Quantum Matter

"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.

Clean and interacting periodically driven quantum systems are believed to exhibit a single, trivial “infinite-temperature” Floquet-ergodic phase. By contrast, I will show that their disordered Floquet many-body localized counterparts can exhibit distinct ordered phases with spontaneously broken symmetries delineated by sharp transitions. Some of these are analogs of equilibrium states, while others are genuinely new to the Floquet setting. I will show that a subset of these novel phases are "time-crystals" in that they spontaneously break the underlying time-translation symmetry of the Floquet drive. Strikingly, the time-crystal phase is remarkably stable to all weak local deformations of the underlying Floquet drives, and the phase simultaneously also spontaneously breaks Hamiltonian dependent emergent spatial symmetries. Thus, the time-crystallinity goes hand in hand with spatial symmetry breaking and, altogether, these phases exhibit a novel form of simultaneous long-range order in space and time. I will describe how this spatiotemporal order can be detected in experiments involving quenches from a broad class of initial states. 

The Kovtun-Son-Starinets conjecture that the ratio of the viscosity to the entropy density was bounded from below by fundamental constants has inspired over a decade of conjectures about fundamental bounds on the hydrodynamic and transport coefficients of strongly interacting quantum systems.  I will present two complementary and (relatively) rigorous approaches to proving bounds on the transport coefficients of strongly interacting systems.   Firstly, I will discuss lower bounds on the conductivities (and thus, diffusion constants) of inhomogeneous fluids, based around the principle that transport minimizes the production of entropy.   I will show explicitly how to use this principle in classical theories, and in theories with a holographic dual. Secondly, I will derive lower bounds on sound velocities and diffusion constants arising from the consistency of hydrodynamics with quantum decoherence and chaos, in large N theories.   I will discuss the possible tension of such bounds with (some) holographic theories, and discuss resolutions to some existing puzzles.

Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics, respectively. In the last decade the study of quantum quenches revealed that these two concepts are intricately intertwined. Although the unitary time evolution ensuing from a pure initial state maintains the system globally at zero entropy, at long time after the quench local properties are captured by an appropriate statistical ensemble with non zero thermodynamic entropy, which can be interpreted as the entanglement accumulated during the dynamics. Therefore, understanding the post-quench entanglement evolution unveils how thermodynamics emerges in isolated quantum systems. An exact computation of the entanglement dynamics has been provided only for non-interacting systems, and it was believed to be unfeasible for genuinely interacting models. Conversely, here we show that the standard quasiparticle picture of the entanglement evolution, complemented with integrability-based knowledge of the asymptotic state, leads to a complete analytical understanding of the entanglement dynamics in the space-time scaling limit. Our framework requires only knowledge about the steady state, and the velocities of the low-lying excitations around it. We provide a thorough check of our result focusing on the spin-1/2 Heisenberg XXZ chain, and considering quenches from several initial states. We compare our results with numerical simulations using both tDMRG and iTEBD, finding always perfect agreement.

Branch point twist fields play an important role in the study of measures of entanglement such as the Rényi entropies and the Negativity. In 1+1 dimensions such measures can be written in terms of multi-point functions of branch point twist fields. For 1+1-dimensional integrable quantum field theories and also in conformal field theory much is known about how to compute correlation functions and, with the help of the twist field, this knowledge can be exploited in order to gain new insights into the properties of various entanglement measures. In this talk I will review some of our main results in this context.

I will then go on to introduce a new (related) class of fields we have recently named conical twist fields. These are fields whose two-point functions have (surprisingly) been found to describe gluon amplitudes in the strong coupling limit of super Yang-Mills theories and therefore have featured in a completely different context from that of entanglement measures. Interestingly, at critical points, branch point and conical twist fields have the same conformal dimension and beyond criticality they also have very similar form factors, however they are different in many other respects.  In my talk I will discuss and justify some of their similarities and differences.

Dataset augmentation, the practice of applying a wide array of domain-specific transformations to synthetically expand a training set, is a standard tool in supervised learning. While effective in tasks such as visual recognition, the set of transformations must be carefully designed, implemented, and tested for every new domain, limiting its re-use and generality. In this talk, I will describe recent methods that  transform data not in input space, but in a feature space found by unsupervised learning. We start with data points mapped to a learned feature space and apply simple transformations such as adding noise, interpolating, or extrapolating between them. Working in the space of context vectors generated by sequence-to-sequence recurrent neural networks, this simple and domain-agnostic technique is demonstrated to be effective for both static and sequential data. 

