Search

We propose a universal framework for classifying quantum states based on their scale-resolved correlation structure. Using the recently introduced information lattice, which provides an operational definition of the total amount of correlations at each scale, we define intrinsic characteristic length scales of quantum states. We analyze ground and midspectrum eigenstates of the disordered interacting Kitaev chain, showing that our framework provides a novel unbiased approach to quantum matter., Comment: 7 pages, 3 figures

The surface of a three dimensional topological insulator (TI) hosts surface states whose properties are determined by a Dirac-like equation. The electronic system on the surface of TI nanowires with polygonal cross-sectional shape adopts the corresponding polygonal shape. In a constant transverse magnetic field, such an electronic system exhibits rich properties as different facets of the polygon experience different values of the magnetic field due to the changing magnetic field projection between facets. We investigate the energy spectrum and transport properties of nanowires where we consider three different polygonal shapes, all showing distinct properties visible in the energy spectrum and transport properties. Here we propose that the wire conductance can be used to differentiate between cross-sectional shapes of the nanowire by rotating the magnetic field around the wire. Distinguishing between the different shapes also works in the presence of impurities as long as conductance steps are discernible, thus revealing the sub-band structure., Comment: 9 pages with 7 figures

We demonstrate that slow growth of the number entropy following a quench from a local product state is consistent with many-body localization. To do this we construct a random circuit l-bit model with exponentially localized l-bits and exponentially decaying interactions between them. We observe an ultraslow growth of the number entropy starting from a N\'eel state, saturating at a value that grows with system size. This suggests that the observation of such growth in microscopic models is not sufficient to rule out many-body localization., Comment: Main: 7 pages, 2 figures. Supplemental material: 7 pages, 9 figures, 1 table, 1 video

During time evolution of many-body systems entanglement grows rapidly, limiting exact simulations to small-scale systems or small timescales. Quantum information tends however to flow towards larger scales without returning to local scales, such that its detailed large-scale structure does not directly affect local observables. This allows for the removal of large-scale quantum information in a way that preserves all local observables and gives access to large-scale and large-time quantum dynamics. To this end, we use the recently introduced information lattice to organize quantum information into different scales, allowing us to define local information and information currents which we employ to systematically discard long-range quantum correlations in a controlled way. Our approach relies on decomposing the system into subsystems up to a maximum scale and time evolving the subsystem density matrices by solving the subsystem von Neumann equations in parallel. Importantly, the information flow needs to be preserved during the discarding of large-scale information. To achieve this without the need to make assumptions about the microscopic details of the information current, we introduce a second scale at which information is discarded while using the state at the maximum scale to accurately obtain the information flow. The resulting algorithm, which we call local information time evolution (LITE), is highly versatile and suitable for investigating many-body quantum dynamics in both closed and open quantum systems with diverse hydrodynamic behaviors. We present results for energy transport in the mixed-field Ising model and magnetization transport in an open XX spin chain where we accurately determine the diffusion coefficients. The information lattice framework employed here promises to offer insightful results about the spatial and temporal behavior of entanglement in many-body systems., Comment: 29 pages, 17 figures

The eigenstate thermalization hypothesis describes how most isolated many-body quantum systems reach thermal equilibrium. However, the hypothesis is violated by phenomena such as many-body localization and quantum many-body scars. In this work, we study a finite, two-dimensional, disordered model hosting a tower of scar states. This construction is a particular instance of a general framework and we demonstrate its generality by constructing two disordered models hosting a different tower of scar states. At weak disorder, we find numerically that the spectra are nonthermal, and the scar states appear as exact eigenstates with high entropy for certain bipartitions. At strong disorder, the spectra localize and the scar states are identified as inverted scars since the scar states are embedded in a localized background as opposed to a thermal background. We argue that, for the considered type of models, the localization is stronger than what would be naively expected, and we show this explicitly for one of the models. The argument also provides guidelines for obtaining similarly strong localization in other scarred models. We study the transition from the thermal phase to localization by observing the adjacent gap ratio shifting from the Wigner surmise to the Poisson distribution with increasing disorder strength. Moreover, the entanglement entropy transitions from volume-law scaling with system size at weak disorder to area-law scaling at strong disorder. Finally, we demonstrate that localization protects scar revivals for initial states with partial support in the scar subspace., Comment: 18 pages, 11 figures, accepted version

While topology is a property of a quantum state itself, most existing methods for characterizing the topology of interacting phases of matter require direct knowledge of the underlying Hamiltonian. We offer an alternative by utilizing the one-particle density matrix formalism to extend the concept of the Chern, chiral, and Chern-Simons markers to include interactions. The one-particle density matrix of a free-fermion state is a projector onto the occupied bands, defining a Brillouin zone bundle of the given topological class. This is no longer the case in the interacting limit, but as long as the one-particle density matrix is gapped, its spectrum can be adiabatically flattened, connecting it to a topologically equivalent projector. The corresponding topological markers thus characterize the topology of the interacting phase. Importantly, the one-particle density matrix is defined in terms of a given state alone, making the local markers numerically favorable, and providing a valuable tool for characterizing topology of interacting systems when only the state itself is available. To demonstrate the practical use of the markers we use the chiral marker to identify the topology of midspectrum eigenstates of the Ising-Majorana chain across the transition between the ergodic and many-body localized phases. We also apply the chiral marker to random states with a known topology, and compare it with the entanglement spectrum degeneracy., Comment: 7 pages, 3 figures + Appendix 2 pages, 1 figure

