4 − 1 2 J 1 S 1 · (ˆ x +iˆ y)] hs|V|↓↓i = 1 √ 2 [ 1 2 J 1 S 1 · (ˆ x−iˆ y)− 1 2 J 3 (ˆ x−iˆ y)·S 4 ] hs|V|↑↓i = 1 √ 2 [ 1 2 J 1 S 1 · ˆ z− 1 2 J 3 ˆ z·S 4 ] hs|V|↓↑i = 1 √ 2 [ 1 2 J 1 S 1 · ˆ z− 1 2 J 3 ˆ z·S 4 ] 2 For simplicity we set~ to 1. 12 From which we can finally calculate P t |hs|V|ti| 2 Es−Et by knowing that E s −E t =−J 2 : X t |hs|V|ti| 2 E s −E t = −3 16J 2 (J 2 1 +J 2 3 ) + J 1 J 3 2J 2 S 1 ·S 4 So, we can take the first term −3 16J 2 (J 2 1 +J 2 3 ) out of the Hamiltonian and alongside with E (0) s the perturbed energy would be E 0 s = −3 4 J 2 − 3 16J 2 (J 2 1 +J 2 3 ). We keep the new term J 1 J 3 2J 2 S 1 ·S 4 in the Hamiltonian which would be a new effective bond J eff between spins 1 and 4. Here we derived the RG rules for a chain with no external field. Though, one can introduce an external field to the problem and to do RG again one needs to find the biggest term of the Hamiltonian at each step and decimate it. In the presence of the external field the biggest term can be the biggest on site field or the biggest bond. At each step of RG one has to choose the biggest term between all the possibilities. A random transverse field Ising spin chain is an example of that situation which has been studied by Daniel Fisher [11]. 2.1.3 Renormalization group flow By now we know how to derive the RG rules and we know how to do decima- tions. In each RG step we are decimating a bond and introducing a new bond to the chain. This means that at each step of RG the distribution of the couplings changes a bit. An intriguing and important question that one can ask is that how does this distribution changes as the RG process goes forward. It is important to know the answer to this question since ultimately we are interested in calculating some averages that are directly related to the distribution of couplings. In each RG step we exclude the largest bond from the hamiltonian alongside with its two neighboring bonds and instead of them we introduce a new effective 13 bond. To keep track of the statistics of this process first we define new variables to make the situation a bit nicer. We had J eff = J i−1 J i+1 2J i . Though, we are more comfortable dealing with linear equations. To get equivalent linear equations one can start by defining Γ = ln Ω 0 Ω to be the RG time parameter in which Ω 0 is the Hamiltonian’s initial energy scale or equivalently Hamiltonian’s largest bond before any decimation and Ω is the reduced energy scale. Furthermore, one can define β = ln Ω J . And since J eff = J i−1 J i+1 2J i , equivalently we have β eff =β i−1 +β i+1 in which a ln 2 factor has been neglected. Keep in mind that both β i−1 and β i+1 are rather big in comparison to ln 2 because bothJ i−1 andJ i+1 are rather small in comparison to J i , so we can safely neglect the ln 2 factor. Now, if we define P Γ (β) to be the distribution of the bonds at RG time Γ the fallowing flow equation can be derived [5, 27]: ∂P Γ (β) ∂Γ = ∂P Γ (β) ∂β +P Γ (0) Z dβ 1 dβ 2 P Γ (β 1 )P Γ (β 2 )δ(β 1 +β 2 −β) (2.1) The first term on the right hand side accounts for the change in the definition of β as the RG time goes forward. The change in the definition happens because Ω is changing and in each RG step it is less than before. So, we need to divide each J by a new Ω to get its equivalent newβ. The second term (integral) on the right hand side accounts for the replacement of the effective bonds after decimations. The fallowing distribution is a fixed point of the RG flow equation and it can be directly checked by substituting it in equation 2.1: P (β) = 1 Γ e −β/Γ (2.2) 14 This fixed point distribution is the infinite randomness fixed point distribution (IRFP) that we mentioned earlier in the introduction 1. Each decimation intro- duces a new bond which is less than half of the bonds that are getting eliminated from the chain. i.e. J eff = J i−1 J i+1 2J i < J i−1 2 , J i+1 2 which is the reason for the fact that the distribution of bonds becomes broader and broader. This means that the disorder is increasing throughout the process of RG towards the IRFP. Any initial distribution with any small amount of disorder would lead to the fixed point of the flow equation, IRFP, after enough number of RG steps. In the RG language disorder is a relevant variable. 2.2 Exact Diagonalization As mentioned before, we are interested studying random antiferromagnetic spin chains. Considering Jordan-Wigner transformation, one can study the equivalent fermionicsysteminstead. TheHamiltonianofthefermionicchainwedoournumer- ics for is a simple tight binding Hamiltonian: H =− X t jk c † j c k in which c † j and c k are creation and annihilation operators and the t jk coefficients come from τ. We choose a subset of sites ,M, of the chain and we calculate the entanglement entropy for that. By definition of von Neumann entropy we have: S(M) =−tr(ρ 2 log 2 (ρ 2 )) 15 In which ρ 2 = tr chainrM (|ψihψ|) is the density matrix of the segment M with the degrees of freedom of the compliment of M traced out. Instead of finding the entanglement entropy directly by using the density matrix we can find the correlation matrix ,C, first and use it to calculateS(M): C ij =hgs|c † i c j |gsi S(M) =−tr[C log 2 (C) + (1−C) log 2 (1−C)] This approach is previously studied and its methodology is well known [24, 25, 20]. Althoughthefocusofthisthesisisnotthetimeevolutionofsystems, wedevelop ournumericalmethodsuchthatitcanbeusedtostudyquenches 3 aswell. Labeling the tight-binding Hamiltonian before and after the quench asH 0 andH, we have: t< 0 :H 0 =−c † τ 0 c t≥ 0 :H =−c † τc We diagonalize τ 0 and τ with matrices U and K respectively: −Uτ 0 U † =E 0 −KτK † =E And we can define our new fermionic operators as: d :=Uc⇒H 0 =d † E 0 d 3 Quenches are sudden changes in the Hamiltonian. 16 f :=Kc⇒H =f † Ef We assume that the system before the quench (t< 0) has been in the ground state of the initial hamiltonian ,|gs 0 i. We have: C ij (t) =hgs 0 |e ıHt c † i c j e −ıHt |gs 0 i =hgs 0 |e ıHt X k f † k K ki X l K † jl f l e −ıHt |gs 0 i = X k,l K ki K † jl hgs 0 |e ıHt f † k e −ıHt e +ıHt f l e −ıHt |gs 0 i = X k,l K ki K † jl hgs 0 |f † k (t)f l (t)|gs 0 i But, for f l (t) we have: d dt f l (t) =ıe ıHt [H,f l ]e −ıHt &[H,f l ] =−E l f l ⇒f l (t) =e −ıE l t f l &f † l (t) =e ıE l t f † l And, by f l we mean f l (0). Therefore, we get: C ij (t) = X k,l K ki K † jl e ıE k t e −ıE l t hgs 0 |f † k f l |gs 0 i We assume we have N electrons. Therefore, the ground state is: |gs 0 i =d † 1 ···d † N |0i 17 And, we also have: Kc =f⇒KU † d =f⇒f l = X q (KU † ) lq d q &f † k = X p d † p (UK † ) pk Therefore: C ij (t) = X k,l,p,q K ki K † jl e ıE k t e −ıE l t (UK † ) pk (KU † ) lq h0|d N ···d 1 d † p d q d † 1 ···d † N |0i We define θ as the fallowing: θ pq :=h0|d N ···d 1 d † p d q d † 1 ···d † N |0i And, it is easy to see thatθ pq =δ pq iffp,q≤N and it is zero otherwise. Therefore, C ij (t) = (K † e −ıEt KU † θUK † e ıEt K) ji i,j∈M And, since the transposition ofC could be used in the same formula to calculate entanglement entropy we can swipe i and j. Therefore: C ij (t) = (K † e −ıEt KU † θUK † e ıEt K) ij i,j∈M In the fallowing we are not concerned about the quench and we will study systems at time t = 0. Therefore,H 0 =H and also U =K. Knowing these we get: C ij (t) = (K † θK) ij i,j∈M 18 To calculate the entanglement entropy for disordered systems we need to find the average of different realizations. The general process for a situation that we have an impurity in our chain would be as the fallowing 4 : • We define the Hamiltonian asH =− P t i c † i c i+1 +h.c.. • We choose all the ’t i ’s from our preferred random distribution 5 , regardless of if we are choosing the impurity link or not. i.e.∀i :t i ∈ [0, 1]. • Then we multiply the hopping link in the middle 6 byλ, which is the impurity strength. i.e. t middle −→λt middle . • We use the formalism explained to calculate the entanglement entropy for section M. • We repeat the calculation for a number of realizations with the same λ and M and a different set of ’t i ’s from the same distribution. and we find the average of entanglement entropy. The code that does this calculation can be found in the appendix A.1. 2.3 Numerical Subtleties Thereareafewnumericalpointsworthmentioningsinceitisimportanttoknow the limit of our machinary. First, we noticed that the diagonalization algorithms 4 The details can change depending on the setup we are studying. For example different impurity types and different segments of M can be studied. 5 Box distribution, Infinite randomness fixed point (IRFP) distribution. 6 Here we have explained single-impurity but the impurity can be of different kinds as we will see in the upcoming material. 19 Figure 2.4: Single impurity with λ = 0. startbecominglessaccurateprettyfastafteraspecificlengthwhichdependsonthe strength of impurity. Typical length range that we can safely monitor is about 500 sites and it goes up to 800 sites for bigger λs 7 . We can monitor this misbehavior by monitoring the returned eigenvalues and checking if they are symmetric. In our code in A.1, sigmacoma is the parameter which keeps track of this. We can improve the situation by choosing the libraries that are more efficient with our problem to use in our code (Fig 2.4 and 2.5) 8 . Though, to increase the length range of the reliable results even further one can do the numerics with RG. And, as long as we are looking for universal behaviors the claim is that RG grasp those characters. One more limit of our machinery comes from the lack of reliability of our slope extraction method at the ends of the data. Our raw data is noisy and even small 7 λ is the strength of impurity that we are going to study in the upcoming chapters. Strictly speaking, the lengths for which we get reliable results both depend on the size of the matrices that we are diagonalizing and the amount of disorder. 