Bio: Graham Taylor is an Associate Professor at the University of Guelph where he leads the Machine Learning Research Group. He is a member of the Vector Insitute for Artificial Intelligence and is an Azrieli Global Scholar with the Canadian Institute for Advanced Research. He received his PhD in Computer Science from the University of Toronto in 2009, where he was advised by Geoffrey Hinton and Sam Roweis. He spent two years as a postdoc at the Courant Institute of Mathematical Sciences, New York University working with Chris Bregler, Rob Fergus, and Yann LeCun. 

Dr. Taylor's research focuses on statistical machine learning, with an emphasis on deep learning and sequential data. Much of his work has focused on "seeing people" in images and video, for example, activity and gesture recognition, pose estimation, emotion recognition, and biometrics.

The hydrodynamic approximation is an extremely powerful tool to describe the behavior of many-body systems such as gases. At the Euler scale (that is, when variations of densities and currents occur only on large space-time scales), the approximation is based on the idea of local thermodynamic equilibrium: locally, within fluid cells, the system is in a Galilean or relativistic boost of a Gibbs equilibrium state. This is expected to arise in conventional gases thanks to ergodicity and Gibbs thermalization, which in the quantum case is embodied by the eigenstate thermalization hypothesis. However, integrable systems are well known not to thermalize in the standard fashion. The presence of infinitely-many conservation laws preclude Gibbs thermalization, and instead generalized Gibbs ensembles emerge. In this talk I will introduce the associated theory of generalized hydrodynamics (GHD), which applies the hydrodynamic ideas to systems with infinitely-many conservation laws. It describes the dynamics from inhomogeneous states and in inhomogeneous force fields, and is valid both for quantum systems such as experimentally realized one-dimensional interacting Bose gases and quantum Heisenberg chains, and classical ones such as soliton gases and classical field theory. I will give an overview of what GHD is, how its main equations are derived, its relation to quantum and classical integrable systems, and some geometry that lies at its core. I will then explain how it reproduces the effects seen in the famous quantum Newton cradle experiment, and how it leads to exact results in transport problems such as Drude weights and non-equilibrium currents.

This is based on various collaborations with Alvise Bastianello, Olalla Castro Alvaredo, Jean-Sébastien Caux, Jérôme Dubail, Robert Konik, Herbert Spohn, Gerard Watts and my student Takato Yoshimura, and strongly inspired by previous collaborations with Denis Bernard, M. Joe Bhaseen, Andrew Lucas and Koenraad Schalm.

Frustrated magnets provide a fertile ground for discovering exotic states of matter, such as those with topologically non-trivial properties. Motivated by several near-ideal material realizations, we focus on aspects of the two-dimensional kagome antiferromagnet. I present two of our works in this area both involving the spin-1/2 XXZ antiferromagnetic Heisenberg model. First, guided by a previous field theoretical study, we explore the XY limit ($J_z=0$) for the case of 2/3 magnetization (i.e. 1/6 filling of hard-core bosons) and perform exact numerical computations to search for a "chiral spin liquid phase". We provide evidence for this phase by analyzing the energetics, determining minimally entangled states and the associated modular matrices, and evaluating the many-body Chern number [1]. The second part of the talk follows from an unexpected outcome of the first work, which realized the existence of an exactly solvable point for the ratio of Ising to transverse coupling $J_z/J=-1/2$. This point in the phase diagram has "three coloring" states as its exact ground states, exists for all magnetizations (fillings) and is found to be the source or "mother" of the observed phases of the kagome antiferromagnet. Using this viewpoint, I revisit certain aspects of the highly contentious Heisenberg case (in zero field) and suggest that it is possibly part of a line of critical points.

[1] K. Kumar, H. J. Changlani, B. K. Clark, E. Fradkin, Phys. Rev. B 94, 134410 (2016)
[2] H. J. Changlani, D. Kochkov, K. Kumar, B. K. Clark, E. Fradkin, under review.