Long-range, multi-particle quantum entanglement plays a fundamental role in our understanding of many modern quantum materials, including the copper-based high temperature superconductors. Hawking’s quantum information puzzle in the quantum theory of black holes also involves non-local entanglement. I will describe a simple model of randomly entangled qubits which has shed light on these distinct fields of physics.

The waves that describe systems in quantum physics can carry information about how their environment has been altered, for example by forces acting on them. This effect is the geometric phase. It also occurs in the optics of polarised light, where it goes back to the 1830s; and it gives insight into the spin-statistics relation for identical quantum particles. The underlying mathematics is geometric: the phenomenon of parallel transport, which also explains how falling cats land on their feet, and why parking a car in a narrow space is difficult. Incorporating the back-reaction of the geometric phase on the dynamics of the changing environment exposes the unsolved problem of how strictly a system can be separated from a slowly-varying environment, and involves different mathematics: divergent infinite series.

Many-body localised (MBL) phases of matter fall outside the conventional paradigm of equilibrium statistical mechanics and thermodynamics. A natural question thus is, what minimal and generic properties must random many-body Hamiltonians possess for a localised phase to be stable? In this talk, I will address the question by exploiting the exact mapping between a many-body Hamiltonian and a tight-binding problem on the Fock-space graph. In particular, I will present a theory for how the strong correlations in the effective Fock-space disorder play a central role in stabilising an MBL phase. The theory is rooted in analytic but approximate calculations of the propagators on the Fock space. To shed further light on the underlying physics, I will also introduce and discuss a classical proxy for the MBL transition in the form of a percolation transition on the Fock space. Finally, I will discuss a novel class of Anderson localisation problems on correlated trees, to understand the effect of such disorder correlations in a more controlled setting.

The ability to create and manipulate optical lattices for cold atoms, with a view towards studying topological matter, has brought renewed focus to the physics of Bloch waves and the role of the lattice in governing their properties. We consider generic tight binding models where particle motion is described in terms of hopping amplitudes between orbitals. The physical attributes of the orbitals, including their locations in space, are independent pieces of information. We identify a notion of geometry-independence: any physical quantity that depends only on the tight-binding parameters (and not on the explicit information about the orbital geometry) is said to be “geometry-independent.” Identification of geometry-dependent vs. independent quantities can be used as a novel principle for constraining a variety of results in both non-interacting and interacting systems. We show, e.g., how Hall measurements based on accelerated lattices or tilted potentials, and those based on applying a chemical potential imbalance between reservoirs, give diffReferenceerent results due to the fact that one is geometry-dependent, while the other is geometry-independent. Similar considerations apply for thermal Hall responses in electronic, cold atomic, and spin systems.

Reference

Steven H. Simon and Mark S. Rudner, Phys. Rev. B 102, 165148, 2020.

This work investigates nematic liquid crystals in three-dimensional curved space, and determines which director deformation modes are compatible with each possible type of non-Euclidean geometry. Previous work by Sethna et al. [1] showed that double twist is frustrated in flat space $\mathbb{R}^3$, but can fit perfectly in the hypersphere $\mathbb{S}^3$. Here, we extend that work to all four deformation modes (splay, twist, bend, and biaxial splay) and all eight Thurston geometries [2]. Each pure mode of director deformation can fill space perfectly, for at least one type of geometry. This analysis shows the ideal structure of each deformation mode in curved space, which is frustrated by the requirements of flat space.

Sethna J. P., Wright D. C. and Mermin N. D., 1983 Phys. Rev. Lett. 51 467–70.

J.-F. Sadoc, R. Mosseri and J. Selinger, New Journal of Physics 22 (2020) 093036.

Quantum chaos, especially when caused by particle interactions, is closely related with topics of high experimental and theoretical interest, from the thermalization of isolated systems to the difficulties to reach a localized phase, and the emergence of quantum scars. In this talk, various indicators of quantum chaos will be compared, including level statistics, structure of eigenstates, matrix elements of observables, out-of-time ordered correlators, and the correlation hole (ramp). These indicators are then employed to identify the minimum number of interacting particles required for the onset of strong chaos in quantum systems with short-range and also with long-range interactions.

Identifying the relevant coarse-grained degrees of freedom in a complex physical system is a key stage in developing effective theories. The renormalization group (RG) provides a framework for this task, but its practical execution in unfamiliar systems is fraught with ad hoc choices. Machine learning approaches, on the other hand, though promising, often lack formal interpretability: it is unclear what relation, if any, the architecture- and training-dependent learned "relevant" features bear to standard objects of physical theory.

