Stochastic Approach to Non-Equilibrium and Dissipative Many-Body Quantum Dynamics

• Karyn Le Hur

Room: 122:026

Entanglement and the fermion sign problem in auxiliary field quantum Monte Carlo simulations

• Simon Trebst

Room: 122:026 Quantum Monte Carlo simulations of fermions are hampered by the notorious sign problem whose most striking manifestation is an exponential growth of sampling errors with the number of particles. With the sign problem known to be an NP-hard problem and any generic solution thus highly elusive, the Monte Carlo sampling of interacting many- fermion systems is commonly thought to be restricted to a small class of model systems for which a sign-free basis has been identified. Here we demonstrate that entanglement measures, in particular the so-called Renyi entropies, can intrinsically exhibit a certain robustness against the sign problem in auxiliary-field quantum Monte Carlo approaches and possibly allow for the identification of global ground-state properties via their scaling behavior even in the presence of a strong sign problem. We corroborate these findings via numerical simulations of fermionic quantum phase transitions of spinless fermions on the honeycomb lattice at and below half-filling.

Universal Signatures of Quantum Critical Points from Finite-Size Torus Spectra: A Window into the Operator Content of Higher-Dimensional Conformal Field Theories.

• Andreas Läuchli

Room: 122:026 The low-energy spectra of many body systems on a torus, of finite size L, are well understood in magnetically ordered and gapped topological phases. However, the spectra at quantum critical points separating such phases are largely unexplored for 2+1D systems. Using a combination of analytical and numerical techniques, we show that the low-energy torus spectrum at criticality provides a universal fingerprint of the underlying quantum field theory, with the energy levels given by universal numbers times 1/L. We highlight the implications of a neighboring topological phase on the spectrum by studying the Ising* transition, in the example of the toric code in a longitudinal field, and advocate a phenomenological picture that provides insight into the operator content of the critical field theory.

Energy projection and modified Laughlin states

• Mikael Fremling

Room: 122:026 We develop a method to efficiently calculate trial wave functions for quantum Hall systems which involve projection onto the lowest Landau level. The method essentially replaces lowest Landau level projection by projection onto the M lowest eigenstates of a suitably chosen hamiltonian acting within the lowest Landau level. The resulting "energy projection" is a controlled approximation to the exact lowest Landau level projection which improves with increasing M. It allows us to study projected trial wave functions for system sizes close to the maximal sizes that can be reached by exact diagonalization and can be straightforwardly applied in any geometry. As a first application and test case, we study a class of trial wave functions first proposed by Girvin and Jach, which are modifications of the Laughlin states involving a single real parameter.

Bridging coupled wires and a lattice Hamiltonian for two-component bosonic quantum Hall states

• Yohei Fuji

Room: 122:026 We investigate a model of hard-core bosons with correlated hopping on the honeycomb lattice in an external magnetic field by means of a coupled-wire approach. It has been numerically shown that this model exhibits at half filling the bosonic integer quantum Hall (BIQH) state, which is a symmetry-protected topological phase protected by the U(1) particle conservation. By combining the bosonization approach and a coupled-wire construction, we analytically confirm this finding and show that it even holds in the strongly anisotropic (quasi-one-dimensional) limit. We further argue that a phase transition between two different BIQH phases is in a deconfined quantum critical point described by two copies of the (2+1)-dimensional O(4) nonlinear sigma model with a topological theta term. Finally we predict a possible fractional quantum Hall state, the Halperin (221) state, at 1/6 filling.

Fractional quantum Hall models in lattice systems from conformal field theory

• Anne Nielsen

Room: 122:026 In recent years, there has been much interest in finding fractional quantum Hall models in lattice systems, both because it may lead to new ways to realize the fractional quantum Hall effect, and because the lattice gives rise to new effects and possibilities. Here we show that conformal field theory is a power full tool to construct fractional quantum Hall models in lattice systems. We build various fractional quantum Hall states with and without anyons as infinite- dimensional-matrix product states of conformal fields. We use conformal field theory to derive few-body Hamiltonians for which the states are exact ground states, and we analyze the properties of the wavefunctions using analytical and numerical tools. Finally, we discuss recent results on how one can obtain models interpolating between lattice fractional quantum Hall systems and continuum fractional quantum Hall systems.

