Quantum Matter

In crystals, quantum electrons can be spatially distributed in a way that the bulk solid supports macroscopic electric multipole moments, which are deeply 

related with emergence of topology insulators in condensed matter systems. However, unlike the classical electric multipoles in open space, 

defining electric multipoles in crystals is a non-trivial task. So far, only the dipolar moment, namely polarization, has been successfully defined and served as a classic example of topological insulators. 

In this talk, we propose the general definition, i.e., many-body invariants, for electric multipoles in crystals, which are related with recently-discovererd higher-order topological insulators. 

Our invariants are designed to measure the distribution of electron charge in unit cells and thus can detect multipole moments purely from the bulk ground state wavefunctions. 

We provide analytic as well as numerical proofs for our invariants. Application of our invariants to spin systems as well as various other aspects of the many-body invariants will be briefly discussed.

This seminar will focus on two cases where the interplay of topology and interactions allows for phases that go beyond simple quasi-particle descriptions. Both models are amenable to sign free auxiliary field quantum Monte Carlo simulations.

First, we design a two-dimensional model consisting of four Dirac-fermion layers on the square lattice. The interaction is given by a four-fermion term where each fermion is from a different layer. In the uncorrelated case, the topology is determined by a Z-valued winding number and previous studies, often using the bulk-boundary correspondence and dimensional reduction arguments, predict the reduction to a Z4 classification in the presence of correlations. An adiabatic path between formerly distinct phases has to visit a strongly interacting state that cannot be described on a mean-field level. We study the phase diagram of the full bulk system and find a symmetry broken state separating topological distinct phases. An attempt to frustrate the ordered state introduces a first order phase transition [Fig. 1 (left)].

Second, we consider Dirac electrons on the honeycomb lattice Kondo coupled to spin-1/2 degrees of freedom on the kagome lattice. The interactions between the spins are chosen along the lines of the Balents-Fisher-Girvin model that is known to host a Z2 spin liquid and a fer- romagnetic phase. While in the ferromagnetic phase the Dirac electrons acquire a gap, they remain massless in the Z2 spin liquid phase due to the breakdown of Kondo screening. Since our model has an odd number of spins per unit cell, this phase is a non-Fermi liquid, also called fractionalized Fermi liquid, that violates the conventional Luttinger theorem, which relates the Fermi volume to the particle density in a Fermi liquid. We probe the Kondo breakdown in this non-Fermi liquid phase via conventional observables such as the spectral function, and also by studying the mutual information between the electrons and the spins [Fig. 1 (right)].

Figure 1: Schematic sketch of the phase diagram discussed during the beginning of the seminar (left). Numerical Results consistent with a FL* to magnetic insulator transition as subject of the second half (right).

In our previous work (Phys. Rev. Lett. 121, 046401 (2018)), we found a quantum spin liquid phase with spinon Fermi surface in the two dimensional spin-1/2 Heisenberg model with four-spin ring exchange on a triangular lattice. In this work we dope the spinon Fermi surface phase by studying the t-J model with four-spin ring exchange. We perform density matrix renormalization group calculations on four-leg cylinders of a triangular lattice and find that the dominant pair correlation function is that of a pair density wave, i.e. it is oscillatory while decaying with distance with a power law. The doping dependence of the period is studied. This is the first example where pair density wave is the dominant pairing in a generic strongly interacting system where the pair density wave cannot be explained as a composite order and no special symmetry is required. Reference: 1. arXiv:1803.00999 [cond-mat.str-el] (Phys. Rev. Lett. 121, 046401 (2018)) 2. arXiv:1811.06538 [cond-mat.str-el]

Systems of strongly interacting particles can give rise to topological phases beyond non-interacting limit.  Although unique features of strongly interacting topological phases, such as fractionalization of quantum degrees of freedom, have important applications in quantum information processing, these topological phases are still far from experimental realizations. In this talk, by presenting constructions of two strongly interacting topological phases, I will argue the key mechanism of their realizations is to add interactions near topological phase transitions.  I will first introduce a model of interacting Majorana fermions that describes a superconducting phase with Fibonacci topological order. Then I will show that a correlated fluid of electrons and holes, dubbed fractional excitonic insulator phase, can exhibit a fractional quantum Hall effect at zero magnetic field. I will present physical evidence and conjecture that this phase can be realized in a higher angular momentum excitonic paired system in the presence of interactions.   

Finding a quantum phase transition between two even well-studied phases of matter can be a hard problem. Free probability theory suggests a new method which can give an answer if the Hamiltonians of two phases satisfy certain conditions. In the latter case, the spectral gap behavior can be calculated without solving the full Hamiltonian but using probabilistic estimation instead. As an example, I will consider generic artificial topological systems created by a periodic drive, including Floquet Majorana modes, and show how FPT can be used to predict and characterize disorder-driven phase transitions.

See also: Phys. Rev. Lett. 121, 126803

Interacting electrons in high magnetic fields exhibit rich physical phenomena including the gapped fractional quantum Hall effects and the gapless states. The composite Fermi liquids (CFLs) are gapless states that can occur at even denominator Landau level fillings. Due to the celebrated work of Halperin, Lee and Read (94), the CFLs were understood as Fermi liquids of composite fermions, which are bound states of electron and electromagnetic flux quanta. However, at 1/2 filling, it is not obvious why the HLR description is consistent with the particle hole symmetry. Motivated by this, recently Son (15) proposed an alternative description for CFLs at 1/2, according to which the composite fermions are instead emergent Dirac fermions. Importantly, Son’s theory predicts a Pi Berry curvature singularity at the composite Fermi sea center. In the first part of this talk [2,3], I will present our numerical work about detecting this Z2 Berry phase at 1/2 filling. In the second part [1], I will present how and why Dirac fermions can emerge at all the other filling fractions (1/2m and 1-1/2m when m is integer) even without the particle hole symmetry.

