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arXiv:1501.06231 · Physical Review C

Analysis of delocalization of clusters in linear-chain α -cluster states with entanglement entropy

Yoshiko Kanada-En’yo

Published Mar 2, 2015• 17 citations
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Abstract

I investigate entanglement entropy of one-dimensional (1D) cluster states to discuss the delocalization of clusters in linear-chain $3\ensuremath{\alpha}$- and $4\ensuremath{\alpha}$-cluster states. In analysis of entanglement entropy of 1D Tohsaki-Horiuchi-Schuck-R\"opke (THSR) and Brink-Bloch cluster wave functions, I show clear differences in the entanglement entropy between localized cluster wave functions and delocalized cluster wave functions. To clarify spatial regions where the entanglement entropy is generated by the delocalization of clusters, I analyze the spatial distribution of entanglement entropy. In the linear-chain $3\ensuremath{\alpha}$-cluster state, the delocalization occurs dominantly in a low-density tail region while it is relatively suppressed in an inner region because of the Pauli blocking effect between clusters. In the linear-chain $4\ensuremath{\alpha}$ state having a larger system size than the linear-chain $3\ensuremath{\alpha}$ state, the delocalization occurs in the whole system. The entanglement entropy is found to be a measure of the delocalization of clusters in the 1D cluster systems.

