Quantum Matter

We  examine 1D spin-chains evolving under random local unitary circuits and prove a number of exact results on the behavior of out-of-time-ordered commutators (OTOCs), and entanglement growth. These results follow from the observation that the spreading of operators in random circuits is described by a ``hydrodynamical'' equation of motion. In this hydrodynamic picture quantum information travels in a front with a `butterfly velocity' $v_{\text{B}}$ that is smaller than the light cone velocity of the system, while the front itself broadens diffusively in time. The OTOC increases sharply after the arrival of the light cone, but we do \emph{not} observe a prolonged exponential regime of the form $\sim e^{\lambda_\text{L}(t-x/v)}$ for a fixed Lyapunov exponent $\lambda_\text{L}$, in disagreement with the existing QFT literature on OTOCs. We find that the diffusive broadening of the front has important consequences for entanglement growth, leading to an entanglement velocity that can be significantly smaller than the butterfly velocity. We conjecture that the hydrodynamical description applies to more generic ergodic systems and support this with numerical simulations. When the circuits are constrained so as to conserve a U$(1)$ charge, we show that the OTOC acquires a diffusively decaying component. 

Low-energy states of quantum spin liquids are thought to involve partons living in a gauge-field background. We study the spectrum of Majorana fermions of the Kitaev honeycomb model on spherical clusters. The gauge field endows the partons with half-integer orbital angular momenta. As a consequence, the multiplicities do not reflect the point-group symmetries of the cluster, but rather its projective symmetries, operations combining physical and gauge transformations. The projective symmetry group of the ground state is the double cover of the point group.

Topological phases of matter entail a prominent research theme, featuring
distinct characteristics that include protected metallic edge states and
the possibility of fractionalized excitations. With the advent of symmetry
protected topological (SPT) phases, many of these phenomena have
effectively become accessible in the form of readily available band
structures. Whereas the role of (anti-)unitary symmetries in such SPT
states has been thoroughly understood, the inclusion of lattice symmetries
provides for an active area of research.

In this talk, I will present a short overview of results on defects in SPT
states that directly motivate the existence of additional physics beyond
the characterization based on (anti-)unitary symmetries. More importantly,
I will then connect these ideas to recent work in which we were able to map
out all different gapped phases of free fermion systems in the presence of
solely lattice symmetries. This revolves around a very simple algorithm
that matches a rather involved mathematical perspective in terms of a
framework called K-theory. I will then discuss the implications of these
combinatorial arguments, such as their impact on the description of Weyl
phases, and sketch related ideas and future perspectives.

Many strongly-correlated systems like high Tc cuprates and heavy fermions have interesting features going beyond quasi-particle description. While Sachdev-Ye-Kitaev(SYK) models are exactly solvable models that can provide a platform to study these physics. In this talk, I will discuss interesting features about the SYK models, including extensive zero temperature entropy and maximally chaos. I will also show some generalization of the SYK models and discuss physical insights from them.

Finite-size spectra and entanglement both characterize nonlocal physics of quantum systems, and are universal properties of a CFT. I discuss the energy spectrum of the Wilson-Fisher CFT on the torus in the \epsilon and 1/N expansions. I also consider a class of deconfined quantum critical points where the torus spectrum contains signatures of proximate Z2 topological order. Finally, I compute the entanglement entropy of the Wilson-Fisher and Gross-Neveu CFTs in the large N limit, where an exact mapping to free field entanglement is obtained. Comparison is made with numerics.

Quantum Monte Carlo methods, when applicable, offer reliable ways to extract the nonperturbative physics of strongly-correlated many-body systems. However, there are some bottlenecks to the applicability of these methods including the sign problem and algorithmic update inefficiencies. Using the t-V model Hamiltonian as the example, I demonstrate how the Fermion Bag Approach--originally developed in the context of lattice field theories--has aided in solving the sign problem for this model as well as aided in developing a more efficient algorithm to study the model. Finally, I discuss some other potential uses for the new algorithm, including for a broader class of models known to be sign-problem-free due to fermion bag ideas.

