Condensed Matter

In this talk I will discuss effective field theories for two classes of non-equilibrium systems, one far and one near equilibrium. The backbone of the approach is the Schwinger-Keldysh formalism, which is the natural starting point for doing field theory in non-equilibrium situations. In the first part of the talk I will present an effective response for topological driven (Floquet) systems, which are inherently far from equilibrium. As an example, I will discuss a chiral Floquet drive coupled to a background $U(1)$ field, which gives rise to a theta term in the response action, and show that this is independent of smooth deformations of the underlying system. In the second part, I will discuss an ongoing project using effective field theories for hydrodynamics. I will show that chiral diffusion for interacting systems in 1+1 dimensions, which may be relevant to edge transport in quantum Hall systems, has an infrared instability. I will then discuss the fate of this instability.

The first part of this talk will introduce generalized Jordan–Wigner
transformation on arbitrary triangulation of any simply connected
manifold in 2d, 3d and general dimensions. This gives a duality
between all fermionic systems and a new class of Z2 lattice gauge
theories. This map preserves the locality and has an explicit
dependence on the second Stiefel–Whitney class and a choice of spin
structure on the manifold. In the Euclidean picture, this mapping is
exactly equivalent to introducing topological terms (Chern-Simon term
in 2d or the Steenrod square term in general) to the Euclidean action.
We can increase the code distance of this mapping, such that this
mapping can correct all 1-qubit and 2-qubits errors and is useful for
the simulation of fermions on the quantum computer. The second part of
my talk is about SPT phases. By the boson-fermion duality, we are able
to show the equivalent between any supercohomology fermionic SPT and
some higher-group bosonic SPT phases. Particularly in (3+1)D, we have
constructed a unitary quantum circuit for any supercohomology
fermionic SPT state with gapped boundary construction. This fermionic
SPT state is derived by gauging higher-form Z2 symmetry in the
higher-group bosonic SPT and apply the boson-fermion duality.

I will briefly review the pseudogap phenomenology in high Tc cuprate superconductor, especially the recent experiments, and propose a unified picture of the phenomenology under only one assumption: the fluctuating pair density wave. By quantum disordering a pair density wave, we found a state composed of insulating antinodal pairs and nodal electron pocket. We compare the theoretical predictions with ARPES results, optical conductivity, quantum oscillation and other experiments.

We derive general results relating revivals in the dynamics of quantum
many-body systems to the entanglement properties of energy eigenstates.
For a D-dimensional lattice system of N sites initialized in a
low-entangled and short-range correlated state, our results show that a
perfect revival of the state after a time at most poly(N) implies the
existence of "quantum many-body scars", whose number grows at least as
the square root of N up to poly-logarithmic factors. These are energy
eigenstates with energies placed in an equally-spaced ladder and with
Rényi entanglement entropy scaling as log(N) plus an area law term for
any region of the lattice. This shows that quantum many-body scars are a
necessary condition for revivals, independent of particularities of the
Hamiltonian leading to them. We also present results for approximate
revivals, for revivals of expectation values of observables and prove
that the duration of revivals of states has to become vanishingly short
with increasing system size.

Work in collaboration with Henrik Wilming (ETH)

It has recently been shown that quenched randomness, via the phenomenon of many-body localization, can stabilize dynamical phases of matter in periodically driven (Floquet) systems, with one example being discrete time crystals. This raises the question: what is the nature of the transitions between these Floquet many-body-localized phases, and how do they differ from equilibrium? We argue that such transitions are generically controlled by infinite randomness fixed points. By introducing a real-space renormalization group procedure for Floquet systems, asymptotically exact in the strong-disorder limit, we characterize the criticality of the periodically driven interacting quantum Ising model, finding forms of (multi-)criticality novel to the Floquet setting. We validate our analysis via numerical simulations of free-fermion models sufficient to capture the critical physics.

It has been realized that anomalies can be classified by topological phases in one higher dimension. Previous studies focus on ’t Hooft anomalies of a theory with a global symmetry that correspond to invertible topological orders and/or symmetry protected topological orders in one higher dimension. In this talk, I will introduce an anomaly that appears on the boundaries of (non-invertible) topological order with anyonic excitations [1]. The anomalous boundary theory is no longer invariant under a re-parametrization of the same spacetime manifold. The anomaly is matched by simple universal topological data in the bulk, essentially the statistics of anyons. The study of non-invertible anomalies opens a systematic way to determine all gapped and gapless boundaries of topological orders, by solving simple eigenvector problems. As an example, we find all conformal field theories (CFT) of so-called ``minimal models’’, except four cases, can be the critical boundary theories of Z_2 topological order (toric code). The matching of non-invertible anomaly have wide applications. For example, we show that the gapless boundary of double-semion topological order must have central charge c_L=c_R >= 25/28. And the gapless boundary of the non-Abelian topological order described by S_3 topological quantum field theory can be three-state Potts CFT, su(2)_4 CFT, etc. [1] WJ, Xiao-Gang Wen, arXiv: 1905.13279, Phys. Rev. Research 1,033054

