Talks

We give an overview of several connections between topics in quantum information theory, graph theory, and statistical mechanics. The central concepts are mappings from statistical mechanical models defined on graphs, to entangled states of multi-party quantum systems. We present a selection of such mappings, and illustrate how they can be used to obtain a cross-fertilization between different research areas. For example, we show how width parameters in graph theory such as \'tree-width\' and \'rank-width\', which may be used to assess the computational hardness of evaluating partition functions, are intimately related with the entanglement measure \'entanglement width\', which is used to asses to computational power of resource states in quantum information. Furthermore, using our mappings we provide simple techniques to relate different statistical mechanical models with each other via basic graph transformations. These techniques can be used to prove that that there exist models which are \'complete\' in the sense that every other model can be viewed as a special instance of such a complete model via a polynomial reduction. Examples of such complete models include the 2D Ising model in an external field, as well as the zero-field 3D Ising model. Joint work with W. Duer, G. de las Cuevas, R. Huebener and H. Briegel

The idea of pseudo-randomness is to use little or no randomness to simulate a random object such as a random number, permutation, graph, quantum state, etc... The simulation should then have some superficial resemblance to a truly random object; for example, the first few moments of a random variable should be nearly the same. This concept has been enormously useful in classical computer science. In my talk, I\'ll review some quantum analogues of pseudo-randomness: unitary k-designs, quantum expanders (and their new cousin, quantum tensor product expanders), extractors. I\'ll talk about relations between them, efficient constructions, and possible applications. Some of the material is joint work with Matt Hastings and Richard Low.

Certain structures arising in Physics (mub\'s and sic-povm\'s) can be viewed as sets of lines in complex space that are as large as possible, given some simple constraints on the angles between distinct lines. The analogous problems in real space have long been of interest in Combinatorics, because of their relation to classical combinatorial structures. In the complex case there seems no reason for any combinatorial connection to exist. will discuss some of the history of the real problems, and describe some recent work that relates the complex problems to some very interesting classes of graphs.

Based on a U(1) gauge theory of the Hubbard model on the triangular lattice, it is argued that a spin liquid phase may exist near the Mott transition in the organic compound κ-(BEDT-TTF)2Cu2(CN)3. In the spin liquid state, low energy excitations are fermionic spinons and an emergent U(1) gauge boson. Highly unusual transport properties are predicted due to the presence of a spinon Fermi surface. Despite rather good agreements with experiments, the stability of the spin liquid state has been continuously questioned because of the fluctuating gauge field which may destabilize the spin liquid state via confinement. In this talk, I will discuss how the presence of spinon Fermi surface can stabilize the spin liquid state against non-perturbative gauge fluctuations.

There are a few examples in the literature of metals that, in the T  0 K limit, show a resistivity that rises with decreasing temperature without any sign of either saturation or a gap. Well known cases include underdoped cuprates in high magnetic fields and some doped uranium heavy fermion compounds. I will review these and some less-well-known cases, before describing the behaviour of FeCrAs [1], in which we find a continuously rising resistivity from 900 K down to below 50 mK, with a brief interruption due to an antiferromagnetic transition at about 100 K. Down to at least 50 mK the resistivity is nearly linear in temperature, but with a negative coefficient. We speculate that this behaviour may be connected to fluctuations of frustrated iron “trimers” that do not order magnetically. 1. W. Wu, A. McCollam, P.M.C. Rourke, D. Rancourt, I. Swainson and S.R. Julian, in preparation.

Calculating universal properties of quantum phase transitions in microscopic Hamiltonians is a challenging task, made possible through large-scale numerical simulations coupled with finite-size scaling analyses. The continuing advancement of quantum Monte Carlo technologies, together with modern high-performance computing infrastructure, has made amenable a new class of quantum Heisenberg Hamiltonian with four-spin exchange, which may harbor a continuous Néel-to-Valence Bond Solid quantum phase transition. Such an exotic quantum critical point would necessarily lie outside of the standard Landau-Ginzburg-Wilson paradigm, and may contain novel physical phenomena such as emergent topological order and quantum number fractionalization. I will discuss efforts to calculate universal critical exponents using large-scale quantum Monte Carlo simulations, and compare them to theoretical predictions, in particular from the recent theory of deconfined quantum criticality.

