Condensed Matter

Operator scrambling is a crucial ingredient of quantum chaos. Specifically, in the quantum chaotic system, a simple operator can become increasingly complicated under unitary time evolution. This can be diagnosed by various measures such as square of the commutator (out-of-time-ordered correlator), operator entanglement entropy etc. In this talk, we discuss operator dynamics in three representative models: a 2-local spin model with all-to-all interaction, a chaotic spin chain with long-range interactions,  and the quantum linear map. In the first two examples, we explore the operator dynamics by using the quantum Brownian circuit approach and transform the operator spreading into a classical stochastic problem. Although the speeds of scrambling are quite different, a simple operator can eventually approach a "highly entangled" operator with operator entanglement entropy taking a volume law value (close to the Page value). Meanwhile, the spectrum of the operator reduced density matrix develops a universal spectral correlation which can be characterized by the Wishart random matrix ensemble.  In contrast, in the third example (the quantum linear map), although the square of commutator can increase exponentially with time, a simple operator does not scramble but performs chaotic motion in the operator basis space determined by the classical linear map. We show that once we modify the quantum linear map such that operator can mix in the operator basis, the operator entanglement entropy can grow and eventually saturate to its Page value, thus making it a truly quantum chaotic model.

This year there appear several amazing experiments  in the graphene moire superlattices.  In this talk I will focus on the ABC trilayer graphene/h-BN system. Mott-like  insulators at 1/4 and 1/2 of the valence band have already been reported by Feng Wang’s group at Berkeley. The sample is dual gated on top and bottom with voltage V_t and V_b.  V_t+V_b controls the density of electrons. Interestingly we find that the displacement field D=V_t-V_b can control both the topology and the bandwidth of the valence band. For one sign of D (for example D>0), there are two narrow Chern bands with opposite Chern numbers C=3,-3 for the two valleys. For D<0, the bands of the two valleys are trivial and have localized Wannier orbitals on a triangular lattice.  As a result, the physics is governed by a spin-valley Hubbard model on a triangular lattice. This talk focuses on the D<0 side and  consists of two parts:  (1) I will provide the details of this spin-valley Hubbard model and discuss some subtleties special to the moire systems.   (2) In the second part I want to show some of our theoretical attempts on this Hubbard model. First I will show that this system is a perfect platform for studying metal-insulating transition. I will provide a theory of continuous Mott transition between a Fermi liquid and a spinon Fermi surface Mott insulator. Second I will discuss some possible metallic phases upon doping away from the Mott insulator.  Unlike the familiar spin 1/2 case, the spin-valley Hubbard model may not be in a conventional Fermi liquid phase even in the over-doped region. I will provide some candidates of possible unconventional metals based on a six-flavor slave boson theory for the hole doped side.

References:

Feng Wang et.al. arxiv: 1803.01985

Ya-Hui Zhang, Dan Mao, Yuan Cao, Pablo Jarillo-Herrero and T. Senthil.  arXiv:1805.08232

Ya-Hui Zhang and T. Senthil, arxiv: 1809.05110

Ya-Hui Zhang and T. Senthil, ongoing work

We implement projective quantum Monte Carlo (PQMC) methods to simulate quantum annealing on classical computers. We show that in the regime where the systematic errors are well controlled, PQMC algorithms are capable of simulating the imaginary-time dynamics of the Schroedinger equation both on continuous space models and discrete basis systems. We also demonstrate that the tunneling time of the PQMC method is quadratically faster than the one of incoherent quantum annealing. It shows remarkable stability when applied to frustrated systems compared to finite-temperature path integral Monte Carlo algorithm, the method mostly chosen to do comparisons with quantum annealers. However, a major drawback of the PQMC method comes from the finite number of random walkers needed
to implement the simulations. It grows exponentially with the system size when no or poor guiding wave-functions are utilized. Nevertheless, we demonstrate that when good enough guiding wave-functions are used – in our case we choose artificial neural networks – the computational complexity seems to go from exponential to polynomial in the system size. We advocate for a search of more efficient guiding wave functions since they could determine when PQMC simulations are feasible on classical computers, a question closely related to a provable need or speed-up of a quantum computer.

References:
- E. M. Inack and S. Pilati, Phys. Rev. E 92, 053304 (2015)
- E. M. Inack, G. Giudici, T. Parolini, G. Santoro and S. Pilati, Phys. Rev. A 97, 032307 (2018)
- E. M. Inack, G. Santoro, L. Dell’Anna, and S. Pilati, arXiv:1809.03562v1

Applying a chemical potential bias to a conductor drives the system out of equilibrium into a current carrying non-equilibrium state.  This current flow is associated with entropy production in the leads, but it remains poorly understood under what conditions the system is driven to local equilibrium by this process.  We investigate this problem using two toy models for coherent quantum transport of diffusive fermions:  Anderson models in the conducting phase and a class of random quantum circuits acting on a chain of qubits, which exactly maps to an interacting fermion problem. Under certain conditions, we find that the long-time states in both models exhibit volume-law mutual information and entanglement, in striking violation of local equilibrium.  Extending this analysis to Anderson metal-insulator transitions, we find that the volume-law entanglement scaling persists at the critical point up to mobility edge effects.  This work points towards a broad class of examples of physical systems where volume-law entanglement can be sustained, and potentially harnessed, despite strong coupling of the system to its surrounding environment. 

Despite much theoretical effort, there is no complete theory of the “strange” metal phase of the high temperature 

superconductors, and its linear-in-temperature resistivity. This phase is believed to be a strongly-interacting metallic 

phase of matter without fermionic quasiparticles, and is virtually impossible to model accurately using traditional

perturbative field-theoretic techniques. Recently, progress has been made using large-N techniques based on the 

solvable Sachdev-Ye-Kitaev (SYK) model, which do not involve expanding about any weakly-coupled limit. I will 

describe constructions of solvable models of strange metals based on SYK-like large-N limits, which can reproduce  

some of the experimentally observed features of strange metals and adjoining phases. These models, and further 

extensions, could possibly pave the way to developing a controlled theoretical understanding of the essential building 

blocks of the electronic state in correlated-electron superconductors near optimal doping.

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.