We investigate the second quantization form of the entanglement Hamiltonian (EH) of various subregions for the ground-state of several interacting lattice fermions and spin models. The relation between the EH and the model Hamiltonian itself is an unsolved problem for the ground-state of generic local Hamiltonians. In this letter, we demonstrate that the EH is practically local and its dominant components are related to the terms present in the model Hamiltonian up to a smooth spatially varying temperature even for (a) discrete lattice systems, (b) systems with no emergent conformal or Lorentz symmetry, and (c) for subsystems with non-flat boundaries, up to relatively strong interactions. We show that the mentioned local temperature at a given point decays inversely proportional to its distance from the boundary between the subsystem and the environment. We find the subdominant terms in the EH as well and show that they are severely suppressed away from the boundaries of subsystem and are relatively small near them.
In this paper, we review how to obtain the central charge of a critical entanglement Hamiltonian through the nested entanglement entropy which was first introduced in [J. Lou et al. PRB 84, 245128 (2011)]. The critical phenomena of the entanglement Hamiltonian can be identified by the central charge obtained by the nested entanglement entropy. We review our previous studies[J. Lou et al. PRB 84, 245128 (2011), S. Tanaka et al. PRA 87, 214401 (2013)] in which we investigated certain entanglement nature of two-dimensional valence-bond-solid (VBS) state and quantum hard-square models on square an…
We discuss the detection of entanglement in interacting quantum spin systems. First, thermodynamic Hamiltonian-based witnesses are computed for a general class of one-dimensional spin-1/2 models. Second, we introduce optimal bipartite entanglement observables. We show that a bipartite entanglement measure can generally be associated to a set of independent two-body spin observables whose expectation values can be used to witness entanglement. The number of necessary observables is ruled by the symmetries of the model. Illustrative examples are presented.
Recent advances in the field of strongly correlated electron systems allow to access the entanglement properties of interacting fermionic models, by means of Monte Carlo simulations. We briefly review the techniques used in this context to determine the entanglement entropies and correlations of the entanglement Hamiltonian. We further apply these methods to compute the spin two-point function of entanglement Hamiltonian for a stripe embedded into a correlated topological insulator. Further we discuss a recent method that allows an unbiased, numerically exact, direct determination of the entan…
We calculate the entanglement entropy of strongly correlated low-dimensional fermions in metallic, superfluid and antiferromagnetic insulating phases. The entanglement entropy reflects the degrees of freedom available in each phase for storing and processing information, but is found not to be a state function in the thermodynamic sense. The role of critical points, smooth crossovers and Hilbert space restrictions in shaping the dependence of the entanglement entropy on the system parameters is illustrated for metallic, insulating and superfluid systems. The dependence of the spin susceptibili…
Recent numerical advances in the field of strongly correlated electron systems allow the calculation of the entanglement spectrum and entropies for interacting fermionic systems. An explicit determination of the entanglement (modular) Hamiltonian has proven to be a considerably more difficult problem, and only a few results are available. We introduce a technique to directly determine the entanglement Hamiltonian of interacting fermionic models by means of auxiliary field quantum Monte Carlo simulations. We implement our method on the one-dimensional Hubbard chain partitioned into two segments…
In a recent article T. Grover [Phys. Rev. Lett. 111, 130402 (2013)] introduced a simple method to compute Renyi entanglement entropies in the realm of the auxiliary field quantum Monte Carlo algorithm. Here, we further develop this approach and provide a stabilization scheme to compute higher order Renyi entropies and an extension to access the entanglement spectrum. The method is tested on systems of correlated topological insulators.
In this paper we discuss behaviors of entanglement entropy between two interacting CFTs and its holographic interpretation using the AdS/CFT correspondence. We explicitly perform analytical calculations of entanglement entropy between two free scalar field theories which are interacting with each other in both static and time-dependent ways. We also conjecture a holographic calculation of entanglement entropy between two interacting $\mathcal{N}=4$ super Yang-Mills theories by introducing a minimal surface in the S$^5$ direction, instead of the AdS$_5$ direction. This offers a possible general…
We present a new algorithm for calculating the Renyi entanglement entropy of interacting fermions using the continuous-time quantum Monte Carlo method. The algorithm only samples interaction correction of the entanglement entropy, which by design ensures efficient calculation of weakly interacting systems. Combined with Monte Carlo reweighting, the algorithm also performs well for systems with strong interactions. We demonstrate the potential of this method by studying the quantum entanglement signatures of the charge-density-wave transition of interacting fermions on a square lattice.
