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arXiv:1402.5437 · Physical Review A

Scaling of entanglement entropy in point-contact, free-fermion systems

Bassir Caravan, Barry Friedman, Gregory Levine

Published May 8, 2014• 4 citations
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Abstract

The scaling of entanglement entropy is computationally studied in several ($1\ensuremath{\le}d\ensuremath{\le}2$)-dimensional free-fermion systems that are connected by one or more point contacts (PCs). For both the $k$-leg Bethe lattice $(d=1)$ and $d=2$ rectangular lattices with a subsystem of ${L}^{d}$ sites, the entanglement entropy associated with a single PC is found to be generically $S\ensuremath{\sim}L$. We argue that the $O(L)$ entropy is an expression of the subdominant $O(L)$ entropy of the bulk entropy-area law. For $d=2$ (square) lattices connected by $m$ PCs, the area law is found to be $S\ensuremath{\sim}a{L}^{d\ensuremath{-}1}+bm\phantom{\rule{0.16em}{0ex}}logL$ and is thus consistent with the anomalous area law for free fermions ($S\ensuremath{\sim}L\phantom{\rule{0.16em}{0ex}}logL$) as $m\ensuremath{\rightarrow}L$. For the Bethe lattice, the relevance of this result to density-matrix renormalization-group schemes for interacting fermions is discussed.