DOI
10.1103/physrevc.91.034303

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I investigate entanglement entropy of one-dimensional (1D) cluster states to discuss the delocalization of clusters in linear-chain 3\ensuremathα3\ensuremath{\alpha}3\ensuremathα- and 4\ensuremathα4\ensuremath{\alpha}4\ensuremathα-cluster states. In analysis of entanglement entropy of 1D Tohsaki-Horiuchi-Schuck-R"opke (THSR) and Brink-Bloch cluster wave functions, I show clear differences in the entanglement entropy between localized cluster wave functions and delocalized cluster wave functions. To clarify spatial regions where the entanglement entropy is generated by the delocalization of clusters, I analyze the spatial distribution of entanglement entropy. In the linear-chain 3\ensuremathα3\ensuremath{\alpha}3\ensuremathα-cluster state, the delocalization occurs dominantly in a low-density tail region while it is relatively suppressed in an inner region because of the Pauli blocking effect between clusters. In the linear-chain 4\ensuremathα4\ensuremath{\alpha}4\ensuremathα state having a larger system size than the linear-chain 3\ensuremathα3\ensuremath{\alpha}3\ensuremathα state, the delocalization occurs in the whole system. The entanglement entropy is found to be a measure of the delocalization of clusters in the 1D cluster systems.
The properties of the entanglement entropy (EE) of low-lying excitations in one-dimensional disordered interacting systems are studied. The ground state EE shows a clear signature of localization, while low-lying excitations show a significantly longer localization length. The dependence of the localization as a function of interaction strength and sample length is studied using the density matrix renormalization group (DMRG). This behavior corresponds to the presence of the predicted many-particle critical energy in the vicinity of the ground state energy. The implications of these results on experiments are discussed.
In the spectrum of many-body quantum systems appearing in condensed matter physics, the low-energy eigenstates were the traditional focus of research. The interest in the statistical properties of the full eigenspectrum has grown more recently, in particular in the context of non-equilibrium questions. Wave functions of interacting lattice quantum systems can be characterized either by local observables or by global properties such as the participation ratio (PR) in a many-body basis or the entanglement between various partitions. We present a study of the PR and of the entanglement entropy (EE) between two roughly equal spatial partitions of the system, in all the eigenfunctions of local Hamiltonians. Motivated by the similarity of the PR and EE—both are generically larger in the bulk and smaller near the edges of the spectrum—we quantitatively analyze the correlation between them. We elucidate the effect of (proximity to) integrability, showing how low-entanglement and low-PR states appear also in the middle of the spectrum as one approaches integrable points. We also determine the precise scaling behaviour of the eigenstate-to-eigenstate fluctuations of the PR and EE with respect to system size and characterize the statistical distribution of these quantities near the middle of the spectrum.
The following thesis focuses on the scaling of entanglement entropy in lower dimensions and is divided into three main parts. Chapter 2 studies the thermal reduced density matrices in fermion and spin systems on ladders. Chapter 3 studies the many-body localization phase transition in a Rokhsar-Kivelson type wave function. Chapter 4 studies the subleading correction term of entanglement entropy in 2+1 dimensional scale invariant systems. Chapter 5 studies the bulk-boundary correspondence in 3 + 1 dimensional topological phases and its entanglement entropy. In chapter 2, we investigate the reduced density matrices for a model of free fermions on a two-leg ladder (gapped by the inter-chain tunneling operator) and in 1/2 spin systems on a ladder with a gapped ground state using exact solutions and several controlled approximations. We calculate the reduced density matrix and the entanglement entropy for a leg of the ladder (i.e. cut made between the chains). In the fermionic system we find the exact form of the reduced density matrix for one of the chains and determine the entanglement spectrum explicitly. Here we find that in the weak tunneling limit of the ladder the entanglement entropy of one chain of the gapped ladder has a simple and universal form dictated by conformal invariance. In the case of the spin system, we consider the strong coupling limit by using perturbation theory and get the reduced density matrix by the Schmidt decomposition. The entanglement entropies of a general gapped system of two coupled conformal field theories (in 1+1 dimensions) is discussed using the replica trick and scaling arguments. We show that 1) for a system with a bulk gap, the reduced density matrix has the form of a thermal density matrix, 2) the long-wavelength modes of one subsystem (a chain) of a gapped coupled system are always thermal, 3) the von Neumann entropy equals to the thermodynamic entropy of one chain, and 4) the bulk gap plays the role of effective temperature. In chapter 3, we construct a family of many-body wave functions to study the many-body localization phase transition. The wave functions have a Rokhsar-Kivelson form, in which the weight for the configurations are chosen from the Gibbs weights of a classical spin glass model, known as the Random Energy Model, multiplied by a random sign structure to represent a highly excited state. These wave functions show a phase transition into an MBL phase. In addition, we see three regimes of entanglement scaling with subsystem size: scaling with entanglement corresponding to an infinite temperature thermal phase, constant scaling, and a sub-extensive scaling between these limits. Near the phase transition point, the fluctuations of the Renyi entropies are non-Gaussian. We find that Renyi entropies with different Renyi index transition into the MBL phase at different points and have different scaling behavior, suggesting a multifractal behavior. In chapter 4, we study the universal scaling behavior of the entanglement entropy of critical theories in 2+1 dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from a) the quasi-one-dimensional conformal field theory, b) the bosonic quantum Lifshitz model, and c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision. In chapter 5, we discuss (2+1)-dimensional gapless surface theories of bulk (3+1)-dimensional topological phases, such as the BF theory at level K, and its generalization. In particular, we put these theories on a flat (2+1) dimensional torus T3 parameterized by its modular parameters, and compute the partition functions obeying various twisted boundary conditions. We show the partition functions are transformed into each other under SL(3, Z) modular transformations, and furthermore establish the bulk-boundary correspondence in (3+1) dimensions by matching the modular S and T matrices computed from the boundary field theories with those computed in the bulk. We propose the three-loop braiding statistics can be studied by constructing the modular S and T matrices from an appropriate boundary field theory. We also study the EE for 3 + 1- dimensional topological phase with or without three-loop braiding phase.
We study the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum model. This entropy is related to the entanglement entropy of a Rokhsar-Kivelson-type wave function built from the corresponding two-dimensional classical model. In both critical and massive cases, we observe that it is composed of an extensive part proportional to the length of the system and a subleading universal constant S0{S}_{0}S0​. In c=1c=1c=1 critical systems (Tomonaga-Luttinger liquids), we find that S0{S}_{0}S0​ is a simple function of the boson compactification radius. This finding is based on a field-theoretical analysis of the Dyson-Gaudin gas related to dimer and Calogero-Sutherland models. We also performed numerical demonstrations in the dimer models and the spin-1/2 XXZXXZXXZ chain. In a massive (crystal) phase, S0{S}_{0}S0​ is related to the ground-state degeneracy. We also examine this entropy in the Ising chain in a transverse field as an example showing a c=1/2c=1/2c=1/2 critical point.
We calculate the von Neumann entanglement entropy and the Schmidt number of one-dimensional (1D) cluster states and show that these are useful measures to estimate entanglement caused by delocalization of clusters. We analyze system size dependence of these entanglement measures in the linear-chain |nαn\alphanα| states given by Tohsaki–Horiuchi–Schuck–Röpke wave functions for 1D cluster gas states. We show that the Schmidt number is an almost equivalent measure to the von Neumann entanglement entropy when the delocalization of clusters occurs in the entire system but it shows different behaviors in a partially delocalized state containing localized clusters and delocalized ones. Thus the Rényi-2 entanglement entropy, which relates to the Schmidt number, is found to be almost equivalent to the von Neumann entanglement entropy for the full delocalized cluster system but it is less sensitive to the partially delocalized cluster system than the von Neumann entanglement entropy. We also propose a new entanglement measure which is a generalized form of the Schmidt number. The sensitivity of these measures of entanglement to the delocalization of clusters in low-density regions is discussed.