Condensed Matter

We know that different systems with the same unbroken global symmetry can nevertheless be in distinct phases of matter.  These different "symmetry-protected topological" phases are characterized by protected (gapless) surface states.  After reviewing this physics in interacting systems with global symmetries, I will describe how a different class of symmetries known as subsystem symmetries, which are neither local nor global, can also lead to protected gapless boundaries.  I will discuss how some of these subsystem-symmetry protected phases are related (though not equivalent) to interacting higher-order topological insulators, with protected gapless modes along corners or hinges in higher dimensional systems.

We propose a new type of critical quantum liquids, dubbed Stiefel liquids, based on 2+1 dimensional Wess-Zumino-Witten models on target space SO(N)/SO(4). We show that the well known deconfined quantum critical point and U(1) Dirac spin liquid are unified as two special examples of Stiefel liquids, with N = 5 and N = 6, respectively. Furthermore, we conjecture that Stiefel liquids with N > 6 are non-Lagrangian, in the sense that the theories do not (at least not easily) admit any weakly-coupled UV completion. Such non-Lagrangian states are beyond the paradigm of parton gauge theory familiar in the study of exotic quantum liquids in condensed matter physics. The intrinsic absence of mean-field construction also makes it difficult to decide whether a non-Lagrangian state can emerge from a specific UV system (such as a lattice spin system). For this purpose we hypothesize that a quantum state is emergible from a lattice system if its quantum anomalies match with the constraints from the (generalized) Lieb-Schultz-Mattis theorems. Based on this hypothesis, we find that some of the non-Lagrangian Stiefel liquids can indeed be realized in frustrated quantum spin systems, for example, on triangular or Kagome lattice, through the intertwinement between non-coplanar magnetic orders and valence-bond-solid orders.

The extension of many-body quantum dynamics to the non-unitary domain has led to a series of exciting developments, including new out-of-equilibrium entanglement phases and phase transitions. We show how a duality transformation between space and time on one hand, and unitarity and non-unitarity on the other, can be used to realize steady state phases of non-unitary dynamics that exhibit a rich variety of behavior in their entanglement scaling with subsystem size --- from logarithmic to extensive to fractal. We show how these outcomes in non-unitary circuits (that are ``spacetime-dual" to unitary circuits) relate to the growth of entanglement in time in the corresponding unitary circuits, and how they differ, through an exact mapping to a problem of unitary evolution with boundary decoherence, in which information gets ``radiated away'' from one edge of the system. In spacetime-duals of chaotic unitary circuits, this mapping allows us to uncover a non-thermal volume-law entangled phase with a logarithmic correction to the entropy distinct from other known examples. Most notably, we also find novel steady state phases with fractal entanglement scaling, $S(\ell) \sim \ell^{\alpha}$ with tunable $0 < \alpha < 1$ for subsystems of size $\ell$ in one dimension. These fractally entangled states add a qualitatively new entry to the families of many-body quantum states that have been studied as energy eigenstates or dynamical steady states, whose entropy almost always displays either area-law, volume-law or logarithmic scaling. We also present an experimental protocol for preparing these novel steady states with only a very limited amount of postselection via a type of ``teleportation" between spacelike and timelike slices of quantum circuits.

Recently, the notion of symmetry has been extended from 0-symmetry described by group to higher symmetry described by higher group. In this talk, we show that the notion of symmetry can be generalized even further to "algebraic higher symmetry". Then we will describe an even more general point of view of symmetry, which puts the (generalized) symmetry charges and topological excitations at equal footing: symmetry can be viewed gravitational anomaly, or symmetry can be viewed as shadow topological order in one higher dimension.

In this work, we introduce and study "hybrid" fracton orders, especially though a family of exactly solvable models. The hybrid fracton orders exhibit both the phenomenology of a conventional 3d topological ordered phase and a fracton phase. There are simple yet non-trivial fusion and braiding between the excitations between the two kinds. One example is the hybrid order of the Z2 topological order with the Z2 Xcube order, in which the fracton excitations fuse into the toric code charge, and in turn, the flux loop of the toric code can fuse into various lineon excitations. In the same way there is a hybrid ordered phase of Haah's code and the 3d toric code. Proliferating certain gapped excitations in these hybrid orders can drive a phase transition into either a fracton order or a conventional 3d topological phase. 

Reference. ArXiv 2102.09555

Fracton models are characterized by an exponentially increasing ground state degeneracy and point excitations with constrained motion. In this talk, I will focus on a prototypical 3D fracton model -- the X-cube model -- and discuss how its ground state degeneracy can be understood from a foliation structure in the model. In particular, we show that there are hidden 2D topological layers in the 3D bulk. To calculate the ground state degeneracy, we can remove the layers until a minimal structure is reached. The ground state degeneracy comes from the combination of the degeneracy of the foliation layers and that associated with the minimal structure. We discuss explicitly how this works for X-cube model with periodic boundary condition, open boundary condition, and even in the presence of screw dislocation defects.

Despite decades of theoretical work, the physical realization of topological order, outside of the fractional quantum Hall effect, has proved to be an elusive goal. Even the simplest example of a time-reversal symmetric topological order, as encountered in the paradigmatic toric code, awaits experimental realization. Key challenges include the lack of physically realistic models in these phases, and of ways to probe their defining properties. I will discuss a simple `Rydberg blockade' model, and describe numerical results that point to (i) a ground state with toric code topological order that could potentially be realized in experiment and (ii) ``smoking gun'' signatures of the phase which be accessed using a dynamic protocol. I will also briefly discuss how a topological qubit can be constructed in this platform by tuning boundaries as well as implications for constructing fault-tolerant quantum memories. Time permitting, a different platform for realizing exotic phases, magic-angle graphene and the special features of its band structure will be described, which make it a prime candidate for realizing fractional quantum Hall topological order even in the absence of a magnetic field.

