The repulsive Hubbard model has been immensely useful in understanding strongly correlated electron systems, and serves as the paradigmatic model of the field. Despite its simplicity, it exhibits a strikingly rich phenomenology which is reminiscent o
f that observed in quantum materials. Nevertheless, much of its phase diagram remains controversial. Here, we review a subset of what is known about the Hubbard model, based on exact results or controlled approximate solutions in various limits, for which there is a suitable small parameter. Our primary focus is on the ground state properties of the system on various lattices in two spatial dimensions, although both lower and higher dimensions are discussed as well. Finally, we highlight some of the important outstanding open questions.
A recently developed formula for the Hall coefficient [A. Auerbach, Phys. Rev. Lett. 121, 66601 (2018)] is applied to nodal line and Weyl semimetals (including graphene), and to spin-orbit split semiconductor bands in two and three dimensions. The ca
lculation reduces to a ratio of two equilibrium susceptibilities, where corrections are negligible at weak disorder. Deviations from Drudes inverse carrier density are associated with band degeneracies, Fermi surface topology, and interband scattering. Experiments which can measure these deviations are proposed.
Magnetotransport theory of layered superconductors in the flux flow steady state is revisited. Longstanding controversies concerning observed Hall sign reversals are resolved. The conductivity separates into a Bardeen-Stephen vortex core contribution
, and a Hall conductivity due to moving vortex charge. This charge, which is responsible for Hall anomaly, diverges logarithmically at weak magnetic field. Its values can be extracted from magetoresistivity data by extrapolation of vortex core Hall angle from the normal phase. Hall anomalies in YBCO, BSCCO, and NCCO data are consistent with theoretical estimates based on doping dependence of London penetration depths. In the appendices, we derive the Streda formula for the hydrodynamical Hall conductivity, and refute previously assumed relevance of Galilean symmetry to Hall anomalies.
We study the time evolution of quantum entanglement for a specific class of quantum dynamics, namely the locally scrambled quantum dynamics, where each step of the unitary evolution is drawn from a random ensemble that is invariant under local (on-si
te) basis transformations. In this case, the average entanglement entropy follows Markovian dynamics that the entanglement property of the future state can be predicted solely based on the entanglement properties of the current state and the unitary operator at each step. We introduce the entanglement feature formulation to concisely organize the entanglement entropies over all subsystems into a many-body wave function, which allows us to describe the entanglement dynamics using an imaginary-time Schrodinger equation, such that various tools developed in quantum many-body physics can be applied. The framework enables us to investigate a variety of random quantum dynamics beyond Haar random circuits and Brownian circuits. We perform numerical simulations for these models and demonstrate the validity and prediction power of the entanglement feature approach.
We show that the topological index of a wavefunction, computed in the space of twisted boundary phases, is preserved under Hilbert space truncation, provided the truncated state remains normalizable. If truncation affects the boundary condition of th
e resulting state, the invariant index may acquire a different physical interpretation. If the index is symmetry protected, the truncation should preserve the protecting symmetry. We discuss implications of this invariance using paradigmatic integer and fractional Chern insulators, $Z_2$ topological insulators, and Spin-$1$ AKLT and Heisenberg chains, as well as its relation with the notion of bulk entanglement. As a possible application, we propose a partial quantum tomography scheme from which the topological index of a generic multi-component wavefunction can be extracted by measuring only a small subset of wavefunction components, equivalent to the measurement of a bulk entanglement topological index.
For weakly disordered fractional quantum Hall phases, the non linear photoconductivity is related to the charge susceptibility of the clean system by a Floquet boost. Thus, it may be possible to probe collective charge modes at finite wavevectors by
electrical transport. Incompressible phases, irradiated at slightly above the magneto-roton gap, are predicted to exhibit negative photoconductivity and zero resistance states with spontaneous internal electric fields. Non linear conductivity can probe composite fermions charge excitations in compressible filling factors.
Recent experiments have reported on controlled nucleation of individual skyrmions in chiral magnets. Here we show that in magnetic ultra-thin films with interfacial Dzyaloshinskii-Moriya interaction, single skyrmions of different radii can be nucleat
ed by creating a local distortion in the magnetic field. In our study, we have considered zero temperature quantum nucleation of a single skyrmion from a ferromagnetic phase. The physical scenario we model is one where a uniform field stabilizes the ferromagnet, and an opposing local magnetic field over a circular spot, generated by the tip of a local probe, drives the skyrmion nucleation. Using spin path integrals and a collective coordinate approximation, the tunneling rate from the ferromagnetic to the single skyrmion state is computed as a function of the tips magnetic field and the circular spot radius. Suitable parameters for the observation of the quantum nucleation of single skyrmions are identified.
Two-dimensional topological phases are characterized by TKNN integers, which classify Bloch energy bands or groups of Bloch bands. However, quantization does not survive thermal averaging or dephasing to mixed states. We show that using Uhlmanns para
llel transport for density matrices (Rep. Math. Phys. 24, 229 (1986)), an integer classification of topological phases can be defined for a finite generalized temperature $T$ or dephasing Lindbladian. This scheme reduces to the familiar TKNN classification for $T<T_{{rm c},1}$, becomes trivial for $T>T_{{rm c},2}$, and exhibits a `gapless intermediate regime where topological indices are not well-defined. We demonstrate these ideas in detail, applying them to Haldanes honeycomb lattice model and the Bernevig-Hughes-Zhang model, and we comment on their generalization to multi-band Chern insulators.
Weyl Semimetals (WS) are a new class of Dirac-type materials exhibiting a phase with bulk energy nodes and an associated vanishing density of states (DOS). We investigate the stability of this nodal DOS suppression in the presence of local impurities
and consider whether or not such a suppression can be lifted by impurity-induced resonances. We find that while a scalar (chemical potential type) impurity can always induce a resonance at arbitrary energy and hence lift the DOS suppression at Dirac/Weyl nodes, for many other impurity types (e.g. magnetic or orbital-mixing), resonances are forbidden in a wide range of energy. We investigate a $4$-band tight-binding model of WS adapted from a physical heterostructure construction due to Burkov, Hook, and Balents, and represent a local impurity potential by a strength $g$ as well as a matrix structure $Lambda$. A general framework is developed to analyze this resonance dichotomy and make connection with the phase shift picture in scattering theory, as well as to determine the relation between resonance energy and impurity strength $g$. A complete classification of impurities based on $Lambda$, based on their effect on nodal DOS suppression, is tabulated. We also discuss the differences between continuum and lattice approaches.
Band insulators appear in a crystalline system only when the filling -- the number of electrons per unit cell and spin projection -- is an integer. At fractional filling, an insulating phase that preserves all symmetries is a Mott insulator, i.e. it
is either gapless or, if gapped, displays fractionalized excitations and topological order. We raise the inverse question -- at an integer filling is a band insulator always possible? Here we show that lattice symmetries may forbid a band insulator even at certain integer fillings, if the crystal is non-symmorphic -- a property shared by a majority of three-dimensional crystal structures. In these cases, one may infer the existence of topological order if the ground state is gapped and fully symmetric. This is demonstrated using a non-perturbative flux threading argument, which has immediate applications to quantum spin systems and bosonic insulators in addition to electronic band structures in the absence of spin-orbit interactions.