Responding electrically to magnetic stimuli and vise versa, multiferroics offer exciting possibilities for applications and challenge our understanding of coupled lattice and spin degrees of freedom in solids. I discuss how multiferroic properties can develop in frustrated magnets where competing interactions produce non-collinear spin order and symmetry breaking lattice distortions. Our experiments in TbMnO3, Ni3V2O8, and RbFe(MoO4)2 show that when the low temperature magnetic order breaks spatial inversion symmetry it is accompanied by ferroelectricity [1-3]. Conversely, the application of an electric field favors one of the two inversion symmetry related antiferromagnetic domains. We infer that inversion symmetry breaking magnetic order acts as an effective electric field through magneto-elastic distortions that relieve frustration. We also present evidence for microscopic correspondence between the ferroelectric and the antiferromagnetic domain structure. The results presented are based on magnetic neutron diffraction, pyrocurrent measurements, and theoretical work by A. B. Harris [4]. [1] M. Kenzelmann, A. B. Harris, S. Jonas, C. Broholm, J. Schefer, S. B. Kim, C. L. Zhang, S.-W. Cheong, O. P. Vajk, and J. W. Lynn, Phys. Rev. Lett. 95, 087206 (2005). [2] G. Lawes, A. B. Harris, T. Kimura, N. Rogado, R. J. Cava, A. Aharony, O. Entin-Wohlman, T. Yildirim, M. Kenzelmann, C. Broholm, and A. P. Ramirez, Phys. Rev. Lett. 95, 087205 (2005). [3] M. Kenzelmann, G. Lawes, A.B. Harris, G. Gasparovic, C. Broholm, A.P. Ramirez, G.A. Jorge, M. Jaime, S. Park, Q. Huang, A.Ya. Shapiro, and L.A. Demianets, Phys. Rev. Lett. 98, 267205 (2007). [4] A. B. Harris, Phys. Rev. B 76 , 054447 (2007).

We analyze the trans-Planckian problem and its formulation in the context of cosmology, black-hole physics, and analogue models of gravity. In particular, we discuss the phenomenological approach to the trans-Planckian problem based on modified, locally Lorentz-breaking, dispersion relations (MDR). The main question is whether MDR leave an detectable imprint on macroscopic physics. In the framework of the semi-classical theory of gravity, this question can be unambiguously answered only through a rigorous formulation of quantum field theory on curved space with MDR. In this context, we propose a momentum-space analysis of the Green\'s function, which will hopefully lead to the correct renormalization of the stress tensor.

Motivated by recent observations of superfluidity of ultracold fermions in optical lattices, we investigate the stability of superfluid flow of paired fermions in the lowest band of a strong optical lattice. For fillings close to one fermion per site, we show that superflow breaks down via a dynamical instability leading to a transient density wave. At lower fillings, there is a distinct dynamical instability, where the superfluid stiffness becomes negative; this evolves, with increasing pairing interaction, from the fermion pair breaking instability to the well-known dynamical instability of lattice bosons. Our most interesting finding is the existence of a transition, over a range of fillings close to one fermion per site, from the fermion depairing instability to the density wave instability as the strength of the pairing interaction is increased. By sharp contrast, the ground state in this regime evolves smoothly with increasing interaction analogous to the BCS-BEC crossover.

I present a short review of recent developments both in experiment and theory in Quantum Hall Effect in Graphene. The emphasis is on the interpretation of the dynamics underlying recently experimentally discovered novel plateaus in strong magnetic fields (B > 20 T).