Condensed Matter

We discuss some algebraic quantum field theory (AQFT) ingredients that should be useful in defining a tensor network describing a Lorentzian space-time.

We look into toy models that approximate Minkowski space and show how Lorentz boosts are approximately recovered, and obtain Rindler modes that can be compared with the entanglement spectrum. 

We make connections of these approximations of Lorentz boosts with the corner transfer matrices in integrable models, and comment on the discrete realization of the Reeh-Schlieder theorem that governs entanglement in a Hilbert space with lower bounded energy. 

Three dimensional fracton phases are new type of phases featuring exotic excitations called fractons. They are gapped point-like excitations constrained to move in sub-dimensional space. In this talk, I will present the gapped fracton topological order discovered in exact solvable models and gapless fracton phase described by U(1) symmetric tensor gauge theories. Their relation with ordinary topological ordered phase would be discussed in detail. Particularly, the fracton topological order exhibited in an exact solvable model called X-cube model can be constructed by coupling toric code layers. And it can also be obtained from a particular rank-2 symmetric tensor gauge theory called scalar charge theory by Higgs mechanism and partial confinement transition.

Searching for a proper set of order parameters which distinguishes different phases of matter sits in the heart of condensed matter physics. In this talk, I discuss topological invariants as (non-local) order parameters for symmetry protected topological (SPT) phases of fermions in the presence of time-reversal symmetry. It turns out that topological invariants provide a natural definition for the partial transpose of density matrices. The partial transpose can then be used to define an entanglement measure (analog of entanglement negativity) for mixed states of fermions. I will show that this quantity captures the mixed state entanglement in fermionic SPTs as well as in a system of free fermions with a Fermi surface.

Classical chaotic systems exhibit exponential divergence of initially infinitesimally close trajectories, which is characterized by the Lyapunov exponent. This sensitivity to initial conditions is popularly known as the  "butterfly effect." Of great recent interest has been to understand how/if the butterfly effect and Lyapunov exponents generalize to quantum mechanics, where the notion of a trajectory does not exist. In this talk, I will introduce the measure of quantum chaoticity  – a so-called out-of-time-ordered four-point correlator (whose semiclassical limit reproduces classical Lyapunov growth), and use it to describe quantum chaotic dynamics and its eventual disappearance in the standard models of classical and quantum chaos – Bunimovich stadium billiard and standard map or kicked rotor [1]. I will describe our recent results on the quantum Lyapunov exponent in these single-particle models as well as results in interacting many-body systems, such as disordered metals [2]. The latter many-body model exhibits an interaction-induced transition from quantum chaotic to non-chaotic dynamics, which may manifest itself in a sharp change of the distribution of energy levels from Wigner-Dyson to Poisson statistics. I will conclude by formulating a many-body analogue of the Bohigas-Giannoni-Schmit conjecture.
References:
[1] "Lyapunov exponent and out-of-time-ordered correlator's growth rate in a chaotic system,"  E. Rozenbaum, S. Ganeshan, and V. Galitski, Physical Review Letters 118, 086801 (2017)
[2] "Non-linear sigma model approach to many-body quantum chaos," Y. Liao and and V. Galitski, arXiv:1807.09799

Entanglement spectrum (ES) contains more information than the entanglement entropy, a single number. For highly excited states, this can be quantified by the ES statistics, i.e. the distribution of the ratio of adjacent gaps in the ES. I will first present examples in both random unitary circuits and Hamiltonian systems, where the ES signals whether a time-evolved state (even if maximally entangled) can be efficiently disentangled without precise knowledge of the time evolution operator. This allows us to define a notion of entanglement complexity that is not revealed by the entanglement entropy.

In the second part, I will discuss how quantum states are scrambled via braiding in systems of non-Abelian anyons through the lens of ES statistics. We define a distance between the entanglement level spacing distribution of a state evolved under random braids and that of a Haar-random state, using the Kullback-Leibler divergence $D_{\mathrm{KL}}$. We study $D_{\mathrm{KL}}$ numerically for random braids of Majorana fermions (supplemented with random local four-body interactions) and Fibonacci anyons. Our results reveal a hierarchy of scrambling among various models --- even for the same amount of entanglement entropy --- at intermediate times, whereas all models exhibit the same late-time behavior. In particular, we find that braiding of Fibonacci anyons scrambles more efficiently than the universal H+T+CNOT set. Our results promote $D_{\mathrm{KL}}$ as a quantifiable metric for scrambling and quantum chaos, which applies to generic quantum systems.

I give an overview of work with Aasen and Mong on topologically invariant defects in two-dimensional classical lattice models, quantum spin chains and tensor networks. We show how to find defects that satisfy commutation relations guaranteeing the partition function depends only on their topological properties. These relations and their solutions can be extended to allow defect lines to branch and fuse, again with properties depending only on topology. These lattice topological defects have a variety of useful applications. In the Ising model, the fusion of duality defects allows Kramers-Wannier duality to be enacted on the torus and higher genus surfaces easily, implementing modular invariance directly on the lattice. These results can be extended to a very wide class of models, giving generalised dualities previously unknown in the statistical-mechanical literature. A consequence is an explicit definition of twisted boundary conditions that yield the precise shift in momentum quantization and thus the spin of the associated conformal field. Other universal quantities we compute exactly on the lattice are the ratios of g-factors for conformal boundary conditions. 

