Entanglement Spectrum Research Articles

Emergent Haldane phase in an alternating-bond Z3 parafermion chain

The Haldane phase represents one of the most important symmetry-protected states in modern physics. This state can be realized using spin-1 and $\mathrm{spin}\ensuremath{-}\frac{1}{2}$ Heisenberg models and bosonic particles. Here we explore the emergent Haldane phase in an alternating bond ${\mathbb{Z}}_{3}$ parafermion chain, which is different from the previous proposals from fundamental statistics and symmetries. We show that this emergent phase can also be characterized by a modified long-range string order, as well as fourfold degeneracy in the ground-state energies and entanglement spectra. This phase is protected by both the charge conjugate and parity symmetry, and the edge modes are shown to satisfy parafermionic statistics in which braiding of the two edge modes yields a $\frac{2\ensuremath{\pi}}{3}$ phase. This model also supports rich phases, including a topological ferromagnetic parafermion (FP) phase, trivial paramagnetic parafermion phase, classical dimer phase, and gapless phase. The boundaries of the FP phase are shown to be gapless and critical with central charge $c=4/5$. Even in the topological FP phase, it is also characterized by long-range string order; thus we observe a drop of string order across the phase boundary between the FP phase and the Haldane phase. This work opens a new way for finding of exotic topological phases with parafermions.

Flat entanglement spectra in fixed-area states of quantum gravity

We use the Einstein-Hilbert gravitational path integral to investigate gravita- tional entanglement at leading order O(1/G). We argue that semiclassical states prepared by a Euclidean path integral have the property that projecting them onto a subspace in which the Ryu-Takayanagi or Hubeny-Rangamani-Takayanagi surface has definite area gives a state with a flat entanglement spectrum at this order in gravitational perturbation theory. This means that the reduced density matrix can be approximated as proportional to the identity to the extent that its Renyi entropies Sn are independent of n at this order. The n-dependence of Sn in more general states then arises from sums over the RT/HRT- area, which are generally dominated by different values of this area for each n. This provides a simple picture of gravitational entanglement, bolsters the connection between holographic systems and tensor network models, clarifies the bulk interpretation of alge- braic centers which arise in the quantum error-correcting description of holography, and strengthens the connection between bulk and boundary modular Hamiltonians described by Jafferis, Lewkowycz, Maldacena, and Suh.

Phase diagram of bipartite entanglement

We investigate the features of the entanglement spectrum (distribution of the eigenvalues of the reduced density matrix) of a large quantum system in a pure state. We consider all Rényi entropies and recover purity and von Neumann entropy as particular cases. We construct the phase diagram of the theory and unveil the presence of two critical lines.

Signatures of information scrambling in the dynamics of the entanglement spectrum

We examine the time evolution of the entanglement spectrum of a small subsystem of a non-integrable spin chain following a quench from a product state. We identify signatures in this entanglement spectrum of the distinct dynamical velocities (related to entanglement and operator spreading) that control thermalization. We show that the onset of level repulsion in the entanglement spectrum occurs on different timescales depending on the "entanglement energy", and that this dependence reflects the shape of the operator front. Level repulsion spreads across the entire entanglement spectrum on timescales close to when the mutual information between individual spins at the ends of the subsystem reaches its maximum; this timescale is much shorter than that for full thermalization of the subsystem. We provide an analytical understanding of this phenomenon and show supporting numerical data for both random unitary circuits and a microscopic Hamiltonian.

Unitary circuits of finite depth and infinite width from quantum channels

We introduce an approach to compute reduced density matrices for local quantum unitary circuits of finite depth and infinite width. Suppose the time-evolved state under the circuit is a matrix-product state with bond dimension $D$; then the reduced density matrix of a half-infinite system has the same spectrum as an appropriate $D\times D$ matrix acting on an ancilla space. We show that reduced density matrices at different spatial cuts are related by quantum channels acting on the ancilla space. This quantum channel approach allows for efficient numerical evaluation of the entanglement spectrum and R\'enyi entropies and their spatial fluctuations at finite times in an infinite system. We benchmark our numerical method on random unitary circuits, where many analytic results are available, and also show how our approach analytically recovers the behaviour of the kicked Ising model at the self-dual point. We study various properties of the spectra of the reduced density matrices and their spatial fluctuations in both the random and translation-invariant cases.