8 From these figures it is obvious that the sparse matrix libraries are giving better results for our problem. 20 Figure 2.5: Single impurity with λ = 1. amounts of noise can drastically change the slope locally, unless we somehow fit a smooth curve to our whole data. We do so by fitting a polynomial of low order. This method is reliable when we are not very close to the ends of our data. 21 Chapter 3 Single-Impurities Introducing disorder to the Kondo physics leads to behaviors that address an area of active research and unanswered questions in the condensed-matter physics commu- nity [3]. This chapter is mainly explaining the ideas in our paper [32] which is an attempt to probe this area. We are going to introduce the concept of healing by studying random antiferromag- netic spin chains with a single-impurity in the middle. 3.1 Background The topic of quantum impurities has been of interest of different physics com- munities and especially condensed matter physicists for more than the last thirty years. The development of technology has made it possible for people to architect single-channel nano-structures. Introducing Coulomb blockade to these systems have yielded intriguing behaviors [22]. Furthermore, solid statements can be used to describe the behavior of these one dimensional systems in comparison to the systems of higher dimension. These very facts alongside some others have been enough motivation for the community to tackle the problem of one dimensional chains with and without impurities. As a very simple example of quantum impurity problems one can imagine a potential scatterer in a Fermi liquid for which, phenomena such as Anderson orthogonality catastrophe can be observed [1]. The simplest situation that would 22 be consider as an example for the potential scatterer in 1D might be the tight- binding model with nearest neighbor hoppings for which a bond is modified -A modified bond is basically a bond which is multiplied by a weak factor λ. The interactions between electrons in a Luttinger liquid at low energy through a weak bond is profoundly different than its counterpart in a Fermi liquid. When the interactions are repulsive, even with a very tiny obstacle (which means 1− λ 1) the electrons get reflected from the scatterer and at temperature T = 0 the conductance vanishes. Though, interestingly for attractive interactions the electrons perfectly transmit through the scatterer even when the scatterer is pretty strong (λ 1) at temperatureT = 0 [18, 17]. The problem is well studied [23, 10, 30, 7] and it can be understood using RG. The system shows a transition between two fixed points, one is the situation in which the system acts as if it is cut in the middle and the other is the situation in which the system acts as if there is no cut and it is called healed [8]. Now, an interesting situation to study is the same model with disorder. From Jordan?Wigner transformation we know that instead of the fermionic chains we can study the equivalent spin chains and in this case we have to study a XXZ antiferromagnetic spin chain with a weak bond: H = −1 X i=−L−1 J i (S i ·S i+1 ) Δ +λJ 0 (S 0 ·S 1 ) Δ + L−1 X i=1 J i (S i ·S i+1 ) Δ in which S is a spin 1 2 and (A·B) Δ = A x B x +A y B y + ΔA z B z and all the J i s including J 0 are chosen from the same random distribution. Also, λ < 1 is the parameter which sets the strength of the weak bond in the middle. Unlike what we saw before, that the type of interactions (attractive, repulsive) would change the behavior of the system, here for all the 0≤ Δ≤ 1 values the healing occurs. 23 3.2 Healing For a system in its pure state |ψi the von Neumann entropy or entangle- ment entropy of a subsystem M with the rest of the system ¯ M is defined as S =− tr(ρ M lnρ M ) in which ρ M = tr ¯ M |ψihψ|. It has been shown that for one dimensional quantum critical spin chains with no disorder the entanglement of a segment of length l 1 with the rest of the chain scales proportional to lnl and the prefactor is defined by the central charge of the field theory associated to the model [16, 33]. The effect of introducing disorder to these chains might be feared to wipe out all the interesting physics of the problem. It turns out that it is not the case and in the existence of disorder the entanglement of a segment of length l 1 with the rest of the chain still scales logarithmically but with a different effective central charge [27]. The key to understand this behavior is through the random singlet phase (RSP) which we studied in the second chapter. We mentioned that the RG scheme is asymptotically correct and regardless of the initial distribution of the bonds 1 after enough number of RG steps the distribution is going to be close enough to the infinite randomness fixed point distribution (IRFP) 2.2: P (β) = 1 Γ e −β/Γ We can assume that we have taken enough number of RG steps that the bonds distribution is already at the fixed point distribution. Having that assumption, one can calculate the rate at which the bonds are getting decimated. It is important to notice that at each decimation step the bond at the very beginning of the 1 In fermionic systems ‘hoppings’ equivalently. 24 distribution with β = 0 is getting decimated. Also, it can be calculated that the averagenumberofRGstepsneededfortwosequentialdecimationsthatarecausing the formation of effective bonds over a specific bond is 3 [27]. In other words, on average it takes 3 RG steps for one singlet to form over a specific bond in the chain. Knowing all of these pieces, one can conclude that d ¯ N = 1 3 P (β = 0)dΓ = 1 3 dΓ Γ in which ¯ N is the average number of bonds (singlets) over a specific bond. This results in ¯ N = 1 3 ln Γ. Now, we want to count the number of singlets forming over the boundaries of our segment in order to find the total number of singlets connecting segmentM and ¯ M. We let RG go forward and we count the number of singlets connecting the two segments. This process breaks down when the length of the singlets exceed the size of the segment of our concern, M. And, for random singlet phase it is known that the length scale of the singlets grow proportional to Γ 2 . Therefore, if we assume that the length of the segment of our concern isL, the number of singlets between the inside and outside of the segment stops growing when the RG parameter is Γ = L 1/2 . Therefore, we have ¯ N = 1 2 1 3 lnL. Using this picture it is now obvious that the entanglement between half of a disordered critical chain of total length 2L with closed boundary conditions with the other half is S half−chain = (2 1 2 1 3 lnL) ln 2 = ln 2 3 lnL because the segment of our concern in this setup has two boundaries (Fig 3.1). Now, if the bond in the very middle of the chain is missing, J middle = 0, then the singlets form over just one end of the segment andS half−chain = ln 2 6 lnL (Fig 3.2). We continue by setting the size of the segmentM to beL and the size of the whole system to be 2L. Our system still is a random antiferromagnetic spin chain with closed boundary conditions, though, instead of the two extreme cases that we just studied, one can first choose all 25 Figure 3.1: Singlets form over both ends of our segment (red box). the couplings from a random distribution 2 , D(x), and then multiply the coupling in the middle of the chain by a factor, 0 6 λ 6 1 which means J middle = λr in which r is chosen from D(x). For simplicity we call the middle coupling J middle the weak bond. Introducing a weak bond to the problem is basically changing the distribution of the middle bond and keeping the distribution of the other bonds unchanged. The effect of this weak bond can be understood through the decimations of the RG process. Before the weak bond gets decimated there are no singlets forming over the weak bond as if the system is cut in the middle and one expects the entanglement to scales with length as S half−chain = ln 2 6 lnL. As soon as the weak bond gets decimated singlets start forming over the weak bond and the entanglement’s scaling function becomes S half−chain = ln 2 3 lnL. More precisely, the weak bond just shifts the RG time and the decimations that are causing effective couplings to form over the weak bond start later. Though, as soon as the decimation process over the weak bond starts it goes on with the rate that we mentioned before, d ¯ N = 1 3 dΓ Γ . This behavior is what we call the crossover behavior and healing 3 . Moreover, we expect the crossover energy in the RG to 2 We have tried box distribution and IRFP 3 By now, the reader must be convinced that a very useful parameter to monitor throughout RG is the average number of singlets forming over a specific bond and another one is the average energy scale. We have a code in appendix A.3 which keeps track of these parameters. 