We investigate the gauging of higher-form finite Abelian symmetries and their sub-groups in quantum spin models in spatial dimensions $d=2$ and 3. Doing so, we naturally uncover gauged models with dual higher-group symmetries and potential mixed 't Hooft anomalies. We demonstrate that the mixed anomalies manifest as the symmetry fractionalization of higher-form symmetries participating in the mixed anomaly. Gauging is realized as an isomorphism or duality between the bond algebras that generate the space of quantum spin models with the dual generalized symmetry structures. We explore the mapping of gapped phases under such gauging related dualities for 0-form and 1-form symmetries in spatial dimension $d=2$ and 3. In $d=2$, these include several non-trivial dualities between short-range entangled gapped phases with 0-form symmetries and 0-form symmetry enriched Higgs and (twisted) deconfined phases of the gauged theory with possible symmetry fractionalizations. Such dualities also imply strong constraints on several unconventional, i.e., deconfined or topological transitions. In $d=3$, among others, we find, dualities between topological orders via gauging of 1-form symmetries. Hamiltonians self-dual under gauging of 1-form symmetries host emergent non-invertible symmetries, realizing higher-categorical generalizations of the Tambara-Yamagami fusion category., Comment: 90 pages, 19 figures

Local topological markers, topological invariants evaluated by local expectation values, are valuable for characterizing topological phases in materials lacking translation invariance. The Chern marker -- the Chern number expressed in terms of the Fourier transformed Chern character -- is an easily applicable local marker in even dimensions, but there are no analogous expressions for odd dimensions. We provide general analytic expressions for local markers for free-fermion topological states in odd dimensions protected by local symmetries: a Chiral marker, a local $\mathbb Z$ marker which in case of translation invariance is equivalent to the chiral winding number, and a Chern-Simons marker, a local $\mathbb Z_2$ marker characterizing all nonchiral phases in odd dimensions. We achieve this by introducing a one-parameter family $P_{\vartheta}$ of single-particle density matrices interpolating between a trivial state and the state of interest. By interpreting the parameter $\vartheta$ as an additional dimension, we calculate the Chern marker for the family $P_{\vartheta}$. We demonstrate the practical use of these markers by characterizing the topological phases of two amorphous Hamiltonians in three dimensions: a topological superconductor ($\mathbb Z$ classification) and a topological insulator ($\mathbb Z_2$ classification)., Comment: 13 pages, 3 figures; V2: Added supplemental material

The 1D Kitaev model in the topological phase, with open boundary conditions, hosts strong Majorana zero modes. These are fermion parity-odd operators that almost commute with the Hamiltonian and manifest in long coherence times for edge degrees of freedom. We obtain higher-dimensional counterparts of such Majorana operators by explicitly computing their closed form expressions in models describing 2D and 3D higher-order superconductors. Due to the existence of such strong Majorana zero modes, the coherence time of the infinite temperature autocorrelation function of the corner Majorana operators in these models diverges with the linear system size. In the presence of a certain class of orbital-selective dissipative dynamics, the coherence times of half of the corner Majorana operators is enhanced, while the time correlations corresponding to the remaining corner Majoranas decay much faster as compared with the unitary case. We numerically demonstrate robustness of the coherence times to the presence of disorder.

We demonstrate that genuinely non-Hermitian topological phases and corresponding topological phase transitions can be naturally realized in monitored quantum circuits, exemplified by the paradigmatic non-Hermitian Su-Schrieffer-Heeger model. We emulate this model by a 1D chain of spinless electrons evolving under unitary dynamics and subject to periodic measurements that are stochastically invoked. The non-Hermitian topology is visible in topological invariants adapted to the context of monitored circuits. For instance, the topological phase diagram of the monitored realization of the non-Hermitian Su-Schrieffer-Heeger model is obtained from the biorthogonal polarization computed from an effective Hamiltonian of the monitored system. Importantly, our monitored circuit realization allows direct access to steady state biorthogonal expectation values of generic observables, and hence, to measure physical properties of a genuine non-Hermitian model. We expect our results to be applicable more generally to a wide range of models that host non-Hermitian topological phases., Comment: 4.5 pages, 3 figures

Quantum many-body dynamics generically results in increasing entanglement that eventually leads to thermalization of local observables. This makes the exact description of the dynamics complex despite the apparent simplicity of (high-temperature) thermal states. For accurate but approximate simulations one needs a way to keep track of essential (quantum) information while discarding inessential one. To this end, we first introduce the concept of the information lattice, which supplements the physical spatial lattice with an additional dimension and where a local Hamiltonian gives rise to well defined locally conserved von Neumann information current. This provides a convenient and insightful way of capturing the flow, through time and space, of information during quantum time evolution, and gives a distinct signature of when local degrees of freedom decouple from long-range entanglement. As an example, we describe such decoupling of local degrees of freedom for the mixed field transverse Ising model. Building on this, we secondly construct algorithms to time-evolve sets of local density matrices without any reference to a global state. With the notion of information currents, we can motivate algorithms based on the intuition that information for statistical reasons flow from small to large scales. Using this guiding principle, we construct an algorithm that, at worst, shows two-digit convergence in time-evolutions up to very late times for diffusion process governed by the mixed field transverse Ising Hamiltonian. While we focus on dynamics in 1D with nearest-neighbor Hamiltonians, the algorithms do not essentially rely on these assumptions and can in principle be generalized to higher dimensions and more complicated Hamiltonians., Comment: 38 pages, 9 figures