DOI
10.1103/physreva.89.052305

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The scaling of entanglement entropy is computationally studied in several (1\ensuremath≤d\ensuremath≤21\ensuremath{\le}d\ensuremath{\le}21\ensuremath≤d\ensuremath≤2)-dimensional free-fermion systems that are connected by one or more point contacts (PCs). For both the kkk-leg Bethe lattice (d=1)(d=1)(d=1) and d=2d=2d=2 rectangular lattices with a subsystem of Ld{L}^{d}Ld sites, the entanglement entropy associated with a single PC is found to be generically S\ensuremath∼LS\ensuremath{\sim}LS\ensuremath∼L. We argue that the O(L)O(L)O(L) entropy is an expression of the subdominant O(L)O(L)O(L) entropy of the bulk entropy-area law. For d=2d=2d=2 (square) lattices connected by mmm PCs, the area law is found to be S\ensuremath∼aLd\ensuremath−1+bmlogLS\ensuremath{\sim}a{L}^{d\ensuremath{-}1}+bm\phantom{\rule{0.16em}{0ex}}logLS\ensuremath∼aLd\ensuremath−1+bmlogL and is thus consistent with the anomalous area law for free fermions (S\ensuremath∼LlogLS\ensuremath{\sim}L\phantom{\rule{0.16em}{0ex}}logLS\ensuremath∼LlogL) as m\ensuremath→Lm\ensuremath{\rightarrow}Lm\ensuremath→L. For the Bethe lattice, the relevance of this result to density-matrix renormalization-group schemes for interacting fermions is discussed.
We show that entanglement entropy of free fermions scales faster than area law, as opposed to the scaling ${L}^{d\ensuremath{-}1}$ for the harmonic lattice, for example. We also suggest and provide evidence in support of an explicit formula for the entanglement entropy of free fermions in any dimension ddd, $S\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}c(\ensuremath{\partial}\ensuremath{\Gamma},\ensuremath{\partial}\ensuremath{\Omega}){L}^{d\ensuremath{-}1}\mathrm{log}L$ as the size of a subsystem $L\ensuremath{\rightarrow}\ensuremath{\infty}$, where $\ensuremath{\partial}\ensuremath{\Gamma}$ is the Fermi surface and $\ensuremath{\partial}\ensuremath{\Omega}$ is the boundary of the region in real space. The expression for the constant $c(\ensuremath{\partial}\ensuremath{\Gamma},\ensuremath{\partial}\ensuremath{\Omega})$ is based on a conjecture due to Widom. We prove that a similar expression holds for the particle number fluctuations and use it to prove a two sided estimate on the entropy SSS.
We study the scaling of the ground-state entanglement entropy of various free fermionic models on one-dimensional lattices, where the hopping and pairing terms decay as a power law. We seek to understand the scaling of entanglement entropy in generic models as the exponent of the power law $\ensuremath{\alpha}$ is varied. We ask if a common ${\ensuremath{\alpha}}_{c}$ exists across different systems governing the transition to area-law scaling found in local systems. We explore several examples numerically and argue that when applicable, the scaling of entanglement entropy in long-range models is constrained by predictions from the low-energy theory. In contrast, disordered models and models without a continuum limit show fractal scaling of entanglement approaching volume-law behavior as $\ensuremath{\alpha}$ approaches zero. These general features are expected to persist on turning on interactions.
We define the entanglement entropy of free fermion quantum states in an arbitrary space-time slice of a discrete set of points and particularly investigate timelike (causal) slices. For one-dimensional lattice free fermions with an energy bandwidth E0{E}_{0}E0​, we calculate the time-direction entanglement entropy SA{S}_{A}SA​ in a time-direction slice of a set of times ${t}_{n}=n\ensuremath{\tau}$ $(1\ensuremath{\le}n\ensuremath{\le}K)$ spanning a time length ttt on the same site. For zero-temperature ground states, we find that SA{S}_{A}SA​ shows volume law when $\ensuremath{\tau}\ensuremath{\gg}{\ensuremath{\tau}}_{0}=2\ensuremath{\pi}/{E}_{0}$; in contrast, ${S}_{A}\ensuremath{\sim}\frac{1}{3}lnt$ when $\ensuremath{\tau}={\ensuremath{\tau}}_{0}$, and ${S}_{A}\ensuremath{\sim}\frac{1}{6}lnt$ when $\ensuremath{\tau}<{\ensuremath{\tau}}_{0}$, resembling the Calabrese-Cardy formula for one flavor of nonchiral and chiral fermion, respectively. For finite-temperature thermal states, the mutual information also saturates when $\ensuremath{\tau}<{\ensuremath{\tau}}_{0}$. For noneigenstates, volume law in ttt and signatures of the Lieb-Robinson bound velocity can be observed in SA{S}_{A}SA​. For generic space-time slices with one point per site, the zero-temperature entanglement entropy shows a clear transition from area law to volume law when the slice varies from spacelike to timelike.
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.
It is known that the long-range quantum entanglement exhibited in free fermion systems is sufficient to "thermalize" a small subsystem in that the subsystem reduced density matrix computed from a typical excited eigenstate of the combined system is approximately thermal. Remarkably, fermions without any interactions are thus thought to satisfy the Eigenstate Thermalization Hypothesis (ETH). We explore this hypothesis when the fermion subsystem is only minimally coupled to a quantum reservoir (in the form of another fermion system) through a quantum point contact (QPC). The entanglement entropy of two 2-d free fermion systems connected by one or more quantum point contacts (QPC) is examined at finite energy and in the ground state. When the combined system is in a typical excited state, it is shown that the entanglement entropy of a subsystem connected by a small number of QPCs is sub-extensive, scaling as the linear size of the subsystem (LAL_ALA​). For sufficiently low energies (EEE) and small subsystems, it is demonstrated numerically that the entanglement entropy SA∼LAES_A \sim L_A ESA​∼LA​E, what one would expect for the thermodynamics of a one-dimensional system. In this limit, we suggest that the entropy carried by each additional QPC is quantized using the one-dimensional finite size/temperature conformal scaling: ΔSA=αlog⁡(1/E)sinh⁡(LAE)ΔS_A = α\log{(1/E)\sinh{(L_AE)}}ΔSA​=αlog(1/E)sinh(LA​E). The sub-extensive entropy in the case of a small number of QPCs should be contrasted with the expectation for both classical, ergodic systems and quantum chaotic systems wherein a restricted geometry might affect the equilibrium relaxation times, but not the equilibrium properties themselves, such as extensive entropy and heat capacity.
Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.