26 Figure 3.2: No singlets form over the middle bond which is missing. Singlets only form over the left edge of our segment (red box). correspond to Γ∼− lnλ. And, we already know that at the crossover we have Γ =L 1/2 . Therefore, for the crossover length we have L∼ (− lnλ) 2 . We show the crossover length by L ? and L ? ∼ (− lnλ) 2 . It can be shown that in the scaling limit Γ→∞, λ→ 0 with Γ lnλ fixed there exists a universal function f such that ∂S ∂ lnL =f( L L ? ) in which lim L→0 f( L L ? ) = ln 2 6 and lim L→∞ f( L L ? ) = ln 2 3 [32]: L ∂S ∂L = ln 2 3 1− 2 π ∞ X n=0 (−1) n 2n + 1 e −π 2 (2n+1) 2 L/4L ? ! . (3.1) Even though the equation 3.1 has been derived for the bipartite entanglement of a system of length 2L, for when we calculate the entanglement of half of the chain with respect to the other half,L can be replaced by` for a subsystem of size` with the impurity at its boundary, with the condition that 1`L. We confirm this with numerics (Fig 3.3) in our paper [32]. 3.3 Some Numerical Checks In [32] we directly check this healing behavior in the scaling regime numerically for a random XX chain with a weak link in the middle for which the couplings are 27 Figure3.3: Thedotsshow thenumericalRGcalculation ofa segment oflength` 2L∼ 4000 with an impurity link with strength λ at its edge in a periodic system. We have for the crossover length` ∗ ∼ (lnλ) 2 . We have shown the equation 3.1 with theblackdashedlineandthegreendashedlineshowsequation3.1withcorrections. The inset shows the scaled bipartite entanglement of a periodic system. chosen from a uniform box distribution. The method we use is based on Jordan- Wigner transformation, i.e. instead of studying the spin chain problem we study the corresponding fermionic chain problem in which the dimension of the matrices thatwehavetodothecalculationforareverymuchreduced 4 . Inthisapproachone needs to calculate the correlation matrix for the segment that we want to calculate the entanglement for,hc † i c j i in which i and j run over the segment sites from 1 to L. We use this method and numerically find different ∂S ∂ lnL versus L curves for different weak link,λ, values and at the same time by changingL ? in equation (3.1) we try to fit each numerical curve the best possible curve from theory. Therefore, we find one L ? for each value of λ and we can plot them versus each other and check the scaling L ? ∼ (− lnλ) 2 [FIG 3.4]. As it is noticeable from FIG 3.4, the curves from numerics seem not to be perfect. There are a few reasons for that 4 We elaborated this method in section 2.2 28 Figure 3.4: (Numerical determination of the entanglement in random XX chain, taken from [32]) The solid lines in the main frame are the result of numerics and in order to find them we average the entanglement over a large number of realizations and afterwards take the derivative. The dashed lines in the main frame come from theory(eq. 3.1). The inset shows the scaling function for L ? versus (− lnλ) 2 that we extract from the main frame. behavior. The first reason to mention is that we are trying to find the derivative of entanglement with respect to lnL from ourS versusL curves and the derivative is very sensitive to error. So, first we try to fit smooth curves to our numerical S versusL data points. In order to do that we fit polynomials of low orders, such as 5, to the data points and then find the derivative of the polynomials. This causes the lower and upper tails of our derivative curves to diverge from the theoretical curves. The other reason is that the diagonalization algorithms of the functions that we use in our code are not stable for very long system sizes. And lastly for very short distances one should not forget possible finite size effects. 29 3.4 Direct Diagonalization versus Renormaliza- tion Group, Box versus Infinite Randomness Fixed Point Distribution As a final note to this chapter we are going to investigate how our results from the two methods of direct diagonalization and RG differ from each other 5 . We will also briefly explore the difference between the entanglement entropy in our finite systems when the bonds (or equivalently hoppings) are initially chosen from a box distribution versus when they are chosen from IRFP. Fig [3.5] shows the bipartite entanglement entropy calculated via direct diag- onalization and RG for a chain with couplings chosen from a box distribution. Fig [3.6] shows the derivative of Fig [3.5] curves. There is a correlation between the slope curves coming from RG and direct diagonalization but still they do not match completely. Finite size effects as well as slope extraction errors are amongst the reasons of the divergence between the two methods. To explore the difference between IRFP distribution and Box distribution we do the numerics for both RG and direct diagonalization for the specific case of λ = 10 −2 with different initial distributions Fig [3.7]. There are a few points worthy of noticing in Fig [3.7]. First, in the RG curves the curve that is coming from IRFP is already healed but the box distribution curve starts in the unhealed regime and ends up in the healed regime (Please notice that the slope of the fitting 5 Please notice that these differences are not contradictory since RG’s claim is that RG is successful in grasping the universal characteristics of the system. In our system the universal characteristics mostly show up in the derivative graphs and we see a better agreement in the slope graphs from different methods. 30 10 1 10 2 10 3 Total Length 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 S a Single Impurity, RG vs Direct Diagonalization numerics, disordered chain with couplings from box distribution, closed B.C. λ =1.0E−1, RG λ =1.0E−2, RG λ =1.0E−3, RG λ =1.0E−4, RG λ =1.0E−1, DD λ =1.0E−2, DD λ =1.0E−3, DD λ =1.0E−4, DD Figure 3.5: Bipartite entanglement from RG vs Direct Diagonalization - (S vsN) curve of the red dots change). The reason is that by increasing the initial disorder we are basically shifting the RG time forward and the RG starts from a situation closer to the healed regime. Second, in the direct diagonalization (DD) curves the curve from IRFP starts diverging from the box distribution curve rather soon. The reason must be the limitations of numerics for the direct diagonalization method. In section 2.3 we mentioned that for long enough chains we always observe this divergence of DD from the theory and the reason is that the diagonalization algo- rithms do not work properly for long enough chains. Now, adding more disorder to the system pushes the system closer to the region that the direct diagonalization numerics does not work properly anymore. The important conclusion is that the direct diagonalization numerics are limited by the system length and the amount of disorder. 31 10 1 10 2 10 3 Total Length (N) 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 S a / lnN Single Impurity, RG vs Direct Diagonalization numerics, disordered chain with couplings from box distribution, closed B.C. λ =1.0E−1, RG λ =1.0E−2, RG λ =1.0E−3, RG λ =1.0E−4, RG λ =1.0E−1, DD λ =1.0E−2, DD λ =1.0E−3, DD λ =1.0E−4, DD Figure 3.6: Bipartite entanglement from RG vs Direct Diagonalization - ( ∂S ∂ lnN vs N) 10 1 10 2 10 3 Total Length 0.5 1.0 1.5 2.0 2.5 3.0 3.5 S a Single Impurity, λ =10 −2 , disordered chain, closed B.C. RG, IRFP(Γ =10) DD, IRFP(Γ =10) RG, Box DD, Box Figure3.7: BipartiteentanglementfromRGvsDirectDiagonalizationforλ = 10 −2 - (S vs N) 32 Chapter 4 Quantum-dot Impurities In this chapter we keep studying XX spin chains with closed boundary conditions, though instead of a single-impurity in the middle, now we have two impurity bonds next to each other. We examine the situation with two different types of q-dots, symmetric and asymmetric. In the case of symmetric q-dots, we first choose all the bonds from a random distribution and we also choose the two middle bonds to be the same and we multiply them by the weak factor λ afterwards. For asymmetric q-dots though, first we choose all the bonds from a random distribution separately and afterwards we multiply the two middle bonds by the weak factor λ. That way the impurity bonds are not necessarily the same anymore. 4.1 Kondo physics 4.1.1 Motivation There are different mechanism that are associated with the scattering of the conduction electrons in metals which explain the behavior of resistivity versus tem- perature. One of these mechanisms that successfully explains the baffling problem of existence of a minimum resistivity at low temperatures [6] due to magnetic impurities is the Kondo effect [19]. In condensed matter physics and especially for many-body systems the scattering from impurities that allow low energy quantum 33 Figure 4.1: (Taken from [6]) The Electrical resistance of Hilger gold versus tem- prature which shows a a minimum resistance at about 6K. degrees of freedom show analogies to the original Kondo effect and fall into the category of Kondo physics. 