Dynamical phase behavior of the single- and multi-lane asymmetric simple exclusion process via matrix product states

The open asymmetric simple exclusion process (ASEP) has emerged as a paradigmatic model of nonequilibrium behavior, in part due to its complex dynamical behavior and wide physical applicability as a model of driven diffusion. We compare the dynamical phase behavior of the one-dimensional (1D) ASEP and the multi-lane ASEP, a previously unstudied extension of the 1D model that may be thought of as a finite-width strip of the fully two-dimensional (2D) system. Our characterization employs large deviation theory (LDT), matrix product states (MPS), and the density matrix renormalization group (DMRG) algorithm, to compute the current cumulant generating function and its derivatives, which serve as dynamical order parameters. We use this measure to show that when particles cannot exit or enter the lattice vertically, the phase behavior of the multi-lane ASEP mimics that of its 1D counterpart, exhibiting the macroscopic and microscopic signatures of the maximal current, shock, and high-density-low-density coexistence phases. Conversely, when particles are allowed to freely enter and exit the lattice, no such transition is observed. This contrast emphasizes the complex interplay between latitudinal and longitudinal hopping rates and the effect of current biasing. Our results support the potential of tensor networks as a framework to understand classical nonequilibrium statistical mechanics.

Entanglement entropy for the valence bond solid phases of two-dimensional dimerized Heisenberg antiferromagnets

We calculate the bipartite von Neumann and second R\'enyi entanglement entropies of the ground states of spin-1/2 dimerized Heisenberg antiferromagnets on a square lattice. Two distinct dimerization patterns are considered: columnar and staggered. In both cases, we concentrate on the valence bond solid (VBS) phase and describe such a phase with the bond-operator representation. Within this formalism, the original spin Hamiltonian is mapped into an effective interacting boson model for the triplet excitations. We study the effective Hamiltonian at the harmonic approximation and determine the spectrum of the elementary triplet excitations. We then follow an analytical procedure, which is based on a modified spin-wave theory for finite systems and was originally employed to calculate the entanglement entropies of magnetic ordered phases, and calculate the entanglement entropies of the VBS ground states. In particular, we consider one-dimensional (line) subsystems within the square lattice, a choice that allows us to consider line subsystems with sizes up to $L' = 1000$. We combine such a procedure with the results of the bond-operator formalism at the harmonic level and show that, for both dimerized Heisenberg models, the entanglement entropies of the corresponding VBS ground states obey an area law as expected for gapped phases. For both columnar-dimer and staggered-dimer models, we also show that the entanglement entropies increase but do not diverge as the dimerization decreases and the system approaches the N\'eel--VBS quantum phase transition. Finally, the entanglement spectra associated with the VBS ground states are presented.

Variational quantum state diagonalization

Variational hybrid quantum-classical algorithms are promising candidates for near-term implementation on quantum computers. In these algorithms, a quantum computer evaluates the cost of a gate sequence (with speedup over classical cost evaluation), and a classical computer uses this information to adjust the parameters of the gate sequence. Here we present such an algorithm for quantum state diagonalization. State diagonalization has applications in condensed matter physics (e.g., entanglement spectroscopy) as well as in machine learning (e.g., principal component analysis). For a quantum state ρ and gate sequence U, our cost function quantifies how far Urho U^dagger is from being diagonal. We introduce short-depth quantum circuits to quantify our cost. Minimizing this cost returns a gate sequence that approximately diagonalizes ρ. One can then read out approximations of the largest eigenvalues, and the associated eigenvectors, of ρ. As a proof-of-principle, we implement our algorithm on Rigetti’s quantum computer to diagonalize one-qubit states and on a simulator to find the entanglement spectrum of the Heisenberg model ground state.