Periodically driven (Floquet) systems can display entirely new many-body phases of matter that have no analog in stationary systems. One such phase is the Floquet time crystal, which spontaneously breaks a discrete time-translation symmetry. In this talk, I will survey the physics of these new phases of matter. I explain how they can be stabilized either through strong quenched disorder (many-body localization), or alternatively in clean systems in a "prethermal" regime which persists until a time that is exponentially long in a small parameter.

"Recently, exactly solvable 3D lattice models have been discovered for a new kind of phase, dubbed fracton topological order, in which the topological excitations are immobile or are bound to lines or surfaces. Unlike liquid topologically ordered phases (e.g. Z_2 gauge theory), which are only sensitive to topology (e.g. the ground state degeneracy only depends on the topology of spatial manifold), fracton orders are also sensitive to the geometry of the lattice. This geometry dependence allows for remarkably new physics which was forbidden in topologically invariant phases of matter.

In this talk, I will review the X-cube model [1] of fracton order. I will then explain how geometry dependence allows for braiding of point-like particles in this 3D phase. I'll summary how the X-cube model can be described by a quantum field theory, which is analogous to a topological quantum field theory (TQFT). [3] We will see that the gauge invariance of the field theory results in the mobility restrictions of the topological excitations by imposing a new kind of geometric charge conservation. I will conclude by briefly discussing current work on the remarkable geometry-dependent phenomenology of fracton order. For example, I will explain why even on a manifold with trivial topology, spatial curvature can induce a robust ground state degeneracy.

[1] Vijay, Haah, Fu 1603.04442

[2] Slagle, Kim 1704.03870"

I will describe the recently developed bimetric theory of fractional quantum Hall states. It is an effective theory that includes the Chern-Simons term that describes the topological properties of the fractional quantum Hall state, and a non-linear, a la bimetric massive gravity action that describes gapped Girvin-MacDonald-Platzman mode at long wavelengths. The theory reproduces the universal features of the chiral lowest Landau level (LLL) FQH states that lie beyond the reach of pure Chern-Simons theory, such as the projected static structure factor and the Girvin-MacDonald-Platzman algebra. The action of particle-hole transformation on the theory is particularly transparent. Familiar quantum Hall observables acquire a geometric interpretation in the bimetric language

Quantum critical points (QCP) beyond the Landau-Ginzburg paradigm are often called unconventional QCPs. There are in general two types of unconventional QCP: type I are QCPs between ordered phases that spontaneously break very different symmetries, and type II involve topological (or quasi-topological) phases on at least one side of the QCP. Recently significant progress has been made in understanding (2+1)-dimensional unconventional QCPs, using the recently developed (2+1)d dualities, i.e., seemingly different theories may actually be identical in the infrared limit. One group of dualities between unconventional QCPs have attracted particular interests in the field of condensed matter theory. This group of dualities include the so called deconfined QCP between the Neel and valence bond solid phases, and the topological transition between a bosonic topological insulator and a trivial Mott insulator. Each of the transitions mentioned above is also "self-dual". This group of dualities make extremely powerful predictions for numerical test. We will review the theoretical aspects and most recent numerical evidences for these new results.

Motivated by the close relations of the renormalization group with both the holography duality and the deep learning, we propose that the holographic geometry can emerge from deep learning the entanglement feature of a quantum many-body state. We develop a concrete algorithm, call the entanglement feature learning (EFL), based on the random tensor network (RTN) model for the tensor network holography. We show that each RTN can be mapped to a Boltzmann machine, trained by the entanglement entropies over all subregions of a given quantum many-body state. The goal is to construct the optimal RTN that best reproduce the entanglement feature. The RTN geometry can then be interpreted as the emergent holographic geometry. We demonstrate the EFL algorithm on 1D free fermion system and observe the emergence of the hyperbolic geometry (AdS_3 spatial geometry) as we tune the fermion system towards the gapless critical point (CFT_2 point).

Two seemingly different quantum field theories may secretly describe the same underlying physics — a phenomenon known as “duality". I will review some recent developments in field theory dualities in (2+1) dimensions and some of their applications in condensed matter physics, in particular in quantum Hall effect and quantum phase transitions.