4.1.2 Minimum resistivity At first glance, one would expect to observe a monotonic increase in the resis- tivity of a metal with the increase in temperature as the result of electrons getting scattered by the nuclei due to increasing lattice vibrations. Though, in 1936 WJ De Haas and GJ Van den Berg observed that there exists a minimum resistivity as they decreased the temperature in the metals they experimented with (Fig 4.1). This remained a puzzle until almost thirty years after that it was discovered that the resistance minimum is associated with the magnetic impurities. A few years later in 1964 Jun Kondo successfully discovered the detailed scattering mechanism 34 that was explaining the resistance minima. Prior to Kondo people used to only account for the first order scattering of the electrons from the magnetic impurities. Kondo studied the second order of the scattering as well in which first the spin of the electron flips in an intermediate step and then it flips again and comes back to normal. It turned out that the contribution of the new correction to scatter- ing which was appearing as a logarithmic term was responsible for the minimum resistivity. 4.1.3 Screening Kondo successfully explained the resistance minima, though, the logarithmic term that he suggested was diverging for temperatures approaching zero, T→ 0. The additional mechanism that fixes this problem is the screening of the magnetic impurity by the conduction electrons which leads to a finite resistance atT = 0 as we expect. The conduction electrons in the metal create a screening cloud around the magnetic impurity that we can look at from different distances. If we look at the impurity from close we see the impurity structure and as we get farther the impurity get lost in the screening cloud as if there has not been an impurity there. This behavior is the parallel of healing in systems with a magneitc impurity instead if a potential scatterer (or a weak bond as we studied in the previous chapters). 4.2 Quantum dots Quantumdots(q-dots)aresemiconductornanostructuresmadeoutofconfined electrons or electron holes. They act like atoms and they have discrete electronic states. When a q-dot is connected to electron baths it can transmit electrons 35 Clean case) Symmetric q-dot) Asymmetric q-dot) Figure 4.2: In the top picture both impurities are set to λ. In the middle picture the two impurities are the same andλ is the weak factor like before andr is chosen from a random distribution. In the bottom picture r 1 and r 2 are chosen from the same distribution and separately, but we multiply both by the same weak factor λ. through if it has states available close to the Fermi level. One would expect to observe Kondo related physics for q-dots in contact with nano wires. In the rest of this chapter we are going to explore q-dot impurities in one dimensional fermionic chains at half-filling or equivalently one dimensional antiferromagnetic spin-chain. A simplified version of a q-dot in a spin-chain is a site with two weak couplings on its right and left. It can be either clean, symmetric, or asymmetric (Fig 4.2). One can calculate the entanglement for different segments as well. In figure 4.3 one can find the situations which we call case a, case b, and case c. 36 In the rest of this chapter first we are going to focus on cases b and c and in specific we are going to studyS c −S b with the idea of extracting the entanglement associated with the central impurity to get a better understanding of if the Kondo effect still takes place. By subtracting the entanglement of segmentb from segment c we basically eliminate the mutual parts of the entanglement of the two geome- tries and what is left is going to be the entanglement between the impurity and the segment of the case c geometry subtracted by the entanglement between the impurity and the compliment of the segment in the geometry of case b (impurity itself excluded) 1 . For the clean case by scaling S c (λ)−S b (λ) for different λs and extracting the crossover length L ∗ we find that L ∗ ∝λ −2 which is the behavior of Kondo physics. As we are going to see by the end of this chapter, our q-dots (double-impurities) exhibit features similar to the single-impurities that we studied in the last chapter, though, the behavior of q-dots is richer, and we observe new intriguing phenomena for them 2 . 4.2.1 Clean chains - case b and c In this section we are going to numerically study case b and c for a non- disordered chain. All the bonds except for the impurities are set to 1 and the q-dot impurities are both λ≤ 1. The only difference between cases b and c is that the very middle spin is included in the segment for which we are calculat- ing the entanglement for in the former one. We are interested in the difference 1 As the system size goes to infinity the second part vanishes. 2 Such as the emergence of bumps in the slope graphs which we will see at the end of this chapter. 37 (N-1)/2 (N+1)/2 (N-1)/4 (N-1)/4 Case a) Case b) Case c) Figure 4.3: The red boxes show the segment for which we are calculating the entanglement. Please notice that the segment do not need to be connected and any subset of the whole chain makes a segment.The length of the whole chain is N in all the cases and each half-chain on each side of the q-dot is (N− 1)/2. between the two situations. Using exact diagonalization (direct diagonalization) we get the entropies, the collapsed slope curves and we extract the scaling law (Figs 4.4,4.5,4.6). The scaling behaves as L ∗ ∝ (λ) −2 which is quit different than L ∗ ∝ (lnλ) 2 . A point worthy of mentioning is that the behavior of the λ = 1 curve in figure 4.4 seem to be a bit strange since the points forλ = 1 are scattered around ln 2. The reason for this scattering is the degeneracy in the ground state of the hamiltonian when all the bonds are equal to one (including the impurity bonds). By doing the same calculation for λ = 1 over and over and taking the average the scattering vanishes and we get a smooth curve for λ = 1. What is 38 10 2 10 3 10 4 Total Length(N=2L+1) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 S c −S b Q-dot, clean system, (total length-1)/2-filing, closed B.C. ln2 λ =1 λ =0.5 λ =0.2 λ =0.1 λ =0.05 λ =0.03 λ =0.015 λ =0.013 λ =0.012 λ =0.011 λ =0.01 λ =0 Figure 4.4: No disorder, direct diagonalization happening is that every time that we run the numerics for the clean system the algorithm chooses a random ground state in the subspace of ground states to do the calculation for and by taking the average we can get rid of the scattering. Though, I kept the numerics for λ = 1 as it is to illustrate this subtlety. 4.2.2 Disordered chain - Symmetric q-dots - Case b and c Now, weintroducedisordertothesystemandwechooseallthehoppingsexcept for the impurities from the unit box distribution. Then we pick another random number from the same box distribution and multiply it by λ and set both of the impurity bonds to be equal to that so the q-dot impurity would be symmetric. The numerics of this section are done with exact diagonalization method. 39 10 0 10 1 10 2 10 3 10 4 10 5 Total Length/L ∗ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 S c −S b Q-dot, clean system, (total length-1)/2-filing, closed B.C. λ =0.5,L ∗ =0.045 λ =0.2,L ∗ =0.26 λ =0.1,L ∗ =1 λ =0.05,L ∗ =3.7 λ =0.03,L ∗ =10 λ =0.015,L ∗ =40 λ =0.013,L ∗ =51 λ =0.012,L ∗ =62 λ =0.011,L ∗ =73 λ =0.01,L ∗ =85 Figure 4.5: No disorder, direct diagonalization First, wecollapsetheS c (λ)−S c (0) versuslengthandS b (λ)−S b (0) versuslength graphs and we observe that the scaling behaves as L ∗ ∝ (lnλ) 2 . Therefore, we expect to see the same scaling behavior for S c (λ)−S b (λ) versus length because: S c (λ,L)−S b (λ,L) = [S c (λ,L)−S c (0,L)]− [S b (λ,L)−S b (0,L)] And we indeed observe that from numerics (Fig 4.7). We notice that even when the collapse of the curves is not very good and the points are rather scattered, we still observe a rather nice scaling behavior. We observed the scaling ofL ? ∝ (lnλ) 2 for symmetric q-dots. A natural question to ask would be: Are we going to observe the same scaling if we replace the symmetric q-dot with an asymmetric one? 40 10 -2 10 -1 λ 10 -1 10 0 10 1 10 2 L ∗ Clean Chain lnL ∗ =−1.94lnλ−4.46 r 2 =0.999942 Figure 4.6: No disorder, direct diagonalization 4.2.3 Disordered chain - Asymmetric q-dots - Case b and c Thistimewechooseallthehoppingsfromarandomboxdistributionlikebefore and then multiply the two impurity bonds by λ. Since the q-dot impurities are chosen separately from the same distribution before getting multiplied by λ we have an asymmetric q-dot now. The numerics is done by the exact diagonalization method again. For our asymmetric q-dot we observe that the scaling behaves the same as before, L ∗ ∝ (lnλ) 2 (Fig 4.8). Next, it is natural to ask how do results from symmetric and asymmetric q- dots compare? From numerics it can be observed that as long asλ is small enough the entanglement for the symmetric and asymmetric q-dot cases are the same (Fig 4.9,4.10 ). 41 0 50 100 150 200 (lnλ) 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 L ∗ Disordered Chain, Symmetric Q-dot Impurity, Scaling curve of the S c (λ)−S b (λ) versus Length L ∗ =0.0180(lnλ) 2 +0.128 r 2 =0.99799 Figure 4.7: Disordered, box distribution, direct diagonalization, symmetric q-dot For the case of asymmetric q-dots 3 we know that the RG rules are the same as what has been explained in section 2.1. Therefore, we can study that specific case with RG in addition to exact diagonalization. For the RG method we choose the bonds from an infinite randomness fixed point (IRFP) distribution instead of a box distribution . Then we scale and collapse the curves . By extracting the scaling law we observe that it is the same as before L ∗ ∝ (lnλ) 2 (Fig 4.11). The numerics from RG and direct diagonalization have different results. Though, we expect RG to grasp the universal features, and we observe that the scaling behaves as before which addresses this point. 