We show how the axial (chiral) anomaly induces a spin torque on the magnetization in magnetic Weyl semimetals. The anomaly produces an imbalance in left- and right-handed chirality carriers when non-orthogonal electric and magnetic fields are applied. Such imbalance generates a spin density which exerts a torque on the magnetization, the strength of which can be controlled by the intensity of the applied electric field. We show how this results in an electric control of the chirality of domain walls, as well as in an improvement of the domain wall dynamics, by delaying the onset of the Walker breakdown. The measurement of the electric field mediated changes in the domain wall chirality would constitute a direct proof of the axial anomaly. Additionally, we show how quantum fluctuations of electronic Fermi arc states bound to the domain wall naturally induce an effective magnetic anisotropy, allowing for high domain wall velocities even if the intrinsic anisotropy of the magnetic Weyl semimetal is small., Comment: 12 pages, 4 figures

We study many-body localization (MBL) in a pair-hopping model exhibiting strong fragmentation of the Hilbert space. We show that several Krylov subspaces have both ergodic statistics in the thermodynamic limit and a dimension that scales much slower than the full Hilbert space, but still exponentially. Such a property allows us to study the MBL phase transition in systems including more than $50$ spins. The different Krylov spaces that we consider show clear signatures of a many-body localization transition, both in the Kullback-Leibler divergence of the distribution of their level spacing ratio and their entanglement properties. But they also present distinct scalings with system size. Depending on the subspace, the critical disorder strength can be nearly independent of the system size or conversely show an approximately linear increase with the number of spins., Comment: 14 + 6 pages, all comments are welcome

We provide an exact analytical technique to obtain within a lattice model the wave functions of the edge states in zigzag- and bearded-edge graphene, as well as of the Fermi-arc surface states in Weyl semimetals described by a minimal bulk model. We model the corresponding boundaries as an infinite scalar potential localized on a line, and respectively within a plane. We use the T-matrix formalism to obtain the dispersion and the spatial distribution of the corresponding boundary modes. Furthermore, to demonstrate the power of our approach, we write down the surface Green's function of the considered Weyl semimetal model, and we calculate the quasiparticle interference patterns originating from an impurity localized at the respective surface., Comment: 6.5 pages, 6 figures (+ appendixes)

The localization landscape gives direct access to the localization of bottom-of-band eigenstates in non-interacting disordered systems. We generalize this approach to eigenstates at arbitrary energies in systems with or without internal degrees of freedom by introducing a modified $\mathcal{L}^2$-landscape, and we demonstrate its accuracy in a variety of archetypal models of Anderson localization in one and two dimensions. This $\mathcal{L}^2$-landscape function can be efficiently computed using hierarchical methods that allow evaluating the diagonal of a well-chosen Green function. We compare our approach to other landscape methods, bringing new insights on their strengths and limitations. Our approach is general and can in principle be applied to both studies of topological Anderson transitions and many-body localization., Comment: 6.5 + 1 pages, all comments are welcome

A space-time dependent node separation in Weyl semimetals acts as an axial vector field. Coupled with domain wall motion in magnetic Weyl semimetals, this induces axial electric and magnetic fields localized at the domain wall. We show how these fields can activate the axial (chiral) anomaly and provide a direct experimental signature of it. Specifically, a domain wall provides a spatially dependent Weyl node separation and an axial magnetic field $\textbf{B}_5$, and domain wall movement, driven by an external magnetic field, gives the Weyl node separation a time dependence, inducing an axial electric field $\textbf{E}_5$. At magnetic fields beyond the Walker breakdown, $\textbf{E}_5\cdot\textbf{B}_5$ becomes nonzero and activates the axial anomaly that induces a finite axial charge density -- imbalance in the number of left- and right-handed fermions -- moving with the domain wall. This axial density, in turn, produces, via the chiral magnetic effect, an oscillating current flowing along the domain wall plane, resulting in a characteristic radiation of electromagnetic waves emanating from the domain wall. A detection of this radiation would constitute a direct measurement of the axial anomaly induced by axial electromagnetic fields., Comment: 10 pages, 4 figures; V2: Added calculations of radiation in near and far field

We study the properties of the entanglement spectrum in gapped non-interacting non-Hermitian systems, and its relation to the topological properties of the system Hamiltonian. Two different families of entanglement Hamiltonians can be defined in non-Hermitian systems, depending on whether we consider only right (or equivalently only left) eigenstates or a combination of both left and right eigenstates. We show that their entanglement spectra can still be computed efficiently, as in the Hermitian limit. We discuss how symmetries of the Hamiltonian map into symmetries of the entanglement spectrum depending on the choice of the many-body state. Through several examples in one and two dimensions, we show that the biorthogonal entanglement Hamiltonian directly inherits the topological properties of the Hamiltonian for line gapped phases, with characteristic singular and energy zero modes. The right (left) density matrix carries distinct information on the topological properties of the many-body right (left) eigenstates themselves. In purely point gapped phases, when the energy bands are not separable, the relation between the entanglement Hamiltonian and the system Hamiltonian breaks down., Comment: 20 pages incl App., 11 figures. To be submitted, all comments are welcome