Entanglement is central to our understanding of many-body quantum matter. In particular, the entanglement spectrum, as eigenvalues of the reduced density matrix of a subsystem, provides a unique footprint of properties of strongly correlated quantum matter from detection of topological order to characterisation of quantum critical systems. However, direct experimental measurement of the entanglement spectrum has so far remained elusive due to lack of direct experimental probes. Here we show that the entanglement spectrum of the ground state of a broad class of Hamiltonians becomes directly acc…
This arXiv repository is a bundle of two closely related papers. Abstract of the first paper: In systems governed by "chaotic" local Hamiltonians, we conjecture the universality of eigenstate entanglement (defined as the average entanglement entropy of all eigenstates) by proposing an exact formula for its dependence on the subsystem size. This formula is derived from an analytical argument based on a plausible assumption, and is supported by numerical simulations. Abstract of the second paper: In systems governed by chaotic local Hamiltonians, the first paper conjectured the universality of t…
The entanglement Hamiltonian $H_E$, defined through the reduced density matrix of a subsystem $ρ_A=\exp(-H_E)$, is an important concept in understanding the nature of quantum entanglement in many-body systems and quantum field theories. In this work, we explore a numerical scheme which explicitly reconstructs the entanglement Hamiltonian using one entangled mode (i.e., an eigenstate) of $ρ_A$. We demonstrate and benchmark this scheme on quantum spin lattice models. The resulting $H_E$ bears a form similar to a physical Hamiltonian with spatially varying couplings, which allows us to make quant…
Correlated many-body problems ubiquitously appear in various fields of physics such as condensed matter physics, nuclear physics, and statistical physics. However, due to the interplay of the large number of degrees of freedom, it is generically impossible to treat these problems from first principles. Thus the construction of a proper model, namely effective Hamiltonian, is essential. Here, we propose a simple scheme of constructing Hamiltonians from given energy or entanglement spectra with machine learning. We apply the proposed scheme to the Hubbard model at the half-filling, and compare t…
We study "field space entanglement" in certain quantum field theories consisting of N number of free scalar fields interacting with each other via kinetic mixing terms. We present exact analytic expressions for entanglement end Renyi entropies between arbitrary number of scalar fields by which we could explore certain entanglement inequalities. Other entanglement measures such as mutual information and entanglement negativity have also been studied. We also give some comments about possible holographic realizations of such models.
We investigate two methods to reconstruct Hamiltonians of quantum matter, using a quantum spin chain to test them. The first method uses correlation functions and the second method uses entanglement spectra. The two methods are not specific to spin chains, and may find wider applicability to quantum matter that is sufficiently well characterized by experiment.
We review results about entanglement (or modular) Hamiltonians of quantum many-body systems in field theory and statistical mechanics models, as well as recent applications in the context of quantum information and quantum simulation.