Machine Learning Phase Diagram in the Half-filled One-dimensional Extended Hubbard Model

We demonstrate that supervised machine learning (ML) with entanglement spectrum can give useful information for constructing phase diagram in the half-filled one-dimensional extended Hubbard model. Combining ML with infinite-size density-matrix renormalization group, we confirm that bond-order-wave phase remains stable in the thermodynamic limit.

Ground state and low-energy excitations of the Kitaev-Heisenberg two-leg ladder

We study the ground state and low-lying excited states of the Kitaev-Heisenberg model on a ladder geometry using the density matrix renormalization group and Lanczos exact diagonalization methods. The Kitaev and Heisenberg interactions are parametrized as $K=\sin\phi$ and $J=\cos\phi$ with an angle parameter $\phi$. Based on the results for several types of order parameters, excitation gaps, and entanglement spectra, the $\phi$-dependent ground-state phase diagram is determined. Remarkably, the phase diagram is quite similar to that of the Kitaev-Heisenberg model on a honeycomb lattice, exhibiting the same long-range ordered states, namely rung-singlet (analog to N\'eel in 3D), zigzag, ferromagnetic, and stripy; and the presence of Kitaev spin liquids around the exactly solvable Kitaev points $\phi=\pm\pi/2$. We also calculate the expectation value of a plaquette operator corresponding to a $\pi$-flux state in order to establish how the Kitaev spin liquid extends away from the $\phi=\pm\pi/2$. Furthermore, we determine the dynamical spin structure factor and discuss the effect of the Kitaev interaction on the spin-triplet dispersion.

Mid-plateau State as a Topological Phase in Dimerized Spin-1/2 Two-Leg Ladders

The spin-1/2 two-leg ladders with dominant spatially modulated rung exchanges are known as the dimerized ladders. In the limit of strong rung exchange, the ground state of the system is in the rung-singlet (RS) gapped phase. Applying an external magnetic field creates two additional new gapped phases in the ground-state phase diagram of the model beside the existence of the two trivial gapless Luttinger liquid (LL) phases. It is already specified that the magnetization curve of the system exhibits a plateau at magnetization which is equal to the half of the saturation value in one of the non-trivial gapped phases known as the mid-plateau state. Here, we calculate the entanglement entropy and entanglement spectrum (ES) for finite ladder systems using the numerical Lanczos method. In fact, it has been suggested that the topological properties of the ground state can be reflected by specific degeneracy of the ES. We investigate the gapped RS and the mid-plateau phases where our outcomes show even degeneracy in the ES in the non-trivial RS and mid-plateau gapped phases is an indication of the symmetry-protected topological phases. In addition, we argue about the difference between the results of ES for various divisions of the ladder system.

Tensor chain and constraints in tensor networks

We develop our recent work on quantum error correction (QEC) and entanglement spectrum (ES) in tensor networks (arXiv:1806.05007). We propose a general framework for planar tensor network state with tensor constraints as a model for AdS3/CFT2 correspondence, which could be viewed as a generalization of hyperinvariant tensor networks recently proposed by Evenbly. We elaborate our proposal on tensor chains in a tensor network by tiling H2 space and provide a diagrammatical description for general multi-tensor constraints in terms of tensor chains, which forms a generalized greedy algorithm. The behavior of tensor chains under the action of greedy algorithm is investigated in detail. In particular, for a given set of tensor constraints, a critically protected (CP) tensor chain can be figured out and evaluated by its average reduced interior angle. We classify tensor networks according to their ability of QEC and the flatness of ES. The corresponding geometric description of critical protection over the hyperbolic space is also given.

