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QM3 Quantum Matter meets Maths   RSS

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Raquel Queiroz 27/07/2020, 17:00 — 18:00 — Online
, Weizmann Institute of Science

Boundary Obstructed Topological Phases

Symmetry protected topological (SPT) phases are gapped phases of matter that cannot be deformed to a trivial phase without breaking the symmetry or closing the bulk gap. Here, we introduce a new notion of a topological obstruction that is not captured by bulk energy gap closings in periodic boundary conditions. More specifically, given a symmetric boundary termination we say two bulk Hamiltonians belong to distinct bound­ary obstructed topological phases (BOTPs) if they can be deformed to each other on a system with periodic boundaries, but cannot be deformed to each other in the open system without closing the gap at at least one high symmetry surface. BOTPs are not topological phases of matter in the standard sense since they are adiabat­ically deformable to each other on a torus but, similar to SPTs, they are associated with boundary signatures in open systems such as surface states or fractional corner charges. In contrast to SPTs, these boundary signatures are not anomalous and can be removed by symmetrically adding lower dimensional SPTs on the boundary, but they are stable as long as the spectral gap at high-symmetry edges/surfaces remains open. We show that the double-mirror quadrupole model of [Science, 357(6346), 2018] is a prototypical example of such phases, and present a detailed analysis of several aspects of boundary obstructions in this model. In addition, we intro­duce several three-dimensional models having boundary obstructions, which are characterized either by surface states or fractional corner charges. We also provide a general framework to study boundary obstructions in free-fermion systems in terms of Wannier band representations (WBR), an extension of the recently-developed band representation formalism to Wannier bands. WBRs capture the notion of topological obstructions in the Wannier bands which can then be used to study topological obstructions in the boundary spectrum by means of the correspondence between the Wannier and boundary spectra. This establishes a form of bulk-boundary correspondence for BOTPs by relating the bulk band representation to the boundary topology.

Christophe Garban 20/07/2020, 17:00 — 18:00 — Online
, Université Lyon 1

A new point of view on topological phase transitions

Topological phase transitions were discovered by Berezinskii-Kosterlitz-Thouless in the 70's. They describe intriguing phase transitions for classical spins systems such as the plane rotator model (or $XY$ model). I will start by reviewing how this phase transition arises in cases such as:

  • the $XY$ model (spins on $\mathbb{Z}^2$ with values in the unit circle)
  • the integer-valued Gaussian Free Field (or $\mathbb{Z}$-ferromagnet)
  • Abelian Yang-Mills on $\mathbb{Z}^4$

I will then connect topological phase transitions to a statistical reconstruction problem concerning the Gaussian Free Field and will show that the feasibility of the reconstruction undergoes a KT transition.

This is a joint work with Avelio Sepúlveda (Lyon) and the talk will be based mostly on the preprint: https://arxiv.org/abs/2002.12284

Giandomenico Palumbo 13/07/2020, 17:00 — 18:00 — Online
, Université Libre de Bruxelles

Four-dimensional semimetals with tensor monopoles: from surface states to topological responses

Quantum anomalies offer a useful guide for the exploration of transport phenomena in topological semimetals. A prominent example is provided by the chiral magnetic effect in three-dimensional Weyl semimetals, which stems from the chiral anomaly. Here, we reveal a distinct quantum effect, coined parity magnetic effect, which is induced by the parity anomaly in a four-dimensional topological semimetal. Upon preserving time-reversal symmetry, the spectrum of our model is doubly degenerate and the nodal (Dirac) points behave like $\mathbb{Z}_2$ monopoles. When time-reversal symmetry is broken, while preserving the sublattice (chiral) symmetry, our system supports spin-3/2 quasiparticles and the corresponding Dirac-like cones host tensor monopoles characterized by a $\mathbb{Z}$ number, the Dixmier-Douady invariant. In both cases, the semimetal exhibits topologically protected Fermi arcs on its boundary. Besides its theoretical implications in both condensed matter and quantum field theory, the peculiar 4D magnetic effect revealed by our model could be measured by simulating higher-dimensional semimetals in synthetic matter.

See also

Slides of the talk

Manuel Asorey 08/07/2020, 11:00 — 12:00 — Online
, University of Zaragoza

Bulk-Edge dualities in Topological Matter

Novel bulk-edge dualities have recently emerged in topological materials from the observation of some phenomenological correspondences. The similarity of these dualities with string theory dualities is very appealing and has boosted a quite significant number of cross field studies.