Large deviation theory gives a general framework for studying nonequilibrium systems which in many ways parallels equilibrium thermodynamics. In transport, according to the large deviation principle, the distribution of rare fluctuations of the total transfer (of charge, energy, etc.) between two baths take a special form encoded by the large deviation function, which plays the role of a free energy. Its Legendre transform is the scaled cumulant generating function (SCGF). For instance, in mesoscopic physics, full counting statistics for charge transport through quantum impurities are SCGFs. In this talk I propose a formalism giving access to SCGFs for ballistic transport in homogeneous, stationary states. The formalism is conjectured to hold for classical and quantum systems alike, and give exact results in terms of the theory of linear fluctuating hydrodynamics. I will explain how it applies to nonequilibrium steady states of integrable models such as the classical hard rod gas, the quantum Lieb-Liniger model and the XXZ chain, using generalised hydrodynamics. This largely extends earlier results for free fermions (Levitov-Lesovik) and for 1+1-dimensional conformal field theory (Bernard-Doyon). The formalism also applies to transport in nonequilibrium critical systems of arbitrary dimension. It is, in a sense, the ballistic counterpart to macroscopic fluctuation theory.

I will present recent results (with Zhen Bi) on novel quantum criticality and a possible field theory duality in 3+1 spacetime dimensions. We describe several examples of Deconfined Quantum Critical Points (DQCP) between Symmetry Protected Topological phases in 3 + 1-D.   We present situations in which the same phase transition allows for multiple universality classes controlled by distinct fixed points. We exhibit the possibility - which we dub “unnecessary quantum critical points” - of stable generic continuous phase transitions within the same phase. We present examples of interaction driven band-theory- forbidden continuous phase transitions between two distinct band insulators. The understanding we develop leads us to suggest an interesting possible 3 + 1-D field theory duality between SU(2) gauge theory coupled to one massless adjoint Dirac fermion and the theory of a single massless Dirac fermion augmented by a decoupled topological field theory.

Many-body localization generalizes the concept of Anderson localization (i.e. single particle localization) to isolated interacting systems, where many-body eigenstates in the presence of sufficiently strong disorder can be localized in a region of Hilbert space even at nonzero temperature. This is an example of ergodicity breaking, which manifests failure of thermalization or more specifically the break down of eigenstate-thermalization hypothesis.

In this talk, I enquire into the quasi many-body localization in topologically ordered states of matter, revolving around the case of Kitaev toric code on the ladder geometry, where different types of anyonic defects carry different masses induced by environmental errors.  Our study verifies that the presence of anyons generates a complex energy landscape solely through braiding statistics, which suffices to suppress the diffusion of defects in such clean, multicomponent anyonic liquid. This nonergodic dynamics suggests a promising scenario for investigation of quasi many-body localization.  Our results unveil how self-generated disorder ameliorates the vulnerability of topological order away from equilibrium. This setting provides a new platform which paves the way toward impeding logical errors by self-localization of anyons in a generic, high energy state, originated exclusively in their exotic statistics.

Three fundamental factors determine the quality of a statistical learning algorithm: expressiveness, generalization and optimization.  The classic strategy for handling these factors is relatively well understood.  In contrast, the radically different approach of deep learning, which in the last few years has revolutionized the world of artificial intelligence, is shrouded by mystery.  This talk will describe a series of works aimed at unraveling some of the mysteries revolving expressiveness, arguably the most prominent factor behind the success of deep learning.  I will begin by showing that state of the art deep learning architectures, such as convolutional networks, can be represented as tensor networks -- a computational model commonly employed in quantum physics.  This connection will inspire the use of quantum entanglement for defining measures of data correlations modeled by deep networks.  Next, I will turn to a quantum max-flow / min-cut theorem characterizing the entanglement captured by tensor networks.  This theorem will give rise to new results that shed light on expressiveness in deep learning, and in addition, provide new tools for deep network design.

Works covered in the talk were in collaboration with Yoav Levine, Or Sharir, David Yakira and Amnon Shashua.

Remarkable recent experiments have observed Mott insulating behavior and superconductivity in moire superlattices of twisted bilayer graphene near a magic twist angle. However, the nature of the Mott insulator, origin of superconductivity and an effective model remain to be determined. I will present our understanding of these phenomena. We propose a Mott insulator with intervalley coherence that spontaneously breaks U(1) valley symmetry, and describe a mechanism that selects this order over the competing magnetically ordered states favored by the Hunds coupling. We also identify symmetry related features of the nearly flat bands that are key to understanding the strong correlation physics and constrain any lattice model description. First, although the charge density is concentrated on the triangular lattice sites of the moire pattern, the Wannier states of a tight-binding model must be centered on different sites which form a honeycomb lattice. Next, spatially localizing electrons derived from the nearly flat bands necessarily breaks valley and other symmetries within any mean-field treatment, which is suggestive of a valley-ordered Mott state, and also dictates that additional symmetry breaking is present to remove symmetry-enforced band contacts. Tight-binding models describing the nearly flat mini-band are derived, which highlight the importance of further neighbor hopping and interactions. We discuss the consequences of this picture.