4.2.4 Disordered chain - Asymmetric q-dots - Case a Going back to figure 4.3, this time we want to explore case a further. We run our numerics using RG for an asymmetric q-dot. We define half of the chain 3 For the symmetric q-dots the degeneracies of the hamiltonian change the RG rules. 42 0 50 100 150 200 (lnλ) 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 L ∗ Disordered Chain, Asymmetric Q-dot Impurity L ∗ =0.0183(lnλ) 2 +0.102 r 2 =0.99941 Figure 4.8: Disordered, box distribution, exact diagonalization, asymmetric q-dot excluding the very middle spin to be the segment (M) of our concern and we calculate the number of singlets forming between M and the rest of the system ( ¯ M) to get the entanglement (Fig 4.12). Figure 4.13 shows the scaled slope curves and the scaling law is L ∗ ∝ (lnλ) 2 as before (Fig 4.14). The data points at the very end tails of the scaled slope curve (Fig 4.13) are not reliable since they are the artifacts of our slope extraction method 4 . Far from the tail ends it seems that there exists a bump after the crossover and there might exist a small pit before the crossover. We re-examine those two areas and approve the bump after the crossover. Though, there is no pits before the crossover (Appendix B). The physics of the emerging bump is very interesting. We did not observe the bump for the case of a single-impurity but we have it for our q-dot (double- impurity). To understand this phenomenon we fallow the same reasoning line as 4 We fit a polynomial of low order to the raw data and extract the slope of the polynomial which workd good for the mid-range data but is not a good approximation at the tails of our data as has been explained in section 2.3 43 10 2 Total Length 1.2 1.4 1.6 1.8 2.0 S b ( ) Symmetric vs Asymmetric Q-dot, disordered chain, (total length-1)/2-filing, closed B.C. = 0, asymmetric = 0, symmetric = 1, asymmetric = 1, symmetric = 0.1, asymmetric = 0.1, symmetric = 0.01, asymmetric = 0.01, symmetric = 0.001, asymmetric = 0.001, symmetric = 0.0001, asymmetric = 0.0001, symmetric = 0.00001, asymmetric = 0.00001, symmetric = 0.000001, asymmetric = 0.000001, symmetric Figure 4.9: Disordered, box distribution, exact diagonalization, symmetric vs asymmetric q-dot 10 2 Total Length 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 S c ( ) Symmetric vs Asymmetric Q-dot, disordered chain, (total length-1)/2-filing, closed B.C. = 0, asymmetric = 0, symmetric = 1, asymmetric = 1, symmetric = 0.1, asymmetric = 0.1, symmetric = 0.01, asymmetric = 0.01, symmetric = 0.001, asymmetric = 0.001, symmetric = 0.0001, asymmetric = 0.0001, symmetric = 0.00001, asymmetric = 0.00001, symmetric = 0.000001, asymmetric = 0.000001, symmetric Figure 4.10: Disordered, box distribution, exact diagonalization, symmetric vs asymmetric q-dot 44 0 10000 20000 30000 40000 50000 60000 (ln ) 2 0 2 4 6 8 10 L * Disordered Chain, Asymmetric Q-dot Impurity, IRFP( = 10), RG numerics for the scaling of S c S b vs length L * = 1.6 * 10 4 (ln ) 2 + 0.2319 r 2 = 0.9973 Figure 4.11: Disordered, IRFP distribution, RG, asymmetric q-dot Impurity Figure4.12: Theimpurityisspecifiedwithblackdashedboxaroundit.Thesegment for which we are calculating the entanglement entropy is specified with the red ellipse and the singlets between that segment and the rest of the system are shown with red curves. 45 10 1 10 2 10 3 10 4 Total Length/L * (N/L * ) 0.10 0.15 0.20 0.25 0.30 S/ lnN Entanglement between two halves, impurity NOT excluded, asymmetric q-dot, RG, initial disorder 0 = 10, disordered chain, closed B.C. ln2/6 ln2/3 = 10 10 , L * = 0.08 ± 0.01 = 10 20 , L * = 0.20 ± 0.02 = 10 30 , L * = 0.37 ± 0.03 = 10 40 , L * = 0.65 ± 0.05 = 10 50 , L * = 1 = 10 60 , L * = 1.4 ± 0.1 = 10 70 , L * = 1.9 ± 0.1 = 10 80 , L * = 2.5 ± 0.2 = 10 90 , L * = 3.2 ± 0.3 = 10 100 , L * = 3.9 ± 0.3 Figure 4.13: The scaled curves for case a (Fig 4.12) our paper [32] and reference [27]. we have to keep track of the probability of the left impurity bond 5 not being decimated at energy scale Γ, p Γ . The probability distribution for the weak link is defined as Q Γ = R ∞ 0 p Γ (β)dβ and since instead of one weak bond we have two neighboring weak bonds, by the symmetry of the situation we have: ∂Q Γ ∂Γ = ∂Q Γ ∂β +P Γ (0)(P Γ ?Q Γ −Q Γ ) +Q Γ (0)(P Γ ?Q Γ −Q Γ ) (4.1) With a parallel logic to our paper, p Γ vanishes for Γ < Γ ∗ ≡− lnλ which means Q Γ (0) = 0. By using a Laplace transformation and knowing Q Γ = 0, it can be derived that Q Γ=− lnλ = P Γ=− lnλ which remarkably means that the distribution of both of the weak links is the fixed point distribution at the crossover scale 5 since we have a symmetric situation for our impurity bonds it does not matter which bond we pick, here we picked the left one. 46 0 10000 20000 30000 40000 50000 60000 (ln ) 2 0 1 2 3 4 L * Entanglement between two halves, impurity NOT excluded, asymmetric q-dot, RG, initial disorder 0 = 10, disordered chain, closed B.C. L * = 7.31 * 10 5 (ln ) 2 + 0.0288 r 2 = 0.9998 Figure 4.14: The scaling for setup in Fig 4.12 Γ ∗ . i.e Q Γ ∗ = 1 − lnλ e β/ lnλ . Though, in the case of the single-impurity we had Q single−impurity Γ ∗ = β (lnλ) 2 e β/ lnλ which is rather different than our q-dot case. Com- paring the two cases we observe that in the single-impurity case, at the crossover scale Γ = Γ ∗ , the impurity is pretty weak and has a little chance of getting deci- mated right away. In contrast to that, for the q-dot case at the crossover scale the weak link is pretty strong and has the same chance of being decimated as the rest of the chain. Now, in a homogeneous chain for bonds that have been decimated many times already, asymptotically we haveh`i = 3 [27] which means that the average RG time (` = ln Γ/Γ 0 ) for two successive decimations is 3. It can be calculated that the average time before the first decimation is even shorter,h`i = 2. This means that the weak bonds for the q-dot get decimated faster than other bonds in the 47 Decimation 1 2 3 4 1 2 4 3 Figure 4.15 chain for a while which is the origin of the emergence of the bump. With a little bit of calculation this leads to the fallowing formula: L ∂S ∂L = ln 2 3 [1− 1 π ∞ X n=0 (−1) n 2n + 1 (3e −(2n+1) 2 π 2 L/(4L ∗ ) −e −(2n+1) 2 π 2 L/(4e 4 L ∗ ) )] (4.2) To put this into perspective intuitively, for the case of a single-impurity when the impurity bond gets decimated for the first time it becomes rather small and even smaller than the typical strength of the couplings in the rest of the chain and does not get decimated for a while afterwards since J eff = J 1 J 3 2J 2 < J 1 , J 3 (Both J 1 and J 3 are rather small in comparison toJ 2 ) (Fig 4.15). Though, for the q-dot case, in contrast to the single-impurity, after one of the weak bonds gets decimated for the first time, the strength of the new effective bond would be the same as the typical strength of the rest of the chain sinceJ 3 andJ 2 are of the same order of magnitude which leads to J eff ∼ J 1 (Fig 4.16) and this means that the impurity bond gets decimated with a faster pace after its first decimation in comparison to the single impurity. This difference in the pace of decimations of the impurities is the reason for why we observe the bump in the q-dot case and not the single-impurity. 48 Decimation 1 2 3 4 1 2 4 3 5 5 Weak Bonds Figure 4.16 10 2 Total Length 0.8 1.0 1.2 1.4 1.6 1.8 2.0 S a ( ) Symmetric vs Asymmetric Q-dot, disordered chain, (total length-1)/2-filling, closed B.C. asymmetric, = 0 asymmetric, = 1 asymmetric, = 0.1 asymmetric, = 0.01 asymmetric, = 0.001 asymmetric, = 0.0001 asymmetric, = 0.00001 asymmetric, = 0.000001 symmetric, = 0 symmetric, = 1 symmetric, = 0.1 symmetric, = 0.01 symmetric, = 0.001 symmetric, = 0.0001 symmetric, = 0.00001 symmetric, = 0.000001 Figure 4.17: Disordered, box distribution, exact diagonalization, symmetric vs asymmetric q-dot On the other hand, we compared symmetric and asymmetric q-dots for case b and c and we observe that their behaviors are very similar as long as λ is not very close to 1. We want to check the same for case a, and we observe the same situation happens. i.e. as long as λ is not very close to 1 both symmetric and asymmetric q-dots behave very similar 4.17. 49 Inanattemptgainfurtherunderstandingofwhatishappeningtoentanglement entropyin our systems when one introduces the q-dot impurities we canask further questions. For example, for the entanglement of case a interestingly we observed that a bump emerges right after the crossover. One might ask what happens if we break the q-dot impurity into two separated impurities? We have explored this question in appendix B. 4.3 Conclusion for observations The first quick observation from the numerics of this chapter is that despite the fact that some collapse of the curves were a bit scattered and not clean, we always observe rather nice scaling. This addresses the fact that the scaling behavior is pretty robust. Second, we observed that the symmetric and asymmetric q-dots act very simil- lar when λ is not close to 1. And, the last but the most intriguing one is the emergence of the bump in case a geometry. As we explained, the reason for this behavior is that right after the first decimation of one of the weak bonds, the q-dot impurity gets decimated with a faster pace for a while and causes a bump to emerge. 