The topological features of quantum many-body wave functions are known to have profound consequences for the physics of ground-states and their low-energy excitations. We describe how topology influences the dynamics of many-body systems when driven far from equilibrium. Our results are succinctly captured by a nonequilibrium topological classification that can be used to predict universal aspects of generic isolated quantum systems as they evolve unitarily in time. By analogy to the classifications used to describe systems in equilibrium, we consider two short-ranged entangled wave functions to be topologically equivalent if they can be interconverted via finite-time unitary evolution governed by a symmetry-respecting Hamiltonian. We demonstrate that this definition captures the salient features of these systems in a broad range of nonequilibrium scenarios. As well as providing conceptual insights into the constraints imposed by topology on many-body dynamics, we discuss the practical implications of our findings. In particular, we show that the characteristic zero-frequency spectroscopic peaks associated with topologically protected edge modes will be broadened by external noise only when the system is trivial in the nonequilibrium classification.

Topological properties of physical systems have attracted tremendous interest in condensed matter and atomic physics in the last decade. Recently, magnetic solid state compounds with and without magnetic order have become a focus. We provide evidence that ${\mathrm{BiCu}}_2{\mathrm{PO}}_6$ is a gapful, disordered quantum antiferromagnet with a non-trivial finite Zak phase, which characterizes one-dimensional systems. Our calculations show that in spite of the non-trivial topology no localized edge modes occur. This unexpected behavior is explained by a generic feature, namely the significance of direct and indirect gaps in topological systems.

Simulations of quantum transport in coherent conductors have evolved into mature techniques that are used in fields of physics ranging from electrical engineering to quantum nanoelectronics and material science. The most efficient general-purpose algorithms have a computational cost that scales as $L^{6,\cdots ,7}$ in three dimensions ($L$: length of the device), which on the one hand is a substantial improvement over older algorithms, but on the other hand still severely restricts the size of the simulation domain, limiting the usefulness of simulations through strong finite-size effects. Here we present a class of algorithms that, for certain systems, allows us to directly access the thermodynamic limit. Our approach, based on the Green's function formalism for discrete models, targets systems that are mostly invariant by translation, i.e., invariant by translation up to a finite number of orbitals and/or quasi-one-dimensional electrodes and/or the presence of edges or surfaces. Our approach is based on an automatic calculation of the poles and residues of series expansions of the Green's function in momentum space. This expansion allows us to integrate analytically in one momentum variable. We illustrate our algorithms with several applications: devices with graphene electrodes that consist of half an infinite sheet, Friedel oscillation calculations of infinite two-dimensional systems in the presence of an impurity, quantum spin Hall physics in the presence of an edge, and the surface of a Weyl semimetal in the presence of impurities and electrodes connected to the surface. In this last example, we study the conduction through the Fermi arcs of the topological material and its resilience to the presence of disorder. Our approach provides a practical route for simulating three-dimensional bulk systems or surfaces as well as other setups that have so far remained elusive.

We study the effects of temperature on the topological nature of ${\mathrm{ZrTe}}_5$, which sits near the phase boundary between strong and weak topological insulating orders. Using first-principles calculations, we show that the band gap and the topological indices of ${\mathrm{ZrTe}}_5$ are extremely sensitive to thermal expansion and that the electron-phonon interaction accounts for more than a third of the total temperature-dependent effects in ${\mathrm{ZrTe}}_5$. We find that the temperature dependence of the band gap has an opposite sign in the strong and weak topological insulator phases. Based on this insight, we propose a robust and unambiguous method to determine the topological nature of any given sample of ${\mathrm{ZrTe}}_5$: If the band gap decreases with temperature it is a strong topological insulator, and if it increases with temperature it is a weak topological insulator. An analogous strategy is expected to be generally applicable to other materials and to become particularly important in the vicinity of topological phase boundaries where other methods provide ambiguous results.