A powerful perspective in understanding non-equilibrium quantum dynamics is through the time evolution of its entanglement content. Yet apart from a few guiding principles for the entanglement entropy, to date, not much else is known about the refined characters of entanglement propagation. Here, we unveil signatures of the entanglement evolving and information propagation out-of-equilibrium, from the view of entanglement Hamiltonian. As a prototypical example, we study quantum quench dynamics of a one-dimensional Bose-Hubbard model by means of time-dependent density-matrix renormalization gro…
The modular (or entanglement) Hamiltonian correspondent to the half-space-bipartition of a quantum state uniquely characterizes its entanglement properties. However, in the context of lattice models, its explicit form is analytically known only for the Ising chain and certain free theories in one-dimension. In this work, we provide a throughout investigation of entanglement Hamiltonians in lattice models obtained via the Bisognano-Wichmann theorem, which provides an explicit functional form for the entanglement Hamiltonian itself in quantum field theory. Our study encompasses a variety of one-…
We investigate of the relationship between the entanglement and subsystem Hamiltonians in the perturbative regime of strong coupling between subsystems. One of the two conditions that guarantees the proportionality between these Hamiltonians obtained by using the nondegenerate perturbation theory within the first order is that the unperturbed ground state has a trivial entanglement Hamiltonian. Furthermore, we study the entanglement Hamiltonian of the Heisenberg ladders in a time-dependent magnetic field using the degenerate perturbation theory, where couplings between legs are considered as a…
Single-particle entanglement entropy (SPEE) is calculated for entanglement Hamiltonian eigen-mode in a one-dimensional free fermion model that undergoes a delocalized-localized phase transition. In this numerical study, we show that SPEE of entanglement Hamiltonian eigen-mode has the same behavior as EPEE of Hamiltonian eigen-mode at the Fermi level: as we go from delocalized phase toward localized phase, SPEE of both modes decreases in the same manner. Furthermore, fluctuations of SPEE of entanglement Hamiltonian eigen-mode -- which can be obtained through the calculation of moments of SPEE -…
In this work, we study the time evolution of the entanglement hamiltonian during the process of thermalization in a (1+1)-dimensional conformal field theory (CFT) after a quantum quench from a special class of initial states. In particular, we focus on a subsystem which is a finite interval at the end of a semi-infinite line. Based on conformal mappings, the exact forms of both entanglement hamiltonian and entanglement spectrum of the subsystem can be obtained. Aside from various interesting features, it is found that in the infinite time limit the entanglement hamiltonian and entanglement spe…
Continuum mechanics of entanglement in noisy interacting fermion chains
Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK（鲁道夫·皮尔斯理论物理中心，克拉伦登实验室）；Laboratoire de Physique de l’École Normale Supérieure, CNRS, ENS & Université PSL, Sorbonne Université, Université Paris Cité, 75005 Paris, France（巴黎高等师范学院物理实验室，法国国家科学研究中心，巴黎文理研究大学，索邦大学，巴黎西岱大学）
Given a specific interacting quantum Hamiltonian in a general spatial dimension, can one access its entanglement properties, such as, the entanglement entropy corresponding to the ground state wavefunction? Even though progress has been made in addressing this question for interacting bosons and quantum spins, as yet there exist no corresponding methods for interacting fermions. Here we show that the entanglement structure of interacting fermionic Hamiltonians has a particularly simple form -- the interacting reduced density matrix can be written as a sum of operators that describe free fermio…
Density functional theory maps an interacting Hamiltonian onto the Kohn-Sham Hamiltonian, an explicitly free model with identical local fermion densities. Using the interaction distance, the minimum distance between the ground state of the interacting system and a generic free fermion state, we quantify the applicability and limitations of the exact Kohn-Sham model in capturing the various properties of the interacting system. As a byproduct, this distance determines the optimal free state that reproduces the entanglement properties of the interacting system as faithfully as possible. The pare…
We propose and test an algorithm to simulate a lattice system of interacting fermions in two spatial dimensions. The approach is an extension of the entanglement renormalization technique [Phys. Rev. Lett. 99, 220405 (2007)] and the related multi-scale entanglement renormalization ansatz. Benchmark calculations for free and interacting fermions on a lattice of $6\times 6$ sites and periodic boundary conditions confirm the validity of this proposal.