References: arXiv:2011.12310. and arXiv:1912.09634
 

In this talk I will begin by introducing symmetry protected topological (SPT) Floquet systems in 1D. I will describe the topological invariants that characterize these systems,  and highlight their differences from SPT phases arising in static systems.  I will also discuss how the entanglement properties of a many-particle wavefunction depend on these  topological invariants. I will then show that the edge modes encountered in free fermion SPTs are remarkably robust to adding interactions, even in disorder-free systems where generic bulk quantities can heat to infinite temperatures due to the periodic driving. This robustness of the edge modes to heating can be understood in the language of strong modes for free fermion SPTs, and  almost strong modes for interacting SPTs.

I will then outline a tunneling calculation for extracting the long lifetimes of these edge modes by mapping the Heisenberg time-evolution of the edge operator to dynamics of a single particle in Krylov space. 

When the twist angle of a bilayer graphene is near the ``magic'' value, there are four narrow bands near the neutrality point, each two-fold spin degenerate. These bands are separated from the rest of the bands by energy gaps. In the first part of the talk, the topology of the narrow bands will be discussed, as well as the associated obstructions --or lack there of -- to construction of a complete localized basis [1,3].

In the second part of the talk, I will present a two stage renormalization group treatment [4] which connects the continuum Hamiltonian at length scales shorter than the moire superlattice period to the Hamiltonian for the active narrow bands only, which is valid at distances much longer than the moire period. Via a progressive numerical elimination of remote bands the relative strength of the one-particle-like dispersion and the interactions within the active narrow band Hamiltonian will be determined, thus quantifying the residual correlations and justifying the strong coupling approach in the final step.

In the last part of the talk, the states favored by electron-electron Coulomb interactions within the narrow bands will be discussed. Analytical and DMRG results based on 2D localized Wannier states [2,5], 1D localized hybrid Wannier states [3] and Bloch states [3,4] will be compared. Topological and symmetry constraints on the spectra of charged and neutral excitation[4] for various ground states, as well as non-Abelian braiding of Dirac nodes[3] , will also be presented.

[1] Jian Kang and Oskar Vafek, Phys. Rev. X 8, 031088 (2018).

[2] Jian Kang and Oskar Vafek, Phys. Rev. Lett. 122, 246401 (2019)

[3] Jian Kang and Oskar Vafek, Phys. Rev. B 102, 035161 (2020)

[4] Oskar Vafek and Jian Kang Phys. Rev. Lett. 125, 257602 (2020)

[5] Bin-Bin Chen, Yuan Da Liao, Ziyu Chen, Oskar Vafek, Jian Kang, Wei Li, Zi Yang Meng arXiv:2011.07602

In this talk I will present a general framework to distinguish different classes of charge insulators based on whether or not they insulate or conduct higher multipole moments (dipole, quadrupole, etc.). This formalism applies to generic many-body systems that support multipolar conservation laws. Applications of this work provide a key link between recently discovered higher order topological phases and fracton phases of matter.

We study the real-time dynamics of a small bubble of "false vacuum'' in a quantum spin chain near criticality, where the low-energy physics is described by a relativistic (1+1)-dimensional quantum field theory. Such a bubble can be thought of as a confined kink-antikink pair (a meson). We carefully construct bubbles so that particle production does not occur until the walls collide. To achieve this in the presence of strong correlations, we extend a Matrix Product State (MPS) ansatz for quasiparticle wavepackets [Van Damme et al., arXiv:1907.02474 (2019)] to the case of confined, topological quasiparticles. By choosing the wavepacket width and the bubble size appropriately, we avoid strong lattice effects and observe relativistic kink-antikink collisions. We use the MPS quasiparticle ansatz to identify scattering outcomes: In the Ising model, with transverse and longitudinal fields, we do not observe particle production despite nonintegrability (supporting recent numerical observations of nonthermalizing mesonic states). With additional interactions, we see production of confined and unconfined particle pairs. Although we simulated these low-energy, few-particle events with moderate resources, we observe significant growth of entanglement with energy and with the number of collisions, suggesting that increasing either will ultimately exhaust our methods. Quantum devices, in contrast, are not limited by entanglement production, and promise to allow us to go far beyond classical methods. We anticipate that kink-antikink scattering in 1+1 dimensions will be an instructive benchmark problem for relatively near-term quantum devices.

Recent years new concepts of symmetries have been developed such as higher form symmetries, and categorical symmetries. The higher form symmetries can be either explicit in a Hamiltonian, or inexplicit as a dual of an ordinary symmetry. The behavior of higher form symmetries are easy to evaluate in phases with gaps. But at quantum criticalities their behaviors are more nontrivial. We evaluate the behaviors of higher form symmetries (either explicit or inexplicit) at various quantum critical points, and demonstrate that for many quantum critical points a universal logarithmic contribution arises, which is analogous to the quantum entanglement entropy. This logarithmic contribution is related to the universal conductance at the quantum critical points, and in some cases can be computed exactly using duality between CFTs developed in last few years. We also evaluate the behavior of categorical symmetries for more exotic cases with subsystem symmetries.