The ground state of the spin-1/2 kagome antiferromagnet remains uncertain despite decades of active research. Here we step aside from this debated question to address the ground-state nature of a related, and potentially just as rich, system made of corner-sharing triangles: the square kagome lattice (SKL). Our work is motivated by the recent synthesis of a distorted SKL compound mentioned by Morita and Tohyama [J. Phys. Soc. Jpn. 87, 043704 (2018)]. We have studied its spin-1/2 $J_1-J_2$ phase diagram with an unrestricted Schwinger boson mean-field theory (SBMFT). Our results agree with previous observations of a plaquette phase ($J_2\ll J_1$) and a ferrimagnet ($J_1\ll J_2$). In addition, three original phases appear: two incommensurate orders and a topological quantum spin liquid with weak nematicity. The topological order is characterized by fluxes on specific gauge-invariant quantities and the phase is stable under anisotropic perturbations relevant for experiments. Finally, we provide dynamical structure factors of the reported phases that could be observed in inelastic neutron scattering.

We propose an exact construction for atypical excited states of a class of nonintegrable quantum many-body Hamiltonians in one, two, and three dimensions that display area-law entanglement entropy. These examples of many-body “scar” states have, by design, other properties, such as topological degeneracies, usually associated with the gapped ground states of symmetry-protected topological phases or topologically ordered phases of matter.

There is a counterintuitive expectation proposed by Huse et al. [Phys. Rev. B 88, 014206 (2013)] and Chandran et al. [Phys. Rev. B 89, 144201 (2014)]: Localization protects the quantum order of the ground state, even in highly excited eigenstates. In this Rapid Communication, we numerically investigate the localization protection for symmetry-protected topological (SPT) order by considering a modified XXZ spin model, related to an interacting Su-Schrieffer-Heeger (SSH) model. We systematically study how many-body localization (MBL) protects SPT for different disorder types and different forms of interactions. A certain disorder leads to the clear degeneracy of the low-lying entanglement spectrum of each excited many-body eigenstate. This fact indicates that the existence of edge modes, which is a hallmark of SPT order, is protected by the MBL in excited many-body eigenstates. In addition, we also report how the MBL-protected edge modes in excited many-body eigenstates fade away for a diagonal-type disorder.

We analyze the localization properties of two-dimensional systems based on partite lattices with a basis. Contrary to standard results, we find that a band of critical states emerges for systems in the unitary class A preserving spin symmetry when disorder is unevenly distributed over the basis atoms. The critical metal arises when the less disordered sublattice is connected and has time-reversal symmetry broken. The unexpected robustness to disorder presented here is an appealing result which can be measured in optical lattices.

The spatial localization of quantum states plays a central role in condensed-matter phenomena, ranging from many-body localization to topological matter. Building on the dissipation-fluctuation theorem, we propose that the localization properties of a quantum-engineered system can be probed by spectroscopy, namely, by measuring its excitation rate upon a periodic drive. We apply this method to various examples that are of direct experimental relevance in ultracold atomic gases, including Anderson localization, topological edge modes, and interacting particles in a harmonic trap. Moreover, inspired by a relation between quantum fluctuations and the quantum metric, we describe how our scheme can be generalized in view of extracting the full quantum-geometric tensor of many-body systems. Our approach opens an avenue for probing localization, as well as quantum fluctuations, geometry and entanglement, in synthetic quantum matter.

We consider a system of weakly coupled Rashba nanowires in the strong spin-orbit interaction (SOI) regime. The nanowires are arranged into two tunnel-coupled layers proximitized by a top and bottom superconductor such that the superconducting phase difference between them is $\pi$. We show that in such a system strong electron-electron interactions can stabilize a helical topological superconducting phase hosting Kramers partners of ${\mathbb{Z}}_{2m}$ parafermion edge modes, where $m$ is an odd integer determined by the position of the chemical potential. Furthermore, upon turning on a weak in-plane magnetic field, the system is driven into a second-order topological superconducting phase hosting zero-energy ${\mathbb{Z}}_{2m}$ parafermion bound states localized at two opposite corners of a rectangular sample. As a special case, zero-energy Majorana corner states emerge in the noninteracting limit $m=1$, where the chemical potential is tuned to the SOI energy of the single nanowires.

