Isaac H. Kim

Constructing local integrals of motion in the many-body localized phase

Many-body localization provides a generic mechanism of ergodicity breaking in quantum systems. In contrast to conventional ergodic systems, many-body localized (MBL) systems are characterized by extensively many local integrals of motion (LIOM), which underlie the absence of transport and thermalization in these systems. Here we report a physically motivated construction of local integrals of motion in the MBL phase. We show that any local operator (e.g., a local particle number or a spin flip operator), evolved with the systems Hamiltonian and averaged over time, becomes a LIOM in the MBL phase. Such operators have a clear physical meaning, describing the response of the MBL system to a local perturbation. In particular, when a local operator represents a density of some globally conserved quantity, the corresponding LIOM describes how this conserved quantity propagates through the MBL phase. Being uniquely defined and experimentally measurable, these LIOMs provide a natural tool for characterizing the properties of the MBL phase, both in experiments and numerical simulations. We demonstrate the latter by numerically constructing an extensive set of LIOMs in the MBL phase of a disordered spin chain model. We show that the resulting LIOMs are quasi-local, and use their decay to extract the localization length and establish the location of the transition between the MBL and ergodic phases.

Operator extension of strong subadditivity of entropy

We prove an operator inequality that extends strong subadditivity of entropy: after taking a trace, the operator inequality becomes the strong subadditivity of entropy.

Spectral tensor networks for many-body localization

Subsystems of strongly disordered, interacting quantum systems can fail to thermalize because of the phenomenon of many-body localization (MBL). In this article, we explore a tensor network description of the eigenspectra of such systems. Specifically, we will argue that the presence of a complete set of local integrals of motion in MBL implies an efficient representation of the entire spectrum of energy eigenstates with a single tensor network, a \emph{spectral} tensor network. Our results are rigorous for a class of idealized systems related to MBL with integrals of motion of finite support. In one spatial dimension, the spectral tensor network allows for the efficient computation of expectation values of a large class of operators (including local operators and string operators) in individual energy eigenstates and in ensembles.

Perturbative analysis of topological entanglement entropy from conditional independence

Topological entanglement entropy is a topological invariant which can detect topological order of quantum many-body ground state. We assume an existence of such order parameter at finite temperature which is invariant under smooth deformation of the subsystems, and study its stability under hamiltonian perturbation. We apply this assumption to a Gibbs state of hamiltonian which satisfies so called ‘strong commuting’ condition, which we shall define in the paper. Interesting models in this category include local hamiltonian models based on quantum error correcting code. We prove a stability of such topologically invariant order parameter against arbitrary perturbation which can be expressed as a sum of geometrically local bounded-norm terms. The first order correction against such perturbation vanishes in the thermodynamic limit.We use the structure of conditionally independent states to analyze the stability of topological entanglement entropy. For the ground state of the quantum double or Levin-Wen model, we obtain a bound on the first-order perturbation of topological entanglement entropy in terms of its energy gap and subsystem size. The bound decreases superpolynomially with the size of the subsystem, provided the energy gap is nonzero. We also study the finite-temperature stability of stabilizer models, for which we prove a stronger statement than the strong subadditivity of entropy. Using this statement and assuming (i) finite correlation length and (ii) small conditional mutual information of certain configurations, first-order perturbation effect for arbitrary local perturbation can be bounded. We discuss the technical obstacles in generalizing these results.

Localization from Superselection Rules in Translationally Invariant Systems

The cubic code model is studied in the presence of arbitrary extensive perturbations. Below a critical perturbation strength, we show that most states with finite energy are localized; the overwhelming majority of such states have energy concentrated around a finite number of defects, and remain so for a time that is near exponential in the distance between the defects. This phenomenon is due to an emergent superselection rule and does not require any disorder. Local integrals of motion for these finite energy sectors are identified as well. Our analysis extends more generally to systems with immobile topological excitations.

Bounds on the concavity of quantum entropy

We give new upper and lower bounds on the concavity of quantum entropy. Comparisons are given with other results in the literature.

Modulus of convexity for operator convex functions

Given an operator convex function f(x), we obtain an operator-valued lower bound for cf(x) + (1 − c)f(y) − f(cx + (1 − c)y), c ∈ [0, 1]. The lower bound is expressed in terms of the matrix Bregman divergence. A similar inequality is shown to be false for functions that are convex but not operator convex.

Long-range entanglement is necessary for a topological storage of quantum information

A general inequality between entanglement entropy and a number of topologically ordered states is derived, even without using the properties of the parent Hamiltonian or the formalism of topological quantum field theory. Given a quantum state |ψ], we obtain an upper bound on the number of distinct states that are locally indistinguishable from |ψ]. The upper bound is determined only by the entanglement entropy of some local subsystems. As an example, we show that log N≤2γ for a large class of topologically ordered systems on a torus, where N is the number of topologically protected states and γ is the constant subcorrection term of the entanglement entropy. We discuss applications to quantum many-body systems that do not have any low-energy topological quantum field theory description, as well as tradeoff bounds for general quantum error correcting codes.

Determining the structure of the real-space entanglement spectrum from approximate conditional independence

We study the ground state of a gapped quantum many-body system whose entanglement entropy S_A can be expressed as S_A=a|∂A|−γ, where a,γ are some constants and |∂A| is an area of the subsystem A. By using a recently proved operator extension of strong subadditivity of entropy [ I. H. Kim J. Math. Phys. 53 122204 (2012)], we show that a certain linear combination of the real-space entanglement spectrum has a small correlation with almost any local operator. Our result implies that there exists a structure relating the real-space entanglement spectrum over different subsystems. Further, this structure is inherited from the generic property of the ground state alone, suggesting that the locality of the entanglement spectrum may be attributed to the area law of entanglement entropy.

Ground state entanglement constrains low-energy excitations

For a general quantum many-body system, we show that its ground-state entanglement imposes a fundamental constraint on the low-energy excitations. For two-dimensional systems, our result implies that any system that supports anyons must have a nonvanishing topological entanglement entropy. We demonstrate the generality of this argument by applying it to three-dimensional quantum many-body systems, and showing that there is a pair of ground state topological invariants that are associated to their physical boundaries. From the pair, one can determine whether the given boundary can or cannot absorb point-like or line-like excitations.