Magnetotransport provides key experimental signatures in Weyl semimetals. The longitudinal magnetoresistance is linked to the chiral anomaly and the transversal magnetoresistance to the dominant charge relaxation mechanism. Axial magnetic fields that act with opposite sign on opposite chiralities facilitate new transport experiments that probe the low-energy Weyl nodes. As recently realized, these axial fields can be achieved by straining samples or adding inhomogeneities to them. Here, we identify a robust signature of axial magnetic fields: an anomalous scaling of the conductance in the diffusive ultraquantum regime. In particular, we demonstrate that the longitudinal conductivity in the ultraquantum regime of a disordered Weyl semimetal subjected to an axial magnetic field increases with both the field strength and sample width due to a spatial separation of charge carriers. We contrast axial magnetic with real magnetic fields to clearly distinguish the different behavior of the conductance. Our results rely on numerical tight-binding simulations and are supported by analytical arguments. We argue that the spatial separation of charge carriers can be used for directed currents in microstructured electronic devices., Comment: 11 pages, 9 figures

In this chapter we review our work on the theory of quantum transport in topological insulator nanowires. We discuss both normal state properties and superconducting proximity effects, including the effects of magnetic fields and disorder. Throughout we assume that the bulk is insulating and inert, and work with a surface-only theory. The essential transport properties are understood in terms of three special modes: in the normal state, half a flux quantum along the length of the wire induces a perfectly transmitted mode protected by an effective time reversal symmetry; a transverse magnetic field induces chiral modes at the sides of the wire, with different chiralities residing on different sides protecting them from backscattering; and, finally, Majorana zero modes are obtained at the ends of a wire in a proximity to a superconductor, when combined with a flux along the wire. Some parts of our discussion have a small overlap with the discussion in the review [Bardarson and Moore, Rep. Prog. Phys., 76, 056501, (2013)]. We do not aim to give a complete review of the published literature, instead the focus is mainly on our own and directly related work., Comment: 22 pages, 8 figures; Chapter in "Topological Matter. Springer Series in Solid-State Sciences, vol 190. Springer"

We study longitudinal transport in Weyl semimetal nanowires, both in the absence and in the presence of a magnetic flux threading the nanowires. We identify two qualitatively different regimes of transport with respect to the chemical potential in the nanowires. In the "surface regime", for low doping, most of the conductance occurs through the Fermi-arc surface states, and it rises in steps of one quantum of conductance as a function of the chemical potential; furthermore, with varying flux the conductance changes in steps of one quantum of conductance with characteristic Fabry-P\'erot interference oscillations. In the "bulk-surface regime", for highly-doped samples, the dominant contribution to the conductance is quadratic in the chemical potential, and mostly conditioned by the bulk states; the flux dependence shows clearly that both the surface and the bulk states contribute. The two aforementioned regimes prove that the contribution of the Fermi-arc surface states is salient and, therefore, crucial for understanding transport properties of finite-size Weyl semimetal systems. Last but not least, we demonstrate that both regimes are robust to disorder., Comment: 13 pages, 6 figures

We address the breakdown of the bulk-boundary correspondence observed in non-Hermitian systems, where open and periodic systems can have distinct phase diagrams. The correspondence can be completely restored by considering the Hamiltonian's singular value decomposition instead of its eigendecomposition. This leads to a natural topological description in terms of a flattened singular decomposition. This description is equivalent to the usual approach for Hermitian systems and coincides with a recent proposal for the classification of non-Hermitian systems. We generalize the notion of the entanglement spectrum to non-Hermitian systems, and show that the edge physics is indeed completely captured by the periodic bulk Hamiltonian. We exemplify our approach by considering the chiral non-Hermitian Su-Schrieffer-Heger and Chern insulator models. Our work advocates a different perspective on topological non-Hermitian Hamiltonians, paving the way to a better understanding of their entanglement structure., Comment: 6+5 pages, 8 figures

The Sachdev-Ye-Kitaev (SYK) model, in its simplest form, describes $k$ Majorana fermions with random all-to-all four-body interactions. We consider the SYK model in the framework of a many-body Altland-Zirnbauer classification that sees the system as belonging to one of eight (real) symmetry classes depending on the value of $k\mod 8$. We show that, depending on the symmetry class, the system may support exact many-body zero modes with the symmetries also dictating whether these may have a nonzero contribution to Majorana fermions, i.e., single-particle weight. These zero modes appear in all but two of the symmetry classes. When present, they leave clear signatures in physical observables that go beyond the threefold (Wigner-Dyson) possibilities for level spacing statistics studied earlier. Signatures we discover include a zero-energy peak or hole in the single-particle spectral function, depending on whether symmetries allow or forbid zero modes to have single-particle weight. The zero modes are also shown to influence the many-body dynamics, where signatures include a nonzero long-time limit for the out-of-time-order correlation function. Furthermore, we show that the extension of the four-body SYK model by quadratic terms can be interpreted as realizing the remaining two complex symmetry classes; we thus demonstrate how the entire tenfold Altland-Zirnbauer classification may emerge in the SYK model., Comment: 15 pages, 6 figures