I will present recent results addressing both issues. We develop a fast algorithm, the RSMI-NE, employing state-of-art results in machine-learning-based estimation of information-theoretic quantities to construct the optimal coarse-graining. We use it to develop a new approach to identifying the most relevant field theory operators describing a statistical system, which we validate on the example of interacting dimer model. I will also discuss formal results underlying the method: we establish equivalence between the information-theoretic notion of relevance defined in the Information Bottleneck (IB) formalism of compression theory, and the field-theoretic relevance of the RG. We show analytically that for statistical physical systems the "relevant" degrees of freedom found using IB compression indeed correspond to operators with the lowest scaling dimensions, providing a dictionary connecting two distinct theoretical toolboxes.

Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. Please do not be scared by the physics though: I will present a very simple operator whose spectral properties are thought to determine which angles are magical. It comes from a recent PR Letter by Tarnopolsky–Kruchkov–Vishwanath. The mathematics behind this is an elementary blend of representation theory (of the Heisenberg group in characteristic three), Jacobi theta functions and spectral instability of non-self-adjoint operators (involving Hoermander’s bracket condition in a very simple setting). The results will be illustrated by colourful numerics which suggest some open problems. This is joint work with M. Embree, J. Wittsten, and M. Zworski.

Spatial dimensionality plays a key role in our understanding of topological physics, with different topological invariants needed to characterise systems with different numbers of spatial dimensions. In a 2D quantum Hall system, for example, a robust quantisation of the Hall response is related to the first Chern number: a 2D topological invariant of the electronic energy bands. Generalising to more spatial dimensions, it was shown that a new type of quantum Hall effect could emerge in four dimensions, but where the quantised response was related to a four-dimensional topological invariant, namely the second Chern number. While systems with four spatial dimensions may seem abstract, recent developments in ultracold atoms and photonics have opened the door to exploring such higher-dimensional topological physics experimentally. In this talk, I will introduce the theory of 4D topological phases of matter, before discussing recent experiments in cold atoms, photonics and electrical circuits that have begun to probe aspects of this physics in the laboratory.

In this talk I will discuss how one can detect quantum chaos in generic interacting models using adiabatic transformations, specifically the fidelity susceptibility. In particular, I will show that it exhibits a very sharp crossover behavior from the algebraic to the exponential scaling form with the system size in the presence of a small integrability breaking parameter. This sensitivity allows one to identify tiny integrability breaking perturbations, not detectable by conventional methods. I will also discuss that generically integrable and chaotic regimes are separated by a universal regime of “maximal chaos” where the fidelity susceptibility saturates its upper bound and the system exhibits exponentially slow, glassy dynamics. I will illustrate how this probe works using several examples of both clean and disordered systems and, in particular, will argue that numerical results indicate absence of a continuous many-body localization transition in the thermodynamic limit.

In the last two decades the field of nonequilibrium quantum many-body physics has seen a rapid development driven, in particular, by the remarkable progress in quantum simulators, which today provide access to dynamics in quantum matter with an unprecedented control. However, the efficient numerical simulation of nonequilibrium real-time evolution in isolated quantum matter still remains a key challenge for current computational methods especially beyond one spatial dimension. In this talk I will present a versatile and efficient machine learning inspired approach. I will first introduce the general idea of encoding quantum many-body wave functions into artificial neural networks. I will then identify and resolve key challenges for the simulation of real-time evolution, which previously imposed significant limitations on the accurate description of large systems and long-time dynamics. As a concrete example, I will consider the dynamics of the paradigmatic two-dimensional transverse field Ising model, where we observe collapse and revival oscillations of ferromagnetic order and demonstrate that the reached time scales are comparable to or exceed the capabilities of state-of-the-art tensor network methods.

Weyl semimetals contain an even number of singular points in their Brillouin zone around which the dispersion is linear and the density of states (DoS) vanishes. How does the density of states change in the (inevitable) presence of impurities? This question has been the subject of an intensive and partially controversial discussion in the recent literature. In particular, it has been suggested that below a critical disorder strength the DoS remains zero, and that the system supports a phase transition separating an intrinsically clean from a disordered phase. In this talk, I discuss this problem on the basis of several effective models. All these support the integrity of the Weyl node and hence are compatible with the above scenario. I will also comment on the (tricky) comparison to numerics and point out a puzzle whose solution invites mathematical research.

To characterize the generic behavior of open quantum many-body systems, we consider random, purely dissipative Liouvillians with a notion of locality. We find that the positivity of the map implies a sharp separation of the relaxation timescales according to the locality of observables. Specifically, we analyze a spin-$1/2$ system of size $\ell$ with up to n-body Lindblad operators, which are n local in the complexity-theory sense. Without locality ($n=l$), the complex Liouvillian spectrum densely covers a “lemon”-shaped support, in agreement with recent findings [S. Denisov et al., Phys. Rev. Lett. 123, 140403 (2019), L. Sa et al., JPA 53, 305303].However, for local Liouvillians ($n < l$), we find that the spectrum is composed of several dense clusters with random matrix spacing statistics, each featuring a lemon-shaped support wherein all eigenvectors correspond to $n$-body decay modes. This implies a hierarchy of relaxation timescales of n-body observables, which we verify to be robust in the thermodynamic limit. Our findings for n locality generalize immediately to the case of spatial locality, introducing further splitting of timescales due to the additional structure.