[1] arXiv 1808.07529. JW.

[2] arXiv 1711.07864. Geraedts, JW, Rezayi, Haldane.

[3] arXiv 1710.09729. JW, Geraedts, Rezayi, Haldane.

We study the chiral anomaly in disordered Weyl semimetals, where the broken translational symmetry prevents the direct application of Nielsen and Ninomiya’s mechanism and disorder is strong enough that quantum effects are important. In the weak disorder regime, there exists rare regions of the random potential where the disorder strength is locally strong, which gives rise to quasi-localized resonances and their effect on the chiral anomaly is unknown. We numerically show that these resonant states do not affect the chiral anomaly only in the case of a single Weyl node. At energies away from the Weyl point, or with strong disorder where one is deep in the diffusive regime, the chiral Landau level itself is not well defined and the semiclassical treatment is not justified. In this limit, we analytically use the supersymmetry method and find that the Chern-Simons (CS) term in the effective action which is not present in non-topological systems gives rise to a non- zero average level velocity which implies chiral charge pumping. We numerically establish that the non-zero average level velocity serves as an indicator of the chiral anomaly in the diffusive limit. 

It is widely known that topological orders have long-range entangled gapped ground states from which nontrivial properties can be extracted. We introduce a new theoretical framework named the information convex, the (convex) set of reduced density matrices of a subsystem in its lowest energy, to characterize topological orders. (1) As a concrete example, we present the calculated topology dependent structure of information convex in the quantum double models and show it reveals properties of bulk anyons and deconfined topological excitations along a gapped boundary, and the condensation rules relating them. (2) As a step towards answering "why the structure of information convex looks that way and whether it has predictive power?" we look into some quantum informational constraints. The topological contribution to von Neumann entropy from each topological excitation type and certain fusion constraints are shown to emerge due to strong subadditivity, assuming fusion multiplicity is encoded in certain topological invariant manner.

It is well-known that sufficiently strong interactions can destabilize some SPT phases of free fermions, while others remain stable even in the presence of interactions. It is also known that certain interacting phases cannot be realized by free fermions. In this talk, we will study both of these phenomena in low dimensions and determine the map from free to interacting SPT phases for an arbitrary unitary symmetry. We will also describe how to compute invariants characterizing interacting phases for free band Hamiltonians with symmetry (in any dimension) using only representation theory.

A recent experiment in the Rydberg atom chain observed unusual oscillatory quench dynamics with a charge density wave initial state, and theoretical works identified a set of many-body ``scar states'' in the Hamiltonian as potentially responsible for the atypical dynamics. In the same nonintegrable Hamiltonian, we discover several eigenstates at infinite temperature that can be represented exactly as matrix product states with finite bond dimension, for both periodic boundary conditions (two exact E = 0 states) and open boundary conditions (two E = 0 states and one each E =± √2). This discovery explicitly demonstrates violation of strong eigenstate thermalization hypothesis in this model. These states show signatures of translational symmetry breaking with period-2 bond-centered pattern, despite being in 1D at infinite temperature. We show that the nearby many-body scar states with energies E ~± 1.33 andE ~ ± 2.66 can be well approximated as ``quasiparticle excitations" on top of our exact E = 0 states, and propose a quasiparticle explanation of the strong oscillations observed in experiments.

Experiments on organic crystals whose structure is well-described by the two-dimensional triangular lattice have found a lack of magnetic ordering down to the lowest accessible temperatures, indicative of a quantum spin liquid phase; however, the precise nature of this phase remains an open question.  In this talk, I present strong evidence that the triangular lattice Hubbard model at half filling, a physically motivated model of these organic crystals, realizes a chiral spin liquid phase.  In particular, I show that the model has a nonmagnetic insulating phase between a metallic phase for weak interactions and a magnetically ordered phase for strong interactions, and that the intermediate phase exhibits the expected properties of a chiral spin liquid: spontaneous breaking of time-reversal symmetry, topological ground state degeneracy, a quantized spin Hall effect, and characteristic level counting in the entanglement spectrum.  These results were obtained using the infinite-system density matrix renormalization group (iDMRG) method in a mixed real- and momentum-space basis; in the talk, I will also discuss the benefits of this mixed-space approach to DMRG in general, including its applicability to systems such as twisted bilayer graphene for which a large unit cell makes real-space DMRG impractical. 

Supersymmetry (SUSY) has not been verified so far as a fundamental symmetry in particle physics. Emergent phenomena in condensed matter physics bring the possibility of realizing SUSY as an IR symmetry. We show that 2+1D N=2 Nf=2 supersymmetric quantum electrodynamics (SQED3) with dynamical gauge bosons and fermionic gauginos emerges naturally at the tricritical point of nematic pair-density-wave (PDW) quantum phase transition on the surface of a correlated topological insulator hosting three Dirac cones, such as the topological Kondo insulator SmB6. It provides a first example of emergent supersymmetric gauge theory in condensed matter systems. We also investigate the possibility of emergent 3+1D SUSY theory in lattice models. By constructing an explicit fermionic lattice model featuring two 3D Weyl nodes, we find a continuous PDW quantum phase transition as a function of attractive Hubbard interaction. We further show that N=1 3+1D SUSY emerges at the PDW transition, which we believe is the first realization of emergent 3+1D space-time SUSY in microscopic lattice models. Supersymmetry allows us to determine certain critical exponents and the optical conductivity at the strongly coupled fixed point exactly, which may be measured in future experiments.