We analyze the bulk-edge dualities in the integer quantum Hall effect, where due to the simpler nature of planar systems the duality can be analyzed by powerful analytic techniques. The results show that the correspondence is less robust than expected. In particular, it is highly dependent of the type of boundary conditions of the topological material. We introduce a formal proof of the equivalence of bulk and edge approaches to the quantization of Hall conductivity for metallic plates with local boundary conditions. However, the proof does not work for non-local boundary conditions, like the Atiyah-Patodi-Singer boundary conditions, due to the appearance of gaps between the bulk and edge states.

See also

Asorey.pdf

Raffaele Resta 29/06/2020, 17:00 — 18:00 — Room P3.10, Mathematics Building Online
, Instituto Officina dei Materiali, CNR, Trieste, Italy

The insulating state of matter: a geometrical theory

The insulating versus conducting behavior of condensed matter is commonly addressed in terms of electronic excitations and/or conductivity. At variance with such wisdom, W. Kohn hinted in 1964 that the insulating state of matter reflects a peculiar organization of the electrons in their ground state, and does not require an energy gap.

Kohn’s theory of the insulating state got a fresh restart in 1999; at the root of these developments is the modern theory of polarization, developed in the early 1990s, and based on a geometrical concept (Berry phase). Since insulators and metals polarize in a qualitatively different way, quantum geometry also discriminates insulators from conductors. A common geometrical “marker”, based on the quantum metric, caracterizes all insulators (band insulators, Anderson insulators, Mott insulators, quantum Hall insulators...); such marker diverges in conductors.

See also

Slides of the talk

Mário Silveirinha 24/06/2020, 11:00 — 12:00 — Online
, Instituto Superior Técnico

Topological theory of non-Hermitian photonic systems

Recently, topological materials and topological effects have elicited a great interest in the photonics community [1]. While condensed-matter phenomena are traditionally described by Hermitian operators, the same is not true in the context of macroscopic electrodynamics where a dissipative response is the rule, not the exception. In this talk, I will discuss how to determine the topological phases of dissipative (non-Hermitian) photonic structures from first principles using a gauge-independent Green function [2, 3]. It is shown that analogous to the Hermitian case, the Chern number can be expressed as an integral of the system Green function over a line parallel to the imaginary-frequency axis. The approach introduces in a natural way the "band-gaps" of non-Hermitian systems as the strips of the complex-frequency plane wherein the system Green function is analytical. I apply the developed theory to nonreciprocal electromagnetic continua and photonic crystals, with lossy and or gainy elements. Furthermore, I discuss the validity of the bulk-edge correspondence in the non-Hermitian case.

  1. L. Lu, J. D. Joannopoulos, M. Soljačić, Topological photonics, Nat. Photonics, 8, 821, (2014).
  2. M. G. Silveirinha, Topological theory of non-Hermitian photonic systems, Phys. Rev. B, 99, 125155, 2019.
  3. F. R. Prudêncio, M. G. Silveirinha, First Principles Calculation of Topological Invariants of non-Hermitian Photonic Crystals.

See also

Silveirinha_slides.pdf

Lucas Sá 17/06/2020, 11:00 — 12:00 — Online
Lucas Sá, Instituto Superior Técnico and CEFEMA

Random matrix theory of dissipative quantum chaos

Describing complex interacting quantum systems is a daunting task. One very fruitful approach to this problem, developed for unitary dynamics, is to represent the Hamiltonian of a system by a large random matrix. This eventually led to the development of the field of quantum chaos. Arguably, one of its most spectacular achievements was the identification of universal signatures of chaos in quantum systems, characterizing the correlations of their energy levels. In this talk, we will focus on the recent application of (non-Hermitian) random matrix theory to open quantum systems, where dissipation and decoherence coexist with unitary dynamics. First, we will discuss a class of stochastic Lindbladians with random Hamiltonian and independent random dissipation channels (jump operators), as a model for the generator of complicated nonunitary dynamics. We will then explain what difficulties arise when combining dissipation with quantum chaos, and how to overcome them. In particular, we discuss a new non-Hermitian random matrix ensemble with eigenvalues on the torus and how it connects to our recent proposal of using complex spacing ratios as a signature of dissipative quantum chaos.

