Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Response under arbitrary groups of point transformations of multipoint-correlation functions of local fluctuating quantities

Alan Martin Wolsky, Melville S. Green

Published Feb 1, 1974• 7 citations
Publisher

Abstract

We derive an identity which determines the infinitesimal response of the cumulant average of the product of $m$ arbitrary local fluctuating quantities (connected $m$-point functions) to changes in its arguments engendered by a member of an arbitrary continuous group of point transformations. We illustrate the use of the identity in a variety of cases, and in particular we show that there is an intimate connection between the covariance of the connected $m$-point functions under the special conformal group and under the group of dilations. By combining our identity with the assumptions of the operator algebra, we find that at the critical point the connected density $m$-point functions are covariant under both groups. We also give another derivation for an expression for the exponent $x=(\frac{1}{2})(5\ensuremath{-}\ensuremath{\eta})$, which was previously found by Green and Gunton.

DOI
10.1103/physreva.9.957

Related papers

Dilation and Conformal Covariance of Multipoint Correlation Functions and Dimensions of Fluctuating Quantities at the Critical Point of Fluids

Alan Martin Wolsky, M. S. Green, J. D. Gunton

The operator algebra and an identity in the statistical mechanics of fluids are used to demonstrate that the cumulant average of arbitrary products of appropriate combinations of local fluctuating energy and density are scale and conformal covariant at the critical point. These combinations are uncorrelated and are found by diagonalizing a matrix whose elements are energy and density derivatives of pressure-density and pressure-energy correlations. The eigenvalues are the dimensions of the appropriate variables.

Physical Review Letters•Nov 5, 1973•4 citations

Conformal covariance in the framework of Wilson's renormalization group approach

Lothar Schäfer

The author constructs a conformal operator in analogy to the generating operator of Wilson's incomplete-integration renormalization group. The invariance of the partition function with respect to that conformal operation yields identities among the cumulants. Evaluating these identities he finds a generalized and corrected form of the selection rule which determines those two-point cumulants which show a long-range tail. A general equation which governs the asymptotic form of the three-point cumulants is established. It is solved for several examples which involve operators of vector- or tensor-type. It is found that surface effects cannot be excluded a priori. However, the asymptotic expressions for the cumulants are consistent with a neglect of surface effects.

Journal of Physics A Mathematical and General•Mar 1, 1976•25 citations

Nonlinear Projection for Ballistic Correlation Functions: a Formula in Terms of Minimal Connected Covers

Benjamin Doyon

Abstract In many-body systems, the dynamics is governed, at large scales of space and time, by the hydrodynamic principle of projection onto the conserved densities admitted by the model. This is formalised as local relaxation of fluctuations in the Ballistic Macroscopic Fluctuation Theory, and is a nonlinear version of the Boltzmann-Gibbs principle. We use it to derive a projection formula, expressing n -point connected correlation functions (cumulants) of local observables at different space-time points, in terms of those of conserved densities. This applies in every $$d\ge 1$$ d ≥ 1 spatial dimensions and under the ballistic scaling of space and time, both in and out of equilibrium. It generalises the well-known linear-response principle for 2-point functions. For higher-point functions, one needs to account for nonlinear fluctuations of conserved densities and, correspondingly, higher derivatives of local averages. Using Malyshev’s formula for the cumulant expansion, and keeping the leading order, the result is a nonlinear projection, expressed as a sum of products of correlation functions of conserved densities with equilibrium multivariances as coefficients. The sum is combinatorially organised via certain covers of the set of space-time points, which we call “minimal connected covers”. We use this in order to get general, explicit formulas for two- and three-point functions in stationary states, expressed in terms of thermodynamic and Euler-scale data.

arXiv:2506.05266•Journal of Statistical Physics•Jul 30, 2026•0 citations

CONFORMAL ALGEBRA AND MULTIPOINT CORRELATION FUNCTIONS IN 2D STATISTICAL MODELS

Vl.S. Dotsenko, V.A. Fateev

Abstract Based on the conformal algebra approach, a general technique is given for the calculation of multipoint correlation functions in 2D statistical models at the critical point. Particular conformal operator algebras are found for operators of the 2D q-component Potts model (1

World Scientific series in 20th century physics•Aug 1, 1996•0 citations

Cumulants and scaling functions of infinite matrix product states

Jason C. Pillay, Ian P. McCulloch

The order parameter cumulants of infinite matrix product ground states are evaluated across a quantum phase transition. A scheme using the Binder cumulant, finite-entanglement scaling, and scaling functions to obtain the critical point and exponents of the correlation length and cumulants is presented. Analogous to the scaling relations that relate the exponents of various thermodynamic quantities, a cumulant exponent relation is derived and used to check the consistency and relationship between the cumulant exponents. This scheme gives a numerically economical way of accurately obtaining the critical exponents. Examples of this scheme are shown for four one-dimensional models---the transverse field Ising model, the topological Kondo insulator, the $S=1$ Heisenberg chain with single-ion anisotropy and the Bose-Hubbard model. A two-dimensional model is also exemplified in the square lattice transverse field Ising model on an infinite cylinder. These exemplary systems portray a variety of local and string order parameters as well as phase transition classes that can be studied with the scaling functions and infinite matrix product states.