The eigenstate entanglement entropy has been recently shown to be a powerful tool to distinguish integrable from generic quantum-chaotic models. In integrable models, a unique feature of the average eigenstate entanglement entropy (over all Hamiltonian eigenstates) is that the volume-law coefficient depends on the subsystem fraction. Hence, it deviates from the maximal (subsystem fraction independent) value encountered in quantum-chaotic models. Using random matrix theory for quadratic Hamiltonians, we obtain a closed-form expression for the average eigenstate entanglement entropy as a functio…
We investigate the entanglement dynamics of two interacting qubits in a spin environment, which is described by an XY model with Dzyaloshinsky-Moriya (DM) interaction. The competing effects of environmental noise and interqubit coupling on entanglement generation for various system parameters are studied. We find that the entanglement generation is suppressed remarkably in weak-coupling region at quantum critical point (QCP). However, the suppression of the entanglement generation at QCP can be compensated both by increasing the DM interaction and by decreasing the anisotropy of the spin chain…
Many-Body Entanglement Properties of Finite Interacting Fermionic Hamiltonians
Purdue University（普渡大学）；Taylor University（泰勒大学）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；University of Waterloo（滑铁卢大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
We investigate the extended Hubbard model as an approximation to the local and spatial entanglement of a one-dimensional chain of nanostructures where the particles interact via a long range interaction represented by a `soft' Coulomb potential. In the process we design a protocol to calculate the particle-particle spatial entanglement for the Hubbard model and show that, in striking contrast with the loss of spatial degrees of freedom, the predictions are reasonably accurate. We also compare results for the local entanglement with previous results found using a contact interaction (PRA, 81 (2…
We study the quench dynamics of entanglement spectra in the Kitaev chain with variable-range pairing quantified by power-law decay rate $α$. Considering the post-quench Hamiltonians with flat bands, we demonstrate that the presence of entanglement-spectrum crossings during its dynamics is able to characterize the topological phase transitions (TPTs) in both short-range ($α$ > 1) or long-range ($α$ < 1) sector. Properties of entanglement-spectrum dynamics are revealed for the quench protocols in the long-range sector or with $α$ as the quench parameter. In particular, when the lowest upper-half…
In quantum statistical mechanics, it is of fundamental interest to understand how close the bipartite entanglement entropy of eigenstates of quantum chaotic Hamiltonians is to maximal. For random pure states in the Hilbert space, the average entanglement entropy is known to be nearly maximal, with a deviation that is, at most, a constant. Here we prove that, in a system that is away from half filling and divided in two equal halves, an upper bound for the average entanglement entropy of random pure states with a fixed particle number and normally distributed real coefficients exhibits a deviat…
We investigate the dynamics of bipartite entanglement after the sudden junction of two leads in interacting integrable models. By combining the quasiparticle picture for the entanglement spreading with Generalised Hydrodynamics we derive an analytical prediction for the dynamics of the entanglement entropy between a finite subsystem and the rest. We find that the entanglement rate between the two leads depends only on the physics at the interface and differs from the rate of exchange of thermodynamic entropy. This contrasts with the behaviour in free or homogeneous interacting integrable syste…
Interacting disordered zigzag graphene nanoribbons have fractional charges, are quasi-one-dimensional, and display an exponentially small gap. Our numerical computations showed that the topological entanglement entropy of these systems has a small finite but universal value, independent of the strength of the interaction and the disorder. The result that was obtained for the topological entanglement entropy shows that the disorder-free phase is critical and becomes unstable in the presence of disorder.
The reduced dynamics of a single or two qubits coupled to an interacting quantum spin bath modeled by a XXZ spin chain is investigated. By using the method of time-dependent density matrix renormalization group (t-DMRG), we go beyond the uniform coupling central spin model and evaluate nonperturbatively the induced decoherence and entanglement. It is shown that both decoherence and entanglement strongly depend on the phase of the underlying spin bath. We show that in general, spin baths can induce entanglement for an initially disentangled pair of qubits. Furthermore, when the spin bath is in …
We introduce a method for the search of parent Hamiltonians of input wave-functions based on the structure of their reduced density matrix. The two key elements of our recipe are an ansatz on the relation between reduced density matrix and parent Hamiltonian that is exact at the field theory level, and a minimization procedure on the space of relative entropies, which is particularly convenient due to its convexity. As examples, we show how our method correctly reconstructs the parent Hamiltonian correspondent to several non-trivial ground state wave functions, including conformal and symmetry…