Feynman Path Integrals over Matrix Product States

• Andrew Green

Room: 122:026 Tensor networks embody deep insights about the entanglement structure of many-body quantum systems. In one dimension, they have led to algorithms that can determine groundstates and follow time evolution with remarkable precision. Entanglement is treated in a very different way in field theories of quantum systems.  These are constructed in such a way that the saddle points do not support entanglement – which is introduced by instanton or fluctuation corrections.  We lift some of the insights about entanglement structure from tensor networks to field theory. Our approach is to explicitly construct a field integral for the partition function over matrix product states, rather than coherent states. The saddle points of such a theory support entanglement in a way that bears interesting comparison with fluctuation and instanton corrections to the usual field theory. In contrast to numerical applications of tensor networks, where the bond order is increased until a certain degree of accuracy is attained, in this field theoretical application, qualitatively new features appear even at low bond order. We demonstrate this by discussing the field theory of certain deconfined quantum critical points.

Spin liquids on kagome lattice and symmetry protected topological phase

• Yin-Chen He

Room: 122:026 In my talk I will introduce the spin liquid phases that occur in kagome antiferromagnets, and discuss their physical origin that are closely related with the newly discovered symmetry protected topological phase (SPT). I will first present our numerical (DMRG) study on the kagome XXZ spin model that exhibits two distinct spin liquid phases, namely the chiral spin liquid and the kagome spin liquid (the groundstate of the nearest neighbor kagome Heisenberg model). Both phases extend from the extreme easy-axis limit, through SU(2) symmetric point, to the pure easy-plane limit. The two phases are separated by a continuous phase transition. Motivated by these numerical results, I will then focus on the easy-axis kagome spin system, and reformulate it as a lattice gauge model. Such formulation enables us to achieve a controlled theoretical description for the spin liquid phases. We then show that the chiral spin liquid is indeed a gauged U(1) SPT phase. On the other hand, we also propose that the kagome spin liquid is a critical spin liquid phase, which can be considered as a gauged deconfined critical point between a SPT and a superfluid phase.

Nonsymmorphic topological crystalline insulators and superconductors: Mobius twists in surface states

• Masatoshi Sato

Room: 122:026 Using the twisted equivariant K-theory, we complete a classification of topological crystalline insulators and superconductors in the presence of additional order-two nonsymmorphic space group symmetries. The order-two nonsymmorphic space groups include half lattice translation with Z2 flip, glide, two-fold screw, and their magnetic space groups. It is pointed out that the nonsymmorphic space groups allow ℤ2 topological phases even in the absence of time-reversal and/or particle-hole symmetries. Furthermore, the coexistence of the nonsymmorphic space group with the time-reversal and/or particle-hole symmetries provides novel ℤ4 topological phases. We argue that the corresponding surface states have Mobius twisted structures in the momentum space.

Steady States of Infinite-Size Dissipative Quantum Chains via Imaginary Time Evolution

• Ying-Jer Kao

Room: 122:026 We show how to use imaginary time evolution of matrix product density operators with the infinite time-evolving block decimation algorithm to determine the nonequilibrium steady states of one-dimensional dissipative quantum lattices in the thermodynamic limit. We provide a demonstration with the transverse field quantum Ising chain. The approach is also amenable to higher dimensions.

Topological phase transitions and anyon condensation in tensor networks: A holographic perspective

• Norbert Schuch

Room: 122:026 We investigate topological phases and phase transitions in the framework of tensor network models. We discuss the role of symmetries in this description, and show how it allows to relate topological phases and transitions between them to symmetry broken and symmetry protected phases exhibited by the transfer operator of the system, i.e., at the boundary.   This is accomplished by translating the string-like topological excitations in the 2D bulk to string order parameters characterizing the different phases under symmetry at the boundary.  We show that by taking into account the constraints arising from complete positivity of the transfer operator, which restricts the possible phases at the boundary, this yields a complete characterization of all possible ways in which topological phase transitions can occur through condensation and confinement of anyons.

