Rahul Nandkishore

Many-Body Localization and Thermalization in Quantum Statistical Mechanics

We review some recent developments in the statistical mechanics of isolated quantum systems. We provide a brief introduction to quantum thermalization, paying particular attention to the eigenstate thermalization hypothesis (ETH) and the resulting single-eigenstate statistical mechanics. We then focus on a class of systems that fail to quantum thermalize and whose eigenstates violate the ETH: These are the many-body Anderson-localized systems; their long-time properties are not captured by the conventional ensembles of quantum statistical mechanics. These systems can forever locally remember information about their local initial conditions and are thus of interest for possibilities of storing quantum information. We discuss key features of many-body localization (MBL) and review a phenomenology of the MBL phase. Single-eigenstate statistical mechanics within the MBL phase reveal dynamically stable ordered phases, and phase transitions among them, that are invisible to equilibrium statistical mechanics and...

Phenomenology of fully many-body-localized systems

Initiative for the Theoretical Sciences, The Graduate Center, CUNY, New York, NY 10016, USA(Dated: August 20, 2014)We consider fully many-body localized systems, i.e. isolated quantum systems where all themany-body eigenstates of the Hamiltonian are localized. We define a sense in which such systems areintegrable, with localized conserved operators. These localized operators are interacting pseudospins,and the Hamiltonian is such that unitary time evolution produces dephasing but not ‘flips’ of thesepseudospins. As a result, an initial quantum state of a pseudospin can in principle be recoveredvia (pseudospin) echo procedures. We discuss how the exponentially decaying interactions betweenpseudospins lead to logarithmic-in-time spreading of entanglement starting from nonentangled initialstates. These systems exhibit multiple different length scales that can be defined from exponentialfunctions of distance; we suggest that some of these decay lengths diverge at the phase transitionout of the fully many-body localized phase while others remain finite.

Chiral superconductivity from repulsive interactions in doped graphene

Chiral superconducting states are expected to support a variety of exotic and potentially useful phenomena. Theoretical analysis suggests that just such a state could emerge in a doped graphene monolayer.

Nonlocal adiabatic response of a localized system to local manipulations

Anderson localization has recently attracted renewed interest in strongly correlated quantum systems. Now, local adiabatic manipulations are shown to lead to a nonlocal response, with implications for quantum control in disordered environments.

Polar Kerr effect and time reversal symmetry breaking in bilayer graphene.

The unique sensitivity of optical response to different types of symmetry breaking can be used to detect and identify spontaneously ordered many-body states in bilayer graphene. We predict a strong response at optical frequencies, sensitive to electronic phenomena at low energies, which arises because of nonzero interband matrix elements of the electric current operator. In particular, the polar Kerr rotation and reflection anisotropy provide fingerprints of the quantum anomalous Hall state and the nematic state, characterized by spontaneously broken time-reversal symmetry and lattice rotation symmetry, respectively. These optical signatures, which undergo a resonant enhancement in the near-infrared regime, lie well within reach of existing experimental techniques.

Mean-field theory of nearly many-body localized metals

We develop a mean-field theory of the metallic phase near the many-body localization (MBL) transition, using the observation that a system near the MBL transition should become an increasingly slow heat bath for its constituent parts. As a first step, we consider the properties of a many-body localized system coupled to a generic ergodic bath whose characteristic dynamical timescales are much slower than those of the system. As we discuss, a wide range of experimentally relevant systems fall into this class; we argue that relaxation in these systems is dominated by collective many-particle rearrangements, and compute the associated timescales and spectral broadening. We then use the observation that the self-consistent environment of any region in a nearly localized metal can itself be modeled as a slowly fluctuating bath to outline a self-consistent mean-field description of the nearly localized metal and the localization transition. In the nearly localized regime, the spectra of local operators are highly inhomogeneous and the typical local spectral linewidth is narrow. The local spectral linewidth is proportional to the DC conductivity, which is small in the nearly localized regime. This typical linewidth and the DC conductivity go to zero as the localized phase is approached, with a scaling that we calculate, and which appears to be in good agreement with recent experimental results.

