Entanglement Cut Research Articles

Numerical investigations of the extensive entanglement Hamiltonian in quantum spin ladders

Entanglement constitutes one of the key concepts in quantum mechanics and serves as an indispensable tool in the understanding of quantum many-body systems. In this work, we perform extensive numerical investigations of extensive entanglement properties of coupled quantum spin chains. This setup has proven useful for e.g. extending the Lieb–Schultz–Mattis theorem to open systems, and contrasts the majority of previous research where the entanglement cut has one lower dimension than the system. We focus on the cases where the entanglement Hamiltonian is either gapless or exhibits spontaneous symmetry breaking behavior. We further employ conformal field theoretical formulae to identify the universal behavior in the former case. The results in our work can serve as a paradigmatic starting point for more systematic exploration of the largely uncharted physics of extensive entanglement, both analytical and numerical.

Characterizing the ambiguity in topological entanglement entropy

Topological entanglement entropy (TEE), the sub-leading term in the entanglement entropy of topological order, is the direct evidence of the long-range entanglement. While effective in characterizing topological orders on closed manifolds, TEE is model-dependent when entanglement cuts intersect with physical gapped boundaries. In this paper, we study the origin of this model-dependence by introducing a model-independent picture of partitioning the topological orders with gapped boundaries. In our picture, the entanglement boundaries (EBs), i.e. the virtual boundaries of each subsystem induced by the entanglement cuts, are assumed to be gapped boundaries with boundary defects. At this model-independent stage, there are two choices one has to make manually in defining the bi-partition: the boundary condition on the EBs, and the coherence between certain boundary states. We show that TEE appears because of a constraint on the defect configurations on the EBs, which is choice-dependent in the cases where the EBs touch gapped boundaries. This choice-dependence is known as the ambiguity in entanglement entropy. Different models intrinsically employ different choices, rendering TEE model-dependent. For D(ℤ2) topological order, the ambiguity can be fully characterized by two parameters that respectively quantifies the EB condition and the coherence. In particular, calculations compatible with the folding trick naturally choose EB conditions that respect electric-magnetic duality and set specific parameter values.

Thermalization of Gauge Theories from their Entanglement Spectrum.

Using dual theories embedded into a larger unphysical Hilbert space along entanglement cuts, we study the entanglement structure of Z_{2} lattice gauge theory in (2+1) spacetime dimensions. We demonstrate Li and Haldane's conjecture, and show consistency of the entanglement Hamiltonian with the Bisognano-Wichmann theorem. Studying nonequilibrium dynamics after a quench, we provide an extensive description of thermalization in Z_{2} gauge theory which proceeds in a characteristic sequence: Maximization of the Schmidt rank and spreading of level repulsion at early times, self-similar evolution with scaling coefficients α=0.8±0.2 and β=0.0±0.1 at intermediate times, and finally thermal saturation of the von Neumann entropy.

Emergence of a thermal equilibrium in a subsystem of a pure ground state by quantum entanglement

By numerically exact calculations of spin-1/2 antiferromagnetic Heisenberg models on small clusters, we demonstrate that quantum entanglement between subsystems $A$ and $B$ in a pure ground state of a whole system $A+B$ can induce thermal equilibrium in subsystem $A$. Here, the whole system is bipartitoned with the entanglement cut that covers the entire volume of subsystem $A$. Temperature ${\cal T}_{A}$ of subsystem $A$ is not a parameter but can be determined from the entanglement von Neumann entropy ${\cal S}_{A}$ and the total energy ${\cal E}_{A}$ of subsystem $A$ calculated for the ground state of the whole system. We show that temperature ${\cal T}_{A}$ can be derived by minimizing the relative entropy for the reduced density matrix operator of subsystem $A$ and the Gibbs state (i.e., thermodynamic density matrix operator) of subsystem $A$ with respect to the coupling strength between subsystems $A$ and $B$. Temperature ${\cal T}_{A}$ is essentially identical to the thermodynamic temperature, for which the entropy and the internal energy evaluated using the canonical ensemble in statistical mechanics for the isolated subsystem $A$ agree numerically with the entanglement entropy ${\cal S}_{A}$ and the total energy ${\cal E}_{A}$ of subsystem $A$.Fidelity calculations ascertain that the reduced density matrix operator of subsystem $A$ for the pure but entangled ground state of the whole system $A+B$ matches, within a maximally $1.5\%$ error in the finite size clusters studied, the thermodynamic density matrix operator of subsystem $A$ at temperature ${\cal T}_{A}$. We argue that quantum fluctuation in an entangled pure state can mimic thermal fluctuation in a subsystem. We also provide two simple but nontrivial analytical examples of free bosons and free fermions for which these statements are exact. We furthermore discuss implications and possible applications of our finding.