Effects of laser phase noise on optomechanical entanglement in the presence of a nonlinear Kerr downconverter

In this paper, we consider an optomechanical system with a Kerr downconversion element in the presence of pump laser phase noise (LPN). In particular, we study the effect of phase fluctuations on entanglement generation in the presence of a nonlinear Kerr downconverter. The steady-state entanglement spectrum depends upon the parametric gain and phase fluctuations associated with the driving field. Interestingly, our results show that optomechanical entanglement exists even for non-zero LPN. It has been shown that the presence of Kerr downconverter nonlinearity improves the robustness of the entanglement against the input LPN.

Correspondence between bulk entanglement and boundary excitation spectra in two-dimensional gapped topological phases

We study the correspondence between boundary spectrum of non-chiral topological orders on an open manifold $\mathcal{M}$ with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed manifold. The closed manifold is bipartitioned into two subsystems, one of which has the same topology as $\mathcal{M}$. Specifically, we focus on the case of generalized string-net models and discuss the cases where $\mathcal{M}$ is a disk or a cylinder. When $\mathcal{M}$ has the topology of a cylinder, different combinations of boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. When both boundaries are charge (smooth) boundaries, the entanglement spectrum can be identified with the boundary excitation distribution spectrum at infinite temperature and constant fugacities. Examples of toric code, $\mathbb{Z}_N$ theories, and the simple non-abelian case of doubled Fibonacci are demonstrated.

Defining a bulk-edge correspondence for non-Hermitian Hamiltonians via singular-value decomposition

We address the breakdown of the bulk-boundary correspondence observed in non-Hermitian systems, where open and periodic systems can have distinct phase diagrams. The correspondence can be completely restored by considering the Hamiltonian's singular value decomposition instead of its eigendecomposition. This leads to a natural topological description in terms of a flattened singular decomposition. This description is equivalent to the usual approach for Hermitian systems and coincides with a recent proposal for the classification of non-Hermitian systems. We generalize the notion of the entanglement spectrum to non-Hermitian systems, and show that the edge physics is indeed completely captured by the periodic bulk Hamiltonian. We exemplify our approach by considering the chiral non-Hermitian Su-Schrieffer-Heger and Chern insulator models. Our work advocates a different perspective on topological non-Hermitian Hamiltonians, paving the way to a better understanding of their entanglement structure.

Locality of edge states and entanglement spectrum from strong subadditivity

We consider two-dimensional states of matter satisfying an uniform area law for entanglement. We show that the topological entanglement entropy is equal to the minimum relative entropy distance from the reduced state to the set of thermal states of local models. The argument is based on strong subadditivity of quantum entropy. For states with zero topological entanglement entropy, in particular, the formula gives locality of the states at the boundary of a region as thermal states of local Hamiltonians. It also implies that the entanglement spectrum of a two-dimensional region is equal to the spectrum of a one-dimensional local thermal state on the boundary of the region.

Quantum criticality on a chiral ladder: An SU(2) infinite density matrix renormalization group study

In this paper we study the ground state properties of a ladder Hamiltonian with chiral $SU(2)$-invariant spin interactions, a possible first step towards the construction of truly two dimensional non-trivial systems with chiral properties starting from quasi-one dimensional ones. Our analysis uses a recent implementation by us of $SU(2)$ symmetry in tensor network algorithms, specifically for infinite Density Matrix Renormalization Group (iDMRG). After a preliminary analysis with Kadanoff coarse-graining and exact diagonalization for a small-size system, we discuss its bosonization and recap the continuum limit of the model to show that it corresponds to a conformal field theory, in agreement with our numerical findings. In particular, the scaling of the entanglement entropy as well as finite-entanglement scaling data show that the ground state properties match those of the universality class of a $c = 1$ conformal field theory (CFT) in $(1+1)$ dimensions. We also study the algebraic decay of spin-spin and dimer-dimer correlation functions, as well as the algebraic convergence of the ground state energy with the bond dimension, and the entanglement spectrum of half an infinite chain. Our results for the entanglement spectrum are remarkably similar to those of the spin-$1/2$ Heisenberg chain, which we take as a strong indication that both systems are described by the same CFT at low energies, i.e., an $SU(2)_1$ Wess-Zumino-Witten theory. Moreover, we explain in detail how to construct Matrix Product Operators for $SU(2)$-invariant three-spin interactions, something that had not been addressed with sufficient depth in the literature.

