Masaki Tezuka

Black Holes and Random Matrices

A bstractWe argue that the late time behavior of horizon fluctuations in large anti-de Sitter (AdS) black holes is governed by the random matrix dynamics characteristic of quantum chaotic systems. Our main tool is the Sachdev-Ye-Kitaev (SYK) model, which we use as a simple model of a black hole. We use an analytically continued partition function |Z(β + it)|2 as well as correlation functions as diagnostics. Using numerical techniques we establish random matrix behavior at late times. We determine the early time behavior exactly in a double scaling limit, giving us a plausible estimate for the crossover time to random matrix behavior. We use these ideas to formulate a conjecture about general large AdS black holes, like those dual to 4D super-Yang-Mills theory, giving a provisional estimate of the crossover time. We make some preliminary comments about challenges to understanding the late time dynamics from a bulk point of view.

Creating and probing the Sachdev–Ye–Kitaev model with ultracold gases: Towards experimental studies of quantum gravity

We suggest that the holographic principle, combined with recent technological advances in atomic, molecular, and optical physics, can lead to experimental studies of quantum gravity. As a specific example, we consider the Sachdev-Ye-Kitaev (SYK) model, which consists of spin-polarized fermions with an all-to-all complex random two-body hopping and has been conjectured to be dual to a certain quantum gravitational system. Achieving low-temperature states of the SYK model is interpreted as a realization of a stringy black hole, provided that the holographic duality is true. We introduce a variant of the SYK model, in which the random two-body hopping is real. This model is equivalent to the origincal SYK model in the large-

Density-Matrix Renormalization Group Study of Trapped Imbalanced Fermi Condensates

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Phase diagram for the one-dimensional Hubbard-Holstein model : A density-matrix renormalization group study

limit. We show that this model can be created in principle by confining ultracold fermionic atoms into optical lattices and coupling two atoms with molecular states via photo-association lasers. This development serves as an important first step towards an experimental realization of such systems dual to quantum black holes. We also show how to measure out-of-time-order correlation functions of the SYK model, which allow for identifying the maximally chaotic property of the black hole.

Ground states and dynamics of population-imbalanced Fermi condensates in one dimension

The density-matrix renormalization group is employed to investigate a harmonically trapped imbalanced Fermi condensate based on a one-dimensional attractive Hubbard model. The obtained density profile shows a flattened population difference of spin-up and spin-down components at the center of the trap, and exhibits phase separation between the condensate and unpaired majority atoms for a certain range of the interaction and population imbalance P. The two-particle density matrix reveals that the sign of the order parameter changes periodically, demonstrating the realization of the Fulde-Ferrell-Larkin-Ovchinnikov phase. The minority spin atoms contribute to the quasicondensate up to at least P approximately 0.8. Possible experimental situations to test our predictions are discussed.

Density-matrix renormalization group study of pairing when electron-electron and electron-phonon interactions coexist: effect of the electronic band structure.

RIKEN, Wako, Saitama 351-0198, Japan(Dated: February 1, 2008)Phase diagram of the Hubbard-Holstein model in the coexistence of electron-electron and electron-phonon interactions has been theoretically obtained with the density-matrix renormalization groupmethod for one-dimensional (1D) systems, where an improved warm-up (the recursive sweep) pro-cedure has enabled us to calculate various correlation functions. We have examined the cases of(i) the systems half-filled by electrons for the full parameter space spanned by the electron-electronand electron-phonon coupling constants and the phonon frequency, (ii) non-half-filled system, and(iii) trestle lattice. For (i), we have detected a region where both the charge and on-site pairingcorrelations decay with power-laws in real space, which suggests a metallic behavior. While pairingcorrelations are not dominant in (i), we have found that they become dominant as the system isdoped in (ii), or as the electronic band structure is modified (with a broken electron-hole symmetry)in (iii) in certain parameter regions.

Onset of Random Matrix Behavior in Scrambling Systems

By using the numerically exact density-matrix renormalization group (DMRG) approach, we investigate the ground states of harmonically trapped one-dimensional (1D) fermions with population imbalance and find that the Larkin-Ovchinnikov (LO) state, which is a condensed state of fermion pairs with nonzero center-of-mass momentum, is realized for a wide range of parameters. The phase diagram comprising the two phases of (i) an LO state at the trap center and a balanced condensate at the periphery and (ii) an LO state at the trap center and a pure majority component at the periphery is obtained. The reduced two-body density matrix indicates that most of the minority atoms contribute to the LO-type quasi-condensate. With the time-dependent DMRG, we also investigate the real-time dynamics of a system of 1D fermions in response to a spin-flip excitation.

Reentrant topological transitions in a quantum wire/superconductor system with quasiperiodic lattice modulation

The density-matrix renormalization group is used to study the pairing when both electron-electron and electron-phonon interactions are strong in the Holstein-Hubbard model at half filling in a region intermediate between the adiabatic (Migdals) and antiadiabatic limits. We have found (i) the pairing correlation obtained for a one-dimensional system is nearly degenerate with the charge density-wave correlation in a region where the phonon-induced attraction is comparable with the electron-electron repulsion, but (ii) pairing becomes dominant when we destroy the electron-hole symmetry in a trestle lattice. This provides an instance in which pairing can arise, in a lattice-structure dependent manner, from coexisting electron-electron and electron-phonon interactions.

Interacting-Holstein and extended-Holstein bipolarons

A bstractThe fine grained energy spectrum of quantum chaotic systems is widely believed to be described by random matrix statistics. A basic scale in such a system is the energy range over which this behavior persists. We define the corresponding time scale by the time at which the linearly growing ramp region in the spectral form factor begins. We call this time tramp. The purpose of this paper is to study this scale in many-body quantum systems that display strong chaos, sometimes called scrambling systems. We focus on randomly coupled qubit systems, both local and k-local (all-to-all interactions) and the Sachdev-Ye-Kitaev (SYK) model. Using numerical results, analytic estimates for random quantum circuits, and a heuristic analysis of Hamiltonian systems we find the following results. For geometrically local systems with a conservation law we find tramp is determined by the diffusion time across the system, order N2 for a 1D chain of N qubits. This is analogous to the behavior found for local one-body chaotic systems. For a k-local system like SYK the time is order log N but with a different prefactor and a different mechanism than the scrambling time. In the absence of any conservation laws, as in a generic random quantum circuit, we find tramp ∼ log N, independent of connectivity.

Topological Properties of Ultracold Bosons in One-Dimensional Quasiperiodic Optical Lattice

We study the condition for a topological superconductor (TS) phase with end Majorana fermions to appear when a quasiperiodic lattice modulation is applied to a one-dimensional quantum wire with strong spin-orbit interaction situated under a magnetic field and in proximity to a superconductor. By density-matrix renormalization group analysis, we find that multiple topological phases with Majorana end modes are realized in finite ranges of the filling factor, showing a sequence of reentrant transitions as the chemical potential is tuned. The locations of these phases reflect the structure of bands in the non-interacting case, which exhibits a distinct self-similar structure. The stability of the TS in the presence of an on-site interaction or a harmonic trap potential is also discussed.