See also

Slides of the talk

Zlatko Papic 10/06/2020, 11:00 — 12:00 — Online
, University of Leeds

Quantum many-body scars: a new form of weak ergodicity breaking in constrained quantum systems

Recent experiments on large chains of Rydberg atoms [1] have demonstrated the possibility of realising one-dimensional, kinetically constrained quantum systems. It was found that such systems exhibit surprising signatures of non-ergodic dynamics, such as robust periodic revivals in global quenches from certain initial states. This weak form of ergodicity breaking has been interpreted as a manifestation of "quantum many-body scars" [2], i.e., the many-body analogue of unstable classical periodic orbits of a single particle in a chaotic stadium billiard. Scarred many-body eigenstates have been shown to exhibit a range of unusual properties which violate the Eigenstate Thermalisation Hypothesis, such as equidistant energy separation, anomalous expectation values of local observables and subthermal entanglement entropy. I will demonstrate that these properties can be understood using a tractable model based on a single particle hopping on the Hilbert space graph, which formally captures the idea that scarred eigenstates form a representation of a large $\operatorname{SU}(2)$ spin that is embedded in a thermalising many-body system. I will show that this picture allows to construct a more general family of scarred models where the fundamental degree of freedom is a quantum clock [3]. These results suggest that scarred many-body bands give rise to a new universality class of constrained quantum dynamics, which opens up opportunities for creating and manipulating novel states with long-lived coherence in systems that are now amenable to experimental study.

  1. H. Bernien et al., Nature 551, 579 (2017).
  2. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, Z. Papic, Nat. Phys. 14, 745 (2018).
  3. Kieran Bull, Ivar Martin, and Z. Papic, Phys. Rev. Lett. 123, 030601 (2019).

See also

Papic_slides.pdf

Johanna Erdmenger 03/06/2020, 11:00 — 12:00 — Online
, University of Würzburg

Turbulent hydrodynamics in strongly correlated Kagome metals

A current challenge in condensed matter physics is the realization of strongly correlated, viscous electron fluids. These fluids are not amenable to the perturbative methods of Fermi liquid theory, but can be described by holography, that is, by mapping them onto a weakly curved gravitational theory via gauge/gravity duality. The canonical system considered for realizations has been graphene, which possesses Dirac dispersions at low energies as well as significant Coulomb interactions between the electrons. In this work, we show that Kagome systems with electron fillings adjusted to the Dirac nodes of their band structure provide a much more compelling platform for realizations of viscous electron fluids, including non-linear effects such as turbulence. In particular, we find that in stoichiometric Scandium (Sc) Herbertsmithite, the fine-structure constant, which measures the effective Coulomb interaction and hence reflects the strength of the correlations, is enhanced by a factor of about 3.2 as compared to graphene, due to orbital hybridization. We employ holography to estimate the ratio of the shear viscosity over the entropy density in Sc-Herbertsmithite, and find it about three times smaller than in graphene. These findings put, for the first time, the turbulent flow regime described by holography within the reach of experiments.

See also

Reference
Slides of the talk

Achilleas Lazarides 27/05/2020, 11:00 — 12:00 — Online
, Loughborough University

Quantum order at infinite temperature, time crystals, and dissipation

Discrete time crystals is the name given to many-body systems displaying long-time dynamics that is sub-harmonic with respect to a driving frequency. While these were first discussed in closed quantum systems a few years ago, recent work (partly motivated by experiments) has focussed on including non-unitary effects such as due to an external environment ("dissipation").

In this talk I will begin by discussing general features of periodically-driven many-body systems, then concentrate on one of the unitary models for discrete time crystals. Time permitting, I will finally discuss a general framework for subharmonic oscillations stabilised by dissipative dynamics.

See also

Lazarides_slides.pdf

Joe Huxford 20/05/2020, 11:00 — 12:00 — Room P1, Mathematics Building Online
, Oxford University

Topological phases in $3+1D$: the Higher Lattice Gauge Theory Model and its excitations

Topological phases in $3+1D$ are less well understood than their lower dimensional counterparts. A useful approach to the study of such phases is to look at toy models that we can solve exactly. In this talk I will present new results for an existing model for certain topological phases in $3+1D$ (the model was first presented in [1]). This model is based on a generalisation of lattice gauge theory known as higher lattice gauge theory, which treats parallel transport of lines as well as points. I will first provide a brief introduction to higher lattice gauge theory and the Hamiltonian model constructed from it. Then we will look at the simple excitations (both point-like and loop-like) that are present in this model and how these excitations can be constructed explicitly using so-called ribbon and membrane operators. Some of the quasi-particles are confined and we discuss how this arises from a condensation-confinement transition. We will then look at the (loop-)braiding relations of the excitations and finish by examining the conserved topological charges realised by the Higher Lattice Gauge Theory Model.