50 Reference List [1] P. W. Anderson. Infrared catastrophe in fermi gases with local scattering potentials. Phys. Rev. Lett., 18:1049–1051, Jun 1967. [2] Pasquale Calabrese. P. calabrese and j. cardy, j. stat. mech.(2004) p06002. J. Stat. Mech., 2004:P06002, 2004. [3] Sudip Chakravarty and Chetan Nayak. Kondo impurity in a disordered metal: Anderson’s theorem revisited. International Journal of Modern Physics B, 14(14):1421–1428, 2000. [4] P. Coleman and A. J. Schofield. Quantum criticality. , 433:226–229, January 2005. [5] Chandan Dasgupta and Shang-keng Ma. Low-temperature properties of the random heisen- berg antiferromagnetic chain. Phys. Rev. B, 22:1305–1319, Aug 1980. [6] WJ De Haas and GJ Van den Berg. The electrical resistance of gold and silver at low temperatures. Physica, 3(6):440–449, 1936. [7] R De-Picciotto, M Reznikov, M Heiblum, V Umansky, G Bunin, and D Mahalu. Direct observation of a fractional charge. Physica B: Condensed Matter, 249:395–400, 1998. [8] Sebastian Eggert and Ian Affleck. Magnetic impurities in half-integer-spin heisenberg anti- ferromagnetic chains. Phys. Rev. B, 46:10866–10883, Nov 1992. [9] J. Eisert, M. Cramer, and M. B. Plenio. Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys., 82:277–306, Feb 2010. [10] P. Fendley, A. W. W. Ludwig, and H. Saleur. Exact conductance through point contacts in the ν = 1/3 fractional quantum hall effect. Phys. Rev. Lett., 74:3005–3008, Apr 1995. [11] Daniel S. Fisher. Random transverse field ising spin chains. Phys. Rev. Lett., 69:534–537, Jul 1992. [12] Daniel S. Fisher. Random antiferromagnetic quantum spin chains. Phys. Rev. B, 50:3799– 3821, Aug 1994. [13] Daniel S. Fisher. Critical behavior of random transverse-field ising spin chains. Phys. Rev. B, 51:6411–6461, Mar 1995. [14] Daniel S. Fisher and A. P. Young. Distributions of gaps and end-to-end correlations in random transverse-field ising spin chains. Phys. Rev. B, 58:9131–9141, Oct 1998. [15] John A. Hertz. Quantum critical phenomena. Phys. Rev. B, 14:1165–1184, Aug 1976. [16] Christoph Holzhey, Finn Larsen, and Frank Wilczek. Geometric and renormalized entropy in conformal field theory. Nuclear Physics B, 424(3):443–467, 1994. [17] C.L.KaneandMatthewP.A.Fisher. Transmissionthroughbarriersandresonanttunneling in an interacting one-dimensional electron gas. Phys. Rev. B, 46:15233–15262, Dec 1992. 51 [18] C. L. Kane and Matthew P. A. Fisher. Transport in a one-channel luttinger liquid. Phys. Rev. Lett., 68:1220–1223, Feb 1992. [19] Jun Kondo. Resistance minimum in dilute magnetic alloys. Progress of Theoretical Physics, 32(1):37–49, 1964. [20] Nicolas Laflorencie. Scaling of entanglement entropy in the random singlet phase. Phys. Rev. B, 72:140408, Oct 2005. [21] Shang-keng Ma, Chandan Dasgupta, and Chin-kun Hu. Random antiferromagnetic chain. Phys. Rev. Lett., 43:1434–1437, Nov 1979. [22] U. Meirav, M. A. Kastner, M. Heiblum, and S. J. Wind. One-dimensional electron gas in gaas: Periodic conductance oscillations as a function of density. Phys. Rev. B, 40:5871–5874, Sep 1989. [23] K. Moon, H. Yi, C. L. Kane, S. M. Girvin, and Matthew P. A. Fisher. Resonant tunneling between quantum hall edge states. Phys. Rev. Lett., 71:4381–4384, Dec 1993. [24] Ingo Peschel. Entanglement entropy with interface defects. Journal of Physics A: Mathe- matical and General, 38(20):4327, 2005. [25] Ingo Peschel and Viktor Eisler. Reduced density matrices and entanglement entropy in free lattice models. Journal of physics a: mathematical and theoretical, 42(50):504003, 2009. [26] Gil Refael and Daniel S. Fisher. Energy correlations in random transverse field ising spin chains. Phys. Rev. B, 70:064409, Aug 2004. [27] Gil Refael and Joel E Moore. Entanglement entropy of random quantum critical points in one dimension. Physical review letters, 93(26):260602, 2004. [28] Shinsei Ryu and Tadashi Takayanagi. Holographic derivation of entanglement entropy from the anti–de sitter space/conformal field theory correspondence. Phys. Rev. Lett., 96:181602, May 2006. [29] Subir Sachdev and Bernhard Keimer. Quantum criticality. arXiv preprint arXiv:1102.4628, 2011. [30] L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne. Observation of the e/3 fractionally charged laughlin quasiparticle. Phys. Rev. Lett., 79:2526–2529, Sep 1997. [31] Mark Srednicki. Entropy and area. Phys. Rev. Lett., 71:666–669, Aug 1993. [32] Romain Vasseur, Arash Roshani, Stephan Haas, and Hubert Saleur. Healing of defects in random antiferromagnetic spin chains. EPL (Europhysics Letters), 119(5):50004, 2017. [33] Guifre Vidal, José Ignacio Latorre, Enrique Rico, and Alexei Kitaev. Entanglement in quantum critical phenomena. Physical review letters, 90(22):227902, 2003. 52 Appendix A Codes A.1 Direct Diagonalization The fallowing Python code is an example of how we use our direct diagonal- ization method to calculate the entanglement entropy. The code can be modified for different impurity types and different sections: """_________________________________________ Q−Dot␣case Direct␣Diagonalization _________________________________________""" import numpy as np from numpy import linalg as LA import scipy . sparse . linalg as spLA import matplotlib . pyplot as plt #import time as tm L_min=250 #minimum half−chain L_max=251 #maximum half−chain L_step=16 L_range=np. arange(L_min,L_max,L_step) 53 lamb=0.0001 #The strength of the impurity tavg=0.5 #t average tsig=1.0 #t sigma eavg=0. #e average esig=eavg∗0. #e sigma numsam=20000 #number of realizations for each lenghth S=np. zeros(L_range. shape [0]) #Entanglement Entropy of the half−chain sigmacoma=np. zeros(L_range. shape [0]) #The error of the correlation matrix detcutoff=1.e−8 comacutoff=1.e−8 def TBHamil(e , t ): """ Tight␣Binging␣Hamiltonian """ n=e. shape [0] Hamil=np. zeros ((n,n) ,dtype=’ float64 ’) for i in range(n): Hamil[ i , i]=e[ i ] for i in range(n−1): Hamil[ i+1,i]=−t [ i ] Hamil[ i , i+1]=−t [ i ] Hamil[n−1,0]=−t [n−1] Hamil[0 ,n−1]=−t [n−1] return Hamil 54 def FullK(preK): """ Finds␣the␣last␣column␣of␣K""" Ksize=preK. shape [0] #final size of the K matrix newK=np. zeros(preK. shape) for j in range(Ksize−1): Kcolumnnorm=(np.dot(np. conjugate(preK[: , j ]) ,preK[: , j ]))∗∗0.5 for i in range(Ksize ): newK[ i , j]=preK[ i , j ]/Kcolumnnorm #normalizes the columns detflag=0 l=0 while detflag==0: if l==Ksize : print ("FullK␣ERROR") return "FullK␣ERROR" v=np. zeros ((Ksize ,1)) v[ l ,0]=1. if np. absolute(LA. det(np.append(newK,v, axis=1)))>detcutoff : detflag=1 v=v−np. dot(np. dot(newK,np. diag(np. dot(v. reshape ((1 , Ksize )) ,newK)[0])) ,np. ones((Ksize−1,1))) v=v/(np. dot(np. conjugate(np. transpose(v)) ,v)[0 ,0])∗∗0.5 newK=np.append(newK,v, axis=1) l=l+1 55 return newK errflag=0 i_L=0 for L in L_range: N=2∗L+1 #length of the total chain for samp in range(numsam): while True: e=(np.random.random(N)−0.5)∗ esig+eavg #site energies t=(np.random.random(N)−0.5)∗ tsig+tavg #hoppings t [L−1]=t [L−1]∗lamb t [L]=t [L−1] H=TBHamil(e , t) eigE , eigV = spLA. eigsh(H, k=N−1) eigVshape=eigV. shape [1] if eigVshape==N−1: eigE=np.append(eigE ,np.sum(e)−np.sum(eigE)) eigV=FullK(eigV) break if eigVshape!=N−1: print("∗∗∗∗∗___________∗∗∗∗∗") print("∗∗∗∗∗___________∗∗∗∗∗") print("∗∗∗∗∗LinAlgError∗∗∗∗∗") 56 print("∗∗∗∗∗___________∗∗∗∗∗") print("∗∗∗∗∗___________∗∗∗∗∗") errflag=errflag+1 idx = eigE . argsort () eigE = eigE [ idx ] #E_i=eigE[ i ] A=eigE eigV = eigV [: , idx ] #K^dagger:=eigV Kdagger=eigV K=np. conjugate(np. transpose(Kdagger)) theta=np. diag(np.append(np. ones(L,dtype=complex) , np. zeros(L+1,dtype=complex))) CoMa=np.dot(np. dot(Kdagger , theta ) ,K) #Correlation Matrix of the whole system #block=np.arange(L) #Specifies the sites which make the block block=np. arange(L/2,N−L/2) #impurity in the middle of the block and included #block=np.append(np.arange(L/2,L),np.arange(L+1,N−L/2)) #impurity in the middle of the block and excluded coma=CoMa[ block ,:] coma=coma[: , block ] #Correlation Matrix of the block . coma=(coma+np. conjugate(np. transpose(coma)))/2. coma=LA. eigvalsh (coma) sigmacoma[i_L]=sigmacoma[i_L]+(np.amax(np. absolute( np. sort (coma)−np. sort(1.−coma))))∗∗2. 57 #B=coma coma=np. extract(np. logical_and(coma>comacutoff ,coma< 1−comacutoff) ,coma) S[i_L]=S[i_L]+np.sum(−coma∗np. log(coma)− (1−coma)∗np. log(1−coma)) print(L) print(samp) sigmacoma[i_L]=(sigmacoma[i_L]/numsam)∗∗0.5 S[i_L]=S[i_L]/numsam i_L=i_L+1 print("Number␣of␣bad␣realizations : ") print( errflag ) plt . plot(2∗L_range ,S, label=’numerics ’) plt . legend( loc=’best ’) plt . xlabel ( ’Total␣Length ’) plt . ylabel ( ’S(half−chain␣at␣t=0)’) plt . title ( ’Q−Dot,␣␣\lambda=0.0001,␣␣uniform␣distribution , #␣of␣samples=20000,␣␣half−filing ,␣␣closed␣B.C. ’) plt . xscale ( ’log ’) plt .show() 58 A.2 Renormalization Group The fallowing Python code is an example of how we use renormalisation group theory to calculate the entanglement entropy. The code can be modified for dif- ferent impurity types and different sections: """_________________________________________ Q−Dot␣case Renormalization␣Group _________________________________________""" import numpy as np import time as tm L_range=np. array([9 ,11 ,13 ,15 ,17 ,19 ,23 ,27 ,31 ,35 ,39 ,43 ,51 ,59 ,67 ,75 ,91 , 107,123,139,155,171,187,219,251,315,379,443,571,699,827,1083,1339, 1595 ,2107]) #number of realizations 1000000 for L<=25, 40000 for 25<L<=77 #and 20000 for L>77 Gamma0=10. #initial disorder strength ( initial RG time) lamb1=lamb2=lamb=1.e−11 #The strength of the impurity #tavg=0.5 #t average #tsig=1. #t sigma #numsam: number of realizations for each length S_RG=np. zeros(L_range. shape [0]) #Entanglement Entropy from RG 59 def OneDecimation(J_hopp,b_seq): J=np.copy(J_hopp) b=np.copy(b_seq) """ Decimates␣the␣largest␣bond ␣␣␣␣J:␣the␣strength␣of␣the␣bonds ␣␣␣␣b:␣the␣destination␣of␣each␣bond ␣␣␣␣C.B.C.