To test our theoretical prediction, we perform experiments on the IBM quantum computing platform, designing different "waiting circuits" to inject two body dissipative interactions by two qubit entangling gates. We find excellent agreement with our theory and observe the predicted hierarchy of timescales.

[1] K. Wang, F. Piazza, D. J. Luitz ” Hierarchy of relaxation timescales in local random Liouvillians “ Phys. Rev. Lett. 124, 100604 (2020)

[2] O. E. Sommer, F. Piazza, and D. J. Luitz “Many-body Hierarchy of Dissipative Timescales in a Quantum Computer” arXiv:2011.08853

We demonstrate that the exact nonequilibrium steady state of the one-dimensional Heisenberg $XXZ$ spin chain driven by boundary Lindblad operators can be constructed explicitly with a matrix product ansatz for the nonequilibrium density matrix. For the isotropic Heisenberg chain, polarized at the boundaries in different directions with a non-zero twist angle, we calculate the exact magnetization profiles and magnetization currents. The in-plane steady-state magnetization profiles are harmonic functions with a frequency proportional to the twist angle. In-plane steady-state magnetization currents are subdiffusive and vanish as the boundary coupling strength increases, while the transverse current is diffusive and saturates as the coupling strength becomes large. The anisotropic chain exhibits spin helix states at special values of the anisotropy where the transverse current is independent of system size, even for non-integrable higher-spin chains.

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 excitation at equal footing: symmetry can be viewed as gravitational anomaly, or symmetry can be viewed as shadow topological order in one higher dimension. This picture allows us to see many duality relations between seeming unrelated symmetries.

Topological states of matter are characterized by a gap in the bulk of the system referring to an insulator or a superconductor and topological edge modes as well which find various applications in transport and spintronics. The bulk-edge correspondence is associated to a topological number. The table of topological states include the quantum Hall effect and the quantum anomalous Hall effect, topological insulators and topological superconductors in various dimensions and lattice geometries. Here, we discuss classes of states which can be understood from mapping onto a spin-1/2 particle in the reciprocal space of wave-vectors. We develop a geometrical approach on the associated Poincare-Bloch sphere, developing smooth fields, which shows that the topology can be encoded from the poles only. We show applications for the light-matter coupling when coupling to circular polarizations and develop a relation with quantum transport and the quantum Hall conductivity. The formalism allows to include interaction effects. We show our recent developments on a stochastic approach to englobe these interaction effects and discuss applications for the Mott transition of the Haldane and Kane-Mele models. Then, we develop a model of coupled spheres and show the possibility of fractional topological numbers as a result of interactions between spheres and entanglement allowing a superposition of two geometries, one encircling a topological charge and one revealing a Bell or EPR pair. Then, we show applications of the fractional topological numbers $C=1/2$ in bilayer honeycomb models describing topological semi-metals characterized by a quantized Berry phase at one Dirac point.

Joel Hutchinson and Karyn Le Hur, arXiv:2002.11823 (under review)

Philipp Klein, Adolfo Grushin, Karyn Le Hur, Phys. Rev. B 103, 035114 (2021)

The eight-vertex model is an useful description that generalizes several spin systems, as well as the more common six-vertex model, and others. In a special "free-fermion" regime, it is known since the work of Fan, Lin, Wu in the late 60s that the model can be mapped to non-bipartite dimers. However, no general theory is known for dimers in the non-bipartite case, contrary to the extensive rigorous description of Gibbs measures by Kenyon, Okounkov, Sheffield for bipartite dimers. In this talk I will show how to transform these non-bipartite dimers into bipartite ones, on generic planar graphs. I will mention a few consequences: computation of long-range correlations, criticality and critical exponents, and their "exact" application to Z-invariant regimes on isoradial graphs.

Chiral fermions have the property that their left-handed and right-handed components transform differently under some symmetry. Folklore suggests that it is impossible to give such fermions a mass without breaking this symmetry. I'll show, through a number of examples, why this folklore is wrong. In particular, I'll show how one generation of fermions in the Standard Model can get a mass without the need for a Higgs boson that breaks electroweak symmetry.