We study the electron and spin transport in a van der Waals system formed by one layer with strong spin-orbit coupling and a second layer without spin-orbit coupling, in the regime when the interlayer tunneling is random. We find that in the layer without intrinsic spin-orbit coupling, spin-charge coupled transport can be induced by two distinct mechanisms. First, the gapless diffusion modes of the two isolated layers hybridize in the presence of tunneling, which constitutes a source of spin-charge coupled transport in the second layer. Second, the random tunneling introduces spin-orbit coupling in the effective disorder-averaged single-particle Hamiltonian of the second layer. This results in nontrivial spin transport and, for sufficiently strong tunneling, in spin-charge coupling. As an example, we consider a van der Waals system formed by a two-dimensional electron gas (2DEG)—such as graphene—and the surface of a topological insulator (TI) and show that the proximity of the TI induces a coupling of the spin and charge transport in the 2DEG. In addition, we show that such coupling can be tuned by varying the doping of the TI's surface. We then obtain, for a simple geometry, the current-induced nonequilibrium spin accumulation (Edelstein effect) caused in the 2DEG by the coupling of charge and spin transport.

We study two-dimensional spinful insulating phases of matter that are protected by time-reversal and crystalline symmetries. To characterize these phases, we employ the concept of corner charge fractionalization: corners can carry charges that are fractions of even multiples of the electric charge. The charges are quantized and topologically stable as long as all symmetries are preserved. We classify the different corner charge configurations for all point groups, and match them with the corresponding bulk topology. For this we employ symmetry indicators and (nested) Wilson loop invariants. We provide formulas that allow for a convenient calculation of the corner charge from Bloch wave functions and illustrate our results using the example of arsenic and antimony monolayers. Depending on the degree of structural buckling, these materials can exhibit two distinct obstructed atomic limits. We present density functional theory calculations for open flakes to support our findings.

We study the single-particle spectrum of three-dimensional Weyl semimetals taking into account electron-phonon interactions that are the result of straining the material. We find that a well-defined fermionic excitation appears in addition to the standard peak corresponding to quasiparticle states as suggested by Landau-Fermi liquid theory. Contrary to the case of Dirac systems interacting via the Coulomb interaction, these satellite peaks appear even at lowest order in perturbation theory. The new excitations are anisotropic, as opposed to the single-particle spectrum, and their behavior is dictated by the Debye frequency, which naturally regulates the electron-phonon coupling.

It is well known that an interface created by two topologically distinct structures could host nontrivial edge states that are immune to defects. In this paper, we introduce a one-dimensional space-time phononic crystal and study the associated anomalous topological edge states when the phononic crystal's density is modulated both in space and time. While preserving the key topological feature of the system, the time modulation also duplicates the edge state mode across the spectrum, both inside and outside the band gap. It is shown that, in contrast to conventional topological edge states which are excited by frequencies in the Bragg regime, the time-modulation-induced frequency conversion can be leveraged to access topological edge states at a deep subwavelength scale where the entire phononic crystal size can be as small as 1/5.1 of the wavelength. This feature could open another route for designing miniature devices that are based on topological physics.

We consider a Floquet triple-layer setup composed of a two-dimensional electron gas with spin-orbit interactions, proximity coupled to an $s$-wave superconductor and to a ferromagnet driven at resonance. The ferromagnetic layer generates a time-oscillating Zeeman field which competes with the induced superconducting gap and leads to a topological phase transition. The resulting Floquet states support a second-order topological superconducting phase with a pair of localized zero-energy Floquet Majorana corner states. Moreover, the phase diagram comprises a Floquet helical topological superconductor, hosting a Kramers pair of Majorana edge modes protected by an effective time-reversal symmetry, as well as a gapless Floquet Weyl phase. The topological phases are stable against disorder as well as against parameter variations and are within experimental reach.

