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arXiv:1605.07085 · Physical review. B./Physical review. B

Entanglement spectrum and Rényi entropies of nonrelativistic conformal fermions

William J. Porter, Joaquín E. Drut

Published Oct 6, 2016• 9 citations
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Abstract

We characterize nonperturbatively the R\'enyi entropies of degree $n=2,\phantom{\rule{0.28em}{0ex}}3,\phantom{\rule{0.28em}{0ex}}4$, and 5 of three-dimensional, strongly coupled many-fermion systems in the scale-invariant regime of short interaction range and large scattering length, i.e., in the unitary limit. We carry out our calculations using lattice methods devised recently by us. Our results show the effect of strong pairing correlations on the entanglement entropy, which modify the subleading behavior for large subsystem sizes (as characterized by the dimensionless parameter $x={k}_{F}{L}_{A}$, where ${k}_{F}$ is the Fermi momentum and ${L}_{A}$ the linear subsystem size), but leave the leading order unchanged relative to the noninteracting case. Moreover, we find that the onset of the subleading asymptotic regime is at surprisingly small $x\ensuremath{\simeq}2--4$. We provide further insight into the entanglement properties of this system by analyzing the spectrum of the entanglement Hamiltonian of the two-body problem from weak to strong coupling. The low-lying entanglement spectrum displays clear features as the strength of the coupling is varied, such as eigenvalue crossing and merging, a sharp change in the Schmidt gap, and scale invariance at unitarity. Beyond the low-lying component, the spectrum appears as a quasicontinuum distribution, for which we present a statistical characterization; we find, in particular, that the mean shifts to infinity as the coupling is turned off, which indicates that that part of the spectrum represents nonperturbative contributions to the entanglement Hamiltonian. In contrast, the low-lying entanglement spectrum evolves to finite values in the noninteracting limit. The scale invariance of the unitary regime guarantees that our results are universal features intrinsic to three-dimensional quantum mechanics and represent a well-defined prediction for ultracold atom experiments, which were recently shown to have direct access to the entanglement entropy.

DOI
10.1103/physrevb.94.165112

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A, by now, well-known signal-to-noise problem plagues Monte Carlo calculations of quantum-information-theoretic observables in systems of interacting fermions, particularly the Renyi entanglement entropies Sn, even in many cases where the infamous sign problem does not appear. Several methods have been put forward to circumvent this affliction including ensemble-switching techniques using auxiliary partition-function ratios. This dissertation presents an algorithm that modifies the recently proposed free-fermion decomposition in an essential way: we incorporate the entanglement-sensitive correlations directly into the probability measure in a natural way. Implementing this algorithm, we demonstrate that it is compatible with the hybrid Monte Carlo algorithm, the workhorse of the lattice quantum chromodynamics community and an essential tool for studying gauge theories that contain dynamical fermions. By studying a simple one-dimensional Hubbard model, we demonstrate that our method does not exhibit the same debilitating numerical difficulties that naive attempts to study entanglement often encounter. Following that, we illustrate some key probabilistic insights, using intuition derived from the previous method and its successes to construct a simpler, better behaved, and more elegant algorithm. Using this method, in combination with new identities which allow us to avoid seemingly necessary numerical difficulties, the inversion of the restricted one-body density matrices, we compute high order Renyi entropies and perform a thorough comparison to this new algorithm's predecessor using the Hubbard model mentioned before. Finally, we characterize non-perturbatively the Renyi entropies of degree n=2,3,4, and 5 of three-dimensional, strongly coupled many-fermion systems in the scale-invariant regime of short interaction range and large scattering length, i.e. in the unitary limit using the algorithms detailed herein. We also detail an exact, few-body projective method which we use to characterize the entanglement properties of the two-body sector across a broad range of attractive couplings. In the many-body case, we determine universal scaling properties of this system, and for the two-body case, we compute the entanglement spectrum exactly, successfully characterizing a vast range of entanglement behavior across the BCS-BEC crossover.
We consider the entanglement entropy for a line segment in the system of noninteracting one-dimensional fermions at zero temperature. In the limit of a large segment length L , the leading asymptotic behavior of this entropy is known to be logarithmic in L . We study finite-size corrections to this asymptotic behavior. Based on an earlier conjecture of the asymptotic expansion for full counting statistics in the same system, we derive a full asymptotic expansion for the von Neumann entropy and obtain first several corrections for the Rényi entropies. Our corrections for the Rényi entropies reproduce earlier results. We also discuss the entanglement spectrum in this problem in terms of single-particle occupation numbers.
A new numerical approach to entanglement entropies of the Rényi type is proposed for one-dimensional quantum field theories. The method extends the truncated conformal spectrum approach and we will demonstrate that it is especially suited to study the crossover from massless to massive behavior when the subsystem size is comparable to the correlation length. We apply it to different deformations of massless free fermions, corresponding to the scaling limit of the Ising model in transverse and longitudinal fields. For massive free fermions the exactly known crossover function is reproduced already in very small system sizes. The new method treats ground states and excited states on the same footing, and the applicability for excited states is illustrated by reproducing Rényi entropies of low-lying states in the transverse field Ising model.
The subsystem-size dependence of ground-state entanglement entropy and its crossover to thermal entropy as a function of temperature are well understood for one-dimensional (1D) gapless systems described by conformal field theory (CFT), and for free fermions with a Fermi surface in any dimension. However, little is known about the entanglement entropy for gapless fermionic systems without quasi-particles, such as a strange metal. Here we study the entanglement entropy of fermions in a solvable large-NNN 1D lattice model akin to the Yukawa-Sachdev-Ye-Kitaev (Yukawa SYK) model. In this model, two Fermi points are coupled to scalar bosons via spatially random Yukawa interactions, providing a solvable model of a strange metal when the bosons become critical at a quantum critical point. We exactly compute the second Rényi entropy of fermions in a spatial subregion in this model. Our results unravel crucial role of intra-subregion entanglement between fermionic and bosonic degrees of freedom along with the inter-subregion entanglement in understanding the ground states of such strongly coupled fermion-boson systems. We show that the crossover from thermal entropy to entanglement entropy, is captured by a single scaling ansatz, that collapses the second Rényi entropy of fermions for different subregion sizes and temperatures into a single universal curve. We further show that the universal scaling curve for the critical strange metal is well described by the standard CFT formula, albeit with an effective central charge substantially larger than the non-interacting value.
We study the R\'enyi entanglement entropy and the Shannon mutual information for a class of spin-1/2 quantum critical XXZ chains with random coupling constants which are partially correlated. In the XX case, distinctly from the usual uncorrelated random case where the system is governed by an infinite-disorder fixed point, the correlated-disorder chain is governed by finite-disorder fixed points. Surprisingly, we verify that, although the system is not conformally invariant, the leading behavior of the R\'enyi entanglement entropies are similar to those of the clean (no randomness) conformally invariant system. In addition, we compute the Shannon mutual information among subsystems of our correlated-disorder quantum chain and verify the same leading behavior as the n=2n=2n=2 R\'enyi entanglement entropy. This result extends a recent conjecture concerning the same universal behavior of these quantities for conformally invariant quantum chains. For the generic spin-1/2 quantum critical XXZ case, the true asymptotic regime is identical to that in the uncorrelated disorder case. However, these finite-disorder fixed points govern the low-energy physics up to a very long crossover length scale, and the same results as in the XX case apply. Our results are based on exact numerical calculations and on a numerical strong-disorder renormalization group.
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.