Open quantum systems composed of atoms interacting with light exhibit behaviour that is akin to that of associative memories [1]. This means that they possess stationary states that can be interpreted as stored memory patterns, which are retrieved when the initial state is inside the basin of attraction of a given pattern [2]. The corresponding pattern retrieval dynamics can be observed in actual experimental settings. In these experiments atoms are confined within an optical cavity whose photons mediate long-range interactions [3]. Stored patterns are encoded in the atom-light coupling constants. This setting offers an interesting opportunity for studying quantum generalisations of associative memories and stored (quantum) patterns in this context [4]. Moreover, it allows to systematically construct scenarios in which quantum effects might be beneficial, e.g., for speeding up the pattern retrieval process [5]. I will talk about recent research of my group on this subject, which builds a bridge between classic machine learning concepts, such as the Hopfield Neural Network, and most recent experimental manifestations of synthetic quantum matter.

E. Fiorelli et al., Physical Review Letters 125, 070604 (2020)

F. Carollo and I. Lesanovsky, arXiv:2009.13932 (2020)

V. D. Vaidya et al., Physical Review X 8, 011002 (2018)

P. Rotondo et al., Journal of Physics A 51, 115301 (2018)

E. Fiorelli et al., Physical Review A 99, 032126 (2019)

Topological states of matter have fascinated physicists since a long time. The notion of topology is however ususally associated with ground states of (many-body)-Hamiltonians, which are pure. So what is left of it at finite temperatures and can topological protection be extended to non-equilibrium steady states (NESS) of open systems? Can suitable observables be constructed that preserve the integrity of topological invariants for mixed states and what are measurable consequences of their existence? Can we classify the topology of finite temperature and NESS using generalized symmetries? Motivated by topological charge pumps, first introduced by Thouless, I will first discuss a topological invariant for systems that break time reversal symmetry based on the many-body polarization, called ensemble geometric phase (EGP) [1]. In contrast to charge transport, the EGP can be used to probe topology in one dimensional non-interacting [2] and interacting [3], closed and open systems alike. Furthermore different from other constructions, such as the Uhlmann phase, it can be extended to two dimensions [4]. I will then extend the definition to systems with time-reversal symmetry and finally talk about measurable consequences of mixed-states topological invariants.

C.E. Bardyn, L. Wawer, A. Altland, M. Fleischhauer, S.Diehl, (PRX 2018).

D. Linzner, L. Wawer, F. Grusdt, M. Fleischhauer, (PRB 2016).

R. Unanyan, M. Kiefer-Emmanouilidis, M. Fleischhauer, (PRL 2020).

Tensor network states provide a comprehensive framework for the analytic and numerical study of strongly correlated many-body systems. In recent years, this framework has been successfully applied to topological phases of matter. In this talk, I will present two dual tensor network representations of the $(3+1)d$ toric code ground state subspace, which are obtained by initially imposing either family of stabilizer constraints. I will discuss topological properties of the model from the point of view of these virtual symmetries, demonstrate that one of these representations is stable to all local tensor perturbations — including those that do not map to local operators on the physical Hilbert space — and explain, both from a physical and category theoretical viewpoint, how the distinguishing properties of these representations are related to the phenomenon of bulk-boundary correspondence.

The "sign problem" (SP) is the fundamental limitation to simulations of strongly correlated materials in condensed matter physics, solving quantum chromodynamics at finite baryon density, and computational studies of nuclear matter. It is often argued that the SP is not intrinsic to the physics of particular Hamiltonians, since the details of how it onsets, and its eventual occurrence, can be altered by the choice of algorithm or many-particle basis. Despite that, I plan to show in this talk that the SP in determinant quantum Monte Carlo (DQMC) is quantitatively linked to quantum critical behavior. This demonstration is done via simulations of a number of fundamental models of condensed matter physics, all of whose critical properties are relatively well understood.

I will begin by defining a canonical family of perturbations of the Dirac equation. These perturbations are complex anti-linear, thus ground states only form a real vector space. A special case of this theory is known as the Jackiw–Rossi theory, which models surface excitations on the surface of a topological insulator placed in proximity to an s-wave superconductor. While the physics of the theory is relatively well-understood, the mathematical side has not been studied, even on surfaces, not to mention its generalizations to higher dimensional and on nontrivial manifolds. I call these equations the generalized Jackiw–Rossi equations.

After the definitions and connections to physics, I will present my current work on the generalized Jackiw–Rossi equations. My main result is a concentration phenomenon which proves the physical expectation that such Majorana fermions concentrate around the vortices of the superconducting order parameter. Moreover, I provide approximate solutions that are exponentially sharp in the large coupling limit.

If time permits, then I will show how these Majorana fermions define a bundle of projective spaces over the simple part of vortex moduli spaces. The holonomies of such bundles give rise to projective representations of (surface) braid groups, and thus, speculatively, can be of interest to quantum information theorists.

Recent years have seen much progress in the mathematical understanding of quantum charge transport under slow driving, in the presence of strong interactions between the charge carriers. I will give an overview of recent results, starting with the adiabatic theorem in an interacting setting, and continuing to topological transport where quantization can be shown to be valid beyond the linear response setting.