Showing papers by "Shivaji Lal Sondhi published in 2010"

Thermal quenches in spin ice.

Abstract: We study the diffusion-annihilation process which occurs when spin ice is quenched from a high temperature paramagnetic phase deep into the spin-ice regime, where the excitations---magnetic monopoles---are sparse. We find that due to the Coulomb interaction between the monopoles, a dynamical arrest occurs, in which nonuniversal lattice-scale constraints impede the complete decay of charge fluctuations. This phenomenon is outside the reach of conventional mean-field theory for a two-component Coulomb liquid. We identify the relevant time scales for the dynamical arrest and propose an experiment for detecting monopoles and their dynamics in spin ice based on this nonequilibrium phenomenon.

Nematic valley ordering in quantum Hall systems

Abstract: The interplay between quantum Hall ordering and spontaneously broken ``internal'' symmetries in two-dimensional electron systems with spin or pseudospin degrees of freedom gives rise to a variety of interesting phenomena, including novel phases, phase transitions, and topological excitations. Here we develop a theory of broken-symmetry quantum Hall states, applicable to a class of multivalley systems, where the symmetry at issue is a point-group element that combines a spatial rotation with a permutation of valley indices. The anisotropy of the dispersion relation, generally present in such systems, favors states where all electrons reside in one of the valleys. In a clean system, the valley ``pseudospin'' ordering occurs via a finite-temperature transition accompanied by a nematic pattern of spatial symmetry breaking. In weakly disordered systems, domains of pseudospin polarization are formed, which prevents macroscopic valley and nematic ordering; however, the resulting state still asymptotically exhibits the quantum Hall effect. We discuss the transport properties in the ordered and disordered regimes, and the relation of our results to recent experiments in AlAs.

Product, generic, and random generic quantum satisfiability

Abstract: We report a cluster of results on $k$-QSAT, the problem of quantum satisfiability for $k$-qubit projectors which generalizes classical satisfiability with $k$-bit clauses to the quantum setting. First we define the $\mathit{NP}$-complete problem of product satisfiability and give a geometrical criterion for deciding when a QSAT interaction graph is product satisfiable with positive probability. We show that the same criterion suffices to establish quantum satisfiability for all projectors. Second, we apply these results to the random graph ensemble with generic projectors and obtain improved lower bounds on the location of the SAT-unSAT transition. Third, we present numerical results on random, generic satisfiability which provide estimates for the location of the transition for $k=3$ and $k=4$ and mild evidence for the existence of a phase which is satisfiable by entangled states alone.

Irrational charge from topological order.

Abstract: Topological or deconfined phases of matter exhibit emergent gauge fields and quasiparticles that carry a corresponding gauge charge. In systems with an intrinsic conserved U(1) charge, such as all electronic systems where the Coulombic charge plays this role, these quasiparticles are also characterized by their intrinsic charge. We show that one can take advantage of the topological order fairly generally to produce periodic Hamiltonians which endow the quasiparticles with continuously variable, generically irrational, intrinsic charges. Examples include various topologically ordered lattice models, the three-dimensional resonating valence bond liquid on bipartite lattices as well as water and spin ice. By contrast, the gauge charges of the quasiparticles retain their quantized values.

Random quantum satisfiabiilty

Abstract: Alongside the effort underway to build quantum computers, it is important to betterunderstand which classes of problems they will find easy and which others even theywill find intractable. We study random ensembles of the QMA1-complete quantum sat-isfiability (QSAT) problem introduced by Bravyi [1]. QSAT appropriately generalizesthe NP-complete classical satisfiability (SAT) problem. We show that, as the density ofclauses/projectors is varied, the ensembles exhibit quantum phase transitions betweenphases that are satisfiable and unsatisfiable. Remarkably, almost all instances of QSATfor any hypergraph exhibit the same dimension of the satisfying manifold. This estab-lishes the QSAT decision problem as equivalent to a, potentially new, graph theoreticproblem and that the hardest typical instances are likely to be localized in a boundedrange of clause density.

AKLT models with quantum spin glass ground states

Abstract: We study AKLT models on locally treelike lattices of fixed connectivity and find that they exhibit a variety of ground states depending upon the spin, coordination, and global (graph) topology. We find (a) quantum paramagnetic or valence-bond solid ground states, (b) critical and ordered N\'eel states on bipartite infinite Cayley trees, and (c) critical and ordered quantum vector spin glass states on random graphs of fixed connectivity. We argue, in consonance with a previous analysis [C. R. Laumann, S. A. Parameswaran, and S. L. Sondhi, Phys. Rev. B 80, 144415 (2009)], that all phases are characterized by gaps to local excitations. The spin glass states we report arise from random long-ranged loops which frustrate N\'eel ordering despite the lack of randomness in the coupling strengths.

Statistical mechanics of classical and quantum computational complexity

Abstract: The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous framework for classifying the hardness of problems according to the computational resources, most notably time, needed to solve them. Its extension to quantum computers allows the relative power of quantum computers to be analyzed. This framework identifies families of problems which are likely hard for classical computers (``NP-complete'') and those which are likely hard for quantum computers (``QMA-complete'') by indirect methods. That is, they identify problems of comparable worst-case difficulty without directly determining the individual hardness of any given instance. Statistical mechanical methods can be used to complement this classification by directly extracting information about particular families of instances---typically those that involve optimization---by studying random ensembles of them. These pose unusual and interesting (quantum) statistical mechanical questions and the results shed light on the difficulty of problems for large classes of algorithms as well as providing a window on the contrast between typical and worst case complexity. In these lecture notes we present an introduction to this set of ideas with older work on classical satisfiability and recent work on quantum satisfiability as primary examples. We also touch on the connection of computational hardness with the physical notion of glassiness.

Non-linear, Finite Frequency Quantum Critical Transport from AdS/CFT

Abstract: Transport at a quantum critical point depends sensitively on the relative magnitudes of temperature, frequency and electric field. Here we used the gauge/gravity correspondence to compute the full temperature and, generally nonlinear, electric field dependence of the electrical conductivity for some model critical theories. In the special case of 2+1 dimensions we are also able to find the full time-dependent response of the system to an arbitrary time dependent external electric field. The response of the system is instantaneous, implying a frequency independent conductivity. We describe a mechanism that rationalizes the instantaneous response.

Renormalization group and the superconducting susceptibility of a Fermi liquid

Abstract: A free Fermi gas has, famously, a superconducting susceptibility that diverges logarithmically at zero temperature. In this paper we ask whether this is still true for a Fermi liquid and find that the answer is that it does not. From the perspective of the renormalization group for interacting fermions, the question arises because a repulsive interaction in the Cooper channel is a marginally irrelevant operator at the Fermi liquid fixed point and thus is also expected to infect various physical quantities with logarithms. Somewhat surprisingly, at least from the renormalization group viewpoint, the result for the superconducting susceptibility is that two logarithms are not better than one. In the course of this investigation we derive a Callan-Symanzik equation for the repulsive Fermi liquid using the momentum-shell renormalization group, and use it to compute the long-wavelength behavior of the superconducting correlation function in the emergent low-energy theory. We expect this technique to be of broader interest.