194,815 research outputs found
Two-time Green's functions and spectral density method in nonextensive quantum statistical mechanics
We extend the formalism of the thermodynamic two-time Green's functions to
nonextensive quantum statistical mechanics. Working in the optimal Lagrangian
multipliers representation, the -spectral properties and the methods for a
direct calculation of the two-time % -Green's functions and the related
-spectral density ( measures the nonextensivity degree) for two generic
operators are presented in strict analogy with the extensive ()
counterpart. Some emphasis is devoted to the nonextensive version of the less
known spectral density method whose effectiveness in exploring equilibrium and
transport properties of a wide variety of systems has been well established in
conventional classical and quantum many-body physics. To check how both the
equations of motion and the spectral density methods work to study the
-induced nonextensivity effects in nontrivial many-body problems, we focus
on the equilibrium properties of a second-quantized model for a high-density
Bose gas with strong attraction between particles for which exact results exist
in extensive conditions. Remarkably, the contributions to several thermodynamic
quantities of the -induced nonextensivity close to the extensive regime are
explicitly calculated in the low-temperature regime by overcoming the
calculation of the grand-partition function.Comment: 48 pages, no figure
Levels of discontinuity, limit-computability, and jump operators
We develop a general theory of jump operators, which is intended to provide
an abstraction of the notion of "limit-computability" on represented spaces.
Jump operators also provide a framework with a strong categorical flavor for
investigating degrees of discontinuity of functions and hierarchies of sets on
represented spaces. We will provide a thorough investigation within this
framework of a hierarchy of -measurable functions between arbitrary
countably based -spaces, which captures the notion of computing with
ordinal mind-change bounds. Our abstract approach not only raises new questions
but also sheds new light on previous results. For example, we introduce a
notion of "higher order" descriptive set theoretical objects, we generalize a
recent characterization of the computability theoretic notion of "lowness" in
terms of adjoint functors, and we show that our framework encompasses ordinal
quantifications of the non-constructiveness of Hilbert's finite basis theorem
Eigenstate thermalization in the Sachdev-Ye-Kitaev model
The eigenstate thermalization hypothesis (ETH) explains how closed unitary
quantum systems can exhibit thermal behavior in pure states. In this work we
examine a recently proposed microscopic model of a black hole in AdS, the
so-called Sachdev-Ye-Kitaev (SYK) model. We show that this model satisfies the
eigenstate thermalization hypothesis by solving the system in exact
diagonalization. Using these results we also study the behavior, in
eigenstates, of various measures of thermalization and scrambling of
information. We establish that two-point functions in finite-energy eigenstates
approximate closely their thermal counterparts and that information is
scrambled in individual eigenstates. We study both the eigenstates of a single
random realization of the model, as well as the model obtained after averaging
of the random disordered couplings. We use our results to comment on the
implications for thermal states of the dual theory, i.e. the AdS black
hole.Comment: 36 pages, many figures; references added; matches published versio
Differential operators, pullbacks, and families of automorphic forms
This paper has two main parts. First, we construct certain differential
operators, which generalize operators studied by G. Shimura. Then, as an
application of some of these differential operators, we construct certain
p-adic families of automorphic forms. Building on the author's earlier work,
these differential operators map automorphic forms on a unitary group of
signature (n,n) to (vector-valued) automorphic forms on the product
of two unitary groups, where denotes
the unitary group associated to a Hermitian form of arbitrary
signature on an n-dimensional vector space. These differential operators have
both a p-adic and a C-infinity incarnation. In the scalar-weight, C-infinity
case, these operators agree with ones studied by Shimura. In the final section
of the paper, we also discuss some generalizations to other groups and
settings.
The results from this paper apply to the author's paper-in-preparation with
J. Fintzen, E. Mantovan, and I. Varma and to her ongoing joint project with M.
Harris, J. -S. Li, and C. Skinner; they also relate to her recent paper with X.
Wan.Comment: Accepted for publication in special issue of Annales Mathematiques du
Quebec in honor of Glenn Stevens's sixtieth birthda
- …