arXiv:1906.03833•Physical review. B./Physical review. B•Dec 24, 2019•10 citations

$$ T\overline{T} $$ deformation of correlation functions

John Cardy

Abstract We study the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the $$ \lambda T\overline{T} $$ λT T ¯ deformation, suitably regularized. We show that this may be viewed in terms of the evolution of each field, with a Dirac-like string being attached at each infinitesimal step. The deformation then acts as a derivation on the whole operator algebra, satisfying the Leibniz rule. We derive an explicit equation which allows for the analysis of UV divergences, which may be absorbed into a non-local field renormalization to give correlation functions which are UV finite to all orders, satisfying a (deformed) operator product expansion and a Callan-Symanzik equation. We solve this in the case of a deformed CFT, showing that the Fourier-transformed renormalized two-point functions behave as k 2∆+2λk2, where ∆ is their IR conformal dimension. We discuss in detail deformed Noether currents, including the energy-momentum tensor, and show that, although they also become non-local, when suitably improved they remain finite, conserved and satisfy the expected Ward identities. Finally, we discuss how the equivalence of the $$ T\overline{T} $$ T T ¯ deformation to a state-dependent coordinate transformation emerges in this picture.

arXiv:1907.03394•Journal of High Energy Physics•Dec 1, 2019•6 citations
We derive an identity which determines the infinitesimal response of the cumulant average of the product of mmm arbitrary local fluctuating quantities (connected mmm-point functions) to changes in its arguments engendered by a member of an arbitrary continuous group of point transformations. We illustrate the use of the identity in a variety of cases, and in particular we show that there is an intimate connection between the covariance of the connected mmm-point functions under the special conformal group and under the group of dilations. By combining our identity with the assumptions of the operator algebra, we find that at the critical point the connected density mmm-point functions are covariant under both groups. We also give another derivation for an expression for the exponent x=(12)(5\ensuremath−\ensuremathη)x=(\frac{1}{2})(5\ensuremath{-}\ensuremath{\eta})x=(21​)(5\ensuremath−\ensuremathη), which was previously found by Green and Gunton.
The operator algebra and an identity in the statistical mechanics of fluids are used to demonstrate that the cumulant average of arbitrary products of appropriate combinations of local fluctuating energy and density are scale and conformal covariant at the critical point. These combinations are uncorrelated and are found by diagonalizing a matrix whose elements are energy and density derivatives of pressure-density and pressure-energy correlations. The eigenvalues are the dimensions of the appropriate variables.
The author constructs a conformal operator in analogy to the generating operator of Wilson's incomplete-integration renormalization group. The invariance of the partition function with respect to that conformal operation yields identities among the cumulants. Evaluating these identities he finds a generalized and corrected form of the selection rule which determines those two-point cumulants which show a long-range tail. A general equation which governs the asymptotic form of the three-point cumulants is established. It is solved for several examples which involve operators of vector- or tensor-type. It is found that surface effects cannot be excluded a priori. However, the asymptotic expressions for the cumulants are consistent with a neglect of surface effects.
Abstract In many-body systems, the dynamics is governed, at large scales of space and time, by the hydrodynamic principle of projection onto the conserved densities admitted by the model. This is formalised as local relaxation of fluctuations in the Ballistic Macroscopic Fluctuation Theory, and is a nonlinear version of the Boltzmann-Gibbs principle. We use it to derive a projection formula, expressing n -point connected correlation functions (cumulants) of local observables at different space-time points, in terms of those of conserved densities. This applies in every d≥1d\ge 1d≥1 d ≥ 1 spatial dimensions and under the ballistic scaling of space and time, both in and out of equilibrium. It generalises the well-known linear-response principle for 2-point functions. For higher-point functions, one needs to account for nonlinear fluctuations of conserved densities and, correspondingly, higher derivatives of local averages. Using Malyshev’s formula for the cumulant expansion, and keeping the leading order, the result is a nonlinear projection, expressed as a sum of products of correlation functions of conserved densities with equilibrium multivariances as coefficients. The sum is combinatorially organised via certain covers of the set of space-time points, which we call “minimal connected covers”. We use this in order to get general, explicit formulas for two- and three-point functions in stationary states, expressed in terms of thermodynamic and Euler-scale data.
The order parameter cumulants of infinite matrix product ground states are evaluated across a quantum phase transition. A scheme using the Binder cumulant, finite-entanglement scaling, and scaling functions to obtain the critical point and exponents of the correlation length and cumulants is presented. Analogous to the scaling relations that relate the exponents of various thermodynamic quantities, a cumulant exponent relation is derived and used to check the consistency and relationship between the cumulant exponents. This scheme gives a numerically economical way of accurately obtaining the critical exponents. Examples of this scheme are shown for four one-dimensional models---the transverse field Ising model, the topological Kondo insulator, the S=1S=1S=1 Heisenberg chain with single-ion anisotropy and the Bose-Hubbard model. A two-dimensional model is also exemplified in the square lattice transverse field Ising model on an infinite cylinder. These exemplary systems portray a variety of local and string order parameters as well as phase transition classes that can be studied with the scaling functions and infinite matrix product states.
TT‾ T\overline{T} TT deformation of correlation functions
Abstract We study the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the λTT‾ \lambda T\overline{T} λTT λT T ¯ deformation, suitably regularized. We show that this may be viewed in terms of the evolution of each field, with a Dirac-like string being attached at each infinitesimal step. The deformation then acts as a derivation on the whole operator algebra, satisfying the Leibniz rule. We derive an explicit equation which allows for the analysis of UV divergences, which may be absorbed into a non-local field renormalization to give correlation functions which are UV finite to all orders, satisfying a (deformed) operator product expansion and a Callan-Symanzik equation. We solve this in the case of a deformed CFT, showing that the Fourier-transformed renormalized two-point functions behave as k 2∆+2λk2, where ∆ is their IR conformal dimension. We discuss in detail deformed Noether currents, including the energy-momentum tensor, and show that, although they also become non-local, when suitably improved they remain finite, conserved and satisfy the expected Ward identities. Finally, we discuss how the equivalence of the TT‾ T\overline{T} TT T T ¯ deformation to a state-dependent coordinate transformation emerges in this picture.