␣so␣far """ n=J. shape [0] J_max=0. i_J_max=0 for i in range(n): if b[b[ i ]]!=b[ i ] and np. absolute(J[ i])>np. absolute(J_max): J_max=J[ i ] i_J_max=i i=0 while b[ i ]!=i_J_max: i=i+1 r_J_max=i J[r_J_max]=J[r_J_max]∗J[b[i_J_max]]/(2.∗J[i_J_max]) b[r_J_max]=b[b[i_J_max]] b[b[i_J_max]]=b[i_J_max] J[b[i_J_max]]=0. return J,b 60 def SingletsCount(tb_hopp,M): """Counts␣the␣singlets␣forming␣over␣subset␣M␣and␣the ␣␣␣␣rest␣of␣the␣chain """ J=np.copy(tb_hopp) n=J. shape [0] b=np.mod(np. arange(n)+1,n) HamLength=n while HamLength>4: J, b = OneDecimation(J,b) HamLength=HamLength−2 Scount=0 for i in range(n): if np. in1d(i ,M) and np. in1d(b[ i ] ,M, invert=True): Scount=Scount+1 if np. in1d(i ,M, invert=True) and np. in1d(b[ i ] ,M): Scount=Scount+1 return Scount def SingletsCount_NOIMPURITY(tb_hopp,M, Impurities ): """Counts␣the␣singlets␣forming␣over␣subset␣M␣and␣the␣rest ␣␣␣␣of␣the␣chain ,␣IMPURITIES␣EXCLUDED""" J=np.copy(tb_hopp) n=J. shape [0] b=np.mod(np. arange(n)+1,n) 61 HamLength=n while HamLength>4: J, b = OneDecimation(J,b) HamLength=HamLength−2 Scount=0 for i in range(n): if np. in1d(i ,M) and np. in1d(b[ i ] ,np.append(M, Impurities ) , invert=True): Scount=Scount+1 if np. in1d(i ,np.append(M, Impurities ) , invert=True) and np. in1d(b[ i ] ,M): Scount=Scount+1 return Scount def SingletsCount_SubSets(tb_hopp,M1,M2): """Counts␣the␣singlets␣forming␣between␣subsets␣M1␣and␣M2""" J=np.copy(tb_hopp) n=J. shape [0] b=np.mod(np. arange(n)+1,n) HamLength=n while HamLength>4: J, b = OneDecimation(J,b) HamLength=HamLength−2 Scount=0 for i in range(n): 62 if np. in1d(i ,M1) and np. in1d(b[ i ] ,M2): Scount=Scount+1 if np. in1d(i ,M2) and np. in1d(b[ i ] ,M1): Scount=Scount+1 return Scount i_L=0 t_i=tm. time() for L in L_range: #N=2∗L #weak−bond − length of the total chain N=2∗L+1 #q−dot − length of the total chain if L<=np.amax(L_range): numsam=20000/20 if L<=77: numsam=40000/20 if L<=25: numsam=1000000/20 #print(L) Impu=np. array ([L−1,L,L+1]) #All the sites that make the impurity blockA_IS=np. arange(L−1) blockA=np. arange(L) blockB=np. arange(L/2,N−L/2) blockC=np.append(np. arange(L/2,L) ,np. arange(L+1,N−L/2)) blockM1=np. array ([L−1]) 63 blockM2=np. arange(L−1) for samp in range(numsam): #t=(np.random.random(N)−0.5)∗tsig+tavg #box distribution hoppings t=np.random.random(N)∗∗Gamma0 #infinite disorder hoppings t [L−1]=t [L−1]∗lamb1 #weak bond & q−dot t [L]=t [L]∗lamb2 #Asymmetric q−dot #t [L]=t [L−1] #Symmetric q−dot #S_RG[i_L]=S_RG[i_L]+np. log (2.)∗ SingletsCount_NOIMPURITY(t ,blockA_IS ,Impu) #S_RG[i_L]=S_RG[i_L]+np. log (2.)∗SingletsCount(t ,blockA) #S_RG[i_L]=S_RG[i_L]+np. log (2.)∗SingletsCount(t ,blockB) #S_RG[i_L]=S_RG[i_L]+np. log (2.)∗SingletsCount(t ,blockC) S_RG[i_L]=S_RG[i_L]+np. log (2.)∗ SingletsCount(t ,blockM1) #S_RG[i_L]=S_RG[i_L]+np. log (2.)∗ SingletsCount_SubSets(t ,blockM1 ,blockM2) #print(L) #print(samp) S_RG[i_L]=S_RG[i_L]/numsam i_L=i_L+1 t_f=tm. time() print("t : ") print(t_f−t_i) print("_______") 64 print(lamb) print(repr(L_range)) print(repr(S_RG)) A.3 Singlet Formation Rate The fallowing Python code can be used to monitor the average number of singlets formed over a specific bond and the average energy scale (Ω) versus RG steps 1 . The code has been written for a chain with no impurities. Though, it can be easily modified for different kinds of impurities. import numpy as np #import matplotlib . pyplot as plt N=2002 #length of the chain Singlet_Place=1 #The place of the singlet on which we are counting the number of #singlets VS number of RG steps Gamma0=10. #initial disorder strength ( initial RG time) numsam=20000 #number of realizations def NewSingletPosition(J_hopp,b_seq): J=np.copy(J_hopp) 1 Here we count each decimation as one RG step. 65 b=np.copy(b_seq) """ Decimates␣the␣largest␣bond ␣␣␣␣J:␣the␣strength␣of␣the␣bonds ␣␣␣␣b:␣the␣destination␣of␣each␣bond ␣␣␣␣C.B.C.␣so␣far """ n=J. shape [0] J_max=0. i_J_max=0 for i in range(n): if b[b[ i ]]!=b[ i ] and np. absolute(J[ i])>np. absolute(J_max): J_max=J[ i ] i_J_max=i i=0 while b[ i ]!=i_J_max: i=i+1 r_J_max=i J[r_J_max]=J[r_J_max]∗J[b[i_J_max]]/(2.∗J[i_J_max]) b[r_J_max]=b[b[i_J_max]] b[b[i_J_max]]=b[i_J_max] J[b[i_J_max]]=0. return J,b,J_max,i_J_max,b[i_J_max] def Singlets_vs_RGsteps(tb_hopp, i_singlet ): """ Returns␣the␣number␣of␣singlets␣forming␣over␣the␣bond ␣␣␣␣ ’ i_singlet ’␣VS␣numbr␣of␣RG␣steps """ 66 J=np.copy(tb_hopp) n=J. shape [0] b=np.mod(np. arange(n)+1,n) HamLength=n n_singlet=1. n_singlet_s=np. array ([ n_singlet ]) EnergyScale_s=np. array ([]) while HamLength>4: J, b, Jmax, i2 , i3= NewSingletPosition(J,b) EnergyScale_s=np.append(EnergyScale_s ,Jmax) #Accounts for the new effective bond if i2<=i_singlet and i3>i_singlet : n_singlet=n_singlet+1. if i2>i_singlet and i3>i_singlet and i3<i2 : n_singlet=n_singlet+1. n_singlet_s=np.append(n_singlet_s , n_singlet) HamLength=HamLength−2 return n_singlet_s , EnergyScale_s AVG_n_singlet_s=0. AVG_EnergyScale_s=0. for samp in range(numsam): t=np.random.random(N)∗∗Gamma0 #infinite disorder hoppings 67 One_n_singlet_s , One_EnergyScale_s= Singlets_vs_RGsteps(t , Singlet_Place) AVG_n_singlet_s=AVG_n_singlet_s+One_n_singlet_s AVG_EnergyScale_s=AVG_EnergyScale_s+One_EnergyScale_s AVG_n_singlet_s=AVG_n_singlet_s/numsam AVG_n_singlet_s=np. delete (AVG_n_singlet_s,−1) #We delete the last element to have all the arrays with the same size . AVG_EnergyScale_s=AVG_EnergyScale_s/numsam """ RGsteps=np. arange(np. shape(AVG_EnergyScale_s)[0]) plt . plot(RGsteps ,AVG_n_singlet_s ,".") #plt . plot(RGsteps ,AVG_EnergyScale_s ,".") #plt . plot(AVG_EnergyScale_s,AVG_n_singlet_s ,".") plt . plot(RGsteps ,RGsteps∗0.01+1) plt . legend( loc=’best ’) plt . xlabel ( ’RG␣steps ’) plt . ylabel ( ’Number␣of␣bonds␣forming␣over␣a␣specific␣bond ’) plt . title ( ’Closed␣B.C. ,␣RG,␣$\Gamma_{0}=10$ ,␣number␣of␣realizations =1000’) #plt . xscale ( ’ log ’) plt .show() """ print("Number␣of␣Singlets : ") print(repr(AVG_n_singlet_s)) print("Energy␣Scale : ") 68 10 46 10 40 10 34 10 28 10 22 10 16 10 10 10 4 10 2 Energy Scale 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Number of bonds forming over a specific bond Closed B.C., RG, 0 = 10 N = 2002 N = 1001 N = 501 1/3ln(ln1/ ) + 0.55 Figure A.1: Bonds coming from IRFP, No impurities - The numerics are done for system sizes N = 501, 1001, 2002. print(repr(AVG_EnergyScale_s)) In the case of a disordered chain with no impurities it can be derived that the number of singlets forming over any specific site is proportional to 1 3 ln ln Ω 0 Ω when Ω is not very close to Ω 0 . We can set Ω 0 = 1 in our numerics and check the scaling(Fig A.1). In addition to a chain with no impurities we do the numerics for a chain with an asymmetric q-dot impurity. We monitor the rate of the singlet formation over the impurity bonds and we observe that the decimation process for each λ starts we the energy scale ,Ω, reaches that λ (Fig A.2) which makes perfect sense. 69 10 50 10 43 10 36 10 29 10 22 10 15 10 8 10 1 (Energy Scale) 1.0 1.2 1.4 1.6 1.8 Number of bonds forming over a specific bond Closed B.C., RG, 0 = 10 N = 2001, = 10 10 N = 2001, = 10 20 N = 2001, = 10 30 N = 2001, = 10 40 Figure A.2: Bonds coming from IRFP, q-dot impurity - The numerics are done for a system size of N = 2001. 70 Appendix B Pit and Bump For the entanglement of case a for the asymmetric q-dot we observe a bump after the crossover and a possible pit before the crossover (Fig C.1). We did not observe such bumps or pits for the single-impurity and we need to make sure about if they are genuine and not simply the artifact of numerics. We claim that the lower pit is not physical and comes from the slope extraction method but the upper bump is physical. To explore the pit and the bump areas further we calculate more points in those areas with more number of realizations which lead to higher precisions in the area of pit and bump. Now the noise in the area of pit and bump is less than before and we can find the slope locally. We fit a line segment to every few sequential points (3, 4,or 5 sequential points) and assign the slope of that line segment to the slope of the corresponding data segment. Doing so shows that the lower bump is not physical (Fig [B.2,B.3,B.4]) and the upper bump is physical (Fig [B.5]). 