Marginal Anderson localization and many-body delocalization

We consider d dimensional systems which are localized in the absence of interactions, but whose single particle (SP) localization length diverges near a discrete set of (single-particle) energies, with critical exponent \nu. This class includes disordered systems with intrinsic- or symmetry-protected- topological bands, such as disordered integer quantum Hall insulators. In the absence of interactions, such marginally localized systems exhibit anomalous properties intermediate between localized and extended including: vanishing DC conductivity but sub-diffusive dynamics, and fractal entanglement (an entanglement entropy with a scaling intermediate between area and volume law). We investigate the stability of marginal localization in the presence of interactions, and argue that arbitrarily weak short range interactions trigger delocalization for partially filled bands at non-zero energy density if \nu \ge 1/d. We use the Harris/Chayes bound \nu \ge 2/d, to conclude that marginal localization is generically unstable in the presence of interactions. Our results suggest the impossibility of stabilizing quantized Hall conductance at non-zero energy density.

Superconductivity from weak repulsion in hexagonal lattice systems

lling for a triangular lattice). We argue that for a generic doping in this range, superconductivity at weak coupling is of Kohn-Luttinger type, and, due to the presence of electronic interactions beyond on-site repulsion, is a threshold phenomenon, with superconductivity emerging only if the attraction generated by the Kohn-Luttinger mechanism exceeds the bare repulsion in some channel. For disconnected Fermi pockets, we predict that Kohn-Luttinger superconductivity, if it occurs, is likely to bef-wave. While the Kohn-Luttinger analysis is adequate over most of the doping range, a more sophisticated analysis is needed near Van Hove doping. We treat Van Hove doping using a parquet renormalization group, the equations for which we derive and analyze. Near this doping level, superconductivity is a universal phenomenon, arising from any choice of bare repulsive interactions. The strongest pairing instability is into a chiral d wave state (d +id). At a truly weak coupling, the strongest competitor is a spin-density-wave instability, however, d wave superconductivity still wins. Moreover, the feedback of the spin density uctuations into the Cooper channel signicantly enhances the critical temperature over the estimates of the Kohn Luttinger theory. We analyze renormalization group equations at stronger couplings and nd that the main competitor to d wave supoerconductivity away from weak coupling is actually ferromagnetism. We also discuss the eect of the edge fermions and show that they are unimportant in the asymptotic weak coupling limit, but may give rise to, e.g., a charge-density-wave order at moderate coupling strengths.

Itinerant half-metal spin-density-wave state on the hexagonal lattice.

We consider electrons on a honeycomb or triangular lattice doped to the saddle point of the band structure. We assume the system parameters are such that spin density wave (SDW) order emerges below a temperature T(N) and investigate the nature of the SDW phase. We argue that at T≤T(N), the system develops a uniaxial SDW phase whose ordering pattern breaks O(3)×Z(4) symmetry and corresponds to an eight-site unit cell with nonuniform spin moments on different sites. This state is a half-metal--it preserves the full original Fermi surface, but has gapless charged excitations in one spin branch only. It allows for electrical control of spin currents and is desirable for nanoscience.

Weyl semimetals with short-range interactions

We construct a low-energy effective field theory of fermions interacting via short-range interactions in a simple two-band model of a Weyl semimetal on the cubic lattice and investigate possible broken-symmetry ground states through a one-loop renormalization group (RG) analysis. Using the symmetries of the noninteracting Hamiltonian to constrain the form of the interaction term leads to four independent coupling constants. We investigate the stability of RG flows towards strong coupling and find a single stable trajectory. In order to explore possible broken-symmetry ground states, we calculate susceptibilities in the particle-hole and particle-particle channels along this trajectory and find that the leading instability is towards a fully gapped spin-density wave (SDW) ground state. The sliding mode of this SDW couples to the external electromagnetic fields in the same way as the Peccei-Quinn axion field of particle physics. We also study the maximally symmetric version of our model with a single independent coupling constant. Possible ground states in this case are either gapless ferromagnetic states where the spin waves couple to the Weyl fermions like the spatial components of a (possibly chiral) gauge field, or a fully gapped spin-singlet Fulde-Ferrell-Larkin-Ovchinnikov superconducting state.