Multipartite entanglement, such as witnessed through the quantum Fisher information (QFI), is a crucial resource for quantum technologies, but its experimental certification is highly challenging. Here, we propose an experimentally friendly protocol to measure the QFI. It relies on recording the short-time dynamics of simple observables after a quench from a thermal state, works for spins, bosons, and fermions, and can be implemented in standard cold-atom experiments and other platforms with temporal control over the system Hamiltonian. To showcase the protocol, we simulate it for the one-dime…
We studied numerically the distribution of the entanglement Hamiltonian eigenvalues in two one-dimensional free fermion models and the typical three-dimensional Anderson model. We showed numerically that this distribution depends on the phase of the system: In the delocalized phase it is centered around very small values and in the localized phase, picks of the distribution goes to larger values. We therefore, based on the distribution of entanglement Hamiltonian eigenvalues, explain the behavior of the entanglement entropy in different phases. In addition we propose the smallest magnitude ent…
We investigate the entanglement between individual field theory modes in finite-density systems of interacting relativistic and non-relativistic fermions in one spatial dimension. We calculate the entanglement entropy for a single field theory mode and the mutual information between any two modes. The calculation is perturbative in the four-fermion (two-body) coupling, with the leading contribution at order lambda^2 log(lambda^2). At this leading order, the perturbative expression for the entanglement entropy of a mode diverges logarithmically as the momentum of the mode approaches the Fermi s…
We study the entanglement spectrum of a Chern insulator on a cylinder geometry, with the cut separating the two partitions taken parallel to the cylinder edge, at varying distances from the edge. In contrast to similar studies on a torus, there is only one cut, and hence only one virtual edge mode in the entanglement spectrum. The entanglement spectrum has a gap when the cut is close enough to the physical edge of the cylinder such that the edge mode spatially extends over the cut. This effect is suppressed for parameter choices where the edge mode is sharply localized at the edge. In the extr…
In a many-body quantum system, local operators in Heisenberg picture $O(t) = e^{i H t} O e^{-i H t}$ spread as time increases. Recent studies have attempted to find features of that spreading which could distinguish between chaotic and integrable dynamics. The operator entanglement - the entanglement entropy in operator space - is a natural candidate to provide such a distinction. Indeed, while it is believed that the operator entanglement grows linearly with time $t$ in chaotic systems, numerics suggests that it grows only logarithmically in integrable systems. That logarithmic growth has alr…
Entanglement measures such as the entanglement entropy have become an indispensable tool to identify the fundamental character of ground states of interacting quantum many-body systems. For systems of interacting spin or bosonic degrees of freedom much recent progress has been made not only in the analytical description of their respective entanglement entropies but also in their numerical classification. Systems of interacting fermionic degrees of freedom however have proved to be more difficult to control, in particular with regard to the numerical understanding of their entanglement propert…
Adding interactions to many-body Hamiltonians of geometrically frustrated lattices often leads to diminished subspaces of localized states. In this paper, we show how to construct interacting many-body Hamiltonians, starting from the non-interacting tight-binding Hamiltonians, that preserve or even expand these subspaces. The methods presented involve modifications in the one-body network representation of the many-body Hamiltonians which generate new interacting terms in these Hamiltonians. The subspace of many-particle localized states can be preserved in the interacting Hamiltonian, by proj…
The entanglement and localization in eigenstates of strongly chaotic subsystems are studied as a function of their interaction strength. Excellent measures for this purpose are the von-Neumann entropy, Havrda-Charv{\' a}t-Tsallis entropies, and the averaged inverse participation ratio. All the entropies are shown to follow a remarkably simple exponential form, which describes a universal and rapid transition to nearly maximal entanglement for increasing interaction strength. An unexpectedly exact relationship between the subsystem averaged inverse participation ratio and purity is derived that…
The properties of the entanglement entropy (EE) in one-dimensional disordered interacting systems are studied. Anderson localization leaves a clear signature on the average EE, as it saturates on length scale exceeding the localization length. This is verified by numerically calculating the EE for an ensemble of disordered realizations using density matrix renormalization group (DMRG). A heuristic expression describing the dependence of the EE on the localization length, which takes into account finite size effects, is proposed. This is used to extract the localization length as function of th…