We construct an exactly solvable lattice model of a fractional Weyl semimetal. The low-energy theory of this strongly interacting state is that of a Weyl semimetal built out of fractionally charged fermions. We show the existence of a universally quantized and fractional circular photogalvanic effect and a violation of the Wiedemann-Franz law in the system. Together with a spectral gap in the single-particle electronic Green's function, they provide strong experimental signatures for this exotic gapless state of matter.

We show how the universal low-energy properties of Weyl semimetals with spatially varying time-reversal (TR) or inversion (I) symmetry breaking are described in terms of chiral fermions experiencing curved-spacetime geometry and synthetic gauge fields. By employing Clifford representations and Schrieffer-Wolff transformations, we present a systematic derivation of an effective curved-space Weyl theory with rich geometric and gauge structure. To illustrate the utility of the formalism, we give a concrete prescription of how to fabricate nontrivial curved spacetimes and event horizons in topological insulators with magnetic textures. Our theory can also account for strain-induced effects, providing a powerful unified framework for studying and designing inhomogeneous Weyl materials.

In recent years, there have been rapid advances in the parallel fields of electronic and photonic topological crystals. Topological photonic crystals in particular show promise for coherent transport of light and quantum information at macroscopic scales. In this work, we apply for the first time the recently developed theory of “topological quantum chemistry” to the study of band structures in photonic crystals. This method allows us to design and diagnose topological photonic band structures using only group theory and linear algebra. As an example, we focus on a family of crystals formed by elliptical rods in a triangular lattice. We show that the symmetry of Bloch states in the Brillouin zone can determine the position of the localized photonic wave packets describing groups of bands. By modifying the crystal structure and inverting bands, we show how the centers of these wave packets can be moved between different positions in the unit cell. Finally, we show that for shapes of dielectric rods, there exist isolated topological bands which do not admit a well-localized description, representing the first physical instance of “fragile” topology in a truly noninteracting system. Our work demonstrates how photonic crystals are the natural platform for the future experimental investigation of fragile topological bands.

The electric field inside typical conductors drops down exponentially with the screening length determined by an intrinsic length scale of the system such as the density of mobile carriers. We show that in a classically conformal system with boundaries, where the intrinsic length scale is absent, the screening of an external electric field is governed by the quantum conformal anomaly associated with the renormalization of the electric charge. The electric field decays algebraically with a fractional power determined by the beta function of the system. We argue that this “anomalous conformal screening effect” is an indirect manifestation of the Schwinger pair production in relativistic field theory. We discuss the experimental feasibility of the proposed phenomenon in Dirac semimetals that would allow a direct experimental access to the beta function.

Many-body spin systems represent a paradigmatic platform for the realization of emergent states of matter in a strongly interacting regime. Spin models are commonly studied in one-dimensional periodic chains, whose lattice constant is on the order of the interatomic distance. However, in cold atomic setups or functionalized twisted van der Waals heterostructures, long-range modulations of the spin physics can be engineered. Here we show that such superlattice modulations in a many-body spin Hamiltonian can give rise to observable topological boundary modes in the excitation spectrum of the spin chain. In the case of an XY spin-$1/2$ chain, these boundary modes stem from a mathematical correspondence with the chiral edge modes of a two-dimensional quantum Hall state. Our results show that the addition of many-body interactions does not close some of the topological gaps in the excitation spectrum, and the topological boundary modes visibly persist in the isotropic Heisenberg limit. These observations carry through when the spin moment is increased and a large-spin limit of the phenomenon is established. Our results show that such spin superlattices provide a promising route to observe many-body topological boundary effects in cold atomic setups and functionalized twisted van der Waals materials.