Multiregion entanglement in locally scrambled quantum dynamics

We study the evolution of multi-region bipartite entanglement entropy under locally scrambled quantum dynamics. We show that the multi-region entanglement can significantly modify the growth of single-region entanglement, whose effect has been largely overlooked in the existing literature. We developed a novel theoretical framework, called the entanglement feature formalism, to organize all the multi-region entanglement systematically as a sign-free many-body state. We further propose a two-parameter matrix product state (MPS) ansatz to efficiently capture the exponentially many multi-region entanglement features. Using these tools, we are able to study the multi-region entanglement dynamics jointly and represent the evolution in the MPS parameter space. By comparing the dynamical constraints on the motion of entanglement cuts, we are able to identify different quantum dynamics models in a unifying entanglement feature Hamiltonian. Depending on the quantum dynamics model, we find that multi-region effects can dominate the single region entanglement growth and only vanish for Haar random circuits. We calculate the operator-averaged out-of-time-order correlator based on the entanglement feature Hamiltonian and extract the butterfly velocity from the result. We show that the previously conjectured bound between the entanglement velocity and the butterfly velocity holds true even under the influence of multi-region entanglement. These developments could enable more efficient numerical simulations and more systematic theoretical understandings of the multi-region entanglement dynamics in quantum many-body systems.

Why Agential Realism Matters to Social Work

AbstractThis article is a theory-building exercise, which focusses on Baradian agential realism to build a framework of key concepts and their interrelationships to show how and why this perspective matters to social work. It makes a statement about the relevance of agential realism, in the writings of Karen Barad and other feminist post-humanist thinkers, to rethink fundamental concepts, which are embedded in social work discourse, especially those humanist notions that relate to agency, subjectivity and reflection. It charts how agential realism can become foundational for social work knowledge. The article is intended both as an introduction to the thought of Barad and as a mapping of current agential realist scholarly trajectories. It guides the reader through the conceptual terrain of agential realist thought by explicating the key concepts of intra-action, entanglement, diffraction and agential cuts. In order to furnish an accessible engagement with Baradian theory, the article marshals case studies and references, drawing on ethnographic and qualitative research, which relate to social work. It concludes by showing how agential realism can provide a rich ontological framework for social work and highlights how it disrupts some of our long-held cherished beliefs about what it means to do social work.

Bulk entanglement entropy for photons and gravitons in AdS$_3$

We study quantum corrections to holographic entanglement entropy in AdS_33/CFT_22; these are given by the bulk entanglement entropy across the Ryu-Takayanagi surface for all fields in the effective gravitational theory. We consider bulk U(1)U(1) gauge fields and gravitons, whose dynamics in AdS_33 are governed by Chern-Simons terms and are therefore topological. In this case the relevant Hilbert space is that of the edge excitations. A novelty of the holographic construction is that such modes live not only on the bulk entanglement cut but also on the AdS boundary. We describe the interplay of these excitations and provide an explicit map to the appropriate extended Hilbert space. We compute the bulk entanglement entropy for the CFT vacuum state and find that the effect of the bulk entanglement entropy is to renormalize the relation between the effective holographic central charge and Newton’s constant. We also consider excited states obtained by acting with the U(1)U(1) current on the vacuum, and compute the difference in bulk entanglement entropy between these states and the vacuum. We compute this UV-finite difference both in the bulk and in the CFT finding a perfect agreement.

Symmetry resolved entanglement: exact results in 1D and beyond

In a quantum many-body system that possesses an additive conserved quantity, the entanglement entropy of a subsystem can be resolved into a sum of contributions from different sectors of the subsystem’s reduced density matrix, each sector corresponding to a possible value of the conserved quantity. Recent studies have discussed the basic properties of these symmetry-resolved contributions, and calculated them using conformal field theory and numerical methods. In this work we employ the generalized Fisher–Hartwig conjecture to obtain exact results for the characteristic function of the symmetry-resolved entanglement (‘flux-resolved entanglement’) for certain 1D spin chains, or, equivalently, the 1D fermionic tight binding and the Kitaev chain models. These results are true up to corrections of order where L is the subsystem size. We confirm that this calculation is in good agreement with numerical results. For the gapless tight binding chain we report an intriguing periodic structure of the characteristic functions, which nicely extends the structure predicted by conformal field theory. For the Kitaev chain in the topological phase we demonstrate the degeneracy between the even and odd fermion parity sectors of the entanglement spectrum due to virtual Majoranas at the entanglement cut. We also employ the Widom conjecture to obtain the leading behavior of the symmetry-resolved entanglement entropy in higher dimensions for an ungapped free Fermi gas in its ground state.

