Emergent Phase Space Description of Unitary Matrix Model

We show that large $N$ phases of a $0$ dimensional generic unitary matrix model (UMM) can be described in terms of topologies of two dimensional droplets on a plane spanned by eigenvalue and number of boxes in Young diagram. Information about different phases of UMM is encoded in the geometry of droplets. These droplets are similar to phase space distributions of a unitary matrix quantum mechanics (UMQM) ($(0 + 1)$ dimensional) on constant time slices. We find that for a given UMM, it is possible to construct an effective UMQM such that its phase space distributions match with droplets of UMM on different time slices at large $N$. Therefore, large $N$ phase transitions in UMM can be understood in terms of dynamics of an effective UMQM. From the geometry of droplets it is also possible to construct Young diagrams corresponding to $U(N)$ representations and hence different large $N$ states of the theory in momentum space. We explicitly consider two examples : single plaquette model with $\text{Tr} U^2$ terms and Chern-Simons theory on $S^3$. We describe phases of CS theory in terms of eigenvalue distributions of unitary matrices and find dominant Young distributions for them.Comment: 52 pages, 15 figures, v2 Introduction and discussions extended, References adde

Holographic entanglement entropy in imbalanced superconductors

We study the behavior of holographic entanglement entropy (HEE) for imbalanced holographic superconductors. We employ a numerical approach to consider the robust case of fully back-reacted gravity system. The hairy black hole solution is found by using our numerical scheme. Then it is used to compute the HEE for the superconducting case. The cases we study show that in presence of a mismatch between two chemical potentials, below the critical temperature, superconducting phase has a lower HEE in comparison to the AdS-Reissner-Nordstrom black hole phase. Interestingly, the effects of chemical imbalance are different in the contexts of black hole and superconducting phases. For black hole, HEE increases with increasing imbalance parameter while it behaves oppositely for the superconducting phase. The implications of these results are discussed.Comment: V2 (Published Version), 19 pages: We replace earlier analytical method by numerical analysis. The effect of imbalance is more cleare

Dynamical Structure Factor of Fulde-Ferrell-Larkin-Ovchinnikov Superconductors

Superconductor with a spatially-modulated order parameter is known as Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductor. Using the time-dependent Ginzburg-Landau (TDGL) formalism we have theoretically studied the temporal behaviour of the equal-time correlation function, or the structure factor, of a FFLO superconductor following a sudden quench from the unpaired, or normal, state to the FFLO state. We find that quenching into the ordered FFLO phase can reveal the existence of a line in the mean-field phase diagram which cannot be accessed by static properties.Comment: 2 pages, Poster presented at 57TH DAE SOLID STATE PHYSICS SYMPOSIUM, 2012. Mainly based on arXiv:1210.220

From Phase Space to Integrable Representations and Level-Rank Duality

We explicitly find representations for different large $N$ phases of Chern-Simons matter theory on $S^2\times S^1$. These representations are characterised by Young diagrams. We show that no-gap and lower-gap phase of Chern-Simons-matter theory correspond to integrable representations of $SU(N)_k$ affine Lie algebra, where as upper-cap phase corresponds to integrable representations of $SU(k-N)_k$ affine Lie algebra. We use phase space description of arXiv:0711.0133 to obtain these representations and argue how putting a cap on eigenvalue distribution forces corresponding representations to be integrable. We also prove that the Young diagrams corresponding to lower-gap and upper-cap representations are related to each other by transposition under level-rank duality. Finally we draw phase space droplets for these phases and show how information about eigenvalue and Young diagram descriptions can be captured in topologies of these droplets in a unified way.Comment: 37 pages, 10 figures, v2 Introduction extended, References adde

Weyl formula and thermodynamics of geometric flow

We study the Weyl formula for the asymptotic number of eigenvalues of the Laplace-Beltrami operator with Dirichlet boundary condition on a Riemannian manifold in the context of geometric flows. Assuming the eigenvalues to be the energies of some associated statistical system, we show that geometric flows are directly related with the direction of increasing entropy chosen. For a closed Riemannian manifold we obtain a volume preserving flow of geometry being equivalent to the increment of Gibbs entropy function derived from the spectrum of Laplace-Beltrami operator. Resemblance with Arnowitt, Deser, and Misner (ADM) formalism of gravity is also noted by considering open Riemannian manifolds, directly equating the geometric flow parameter and the direction of increasing entropy as time direction.Comment: 7 page