71 10 1 10 2 10 3 10 4 Total Length/L * (N/L * ) 0.10 0.15 0.20 0.25 0.30 S/ lnN Entanglement between two halves, impurity NOT excluded, asymmetric q-dot, RG, initial disorder 0 = 10, disordered chain, closed B.C. ln2/6 ln2/3 = 10 10 , L * = 0.08 ± 0.01 = 10 20 , L * = 0.20 ± 0.02 = 10 30 , L * = 0.37 ± 0.03 = 10 40 , L * = 0.65 ± 0.05 = 10 50 , L * = 1 = 10 60 , L * = 1.4 ± 0.1 = 10 70 , L * = 1.9 ± 0.1 = 10 80 , L * = 2.5 ± 0.2 = 10 90 , L * = 3.2 ± 0.3 = 10 100 , L * = 3.9 ± 0.3 Figure B.1: Scaled slope curves for case a. 10 2 10 3 10 4 Total Length (N) 0.10 0.15 0.20 0.25 0.30 0.35 S a / lnN Asymmetric q-dot, RG, initial disorder Γ 0 =10, disordered chain, closed B.C. λ =1.0E−45, polynomial derivative λ =1.0E−45, segment derivative Figure B.2: blue dots: polynomial derivative, red pluses: segment derivative 72 10 1 10 2 10 3 10 4 Total Length (N) 0.05 0.10 0.15 0.20 0.25 0.30 S a / lnN Asymmetric q-dot, RG, initial disorder Γ 0 =10, disordered chain, closed B.C. λ =1.0E−70 λ =1.0E−70 Figure B.3: blue dots: polynomial derivative, red pluses: segment derivative 10 1 10 2 10 3 10 4 Total Length (N) 0.10 0.15 0.20 0.25 S a / lnN Asymmetric q-dot, RG, initial disorder Γ 0 =10, disordered chain, closed B.C. λ =1.0E−100 λ =1.0E−100 Figure B.4: blue dots: polynomial derivative, yellow dots: segment derivative 73 10 1 10 2 10 3 10 4 Total Length (N) 0.05 0.10 0.15 0.20 0.25 0.30 S a / lnN Asymmetric q-dot, RG, initial disorder Γ 0 =10, disordered chain, closed B.C. λ =1.0E−40 λ =1.0E−40 Figure B.5: green dots: polynomial derivative, red dots: segment derivative 74 Appendix C Broken Q-dots Foradisorderedspinchainwithanasymmetricq-dotinthemiddleweobserved a genuine bump after the crossover [Fig C.1]. In this appendix we want to investi- gate that if the bump survives if we break the q-dots. So, we are going to study the RG numerics for our new setup with the two impurity bonds separated[Fig C.2]. We keep the segment for which we calculate the entanglement (M) same as M in case a, i.e. half of the whole chain. So, the weak bonds are now inside M and ¯ M instead of their edges. Also, the weak bonds are placed symmetrically with respect to the center of the chain [Fig C.2]. The distance between the weak bonds 10 1 10 2 10 3 10 4 Total Length/L * (N/L * ) 0.10 0.15 0.20 0.25 0.30 S/ lnN Entanglement between two halves, impurity NOT excluded, asymmetric q-dot, RG, initial disorder 0 = 10, disordered chain, closed B.C. ln2/6 ln2/3 = 10 10 , L * = 0.08 ± 0.01 = 10 20 , L * = 0.20 ± 0.02 = 10 30 , L * = 0.37 ± 0.03 = 10 40 , L * = 0.65 ± 0.05 = 10 50 , L * = 1 = 10 60 , L * = 1.4 ± 0.1 = 10 70 , L * = 1.9 ± 0.1 = 10 80 , L * = 2.5 ± 0.2 = 10 90 , L * = 3.2 ± 0.3 = 10 100 , L * = 3.9 ± 0.3 Figure C.1: Qgap=0, there is no bonds between the two weak bonds - case a 75 Qgap Qgap L=(N-1)/2 Figure C.2: Segment M is shown by the solid red box. The impurities are shown with red dashed line segments. is 2Qgap. We do the numerics for Qgap = 1, 5, 9, 50. As we increase the gap between the weak bonds from 2∗ 1 to 2∗ 9 we observe that the curves become less and less scalable [Fig C.3, C.4, C.5] and for the Qgap = 50 it is not really possible to scale the curves anymore so we do not include the result for Qgap = 50 here. There are a few points worthy of noticing. First, on the left tail of the Qgap = 1 curves [Fig C.3] there exists a plateau region which lowers later on. The plateau is even more noticeable for Qgap = 5 curves [Fig C.4]. For Qgap = 9 [Fig C.5], the data is very scattered, though it seems that the plateau is shifting towards the crossover region. λ = 10 −40 curve in the Qgap = 9 curves reflects this behavior clearly [Fig C.6]. In all of the situations discussed in this appendix we observe the upper bump that we were concerned about initially. Though, it seems that as we increase the gap, the bump get tamed a bit and it is not as obvious anymore. We can also try increasing the number of impurity bonds from two to three neighboring impurities instead of breaking them. That also would not erase the bump [Fig C.7]. It seems that the bump for impurities that are longer than a single-impurity is here to stay! 76 10 1 10 2 10 3 10 4 Total Length/L ∗ (N/L ∗ ) 0.15 0.20 0.25 S a / lnN Broken q-dot impurity with Qgap=1, RG, initial disorder Γ 0 =10, disordered chain, closed B.C. ln2/6 ln2/3 λ =10 −10 , L ∗ =0.08±0.01 λ =10 −20 , L ∗ =0.21±0.02 λ =10 −30 , L ∗ =0.40±0.03 λ =10 −40 , L ∗ =0.65±0.05 λ =10 −50 , L ∗ =1 λ =10 −60 , L ∗ =1.3±0.1 λ =10 −70 , L ∗ =1.6±0.1 λ =10 −80 , L ∗ =2.2±0.2 λ =10 −90 , L ∗ =3.0±0.2 λ =10 −100 , L ∗ =3.2±0.2 Figure C.3: Qgap = 1 10 1 10 2 10 3 10 4 Total Length/L ∗ (N/L ∗ ) 0.15 0.20 0.25 S a / lnN Broken q-dot impurity with Qgap=5, RG, initial disorder Γ 0 =10, disordered chain, closed B.C. ln2/6 ln2/3 λ =10 −10 , L ∗ =0.19±0.02 λ =10 −20 , L ∗ =0.26±0.02 λ =10 −30 , L ∗ =0.48±0.03 λ =10 −40 , L ∗ =0.65±0.05 λ =10 −50 , L ∗ =1 λ =10 −60 , L ∗ =1.3±0.1 λ =10 −70 , L ∗ =1.6±0.1 λ =10 −80 , L ∗ =2.3±0.2 λ =10 −90 , L ∗ =2.6±0.2 λ =10 −100 , L ∗ =3.2±0.2 Figure C.4: Qgap = 5 77 10 2 10 3 10 4 Total Length/L ∗ (N/L ∗ ) 0.10 0.15 0.20 0.25 0.30 S a / lnN Broken q-dot impurity with Qgap=9, RG, initial disorder Γ 0 =10, disordered chain, closed B.C. ln2/6 ln2/3 λ =10 −10 , L ∗ =0.16±0.02 λ =10 −20 , L ∗ =0.37±0.03 λ =10 −30 , L ∗ =0.5±0.05 λ =10 −40 , L ∗ =0.8±0.1 λ =10 −50 , L ∗ =1 λ =10 −60 , L ∗ =1.4±0.1 λ =10 −70 , L ∗ =1.9±0.2 λ =10 −80 , L ∗ =2.3±0.2 λ =10 −90 , L ∗ =2.7±0.2 λ =10 −100 , L ∗ =3.3±0.3 Figure C.5: Qgap = 9 10 3 Total Length (N) 0.15 0.20 0.25 S a / lnN Broken q-dot impurity with Qgap=9, RG, initial disorder Γ 0 =10, disordered chain, closed B.C. λ =1.0E−40 Figure C.6: Qgap = 9,λ = 10 −40 78 10 1 10 2 10 3 10 4 Total Length/L * (N/L * ) 0.05 0.10 0.15 0.20 0.25 0.30 S a / lnN Triple-impurity, RG, initial disorder 0 = 10, disordered chain, closed B.C. ln2/6 ln2/3 = 10 10 , L * = 0.10 ± 0.02 = 10 20 , L * = 0.23 ± 0.03 = 10 30 , L * = 0.41 ± 0.04 = 10 40 , L * = 0.68 ± 0.05 = 10 50 , L * = 1 = 10 60 , L * = 1.4 ± 0.1 = 10 70 , L * = 2.0 ± 0.1 = 10 80 , L * = 2.5 ± 0.2 = 10 90 , L * = 3.1 ± 0.2 = 10 100 , L * = 3.6 ± 0.3 Figure C.7: Triple impurities - The three neighboring impurity bonds are not necessarily the same. 79
Abstract (if available)
Abstract
In this thesis we study the interplay of the physics of randomness and the physics of impurities (Kondo physics) in one dimensional spin-chains. Particularly, we study the effects of a weakened link in random antiferromagnetic spin chains. We show that healing occurs, and that homogeneity is restored at low energy, in a way that is qualitatively similar to the fate of impurities in clean ferromagnetic spin chains, or in Luttinger liquids with attractive interactions. Healing in the random case occurs even without interactions, and is characteristic of the random singlet phase. Using real-space renormalization group and exact diagonalization methods, we characterize this universal healing crossover by studying the entanglement across the weak link. We identify a crossover healing length L ⃰ that separates a regime where the system is cut in half by the weak link from a fixed point where the spin chain is healed.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
Disordered quantum spin chains with long-range antiferromagnetic interactions
PDF
Entanglement parity effects in quantum spin chains
PDF
Phase diagram of disordered quantum antiferromagnets
PDF
Dissipation as a resource for quantum information processing: harnessing the power of open quantum systems
PDF
Topological protection of quantum coherence in a dissipative, disordered environment
PDF
Tunneling, cascades, and semiclassical methods in analog quantum optimization
PDF
Out-of-equilibrium dynamics of inhomogeneous quantum systems
PDF
Open-system modeling of quantum annealing: theory and applications
PDF
Topics in quantum information and the theory of open quantum systems
PDF
Coherence generation, incompatibility, and symmetry in quantum processes
PDF
Coulomb interactions and superconductivity in low dimensional materials
PDF
Entanglement in strongly fluctuating quantum many-body states
PDF
Plasmonic excitations in quantum materials: topological insulators and metallic monolayers on dielectric substrates
PDF
Topics in quantum information -- Continuous quantum measurements and quantum walks
PDF
Explorations in semi-classical and quantum gravity
PDF
Signatures of topology in a quasicrystal: a case study of the non-interacting and superconducting Fibonacci chain
PDF
Destructive decomposition of quantum measurements and continuous error detection and suppression using two-body local interactions
PDF
Developmnt of high-frequency electron paramagnetic resonance (EPR) spectrometer and investigation of paramagnetic defects and impurities in diamonds by multi-frequency EPR spectroscopy
PDF
Applications in optical communications: quantum communication systems and optical nonlinear device
PDF
Advancing the state of the art in quantum many-body physics simulations: Permutation Matrix Representation Quantum Monte Carlo and its Applications
Asset Metadata
Creator
Roshani, Arash
(author)
Core Title
Healing of defects in random antiferromagnetic spin chains
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Publication Date
10/04/2019
Defense Date
03/01/2019
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
entanglement entropy,Healing,OAI-PMH Harvest,quantum criticality,quantum many-body systems,randomness,renormalization group theory,spin chain
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Haas, Stephan (
committee chair
), Nakano, Aiichiro (
committee member
), Saleur, Hubert (
committee member
), Venuti, Lorenzo (
committee member
), Zanardi, Paolo (
committee member
)
Creator Email
arashroshani@gmail.com,roshani@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c89-224204
Unique identifier
UC11673751
Identifier
etd-RoshaniAra-7849.pdf (filename),usctheses-c89-224204 (legacy record id)
Legacy Identifier
etd-RoshaniAra-7849.pdf
Dmrecord
224204
Document Type
Dissertation
Rights
Roshani, Arash
Type
texts
Source
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 a...
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
Tags
entanglement entropy
quantum criticality
quantum many-body systems
randomness
renormalization group theory
spin chain