scholarly journals A basis of analytic functionals for CFTs in general dimension

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Dalimil Mazáč ◽  
Leonardo Rastelli ◽  
Xinan Zhou
Keyword(s):  
Dimensional Space ◽  
Mean Field Theory ◽  
Mean Field ◽  
Linear Functionals ◽  
Spacetime Dimension ◽  
Boundedness Condition ◽  
Conformal Blocks ◽  
Infinite Dimensional ◽  
General Dimension ◽  
Twist Operators

Abstract We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition at infinity. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot.

RECENT APPLICATIONS OF THE DMRG METHOD

2006 ◽  
Vol 20 (19) ◽  
pp. 2624-2635
Author(s):  
KAREN HALLBERG
Keyword(s):  
Degrees Of Freedom ◽  
Mean Field Theory ◽  
Mean Field ◽  
Finite Size ◽  
Strongly Correlated Systems ◽  
Strongly Correlated ◽  
Correlated Systems ◽  
Infinite Dimensional ◽  
Low Dimensional ◽  
Time Dependent Problems

Since its inception, the DMRG method has been a very powerful tool for the calculation of physical properties of low-dimensional strongly correlated systems. It has been adapted to obtain dynamical properties and to consider finite temperature, time-dependent problems, bosonic degrees of freedom, the treatment of classical problems and non-equilibrium systems, among others. We will briefly review the method and then concentrate on its latest developments, describing some recent successful applications. In particular we will show how the dynamical DMRG can be used together with the Dynamical Mean Field Theory (DMFT) to solve the associated impurity problem in the infinite-dimensional Hubbard model. This method is used to obtain spectral properties of strongly correlated systems. With this algorithm, more complex problems having a larger number of degrees of freedom can be considered and finite-size effects can be minimized.

PSEUDOGAP BEHAVIOR IN THE INFINITE DIMENSIONAL ATTRACTIVE HUBBARD MODEL

2011 ◽  
Vol 25 (12n13) ◽  
pp. 973-978 ◽  
Author(s):  
AKIHISA KOGA ◽  
PHILIPP WERNER
Keyword(s):  
Hubbard Model ◽  
Quantum Monte Carlo ◽  
Mean Field Theory ◽  
Mean Field ◽  
Pair Potential ◽  
Infinite Dimensional ◽  
Spatial Dimensions ◽  
Time Quantum ◽  
Attractive Hubbard Model ◽  
The Stability

We study the attractive Hubbard model in infinite spatial dimensions by means of dynamical mean-field theory with continuous-time quantum Monte Carlo simulations. Calculating the pair potential and the spectral function, we discuss the stability of the superfluid state at low temperatures. The pseudogap behavior above the critical temperature is also addressed.

scholarly journals A Geometrical Analysis of Global Stability in Trained Feedback Networks

2019 ◽  
Vol 31 (6) ◽  
pp. 1139-1182 ◽  
Author(s):  
Francesca Mastrogiuseppe ◽  
Srdjan Ostojic
Keyword(s):  
Global Stability ◽  
Dimensional Space ◽  
Mean Field Theory ◽  
Mean Field ◽  
Finite Size ◽  
Analytical Description ◽  
Global Dynamics ◽  
Stability Properties ◽  
Starting Point ◽  
Feedback Networks

Recurrent neural networks have been extensively studied in the context of neuroscience and machine learning due to their ability to implement complex computations. While substantial progress in designing effective learning algorithms has been achieved, a full understanding of trained recurrent networks is still lacking. Specifically, the mechanisms that allow computations to emerge from the underlying recurrent dynamics are largely unknown. Here we focus on a simple yet underexplored computational setup: a feedback architecture trained to associate a stationary output to a stationary input. As a starting point, we derive an approximate analytical description of global dynamics in trained networks, which assumes uncorrelated connectivity weights in the feedback and in the random bulk. The resulting mean-field theory suggests that the task admits several classes of solutions, which imply different stability properties. Different classes are characterized in terms of the geometrical arrangement of the readout with respect to the input vectors, defined in the high-dimensional space spanned by the network population. We find that such an approximate theoretical approach can be used to understand how standard training techniques implement the input-output task in finite-size feedback networks. In particular, our simplified description captures the local and the global stability properties of the target solution, and thus predicts training performance.

scholarly journals Mean-field sparse optimal control

2014 ◽  
Vol 372 (2028) ◽  
pp. 20130400 ◽  
Author(s):  
Massimo Fornasier ◽  
Benedetto Piccoli ◽  
Francesco Rossi
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Mean Field Theory ◽  
Mean Field ◽  
Control Problems ◽  
Mean Field Limit ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Leaders And Followers ◽  
Sparse Optimal Control

We introduce the rigorous limit process connecting finite dimensional sparse optimal control problems with ODE constraints, modelling parsimonious interventions on the dynamics of a moving population divided into leaders and followers, to an infinite dimensional optimal control problem with a constraint given by a system of ODE for the leaders coupled with a PDE of Vlasov-type, governing the dynamics of the probability distribution of the followers. In the classical mean-field theory, one studies the behaviour of a large number of small individuals freely interacting with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect. In this paper, we address instead the situation where the leaders are actually influenced also by an external policy maker , and we propagate its effect for the number N of followers going to infinity. The technical derivation of the sparse mean-field optimal control is realized by the simultaneous development of the mean-field limit of the equations governing the followers dynamics together with the Γ -limit of the finite dimensional sparse optimal control problems.