[1] A Bullivant, M. Calcada et al., Topological phases from higher gauge symmetry in 3+1D, Phys. Rev. B 95, 155118 (2017).

See also

Slides of the talk

Bruno Amorim 13/05/2020, 11:00 — 12:00 — Room P1, Mathematics Building Online
, Universidade do Minho

Strain in two-dimensional materials

Graphene is the prototypical two-dimensional material. One of main features of two-dimensional materials is the ease with which their properties can be externally modified. Application of strain is one possible way. In this seminar we will review the geometrical description of strains in crystalline materials, with a focus on graphene. Using this method, we will study the form of the electron-lattice interaction. We will compare this model with the description of electrons in strained graphene in terms of a Dirac equation in curved space. An overview of anharmonic lattice effects in two-dimensional materials will also be made.

See also

Physics Reports 617, 1 - 54 (2016)
Slides of the talk

Andrea de Luca 06/05/2020, 11:00 — 12:00 — Online
, University of Cergy-Pontoise, CNRS

Generalized hydrodynamics with dephasing noise

I review recent advances in the development of generalized hydrodynamics, a flexible approach to the out-of-equilibrium dynamics of integrable quantum systems. I explain how this methodology has allowed exact calculations of transport in $1D$ system. Then, I consider the out-of-equilibrium dynamics of an interacting integrable system in the presence of an external dephasing noise. In the limit of large spatial correlation of the noise, we developed an exact description of the dynamics of the system based on a hydrodynamic formulation. This results in an additional term to the standard generalized hydrodynamics theory describing diffusive dynamics in the momentum space of the quasiparticles of the system, with a time- and momentum-dependent diffusion constant. Our analytical predictions are then benchmarked in the classical limit by comparison with a microscopic simulation of the non-linear Schrodinger equation, showing perfect agreement. In the quantum case, our predictions agree with state-of-the-art numerical simulations of the anisotropic Heisenberg spin in the accessible regime of times and with bosonization predictions in the limit of small dephasing times and temperatures.

See also

Reference

Semyon Klevtsov 29/04/2020, 11:00 — 12:00 — Online
, IRMA, Université de Strasbourg

Laughlin states on Riemann surfaces

Laughlin state is an $N$-particle wave function, describing the fractional quantum Hall effect (FQHE). We define and construct Laughlin states on genus-$g$ Riemann surface, prove topological degeneracy and discuss adiabatic transport on the corresponding moduli spaces. Mathematically, the problems around Laughlin states involve subjects as asymptotics of Bergman kernels for higher powers of line bundle on a surface, large-$N$ asymptotics of Coulomb gas-type integrals, vector bundles on moduli spaces.

See also

Slides of the talk

Tomaž Prosen 22/04/2020, 11:00 — 12:00 — Online
, University of Ljubljana

The Rule 54: Completely Solvable Statistical Mechanics Model of Deterministic Interacting Dynamics

Derivation of macroscopic statistical laws, such as Fourier's, Ohm's or Fick's laws, from reversible microscopic equations of motion is one of the central fundamental problems of statistical physics. In recent years we have witnessed a remarkable progress in understanding the dynamics and nonequilibrium statistical physics of integrable systems. This encourages us to attempt to understand the aforementioned connection at least in specific classes of nontrivial integrable systems with strong interactions. In my talk I will introduce a family of reversible cellular automata, which model systems of interacting particles, and for which we can prove the existence of diffusion and exactly solve several interesting paradigms of statistical physics, e.g.: nonequilibrium steady states of the system between two stochastic reservoirs, the problem of relaxation to the nonequilibrium steady state, or even the problem of explicit time evolution of macroscopic states, for instance, the solution of inhomogeneous quench problems and the calculation of dynamical structure factor in highly entropic equilibrium states.

