scholarly journals Extremal metrics for spectral functions of Dirac operators in even and odd dimensions

2012 ◽  
Vol 229 (2) ◽  
pp. 1001-1046 ◽  
Author(s):  
Niels Martin Møller
Keyword(s):  
Dirac Operators ◽  
Extremal Metrics ◽  
Spectral Functions ◽  
Odd Dimensions

scholarly journals High orders of Weyl series for the heat content

2011 ◽  
Vol 467 (2133) ◽  
pp. 2479-2499 ◽  
Author(s):  
Igor Travěnec ◽  
Ladislav Šamaj
Keyword(s):  
Asymptotic Form ◽  
Smooth Boundary ◽  
Heat Content ◽  
Dirichlet Laplacian ◽  
Spectral Functions ◽  
Circular Symmetry ◽  
Exactly Solvable ◽  
Elliptic Domain ◽  
Dimensional Domain ◽  
Odd Dimensions

This article concerns the Weyl series of spectral functions associated with the Dirichlet Laplacian in a d -dimensional domain with a smooth boundary. In the case of the heat kernel, Berry and Howls predicted the asymptotic form of the Weyl series characterized by a set of parameters. Here, we concentrate on another spectral function, the (normalized) heat content. We show on several exactly solvable examples that, for even d , the same asymptotic formula is valid with different values of the parameters. The considered domains are d -dimensional balls and two limiting cases of the elliptic domain with eccentricity ε : a slightly deformed disk ( ε →0) and an extremely prolonged ellipse ( ε →1). These cases include two-dimensional domains with circular symmetry and those with only one shortest periodic orbit for the classical billiard. We also analyse the heat content for the balls in odd dimensions d for which the asymptotic form of the Weyl series changes significantly.

scholarly journals Low-energy behavior of spin-liquid electron spectral functions

2013 ◽  
Vol 87 (4) ◽  
Author(s):  
Evelyn Tang ◽  
Matthew P. A. Fisher ◽  
Patrick A. Lee
Keyword(s):  
Low Energy ◽  
Spin Liquid ◽  
Spectral Functions ◽  
Energy Behavior

scholarly journals Hardy-type estimates for Dirac operators

2007 ◽  
Vol 40 (6) ◽  
pp. 885-900 ◽  
Author(s):  
J DOLBEAULT ◽  
M ESTEBAN ◽  
J DUOANDIKOETXEA ◽  
L VEGA
Keyword(s):  
Dirac Operators ◽  
Hardy Type

scholarly journals Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 477
Author(s):  
Katarzyna Górska ◽  
Andrzej Horzela
Keyword(s):  
Stochastic Processes ◽  
Spectral Function ◽  
Evolution Equations ◽  
Memory Function ◽  
Relaxation Phenomena ◽  
Stieltjes Function ◽  
Infinitely Divisible ◽  
Spectral Functions ◽  
Debye Relaxation ◽  
Relaxation Functions

In this paper, we show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semi-axis. Using only this property, it can be shown that the response and relaxation functions are non-negative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function M(t), which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes-based approach to the relaxation phenomena gives the possibility to identify the memory function M(t) with the Laplace (Lévy) exponent of some infinitely divisible stochastic processes and to introduce its partner memory k(t). Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process.

scholarly journals Charge algebra in Al(A)dSn spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Adrien Fiorucci ◽  
Romain Ruzziconi
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Boundary Structure ◽  
Asymptotic Symmetry ◽  
Renormalization Procedure ◽  
Field Dependent ◽  
Odd Dimensions

Abstract The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in n dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent 2-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the 2-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the Λ-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in n dimensions.

scholarly journals Solving the 2D SUSY Gross-Neveu-Yukawa model with conformal truncation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
A. Liam Fitzpatrick ◽  
Emanuel Katz ◽  
Matthew T. Walters ◽  
Yuan Xin
Keyword(s):  
Phase Transition ◽  
Critical Point ◽  
Fine Tuning ◽  
Yukawa Model ◽  
Spectral Functions ◽  
Scalar Superfield ◽  
Local Operators ◽  
Dimensionless Coupling ◽  
Zumino Model ◽  
Yukawa Theory

Abstract We use Lightcone Conformal Truncation to analyze the RG flow of the two-dimensional supersymmetric Gross-Neveu-Yukawa theory, i.e. the theory of a real scalar superfield with a ℤ2-symmetric cubic superpotential, aka the 2d Wess-Zumino model. The theory depends on a single dimensionless coupling $$ \overline{g} $$ g ¯ , and is expected to have a critical point at a tuned value $$ {\overline{g}}_{\ast } $$ g ¯ ∗ where it flows in the IR to the Tricritical Ising Model (TIM); the theory spontaneously breaks the ℤ2 symmetry on one side of this phase transition, and breaks SUSY on the other side. We calculate the spectrum of energies as a function of $$ \overline{g} $$ g ¯ and see the gap close as the critical point is approached, and numerically read off the critical exponent ν in TIM. Beyond the critical point, the gap remains nearly zero, in agreement with the expectation of a massless Goldstino. We also study spectral functions of local operators on both sides of the phase transition and compare to analytic predictions where possible. In particular, we use the Zamolodchikov C-function to map the entire phase diagram of the theory. Crucial to this analysis is the fact that our truncation is able to preserve supersymmetry sufficiently to avoid any additional fine tuning.

scholarly journals Local topological markers in odd dimensions

2021 ◽  
Vol 103 (15) ◽  
Author(s):  
Joseph Sykes ◽  
Ryan Barnett
Keyword(s):  
Odd Dimensions
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