scholarly journals Mean Field Game with Delay: A Toy Model

Risks ◽  
2018 ◽  
Vol 6 (3) ◽  
pp. 90
Author(s):  
Jean-Pierre Fouque ◽  
Zhaoyu Zhang
Keyword(s):  
Master Equation ◽  
Dimensional Space ◽  
Mean Field ◽  
Kolmogorov Equation ◽  
Linear Quadratic ◽  
Quadratic Mean ◽  
Mean Field Game ◽  
Infinite Dimensional ◽  
Player Game ◽  
Hamilton Jacobi Bellman

We study a toy model of linear-quadratic mean field game with delay. We “lift” the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We identify the corresponding master equation. A solution to this master equation is computed, and we show that it provides an approximation to a Nash equilibrium of the finite player game.

scholarly journals LOCAL ENTANGLEMENT ENTROPY AT THE MOTT METAL-INSULATOR TRANSITION IN INFINITE DIMENSIONS

2013 ◽  
Vol 27 (05) ◽  
pp. 1350034 ◽  
Author(s):  
DAN-DAN SU ◽  
XI DAI ◽  
NING-HUA TONG
Keyword(s):  
Entanglement Entropy ◽  
Mean Field Theory ◽  
Mean Field ◽  
Insulator Transition ◽  
Single Site ◽  
Infinite Dimensions ◽  
Metal Insulator Transition ◽  
Metal Insulator ◽  
Infinite Dimensional ◽  
Two Sides

We study the critical behavior of the single-site entanglement entropy S at the Mott metal-insulator transition in infinite-dimensional Hubbard model. For this model, the entanglement between a single site and rest of the lattice can be evaluated exactly, using the dynamical mean-field theory (DMFT). Both the numerical solution using exact diagonalization and the analytical one using two-site DMFT gives S - Sc∝ α log2[(1/2 - Dc)/Dc](U - Uc), with Dcthe double occupancy at Ucand α < 0 being different on two sides of the transition.

Microscopic description in the Gaussian approximation of the nematic-isotropic phase transition

1991 ◽  
Vol 69 (2) ◽  
pp. 154-160
Author(s):  
P. A. Pashkov ◽  
S. I. Vasiliev
Keyword(s):  
Field Theory ◽  
Phase Transition ◽  
Gaussian Approximation ◽  
Dimensional Space ◽  
Mean Field Theory ◽  
Mean Field ◽  
Isotropic Phase ◽  
Cluster Approximation ◽  
Field Fluctuations ◽  
The Mean

The nematic-isotropic phase transition of a system of nonpolar rodlike molecules is considered. The mean-field theory is extended to take into account the local field fluctuations in the Gaussian approximation. The calculations are performed in the five-dimensional space of the order-parameter components. The numerical results are obtained for the lattice system of molecules and compared with the calculations in the mean-field theory, cluster approximation, and Monte-Carlo simulation data.

Self-assembly of ABC dendrimer by real-space self-consistent mean field theory in a two-dimensional space

Soft Matter ◽  
2011 ◽  
Vol 7 (21) ◽  
pp. 10076 ◽  
Author(s):  
Bo Lin ◽  
Hongdong Zhang ◽  
Ping Tang ◽  
Feng Qiu ◽  
Yuliang Yang
Keyword(s):  
Field Theory ◽  
Self Assembly ◽  
Dimensional Space ◽  
Mean Field Theory ◽  
Mean Field ◽  
Real Space ◽  
Two Dimensional ◽  
Self Consistent

scholarly journals Dispersive CFT sum rules

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Simon Caron-Huot ◽  
Dalimil Mazáč ◽  
Leonardo Rastelli ◽  
David Simmons-Duffin
Keyword(s):  
Field Theory ◽  
Sum Rules ◽  
Mean Field Theory ◽  
Mean Field ◽  
Dispersion Relations ◽  
Position Space ◽  
Central Idea ◽  
Conformal Field ◽  
Regge Behavior ◽  
Twist Operators

Abstract We give a unified treatment of dispersive sum rules for four-point correlators in conformal field theory. We call a sum rule “dispersive” if it has double zeros at all double-twist operators above a fixed twist gap. Dispersive sum rules have their conceptual origin in Lorentzian kinematics and absorptive physics (the notion of double discontinuity). They have been discussed using three seemingly different methods: analytic functionals dual to double-twist operators, dispersion relations in position space, and dispersion relations in Mellin space. We show that these three approaches can be mapped into one another and lead to completely equivalent sum rules. A central idea of our discussion is a fully nonperturbative expansion of the correlator as a sum over Polyakov-Regge blocks. Unlike the usual OPE sum, the Polyakov-Regge expansion utilizes the data of two separate channels, while having (term by term) good Regge behavior in the third channel. We construct sum rules which are non-negative above the double-twist gap; they have the physical interpretation of a subtracted version of “superconvergence” sum rules. We expect dispersive sum rules to be a very useful tool to study expansions around mean-field theory, and to constrain the low-energy description of holographic CFTs with a large gap. We give examples of the first kind of applications, notably we exhibit a candidate extremal functional for the spin-two gap problem.

Sign in / Sign up

Export Citation Format

Share Document