Self-organized topological superconductivity in a Yu-Shiba-Rusinov chain

• Jens Paaske

Room: 122:026 In this talk I show how a chain of magnetic moments exchange coupled to a conventional three-dimensional superconductor is unstable towards the formation of a magnetic spiral state. Beyond weak exchange coupling the spiral wave vector greatly exceeds the inverse superconducting coherence length as a result of the strong spin-spin interaction mediated through the subgap band of Yu-Shiba-Rusinov states. Moreover, the simple spin-spin exchange description breaks down as the subgap band crosses the Fermi energy, wherein the spiral phase becomes stabilized by the spontaneous opening of a p-wave superconducting gap within the band. This leads to the possibility of electron-driven topological superconductivity with Majorana boundary modes using magnetic atoms on superconducting surfaces.

Fractional chiral metals

• Jens Bardarson

Room: 122:026

Turning a corner on entanglement entropy

• Roger Melko

Room: 122:026 he entanglement entropy of a quantum critical system receives a logarithmic contribution when the entangling boundary contains a sharp corner.  Numerical calculations indicate that for the Wilson-Fisher fixed point in 2+1 dimensions, the coefficient of this logarithm is universal and contains low-energy information, scaling for example with the number of vector components of the field theory.  Recently, these numerical results have been confirmed analytically, revealing a relationship between the corner coefficient and a central charge defined from the stress tensor two-point function. The combination of analytical understanding and easy numerical accessibility promises to make the corner entanglement an important theoretical tool, providing a new window on universality for free and interacting systems alike.

Topological Quantum Infidelity

• Kirill Shtengel

Room: 122:026

New Fermions

• Bogdan Bernevig

Room: 122:026 In quantum field theory, we learn that fermions come in three varieties: Majorana, Weyl, and Dirac. In this paper, we show that this is not a complete classification. We find the types of crystal symmetry-protected free fermionic excitations that can occur in condensed matter systems, going beyond the classification of Majorana, Weyl, and Dirac particles. We exhaustively classify linear and quadratic 3-, 6- and 8- band crossings stabilized by space group symmetries in solid state systems with spin-orbit coupling and time-reversal symmetry. Several distinct types of fermions arise, differentiated by their degeneracies at and along high symmetry points, lines, and surfaces. For each new class of fermion, we analyze its topological properties by constructing the low-energy effective Hamiltonian and comment on any possible experimental signatures. Some notable consequences of these fermions are the presence of Fermi arcs in non-Weyl systems, the fermionic spin-1 generalization of a Weyl fermion, and the existence of Dirac lines. In addition, we present 18 candidate materials that should realize these exotic fermions, as verified by ab-initio calculations. We also present holographic fermions - fermions that can appear only at the boundary of higher dimensional insulators, and show that their connectivity in the Brillouin zone is described by an extension of Group Cohomology to the Brillouin zone. For all these systems we present realistic materials.

Quantum Hall Edges with Hard Confinement:  Exact Solutions and Numerics beyond Luttinger Liquid

• Steve Simon

Room: 122:026 We consider a Laughlin droplet in a confining potential which is very steep but also weak compared to the ultra-short ranged inter-particle interactions.  We find that the eigenstates have a Jack polynomial structure, and have an energy spectrum which is extremely different from the well- known Luttinger liquid edge.

Many body localization and thermalization: insights from the entanglement spectrum

• Nicolas Regnault

Room: 122:026

Chiral edge modes of a critical spin liquid

• Didier Poilblanc

Room: 122:026 Protected chiral edge modes are a well-known signature of topologically ordered phases like the Fractional Quantum Hall States (FQHS). Using the framework of projected entangled pair states (PEPS) on the square lattice, we construct a family of chiral spin-1/2 quantum spin liquids with $\mathbb{Z}_2$ gauge symmetry and analyze in full details the properties of the edge modes. Surprisingly, we show that the latter can be well described by a chiral Conformal Field Theory of free bosons (SU(2)$_1$), as for the $\nu=1/2$ (bosonic) gapped Laughlin state, despite the fact that our numerical data suggest a critical bulk. We propose that our family of PEPS physically describes a boundary between a chiral topological phase and a trivial phase and might be closely connected to an (unknown) analogous FQHS.

