Many-body localization beyond eigenstates in all dimensions

Isolated quantum systems with quenched randomness exhibit many-body localization (MBL), wherein they do not reach local thermal equilibrium even when highly excited above their ground states. It is widely believed that individual eigenstates capture this breakdown of thermalization at finite size. We show that this belief is false in general and that a MBL system can exhibit the eigenstate properties of a thermalizing system. We propose that localized approximately conserved operators (l$^*$-bits) underlie localization in such systems. In dimensions $d>1$, we further argue that the existing MBL phenomenology is unstable to boundary effects and gives way to l$^*$-bits. Physical consequences of l$^*$-bits include the possibility of an eigenstate phase transition within the MBL phase unrelated to the dynamical transition in $d=1$ and thermal eigenstates at all parameters in $d>1$. Near-term experiments in ultra-cold atomic systems and numerics can probe the dynamics generated by boundary layers and emergence of l$^*$-bits.Comment: 12 pages, 5 figure

Boxicity and separation dimension

A family $\mathcal{F}$ of permutations of the vertices of a hypergraph $H$ is called 'pairwise suitable' for $H$ if, for every pair of disjoint edges in $H$, there exists a permutation in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for $H$ is called the 'separation dimension' of $H$ and is denoted by $\pi(H)$. Equivalently, $\pi(H)$ is the smallest natural number $k$ so that the vertices of $H$ can be embedded in $\mathbb{R}^k$ such that any two disjoint edges of $H$ can be separated by a hyperplane normal to one of the axes. We show that the separation dimension of a hypergraph $H$ is equal to the 'boxicity' of the line graph of $H$. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to WG-2014. Some results proved in this paper are also present in arXiv:1212.6756. arXiv admin note: substantial text overlap with arXiv:1212.675

Bulk-Edge Correspondence in the Entanglement Spectra

Li and Haldane conjectured and numerically substantiated that the entanglement spectrum of the reduced density matrix of ground-states of time-reversal breaking topological phases (fractional quantum Hall states) contains information about the counting of their edge modes when the ground-state is cut in two spatially distinct regions and one of the regions is traced out. We analytically substantiate this conjecture for a series of FQH states defined as unique zero modes of pseudopotential Hamiltonians by finding a one to one map between the thermodynamic limit counting of two different entanglement spectra: the particle entanglement spectrum, whose counting of eigenvalues for each good quantum number is identical (up to accidental degeneracies) to the counting of bulk quasiholes, and the orbital entanglement spectrum (the Li-Haldane spectrum). As the particle entanglement spectrum is related to bulk quasihole physics and the orbital entanglement spectrum is related to edge physics, our map can be thought of as a mathematically sound microscopic description of bulk-edge correspondence in entanglement spectra. By using a set of clustering operators which have their origin in conformal field theory (CFT) operator expansions, we show that the counting of the orbital entanglement spectrum eigenvalues in the thermodynamic limit must be identical to the counting of quasiholes in the bulk. The latter equals the counting of edge modes at a hard-wall boundary placed on the sample. Moreover, we show this to be true even for CFT states which are likely bulk gapless, such as the Gaffnian wavefunction.Comment: 20 pages, 6 figure

Real-Space Entanglement Spectrum of Quantum Hall States

We investigate the entanglement spectra arising from sharp real-space partitions of the system for quantum Hall states. These partitions differ from the previously utilized orbital and particle partitions and reveal complementary aspects of the physics of these topologically ordered systems. We show, by constructing one to one maps to the particle partition entanglement spectra, that the counting of the real-space entanglement spectra levels for different particle number sectors versus their angular momentum along the spatial partition boundary is equal to the counting of states for the system with a number of (unpinned) bulk quasiholes excitations corresponding to the same particle and flux numbers. This proves that, for an ideal model state described by a conformal field theory, the real-space entanglement spectra level counting is bounded by the counting of the conformal field theory edge modes. This bound is known to be saturated in the thermodynamic limit (and at finite sizes for certain states). Numerically analyzing several ideal model states, we find that the real-space entanglement spectra indeed display the edge modes dispersion relations expected from their corresponding conformal field theories. We also numerically find that the real-space entanglement spectra of Coulomb interaction ground states exhibit a series of branches, which we relate to the model state and (above an entanglement gap) to its quasiparticle-quasihole excitations. We also numerically compute the entanglement entropy for the nu=1 integer quantum Hall state with real-space partitions and compare against the analytic prediction. We find that the entanglement entropy indeed scales linearly with the boundary length for large enough systems, but that the attainable system sizes are still too small to provide a reliable extraction of the sub-leading topological entanglement entropy term.Comment: 13 pages, 11 figures; v2: minor corrections and formatting change

Thermal inclusions: how one spin can destroy a many-body localized phase

Many-body localized (MBL) systems lie outside the framework of statistical mechanics, as they fail to equilibrate under their own quantum dynamics. Even basic features of MBL systems such as their stability to thermal inclusions and the nature of the dynamical transition to thermalizing behavior remain poorly understood. We study a simple model to address these questions: a two level system interacting with strength $J$ with $N\gg 1$ localized bits subject to random fields. On increasing $J$, the system transitions from a MBL to a delocalized phase on the \emph{vanishing} scale $J_c(N) \sim 1/N$, up to logarithmic corrections. In the transition region, the single-site eigenstate entanglement entropies exhibit bi-modal distributions, so that localized bits are either "on" (strongly entangled) or "off" (weakly entangled) in eigenstates. The clusters of "on" bits vary significantly between eigenstates of the \emph{same} sample, which provides evidence for a heterogenous discontinuous transition out of the localized phase in single-site observables. We obtain these results by perturbative mapping to bond percolation on the hypercube at small $J$ and by numerical exact diagonalization of the full many-body system. Our results imply the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions $d$ that are high but finite.Comment: 17 pages, 12 figure

Haldane Statistics in the Finite Size Entanglement Spectra of Laughlin States

We conjecture that the counting of the levels in the orbital entanglement spectra (OES) of finite-sized Laughlin Fractional Quantum Hall (FQH) droplets at filling $\nu=1/m$ is described by the Haldane statistics of particles in a box of finite size. This principle explains the observed deviations of the OES counting from the edge-mode conformal field theory counting and directly provides us with a topological number of the FQH states inaccessible in the thermodynamic limit- the boson compactification radius. It also suggests that the entanglement gap in the Coulomb spectrum in the conformal limit protects a universal quantity- the statistics of the state. We support our conjecture with ample numerical checks.Comment: 4.1 pages, published versio