See also

Slides of the talk

Eva-Maria Graefe 15/04/2020, 11:00 — 12:00 — Online
, Imperial College London

Evolution of Gaussian wave packets generated by a non-Hermitian Hamiltonian in the semiclassical limit

In recent years there has been growing interest in open quantum systems described by non-Hermitian Hamiltonians in various fields. In this talk I present results on the quantum evolution of Gaussian wave packets generated by a non-Hermitian Hamiltonian in the semiclassical limit of small $\hbar$. This yields a generalisation of the Ehrenfest theorem for the dynamics of observable expectation values. The resulting equations of motion for dynamical variables are coupled to an equation of motion for the phase-space metric — a phenomenon having no analogue in Hermitian theories. The insight that can be gained by this classical description will be demonstrated for a number of example systems.

See also

Slides

Bruno Mera 08/04/2020, 11:00 — 12:00 — Online
, Security and Quantum Information Group of Instituto de Telecomunicações

Localization anisotropy and complex geometry in two-dimensional insulators

The localization tensor is a measure of distinguishability between insulators and metals. This tensor is related to the quantum metric tensor associated with the occupied bands in momentum space. In two dimensions and in the thermodynamic limit, it defines a flat Riemannian metric over the twist-angle space, topologically a torus, which endows this space with a complex structure, described by a complex parameter τ . It is shown that the latter is a physical observable related to the anisotropy of the system. The quantity τ and the Riemannian volume of the twist-angle space provide an invariant way to parametrize the flat quantum metric obtained in the thermodynamic limit. Moreover, if by changing the couplings of the theory, the system undergoes quantum phase transitions in which the gap closes, the complex structure τ is still well defined, although the metric diverges (metallic state), and it is fixed by the form of the Hamiltonian near the gap closing points. The Riemannian volume is responsible for the divergence of the metric at the phase transition.

[1] Bruno Mera. Localization anisotropy and complex geometry in two-dimensional insulators. Phys. Rev. B, 101:115128, Mar 2020.

See also

Slides of the talk

Paul A. McClarty 01/04/2020, 11:00 — 12:00 — Online
Paul A. McClarty, Max Planck Institute for the Physics of Complex Systems

Topological Magnons

I give an overview of the insights we and other people have had into the band structure of magnons and discuss in some detail three main topics from our work: (i) the robustness of topological edge states in the presence of magnon interactions (ii) visualization of spin-momentum locking in magnon systems (iii) the non-Hermitian topology of spontaneous magnon decay.

Alex Bullivant 29/11/2019, 11:30 — 12:30 — Room P3.10, Mathematics Building
, University of Leeds

Topological Quantum Computing with loops

Non-abelian statistics of anyons in two dimensions have attracted considerable interest in the past 20 years, in part due to the potential for realising fault-tolerant quantum computation. In comparison, in three dimensions there exists a no-go theorem for point particles realising non-abelian statistics. However, three-dimensional condensed matter systems naturally support spatially extended excitations, such as loops, which can admit non-abelian statistics.

In this talk I will give a brief overview of topological quantum computing with anyons, utilising the connections between the mathematics of topological quantum field theories and Hamiltonian models of topological phases of matter. Building on this connection I will then discuss the ongoing work of describing non-abelian exchange statistics of loop excitations in three dimensions and the potential applications to quantum computation.

Angelo Carollo 15/11/2019, 11:30 — 12:30 — Room P3.10, Mathematics Building
, University of Palermo

On quantumness in multi-parameter quantum critical metrology

I will introduce a measure of quantum-ness in quantum multi-parameter estimation problems. One can show that the ratio between the mean Uhlmann Curvature and the Fisher Information provides a figure of merit which estimates the amount of incompatibility arising from the quantum nature of the underlying physical system. This ratio accounts for the discrepancy between the attainable precision in the simultaneous estimation of multiple parameters and the precision predicted by the Cramér-Rao bound. We apply this measure to quantitatively assess the quantum character of phase transition phenomena in peculiar quantum critical models. We consider a paradigmatic class of lattice fermion systems, which shows equilibrium quantum phase transition and dissipative non-equilibrium steady-state phase transitions.

See also

Carollo on quantumness...

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The goal of QM3 is to discuss recent topics in quantum matter from a mathematical perspective while building bridges between the physics and mathematics communities.


Organizers: Bruno Mera, João Pimentel Nunes, José Mourão, Pedro Ribeiro, Roger Picken, Vitor Rocha Vieira
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