Topological classification of molecules and chemical reactions with a perplectic structure

In this paper, a topological classification of molecules and their chemical reactions is proposed on a single particle level . We consider zero-dimensional electronic Hamiltonians in a real-space tight-binding basis with spinless time-reversal symmetry and an additional spatial reflection symmetry. The symmetry gives rise to a perplectic structure and suggests a $\mathbb{Z}_2$ invariant in form of a pfaffian, which can be captured by an entanglement cut. We apply our findings to a class of chemical reactions studied by Woodward and Hoffmann, where a reflection symmetry is preserved along a one-dimensional reaction path and argue that the topological classification should contribute to the rate constants of these reactions. More concretely, we find that a reaction takes place experimentally whenever the reactants and products can be adiabatically deformed into each other, while reactions that require a change of topological invariants have not been observed experimentally.

Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part II. Cutting through the boundary

We compute the entanglement entropy in a 2+1 dimensional topological order in the presence of gapped boundaries. Specifically, we consider entanglement cuts that cut through the boundaries. We argue that based on general considerations of the bulk- boundary correspondence, the “twisted characters” feature in the Renyi entropy, and the topological entanglement entropy is controlled by a “half-linking number” in direct analogy to the role played by the S-modular matrix in the absence of boundaries. We also construct a class of boundary states based on the half-linking numbers that provides a “closed-string” picture complementing an “open-string” computation of the entanglement entropy. These boundary states do not correspond to diagonal RCFT’s in general. These are illustrated in specific Abelian Chern-Simons theories with appropriate boundary conditions.

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.

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.

Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part I

In this paper, we study gapped edges/interfaces in a 2+1 dimensional bosonic topological order and investigate how the topological entanglement entropy is sensitive to them. We present a detailed analysis of the Ishibashi states describing these edges/interfaces making use of the physics of anyon condensation in the context of Abelian Chern-Simons theory, which is then generalized to more non-Abelian theories whose edge RCFTs are known. Then we apply these results to computing the entanglement entropy of different topological orders. We consider cases where the system resides on a cylinder with gapped boundaries and that the entanglement cut is parallel to the boundary. We also consider cases where the entanglement cut coincides with the interface on a cylinder. In either cases, we find that the topological entanglement entropy is determined by the anyon condensation pattern that characterizes the interface/boundary. We note that conditions are imposed on some non-universal parameters in the edge theory to ensure existence of the conformal interface, analogous to requiring rational ratios of radii of compact bosons.

Symmetry-protected topological interfaces and entanglement sequences

Gapped interfaces (and boundaries) of two-dimensional (2D) Abelian topological phases are shown to support a remarkably rich sequence of 1D symmetry-protected topological (SPT) states. We show that such interfaces can provide a physical interpretation for the corrections to the topological entanglement entropy of a 2D state with Abelian topological order found in Ref.~\onlinecite{cano-2015}. The topological entanglement entropy decomposes as $\gamma = \gamma_a + \gamma_s$, where $\gamma_a > 0$ only depends on universal topological properties of the 2D state, while a correction $\gamma_s > 0$ signals the emergence of the 1D SPT state that is produced by interactions along the entanglement cut and provides a direct measure of the stabilizing symmetry of the resulting SPT state. A correspondence is established between the possible values of $\gamma_s$ associated with a given interface - which is named Boundary Topological Entanglement Sequence - and classes of 1D SPT states. We show that symmetry-preserving domain walls along such 1D interfaces (or boundaries) generally host localized parafermion-like excitations that are stable to local symmetry-preserving perturbations.

Phase space entanglement spectrum

The study of quantum entanglement involves the choice of an entanglement cut, often corresponding to a partitioning of the system in position space. We generalize these partitions to a family of entanglement cuts corresponding to arbitrary directions in the classical phase space. For one dimensional free fermionic many-body wavefunctions, we analytically compute the full entanglement spectra for a family of entanglement cuts corresponding to a rotation in the classical phase space. These cuts continuously interpolate between the commonly studied position- and momentum-space cuts, as well as provide an implementation of the inversion symmetry smoothly connected to identity. The inversion symmetry of the wavefunctions is reflected in the chiral symmetry of the phase space entanglement spectra, and we use the associated chiral invariant to classify the inversion symmetric fermionic many-body wavefunctions.

Entanglement evolution across a conformal interface

For two-dimensional conformal field theories (CFTs) in the ground state, it is known that a conformal interface along the entanglement cut can suppress the entanglement entropy from to , where L is the length of the subsystem A, and is the effective central charge which depends on the transmission property of the conformal interface. In this work, by making use of conformal mappings, we show that a conformal interface has the same effect on entanglement evolution in non-equilibrium cases, including global, local and certain inhomogeneous quantum quenches. I.e. a conformal interface suppresses the time evolution of entanglement entropy by effectively replacing the central charge c with , where is exactly the same as that in the ground state case. We confirm this conclusion by a numerical study on a critical fermion chain. Furthermore, based on the quasi-particle picture, we conjecture that this conclusion holds for an arbitrary quantum quench in CFTs, as long as the initial state can be described by a regularized conformal boundary state.