Many of the most mysterious phenomena in condensed matter physics involve systems near quantum phase transitions where the interplay of quenched disorder and strong interactions likely plays an essential role. Examples include the appearance of "anomalous" 2d metallic phases and the sharing of critical exponents between seemingly different quantum Hall plateau transitions. However, few organizing principles have been developed for understanding quantum critical systems with interactions and disorder, and analytically tractable models have proven rare. In this talk, I will describe some of the first examples of quantum critical theories in 2d characterized by finite disorder and interaction strengths, focusing on particular examples of quenched disorder in systems of (i) Dirac fermions coupled to an emergent gauge field (QED3) and (ii) scalar bosons at their Wilson-Fisher fixed point. Both of these examples exhibit universal features not found together in systems with interactions or disorder alone, such as vanishing density of states, finite DC conductivities, and novel critical exponents.

QFTs in 2+1 dimensions are powerful systems to understand the emergence of mass-gap and particle spectrum in QCD-like theories that describe our 3+1 dimensional world. Recently, these 2+1 dimensional systems have attracted even more attention due to conjectured dualities between seemingly very different theories and due to their applications to condensed matter systems. In this talk, I will describe our numerical investigations of the infrared behaviors of 2+1 dimensional U(1) and SU(N) gauge theories coupled to many favors of massless fermions using lattice regularization. I will also explain how lattice formulation is a potential tool to study and check particle-vortex dualities.

A tremendous amount of recent attention has focused on characterizing the dynamical properties of periodically driven many-body systems. Here, we use a novel numerical tool termed ‘density matrix truncation’ (DMT) to investigate the long-time dynamics of large-scale Floquet systems. By implementing a spatially inhomogeneous drive to a 1D quantum chain, we demonstrate that an interplay between Floquet heating and diffusive transport is crucial to understanding the system’s dynamics. We find that DMT accurately captures two essential pieces of Floquet physics, namely prethermalization and late-time heating to infinite temperature. Moreover, we show that these two aspects are driven by different microscopic mechanisms.

Many free-fermion topological phases can be diagnosed by analyzing a suitable collection of symmetry data. While the Fu-Kane parity criterion for topological insulators is an early example, the systematic generalization to cover all possible crystalline symmetries and their associated topological phases has only recently been achieved. Interestingly, a key step in the generalization is to recognize the importance of understanding the entanglement-free product states: once these trivial phases are completely mapped, the topological ones are naturally exposed as the “remainders.” In this talk, we will first introduce the theory of symmetry indicators for topological (crystalline) insulators and semimetals, and then discuss how it can be generalized to cover superconductors.

Thanks to the Lanczos algorithm, the Hamiltonian dynamics of any operator can be written as a hopping problem on a semi-infinite one-dimensional chain. Our hypothesis states that the hopping strength grows linearly down the chain, with a universal growth rate $\alpha$ that is an intrinsic property of the system. This leads to an exponential motion of the operator down the chain, capturing the irreversible process of simple operators inevitably evolving into complex ones. This exponential growth exists for generic quantum systems, even away from large-$N$ or semiclassical limits. In fact, $\alpha$ gives an upper bound for the exponential growth rate of a large class of operator complexity measures, including out-of-time-order correlations. As a result, we conjecture a new bound on Lyapunov exponents $\lambda_L \leq 2 \alpha$, which generalizes the known universal low-temperature bound $\lambda_L \leq 2 \pi T$. We illustrate the hypothesis in paradigmatic examples such as non-integrable spin chains, the $q$-SYK model, and chaotic coupled top models, and show that some of them saturate the conjectured bound.

The discrete time-translation symmetry of a periodically-driven (Floquet) system allows for the existence of novel, nonequilibrium interacting phases of matter. A well-known example is the discrete time crystal, a phase characterized by the spontaneous breaking of this time-translation symmetry. In this talk, I will show that the presence of *multiple* time-translational symmetries, realized by quasiperiodically driving a system with two or more incommensurate frequencies, leads to a panoply of novel non-equilibrium phases of matter, both spontaneous symmetry breaking ("discrete time quasi-crystals") and topological. In order to stabilize such phases, I will outline rigorous mathematical results establishing slow heating of systems driven quasiperiodically at high frequencies. As a byproduct, I will introduce the notion of many-body localization (MBL) in quasiperiodically driven systems.