Ergodicity, entanglement, and many-body localization

• Dmitry Abanin

Room: 122:026 We are used to describing systems of many particles by statistical mechanics. However, recently it was realized that the basic postulate of statistical mechanics – ergodicity -- breaks down in so-called many-body localized systems, where disorder prevents particle transport and thermalization. In this talk, I will describe a theory of the many-body localized (MBL) phase, based on new insights from quantum entanglement. I will argue that, in contrast to ergodic systems, MBL eigenstates are not highly entangled, but rather obey so-called area law, typical of ground states in gapped systems. I will use this fact to show that MBL phase is characterized by an infinite number of emergent local conservation laws, in terms of which the Hamiltonian acquires a universal form. Turning to the experimental implications, I will show that MBL systems exhibit a universal response to quantum quenches: surprisingly, entanglement shows logarithmic in time growth, reminiscent of glasses, while local observables exhibit power-law approach to “equilibrium” values. I will also introduce a criterion for the transition between many-body localized and ergodic phases, based on the response of the system to a local perturbation.

Matrix product state representation of quasielectron wave functions

• Jonas Kjäll

Room: 122:026 Recently it has been shown that the model wave functions describing the fractional quantum Hall (FQH) effect have an exact representation as matrix product states (MPS). This representation is extremely useful for numerical calculation of FQH physics. However, a crucial piece, the implementation of the non-local quasielectron operator has been missing. We show how to implement the quasi-electron as an MPS for the Laughlin state and numerically compute the Berry phases for different excitations.

Glassy dynamics as a quantum problem

• Zohar Nussinov

Room: 122:026 We review the underpinning of the microcanonical ensemble and the more refined (and explicitly quantum) "Eigenstate Thermalization Hypothesis". We then find and apply a simple corollary of these to analyze the evolution of a liquid upon supercooling to form a structural glass. Simple theoretical considerations lead to predictions for general properties of supercooled liquids. Amongst other things, a collapse of the viscosity of glass formers is predicted from this theory. This collapse indeed occurs over 16 decades of relaxation times for all known types of glass formers.

Projective construction of the Zk Read-Rezayi fractional quantum Hall states

• Cecile Repellin

Room: 122:026 The Zk Read-Rezayi series1 is a well known sequence of fractional quantum Hall (FQH) states with non-Abelian topological order. The k = 1 member of this series is the (Abelian) Laughlin state. The k = 2 member of this series is the Moore-Read3 state supporting Majorana excitations and a well studied candidate state for the FQH plateau of electrons at ν = 5/2. The Zk=3 Read- Rezayi state supports Fibonacci anyons, which can in principle be used to perform the operations of a universal topological quantum computer. The Zk Read-Rezayi series can be obtained using projective constructions, by starting from the Laughlin state as a parent state. In the simplest version of the projective construction, the bosonic Moore-Read state (Zk=2) is written by symmetrizing the product of two bosonic Laughlin wave functions. On the torus, this construction does not yield the Moore-Read state for an odd number of particles in finite size. We show that introducing a defect allowing the bosons to go from one layer to the next can remedy this problem. This obstruction also appears in the more generic case of the projective construction of the bosonic Zk Read-Rezayi states, and is resolved similarly. The projective construction could provide an avenue to write higher k members of the Zk series in the matrix product state language. In this context, our new construction scheme should allow us to recover all topological sectors of the SU(2)k theory starting only from the Laughlin state. Beyond model states, the projective construction can also be used to describe the neutral excitations above the Read-Rezayi states. I will explore the accuracy of this approximation in the case of the Moore-Read state neutral mode. Finally, I will discuss the possibility that a microscopic interlayer coupling term might physically realize the symmetrization in a bilayer system.

Symmetry Protected Topological Phases of fermions

• Lukasz Fidkowski

Room: 122:026 Topological insulators constitute a different phase of quantum matter from ordinary insulators, despite not breaking any symmetries. They are an example of a `symmetry protected topological' phase, in that they exhibit exotic phenomena, namely gapless surface modes, that are entirely protected by time reversal symmetry. Motivated by this, one can classify all possible non-trivial topological insulators in all dimensions and symmetry classes at the level of band structures. However, the role of interactions in this scheme is still not completely understood. In this talk I'll describe our recent work on classifying interacting analogues of topological insulators, and constructing exactly solvable lattice realizations of them, which may allow them to be targeted by numerical algorithms.