A system of fermions forming a Fermi surface exhibits a large degree of quantum entanglement, even in the absence of interactions. In particular, the usual case of a codimension one Fermi surface leads to a logarithmic violation of the area law for entanglement entropy, as dictated by the Widom formula. We here generalize this formula to the case of arbitrary codimension, which is of particular interest for nodal lines in three dimensions. We first rederive the standard Widom formula by calculating an entanglement Hamiltonian for Fermi surface systems, obtained by repurposing a trick commonly …
In this work, we make a connection between two seemingly different problems. The first problem involves characterizing the properties of entanglement in the ground state of gapped local Hamiltonians, which is a central topic in quantum many-body physics. The second problem is on the quantum communication complexity of testing bipartite states with EPR assistance, a well-known question in quantum information theory. We construct a communication protocol for testing (or measuring) the ground state and use its communication complexity to reveal a new structural property for the ground state entan…
Dramatically different from the Hermitian systems, the conventional Bulk-Boundary Correspondence (BBC) is broken in the non-Hermitian systems. In this article, we use edge entanglement entropy to characterize the topological properties of non-Hermitian Su-Schrieffer-Heeger Hubbard model. For free Fermions, we study the scaling behavior of entanglement entropy and demonstrate that the edge entanglement entropy is a good indicator to delimit different phases of non-Hermitian systems. We further generalize the edge entanglement entropy to the non-Hermitian interacting Hubbard chain, and obtain th…
We study in this work the ground state entanglement properties of two models of non-interacting fermions moving in one-dimension (1D), that exhibit metal-insulator transitions. We find that entanglement entropy grows either logarithmically or in a power-law fashion in the metallic phase, thus violating the (1D version of) entanglement area law. No such violation is found in the insulating phase. We further find that characteristics of {\em single fermion} states at the Fermi energy (which can {\em not} be obtained from the many-fermion Slater determinant) is captured by the lowest energy singl…
Relatively local Hamiltonians are a class of background independent non-local Hamiltonians from which local theories emerge within a set of short-range entangled states. The dimension, topology and geometry of the emergent local theory is determined by the initial state to which the Hamiltonian is applied. In this paper, we study dynamical properties of a simple relatively local Hamiltonian for N scalar fields in the large N limit. It is shown that the coordinate speeds at which entanglement spreads and local disturbance propagates in space strongly depend on state in the relatively local Hami…
A general method to build the entanglement renormalization (cMERA) for interacting quantum field theories is presented. We improve upon the well-known Gaussian formalism used in free theories through a class of variational non-Gaussian wavefunctionals for which expectation values of local operators can be efficiently calculated analytically and in a closed form. The method consists of a series of scale-dependent nonlinear canonical transformations on the fields of the theory under consideration. Here, the $λ\, ϕ^4$ and the sine-Gordon scalar theories are used to illustrate how non-perturbative…
We study the fractal properties of single-particle eigen-modes of entanglement Hamiltonian in free fermion models. One of these modes that has the highest entanglement information and thus called maximally entangled mode (MEM) is specially considered. In free Fermion models with Anderson localization, fractality of MEM is obtained numerically and compared with the fractality of Hamiltonian eigen-mode at Fermi level. We show that both eigen-modes have similar fractal properties: both have same single fractal dimension in delocalized phase which equals the dimension of the system, and both show …
We show that the two recently proposed methods to compute Renyi entanglement entropies in the realm of determinant quantum Monte Carlo methods for fermions are in principle equivalent, but differ in sampling strategies. The analogy allows to formulate a numerically stable calculation of the entanglement spectrum at strong coupling. We demonstrate the approach by studying static and dynamical properties of the entanglement hamiltonian across the interaction driven quantum phase transition between a topological insulator and quantum antiferromagnet in the Kane-Mele Hubbard model. The formulation…
The existence of degenerate or gapless edge states is a characteristic feature of topological insulators, but is difficult to detect in the presence of interactons. We propose a new method to obtain the degeneracy of the edge states from the perspective of entanglement entropy, which is very useful to identify interacting topological states. Employing the determinant quantum Monte Carlo technique, we investigate the interaction effect on two representative models of fermionic topological insulators in one and two dimensions, respectively. In the two topologically nontrivial phases, the edge de…