We study numerically the formation of entanglement clusters across the many-body localization phase transition. We observe a crossover from strong many-body entanglement in the ergodic phase to weak local correlations in the localized phase, with contiguous clusters throughout the phase diagram. Critical states close to the transition have a structure compatible with fractal or multiscale-entangled states, characterized by entanglement at multiple levels: small strongly entangled clusters are weakly entangled together to form larger clusters. The critical point therefore features subthermal entanglement and a power-law distributed cluster size, while the localized phase presents an exponentially decreasing cluster distribution. These results are consistent with some of the recently proposed phenomenological renormalization-group schemes characterizing the many-body localized critical point, and may serve to constrain other such schemes., Comment: 8 pages, 7 figures + Appendices (3 pages, 6 figures)

Among the different platforms to engineer Majorana fermions in one-dimensional topological superconductors, topological insulator nanowires remain a promising option. Threading an odd number of flux quanta through these wires induces an odd number of surface channels, which can then be gapped with proximity induced pairing. Because of the flux and depending on energetics, the phase of this surface pairing may or may not wind around the wire in the form of a vortex. Here we show that for wires with discrete rotational symmetry, this vortex is necessary to produce a fully gapped topological superconductor with localized Majorana end states. Without a vortex the proximitized wire remains gapless, and it is only if the symmetry is broken by disorder that a gap develops, which is much smaller than the one obtained with a vortex. These results are explained with the help of a continuum model and validated numerically with a tight binding model, and highlight the benefit of a vortex for reliable use of Majorana fermions in this platform., Comment: 22 pages, 6 figures. Substantial rewriting in several sections after reviewers comments. Resubmission to SciPost

We numerically study the dynamics on the ergodic side of the many-body localization transition in a periodically driven Floquet model with no global conservation laws. We describe and employ a numerical technique based on the fast Walsh-Hadamard transform that allows us to perform an exact time evolution for large systems and long times. As in models with conserved quantities (e.g., energy and/or particle number) we observe a slowing down of the dynamics as the transition into the many-body localized phase is approached. More specifically, our data is consistent with a subballistic spread of entanglement and a stretched-exponential decay of an autocorrelation function, with their associated exponents reflecting slow dynamics near the transition for a fixed system size. However, with access to larger system sizes, we observe a clear flow of the exponents towards faster dynamics and can not rule out that the slow dynamics is a finite-size effect. Furthermore, we observe examples of non-monotonic dependence of the exponents with time, with dynamics initially slowing down but accelerating again at even larger times, consistent with the slow dynamics being a crossover phenomena with a localized critical point., Comment: 9 pages, 8 figures; added details on the level statistics and the energy absorption

We show that Weyl semimetals exhibit a mixed axial-torsional anomaly in the presence of axial torsion, a concept exclusive of these materials with no known natural fundamental interpretation in terms of the geometry of spacetime. This anomaly implies a nonconservation of the axial current---the difference in current of left- and right-handed chiral fermions---when the torsion of the spacetime in which the Weyl fermions move couples with opposite sign to different chiralities. The anomaly is activated by driving transverse sound waves through a Weyl semimetal with a spatially varying tilted dispersion, which can be engineered by applying strain. This leads to sizable alternating current in presence of a magnetic field that provides a clear-cut experimental signature of our predictions.

Condensed matter systems realizing Weyl fermions exhibit striking phenomenology derived from their topologically protected surface states as well as chiral anomalies induced by electromagnetic fields. More recently, inhomogeneous strain or magnetization were predicted to result in chiral electric $\mathbf{E}_5$ and magnetic $\mathbf{B}_5$ fields, which modify and enrich the chiral anomaly with additional terms. In this work, we develop a lattice-based approach to describe the chiral anomaly, which involves Landau and pseudo-Landau levels and treats all anomalous terms on equal footing, while naturally incorporating Fermi arcs. We exemplify its potential by physically interpreting the largely overlooked role of Fermi arcs in the covariant (Fermi level) contribution to the anomaly and revisiting the factor of $1/3$ difference between the covariant and consistent (complete band) contributions to the $\mathbf{E}_5\cdot\mathbf{B}_5$ term in the anomaly. Our framework provides a versatile tool for the analysis of anomalies in realistic lattice models as well as a source of simple physical intuition for understanding strained and magnetized inhomogeneous Weyl semimetals., Comment: 10 pages, 7 figures

Entanglement has developed into an essential concept for the characterization of phases and phase transitions in ground states of quantum many-body systems. In this work, we use the logarithmic negativity to study the spatial entanglement structure in the transverse-field Ising chain both in the ground state and at nonzero temperatures. Specifically, we investigate the entanglement between two disjoint blocks as a function of their separation, which can be viewed as the entanglement analog of a spatial correlation function. We find sharp entanglement thresholds at a critical distance beyond which the logarithmic negativity vanishes exactly and thus the two blocks become unentangled, which holds even in the presence of long-ranged quantum correlations, i.e., at the system's quantum critical point. Using Time-Evolving Block Decimation (TEBD), we explore this feature as a function of temperature and size of the two blocks and present a simple model to describe our numerical observations., Comment: 12 pages, 7 figures