Entanglement spectra of stabilizer codes: A window into gapped quantum phases of matter

The entanglement spectrum (ES) provides a barometer of quantum entanglement and encodes physical information beyond that contained in the entanglement entropy. In this paper, we explore the ES of stabilizer codes, which furnish exactly solvable models for a plethora of gapped quantum phases of matter. Studying the ES for stabilizer Hamiltonians in the presence of arbitrary weak local perturbations thus allows us to develop a general framework within which the entanglement features of gapped topological phases can be computed and contrasted. In particular, we study models harboring fracton order, both type-I and type-II, and compare the resulting ES with that of both conventional topological order and of (strong) subsystem symmetry protected topological (SSPT) states. We find that non-local surface stabilizers (NLSS), a set of symmetries of the Hamiltonian which form on the boundary of the entanglement cut, act as purveyors of universal non-local features appearing in the entanglement spectrum. While in conventional topological orders and fracton orders, the NLSS retain a form of topological invariance with respect to the entanglement cut, subsystem symmetric systems---fracton and SSPT phases---additionally show a non-trivial geometric dependence on the entanglement cut, corresponding to the subsystem symmetry. This sheds further light on the interplay between geometric and topological effects in fracton phases of matter and demonstrates that strong SSPT phases harbour a measure of quasi-local entanglement beyond that encountered in conventional SPT phases. We further show that a version of the edge-entanglement correspondence, established earlier for gapped two-dimensional topological phases, also holds for gapped three-dimensional fracton models.

Entanglement entropy, quantum fluctuations, and thermal entropy in topological phases

Entanglement entropy in topologically ordered matter phases has been computed extensively using various methods. In this paper, we study the entanglement entropy of topological phases in two-spaces from a new perspective — the perspective of quasiparticle fluctuations. In this picture, the entanglement spectrum of a topologically ordered system encodes the quasiparticle fluctuations of the system, and the entanglement entropy measures the maximal quasiparticle fluctuations on the EB. As a consequence, entanglement entropy corresponds to the thermal entropy of the quasiparticles at infinite temperature on the entanglement boundary. We corroborates our results with explicit computation in the quantum double model with/without boundaries. We then systematically construct the reduced density matrices of the quantum double model on generic 2-surfaces with boundaries.

Localization in Fractonic Random Circuits

We study the spreading of initially-local operators under unitary time evolution in a 1d random quantum circuit model which is constrained to conserve a $U(1)$ charge and its dipole moment, motivated by the quantum dynamics of fracton phases. We discover that charge remains localized at its initial position, providing a crisp example of a non-ergodic dynamical phase of random circuit dynamics. This localization can be understood as a consequence of the return properties of low dimensional random walks, through a mechanism reminiscent of weak localization, but insensitive to dephasing. The charge dynamics is well-described by a system of coupled hydrodynamic equations, which makes several nontrivial predictions in good agreement with numerics. Importantly, these equations also predict localization in 2d fractonic circuits. Immobile fractonic charge emits non-conserved operators, whose spreading is governed by exponents distinct to non-fractonic circuits. Fractonic operators exhibit a short time linear growth of observable entanglement with saturation to an area law, as well as a subthermal volume law for operator entanglement. The entanglement spectrum follows semi-Poisson statistics, similar to eigenstates of MBL systems. The non-ergodic phenomenology persists to initial conditions containing non-zero density of dipolar or fractonic charge. Our work implies that low-dimensional fracton systems preserve forever a memory of their initial conditions in local observables under noisy quantum dynamics, thereby constituting ideal memories. It also implies that 1d and 2d fracton systems should realize true MBL under Hamiltonian dynamics, even in the absence of disorder, with the obstructions to MBL in translation invariant systems and in d>1 being evaded by the nature of the mechanism responsible for localization. We also suggest a possible route to new non-ergodic phases in high dimensions.