Topological Defects on the Lattice

• Roger Mong

Room: 122:026 We construct topological defects in 2D classical lattice models and 1D quantum chains.  The defects satisfy commutation relations guaranteeing the partition function depends only on topological properties of the defects.  One useful consequence is a generalization of Kramers-Wannier duality to a wide class of height models, applicable on any surfaces.  This enables us to derive modular transformations of conformal field theories (CFTs) directly from lattice considerations.  We will also discuss how to write chiral CFT operators on a lattice.

Spontaneous currents and Majorana fermions from magnetic impurities on spin-orbit coupled superconductors

• Annica Black-Schaffer

Room: 122:026 Depositing magnetic impurities or islands on top of spin-orbit coupled superconductors can generate a topological superconducting state. For one-dimensional impurity wires, a non-trivial topology will give rise to Majorana fermions appearing at the end points of the wire, whereas for magnetic islands edge states disperse across the bulk energy gap. In this talk I will focus on several recent results for these systems. I will show that persistent currents are generated around magnetic impurities. The currents are independent on topology, but still generally peak around the topological phase transition. I will also show how a self-consistent solution for the superconducting state results in a heavily suppressed superconducting order parameter at the impurity sites, which can even experience a pi-shift. This pi-shift can be explained by impurity-induced quasiparticles being completely out of phase with the condensate.

"Bosonic" quantum phase transition of strongly interacting topological insulators: field theory, Numerics, and experimental platform

• Cenke Xu

Room: 122:026 We discuss our recent works on strongly interacting bilayer quantum Spin Hall insulators with Sz conservation. We demonstrate that interaction will drive the system into a "bosonic" symmetry protected topological phase, in the sense that bosonic modes are gapless at the boundary, while local fermion excitations are gapped. Also, at the bulk quantum critical point between the topological and trivial states, bosonic modes close their gap, while fermion modes remain gapped. Thus this quantum critical point is a purely "bosonic" conformal field theory, which is described by a 2+1d O(4) nonlinear sigma model with a Theta term at Theta = Pi. All these results are confirmed by both field theory analysis and also quantum Monte Carlo simulation. We also propose that these physics can be well realized in bilayer graphene under (strong) out of plane magnetic field.

Topological phenomena in periodically driven systems: the role of disorder and interactions

• Erez Berg

Room: 122:026 Periodically driven quantum systems, such as semiconductors subject to light and cold atoms in optical lattices, provide a noveland versatile platform for realizing topological phenomena. Some of these are analogs of topological insulators andsuperconductors, attainable also in static systems; others are unique to the periodically driven case. I will describe how periodic driving, disorder, and interactions can conspire to give rise to new robust steady states, with no analogues in static systems. In disordered two-dimensional driven systems, a phase with chiral edge states and fully localized bulk states is possible; this phase can realize a non-adiabatic quantized charge pump. In interacting one dimensional driven systems, current carrying states with excessively long life times can arise.

Variational Tensor Network Renormalization in Imaginary Time: Benchmark results in the  quantum compass model and the Hubbard model

• Piotr Czarnik

Room: 122:026 The Gibbs operator $e^{−\beta H}$ for a two-dimensional (2D) lattice system with a Hamiltonian H can be represented by a three-dimensional tensor network, the third dimension being the imaginary time (inverse temperature) $\beta$. Coarse-graining the network along $beta$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension D. The coarse-graining is performed by a tree tensor network of isometries. The isometries are optimized variationally --- taking into account full tensor environment -- - to maximize the accuracy of the PEPO. The algorithm is applied to the isotropic quantum compass model on an infinite square lattice near a symmetry-breaking phase transition at finite temperature. From the linear susceptibility in the symmetric phase and the order parameter in the symmetry-broken phase the critical temperature is estimated at $T_c=0.0606(4)J$, where J is the isotropic coupling constant between S=1/2 pseudospins. The algorithm is also applied to the two-dimensional Hubbard model on an infinite square lattice. Benchmark results are obtained that are consistent with the best dynamical mean field theory (DCA - dynamical cluster approximation)  and power series expansion (NLCE - numerically linked cluster expansion)  in the regime of parameters where these more conventional methods yield mutually consistent results.