In this paper we consider charge current generated by maintaining a temperature difference over a nanowire at zero voltage bias. For topological insulator nanowires in a perpendicular magnetic field the current can change sign as the temperature of one end is increased. Here we study how this thermoelectric current sign reversal depends on magnetic field and how impurities affect the size of the thermoelectric current. We consider both scalar and magnetic impurities and show that their influence on the current are quite similar, although the magnetic impurities seem to be more effective in reducing the effect. For moderate impurity concentration the sign reversal persists., Comment: 8 pages, 3 figures

Topological insulator nanowires with uniform cross section, combined with a magnetic flux, can host both a perfectly transmitted mode and Majorana zero modes. Here we consider nanowires with rippled surfaces---specifically, wires with a circular cross section with a radius varying along its axis---and calculate their transport properties. At zero doping, chiral symmetry places the clean wires (no impurities) in the AIII symmetry class, which results in a $\mathbb{Z}$ topological classification. A magnetic flux threading the wire tunes between the topologically distinct insulating phases, with perfect transmission obtained at the phase transition. We derive an analytical expression for the exact flux value at the transition. Both doping and disorder breaks the chiral symmetry and the perfect transmission. At finite doping, the interplay of surface ripples and disorder with the magnetic flux modifies quantum interference such that the amplitude of Aharonov-Bohm oscillations reduces with increasing flux, in contrast to wires with uniform surfaces where it is flux-independent., Comment: 12 pages, 6 figures. v2. 2 new figures and a new appendix

Purification is a tool that allows to represent mixed quantum states as pure states on enlarged Hilbert spaces. A purification of a given state is not unique and its entanglement strongly depends on the particular choice made. Moreover, in one-dimensional systems, the amount of entanglement is linked to how efficiently the purified state can be represented using matrix-product states (MPS). We introduce an MPS based method that allows to find the minimally entangled representation by iteratively minimizing the second Renyi entropy. First, we consider the thermofield double purification and show that its entanglement can be strongly reduced especially at low temperatures. Second, we show that a slowdown of the entanglement growth following a quench of an infinite temperature state is possible., Comment: 8 pages, 7 figures, enhanced discussion of the algorithm

The bulk-boundary correspondence, a topic of intensive research interest over the past decades, is one of the quintessential ideas in the physics of topological quantum matter. Nevertheless, it has not been proven in all generality and has in certain scenarios even been shown to fail, depending on the boundary profiles of the terminated system. Here, we introduce bulk numbers that capture the exact number of in-gap modes, without any such subtleties in one spatial dimension. Similarly, based on these 1D bulk numbers, we define a new 2D winding number, which we call the pole winding number, that specifies the number of robust metallic surface bands in the gap as well as their topological character. The underlying general methodology relies on a simple continuous extrapolation from the bulk to the boundary, while tracking the evolution of Green's function's poles in the vicinity of the bulk band edges. As a main result we find that all the obtained numbers can be applied to the known insulating phases in a unified manner regardless of the specific symmetries. Additionally, from a computational point of view, these numbers can be effectively evaluated without any gauge fixing problems. In particular, we directly apply our bulk-boundary correspondence construction to various systems, including 1D examples without a traditional bulk-boundary correspondence, and predict the existence of boundary modes on various experimentally studied graphene edges, such as open boundaries and grain boundaries. Finally, we sketch the 3D generalization of the pole winding number by in the context of topological insulators., Comment: 19 pages, 11 figures

Transport properties of disordered quantum confined helical Dirac systems are investigated in the large energy limit. As long as the 2D transport length is larger than the perimeter of the nanowire, the conductance and the Fano factor are sensitive to disorder only when the Fermi energy is close to an opening of a transverse mode. In the limit of a large number of transverse modes, transport properties are insensitive to the geometry of the nanowire or the nature and strength of the disorder but, instead, are dominated by the properties of the interface between the ohmic contact and the nanowire. In the case of a heavily doped Dirac metallic contact, the conductance is proportional to the energy with an average transmission $\mathcal{T}=\pi/4$ and a Fano factor of $F\simeq 0.13$. Those results can be generalized to a much broader class of contacts, the exact values of $\mathcal{T}$ and $F$ depending on the model used for the contacts. The energy dependence of Aharonov-Bohm oscillations is determined, when a magnetic flux is threaded through the cross section of the nanowire.