Remarks on entanglement entropy in string theory

Entanglement entropy for spatial subregions is difficult to define in string theory because of the extended nature of strings. Here we propose a definition for Bosonic open strings using the framework of string field theory. The key difference (compared to ordinary quantum field theory) is that the subregion is chosen inside a Cauchy surface in the "space of open string configurations". We first present a simple calculation of this entanglement entropy in free light-cone string field theory, ignoring subtleties related to the factorization of the Hilbert space. We reproduce the answer expected from an effective field theory point of view, namely a sum over the one-loop entanglement entropies corresponding to all the particle-excitations of the string, and further show that the full string theory regulates the ultraviolet divergences in the entanglement entropy. We then revisit the question of factorization of the Hilbert space by analyzing the covariant phase-space associated with a subregion in Witten's covariant string field theory. We show that the pure gauge (i.e., BRST exact) modes in the string field become dynamical at the entanglement cut. Thus, a proper definition of the entropy must involve an extended Hilbert space, with new stringy edge modes localized at the entanglement cut.

Entanglement entropy from tensor network states for stabilizer codes

In this paper, we present the construction of tensor network states (TNS) for some of the degenerate ground states of 3D stabilizer codes. We then use the TNS formalism to obtain the entanglement spectrum and entropy of these ground-states for some special cuts. In particular, we work out the examples of the 3D toric code, the X-cube model and the Haah code. The latter two models belong to the category of "fracton" models proposed recently, while the first one belongs to the conventional topological phases. We mention the cases for which the entanglement entropy and spectrum can be calculated exactly: for these, the constructed TNS is the singular value decomposition (SVD) of the ground states with respect to particular entanglement cuts. Apart from the area law, the entanglement entropies also have constant and linear corrections for the fracton models, while the entanglement entropies for the toric code models only have constant corrections. For the cuts we consider, the entanglement spectra of these three models are completely flat. We also conjecture that the negative linear correction to the area law is a signature of extensive ground state degeneracy. Moreover, the transfer matrices of these TNS can be constructed. We show that the transfer matrices are projectors whose eigenvalues are either 1 or 0. The number of nonzero eigenvalues is tightly related to the ground state degeneracy.

Entanglement entropy of local operators in quantum Lifshitz theory

We study the growth of entanglement entropy (EE) of local operator excitation in the quantum Lifshitz model with dynamic exponent z = 2. Specifically, we apply a local vertex operator to the groundstate at a distance l to the entanglement cut and calculate the EE as a function of time for the state’s subsequent time evolution. We find that the excess EE compared with the groundstate is a monotonically increasing function which is vanishingly small before the onset at and eventually saturates at a constant value proportional to the scaling dimension of the vertex operator. The quasi-particle picture can interpret the final saturation as the exhaustion of the quasi-particle pairs, while the diffusive nature of the time scale replaces the common causality constraint in CFT calculations. To further understand this property, we compute the excess EE of a small disk probe far from the excitation point and find chromatography patterns in EE generated by quasi-particles of different propagation speeds.

Spurious long-range entanglement and replica correlation length

Topological entanglement entropy has been regarded as a smoking-gun signature of topological order in two dimensions, capturing the total quantum dimension of the topological particle content. An extrapolation method on cylinders has been used frequently to measure the topological entanglement entropy. Here, we show that a class of short-range entangled 2D states, when put on an infinite cylinder of circumference $L$, exhibits the entanglement R\'enyi entropy of any integer index $\alpha \ge 2$ that obeys $S_\alpha = a L - \gamma$ where $a, \gamma > 0 $. Under the extrapolation method, the subleading term $\gamma$ would be identified as the topological entanglement entropy, which is spurious. A nonzero $\gamma$ is always present if the 2D state reduces to a certain symmetry-protected topological 1D state, upon disentangling spins that are far from the entanglement cut. The internal symmetry that stabilizes $\gamma > 0$ is not necessarily a symmetry of the 2D state, but should be present after the disentangling reduction. If the symmetry is absent, $\gamma$ decays exponentially in $L$ with a characteristic length, termed as a replica correlation length, which can be arbitrarily large compared to the two-point correlation length of the 2D state. We propose a simple numerical procedure to measure the replica correlation length through replica correlation functions. We also calculate the replica correlation functions for representative wave functions of abelian discrete gauge theories and the double semion theory in 2D, to show that they decay abruptly to zero. This supports a conjecture that the replica correlation length being small implies that the subleading term from the extrapolation method determines the total quantum dimension.