Recent advances with iPEPS: simulations of the 2D Hubbard model, improved energy extrapolations, and variational optimization

• Philippe Corboz

Room: 122:026 In this talk I report on recent progress with infinite projected entangled-pair states (iPEPS) which is a tensor network ansatz for 2D wave functions in the thermodynamic limit. I present simulation results for the 2D t-J and Hubbard models, for bond dimensions up to D=16, which reveal an extremely close competition between a uniform d-wave superconducting state and different types of stripe states. The iPEPS variational energies are lower than previous state-of-the-art variational results for large 2D systems which demonstrates the competitiveness of the approach. A key factor to determine the true ground state among several competing states is to have an accurate estimate of their energies in the infinite bond dimension D limit. However, a simple extrapolation in 1/D often does not provide an accurate result. We show how to improve these estimates by an extrapolation based on a truncation error which is computed within the iPEPS imaginary time evolution algorithm. Finally, I present a variational optimization scheme for iPEPS which yields a higher accuracy than previous ground state algorithms.

Realizing all so(N)_1 quantum criticalities in symmetry protected cluster models

• Ville Lahtinen

Room: 122:026 We show that all so(N)1 universality class quantum criticalities emerge when one-dimensional generalized cluster models are perturbed with Ising or Zeeman terms. Each critical point is described by a low-energy theory of N linearly dispersing fermions, whose spectrum we show to precisely match the prediction by so(N)1 conformal field theory. Furthermore, by an explicit construction we show that all the cluster models are dual to non-locally coupled transverse field Ising chains, with the universality of the so(N)1 criticality manifesting itself as N of these chains becoming critical. This duality also reveals that the symmetry protection of cluster models arises from the underlying Ising symmetries and it enables the identification of local representations for the primary fields of the so(N)1conformal field theories. For the simplest and experimentally most realistic case that corresponds to the original one-dimensional cluster model with local three-spin interactions, our results show that the su(2)2≃so(3)1 Wess-Zumino-Witten model can emerge in a local, translationally invariant and Jordan-Wigner solvable spin-1/2 model. Our results pave the way for classifying criticalities between symmetry-protected topological states.

Quantum critical point of Dirac fermion mass generation without spontaneous symmetry

• Zi-Yang Meng

Room: 122:026 We study a lattice model of interacting Dirac fermions in (2+1) dimension space-time with an SU(4) symmetry. While increasing interaction strength, this model undergoes a continuous quantum phase transition from the weakly interacting Dirac semimetal to a fully gapped and nondegenerate phase without condensing any Dirac fermion bilinear mass operator. This unusual mechanism for mass generation is consistent with recent studies of interacting topological insulators/superconductors, and also consistent with recent progresses in lattice QCD community.

Self-organized semi-metals on grain boundaries in topological band insulators

• Robert-Jan Slager

Room: 122:026 Semi-metals are characterized by nodal band structures that give rise to exotic electronic properties. The stability of Dirac semi-metals, such as graphene in two spatial dimensions (2D), requires the presence of lattice symmetries, while akin to the surface states of topological band insulators, Weyl semi-metals in three spatial dimensions (3D) are protected by band topology. I will show that in the bulk of topological band insulators, self-organized topologically protected semi- metals can emerge along a grain boundary, a ubiquitous extended lattice defect in any crystalline material. In addition to experimentally accessible electronic transport measurements, these states exhibit a valley anomaly that influences the edge spin transport in two spatial dimensions (2D), whereas in 3D they appear as graphene-like states that should exhibit an odd-integer quantum Hall effect. The general mechanism underlying these novel semi-metals, being the hybridization of spinon modes bound to the grain boundary, suggests  that topological semi-metals can emerge in any topological material where lattice dislocations bind localized topological modes.

Symmetry twisting in entanglement

• Shunji Matsuura

Room: 122:026 Entanglement and symmetries are two fundamental concepts in classifying quantum states. We consider a new type of entanglement entropy which is enriched by symmetry twisting, and show how the entropy distinguishes quantum states, including symmetry protected and enriched topological phases. This is based on works with Wen, Ryu, Hung, and Caputa.