The coherent charge transport through an illuminated graphene ribbon is studied as function of electronic doping, frequency and strength of the electromagnetic driving, for monochromatic circularly polarized light. We focus on the DC current carried by 2D bulk carriers which is dominant (over edge transport) for short and wide enough samples. Broad dips in conductance are predicted for one-photon and multi-photon resonances between the valence and conductance bands. The residual conductance can be associated with evanescent states and related to dynamical gaps in the Floquet quasi-energy spectrum., Comment: 15 pages, 11 figures; Fig 11 updated in v2

The existence and topological classification of lower-dimensional Fermi surfaces is often tied to the crystal symmetries of the underlying lattice systems. Artificially engineered lattices, such as heterostructures and other superlattices, provide promising avenues to realize desired crystal symmetries that protect lower-dimensional Fermi surface, such as nodal lines. In this work, we investigate a Weyl semimetal subjected to spatially periodic onsite potential, giving rise to several phases, including a nodal-line semimetal phase. In contrast to proposals that purely focus on lattice symmetries, the emergence of the nodal line in this setup does not require small spin-orbit coupling, but rather relies on its presence. We show that the stability of the nodal line is understood from reflection symmetry and a combination of a fractional lattice translation and charge-conjugation symmetry. Depending on the choice of parameters, this model exhibits drumhead surface states that are exponentially localized at the surface, or weakly localized surface states that decay into the bulk at all energies., Comment: 11 pages, 8 figures, Editors' Suggestion

We study the anomalous Nernst and thermal Hall effects in a linearized low-energy model of a tilted Weyl semimetal, with two Weyl nodes separated in momentum space. For inversion symmetric tilt, we give analytic expressions in two opposite limits: for a small tilt, corresponding to a type-I Weyl semimetal, the Nernst conductivity is finite and independent of the Fermi level, while for a large tilt, corresponding to a type-II Weyl semimetal, it acquires a contribution depending logarithmically on the Fermi energy. This result is in a sharp contrast to the nontilted case, where the Nernst response is known to be zero in the linear model. The thermal Hall conductivity similarly acquires Fermi surface contributions, which add to the Fermi level independent, zero tilt result, and is suppressed as one over the tilt parameter at half filling in the Type-II phase. In the case of inversion breaking tilt, with the tilting vector of equal modulus in the two Weyl cones, all Fermi surface contributions to both anomalous responses cancel out, resulting in zero Nernst conductivity. We discuss two possible experimental setups, representing open and closed thermoelectric circuits.

We calculate the charge current generated by a temperature bias between the two ends of a tubular nanowire. We show that in the presence of a transversal magnetic field the current can change sign, i.e., electrons can either flow from the hot to the cold reservoir, or in the opposite direction, when the temperature bias increases. This behavior occurs when the magnetic field is sufficiently strong, such that Landau and snaking states are created, and the energy dispersion is non-monotonic with respect to the longitudinal wave vector. The sign reversal can survive in the presence of impurities. We predict this result for core/shell nanowires, for uniform nanowires with surface states due to the Fermi level pinning, and for topological insulator nanowires., Comment: Main text and Supplemental Material (8 pages, 7 figures)

The chiral anomaly in Weyl semimetals states that the left- and right-handed Weyl fermions, constituting the low energy description, are not individually conserved, resulting, for example, in a negative magnetoresistance in such materials. Recent experiments see strong indications of such an anomalous resistance response; however, with a response that at strong fields is more sharply peaked for parallel magnetic and electric fields than expected from simple theoretical considerations. Here, we uncover a mechanism, arising from the interplay between the angle-dependent Landau level structure and long-range scalar disorder, that has the same phenomenology. In particular, we ana- lytically show, and numerically confirm, that the internode scattering time decreases exponentially with the angle between the magnetic field and the Weyl node separation in the large field limit, while it is insensitive to this angle at weak magnetic fields. Since, in the simplest approximation, the internode scattering time is proportional to the anomaly-related conductivity, this feature may be related to the experimental observations of a sharply peaked magnetoresistance., Comment: 8 pages, 4 figures

The emergent integrability of the many-body localized phase is naturally understood in terms of localized quasiparticles. As a result, the occupations of the one-particle density matrix in eigenstates show a Fermi-liquid-like discontinuity. Here we show that in the steady state reached at long times after a global quench from a perfect density-wave state, this occupation discontinuity is absent, reminiscent of a Fermi liquid at a finite temperature, while the full occupation function remains strongly nonthermal. We discuss how one can understand this as a consequence of the local structure of the density-wave state and the resulting partial occupation of quasiparticles. This partial occupation can be controlled by tuning the initial state and can be described by an effective temperature., Comment: 8 pages, 9 figures; added a figure with different initial states

We study the dynamics of thermalization following a quantum quench using tensor-network methods. Contrary to the common belief that the rapid growth of entanglement and the resulting exponential growth of the bond dimension restricts simulations to short times, we demonstrate that the long time limit of local observables can be well captured using the time-dependent variational principle. This allows to extract transport coefficients such as the energy diffusion constant from simulations with rather small bond dimensions. We further study the characteristic of the chaotic wave that precedes the emergence of hydrodynamics, to find a ballistic diffusively-broadening wave-front., Comment: New version: 10 pages including appendices. Added an author. Used a different quantity to diagnose chaos, for which results are independent of bond dimension. Leads to a modified conclusion concerning the chaotic dynamics

We study a model of interacting fermions in one dimension subject to random, uncorrelated onsite disorder. The model realizes an interaction-driven quantum phase transition between an ergodic and a many-body localized phase (MBL). We propose a single-particle framework to characterize these phases by the eigenstates (the natural orbitals) and the eigenvalues (the occupation spectrum) of the one-particle density matrix (OPDM) in many-body eigenstates. As a main result, we find that the natural orbitals are localized in the MBL phase, but delocalized in the ergodic phase. This qualitative change in these single-particle states is a many-body effect, since without interactions the single-particle energy eigenstates are all localized. The occupation spectrum in the ergodic phase is thermal in agreement with the eigenstate thermalization hypothesis, while in the MBL phase the occupations preserve a discontinuity at an emergent Fermi edge. This suggests that the MBL eigenstates are weakly dressed Slater determinants, with the eigenstates of the underlying Anderson problem as reference states. We discuss the statistical properties of the natural orbitals and of the occupation spectrum in the two phases and as the transition is approached. Our results are consistent with the existing picture of emergent integrability and localized integrals of motion, or quasiparticles, in the MBL phase. We emphasize the close analogy of the MBL phase to a zero-temperature Fermi liquid: in the studied model, the MBL phase is adiabatically connected to the Anderson insulator and the occupation-spectrum discontinuity directly indicates the presence of quasiparticles localized in real space. Finally, we show that the same picture emerges for interacting fermions in the presence of an experimentally-relevant bichromatic lattice and thereby demonstrate that our findings are not limited to a specific model., Comment: 19 pages, 20 figures

The Zak phase $\gamma$, the generalization of the Berry phase to Bloch wave functions in solids, is often used to characterize inversion-symmetric 1D topological insulators; however, since its value can depend on the choice of real-space origin and unit cell, only the difference between the Zak phase of two regions is believed to be relevant. Here, we show that one can extract an origin-independent part of $\gamma$, the so-called inter-cellular Zak phase $\gamma^{\mathrm{inter}}$, which can be used as a bulk quantity to predict the number of surface modes as follows: a neutral finite 1D tight-binding system has $n_s = \gamma^{\mathrm{inter}}/\pi$ (mod 2) number of in-gap surface modes below the Fermi level if there exists a commensurate bulk unit cell that respects inversion symmetry. We demonstrate this by first verifying that $\pm e\gamma^{\mathrm{inter}}/2\pi$ (mod $e$) is equal to the extra charge accumulation in the surface region for a general translationally invariant 1D insulator, while the remnant part of $\gamma$, the intra-cellular Zak phase $\gamma^{\mathrm{intra}}$, corresponds to the electronic part of the dipole moment of the bulk's unit cell. Second, we show that the extra charge accumulation can be related to the number of surface modes when the unit cell is inversion symmetric. This bulk-boundary correspondence using $\gamma^{\mathrm{inter}}$ reduces to the conventional one using $\gamma$ when the real-space origin is at the inversion center. Our work thereby clarifies the usage of $\gamma$ in the bulk-boundary correspondence. We study several tight binding models to quantitatively check the relation between the extra charge accumulation and the inter-cellular Zak phase as well as the bulk-boundary correspondence using the inter-cellular Zak phase., Comment: 15 pages, 9 figures

We demonstrate that the quantum mutual information (QMI) is a useful probe to study many-body localization (MBL). First, we focus on the detection of a metal--insulator transition for two different models, the noninteracting Aubry-Andr\'e-Harper model and the spinless fermionic disordered Hubbard chain. We find that the QMI in the localized phase decays exponentially with the distance between the regions traced out, allowing us to define a correlation length, which converges to the localization length in the case of one particle. Second, we show how the QMI can be used as a dynamical indicator to distinguish an Anderson insulator phase from an MBL phase. By studying the spread of the QMI after a global quench from a random product state, we show that the QMI does not spread in the Anderson insulator phase but grows logarithmically in time in the MBL phase., Comment: 4+2 pages, 5+5 figures

Clean topological insulators exposed to a magnetic field develop Landau levels accompanied by a nonzero Hall conductivity for the infinite slab geometry. In this work we consider the case of disordered topological insulator nanowires and find, in contrast, that a zero Hall plateau emerges within a broad energy window close to the Dirac point. We numerically calculate the conductance and its distribution for a statistical ensemble of disordered nanowires, and use the conductance fluctuations to study the dependence of the insulating phase on system parameters, such as the nanowire length, disorder strength and the magnetic field., Comment: 7 pages, 7 figures

Formulating consistent theories describing strongly correlated metallic topological phases is an outstanding problem in condensed matter physics. In this work we derive a theory defining a fractionalized analogue of the Weyl semimetal state: the fractional chiral metal. Our approach is to construct a 4+1D quantum Hall insulator by stacking 3+1D Weyl semimetals in a magnetic field. In a strong enough field the low-energy physics is determined by the lowest Landau level of each Weyl semimetal, which is highly degenerate and chiral, motivating us to use a coupled-wire approach. The one-dimensional dispersion of the lowest Landau level allows us to model the system as a set of degenerate 1+1D quantum wires that can be bosonized in the presence of electron-electron interactions and coupled such that a gapped phase is obtained, whose response to an electromagnetic field is given in terms of a Chern-Simons field theory. At the boundary of this phase we obtain the field theory of a 3+1D gapless fractional chiral state, which we show is consistent with a previous theory for the surface of a 4+1D Chern-Simons theory. The boundary's response to an external electromagnetic field is determined by a chiral anomaly with a fractionalized coefficient. We suggest that such anomalous response can be taken as a working definition of a fractionalized strongly correlated analogue of the Weyl semimetal state., Comment: 15 pages, 6 figures, extended discussion about potential realizations of 3+1D fractional chiral metals, final version