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

arXiv:hep-lat/9402012

Variational description of statistical field theories using Daubechies' wavelets

Best, Christoph, Schaefer, Andreas

Published Feb 16, 1994• 7 citations
arXivHTMLPDF

Abstract

We investigate the description of statistical field theories using Daubechies' orthonormal compact wavelets on a lattice. A simple variational approach is used to extend mean field theory and make predictions for the fluctuation strengths of wavelet coefficients and thus for the correlation function. The results are compared to Monte Carlo simulations. We find that wavelets provide a reasonable description of critical phenomena with only a small number of variational parameters. This lets us hope for an implementation of the renormalization group in wavelet space.

DOI
10.48550/arxiv.hep-lat/9402012

Related papers

The application of wavelets in field theory

Somchai Ruangpermpool, Ahpisit Ungkitchanukit

In this thesis, it has been studied and repeated C. Best and A. Schafer’s work on the application of wavelet to the Gaussian model and the Landau-Ginzburg model. The fields of the Gaussian model are expanded in wavelet representation and with the fluctuation strengths of wavelet coefficients as variational parameter, the free energy of the model in Gaussian ensemble is minimized analytically. We then numerically recalculate the fluctuation strengths of wavelet coefficients and thus the correlation function. The result confirms that wavelets provide a reasonable description of the critical phenomena with only a small number of variational parameters, which is the same as the application of wavelet in other fields. In Landau-Ginzburg model, by analytic approximation, it is shown how the phase-transition is represented in wavelet space and how critical exponents can be computed from the wavelet expansion.

•Jan 1, 1996•0 citations

Looking through wavelets to view the Ising problem

Gagan Mirchandani, John T. Evans, Robert R. Snapp +1 more

In the theoretical study of phase transitions and critical phenomena, renormalization groups (RG) provide a course-grain representation of a many particle system and aid in the determination of the critical exponents associated with phase transitions. Monte Carlo and Molecular Dynamic simulations are often the simulation tools used for verification. However, the rich structure and multiscale properties of wavelets have found limited use in these applications. To promote exploration here, we apply and investigate wavelets to the solution of the simplest of molecular models, the Ising model and evaluate the most fundamental of objects in statistical mechanics, the partition function, now based on a wavelet-modifed Hamiltonian. As in we iteratively compute the waveletbased coarse grain 2-D finite-size Ising model. Thermodynamic properties are calculated to predict, given the finite size effect, potential for phase transition. Exact calculations are confirmed using Monte Carlo simulations. Based on this investigation and corresponding insight gained, we propose a wavelet-based investigation in three open areas: Wavelet Bases, Critical Exponents and Phase Transitions, Renormalization Groups and Continuous Wavelets.

IEEE Statistical Signal Processing Workshop (SSP)•Aug 1, 2009•1 citation

Wavelet Matrix Product States for Quantum Fields

Molly Kaplan, Antoine Tilloy

We introduce a variational method to solve continuum quantum models with discrete tensor network techniques. The method leverages wavelet matrix product states (wMPS): matrix product states built on top of sufficiently regular ($N\geq 6$) Daubechies scaling functions. These states live in the continuum field theory Fock space, have finite energy density, and can be optimized with standard algorithms, without restriction to free theories. Further, exploiting the multi-resolution analysis built into wavelets, and its quantum circuit description, we can iteratively refine wMPS to obtain accurate approximations at arbitrarily fine length-scales. We showcase the efficiency of the method on the Lieb-Liniger model, computing energy density and correlation functions.

quant-pharXiv:2606.23823•Jun 24, 2026•0 citations

Variational theorem for vector-mean-field theories of statistical transmutation

Claudius Gros, S. M. Girvin, Geoffrey Canright +1 more

We examine the validity of vector-mean-field theory (VMFT) for statistical transmutation on large lattices with a high density of particles per site (1/2 and 1/4). We take as a difficult test case the representation of hard-core bosons as fermions plus attached flux tubes. We use a variational Monte Carlo method to test the variational properties of the mean-field ground-state wave function against the predictions of the VMFT. We find a discrepancy of order 1 in the thermodynamic limit. This leads us to postulate that a better formulation on a lattice may be that of a renormalized vector-mean-field theory. We show that the renormalization coefficients can be understood by an analysis of the phase fluctuations (whose magnitude we estimate) of the long-range gauge interaction. These phase fluctuations are of order \ensuremath{\pi} on the lattice (thus leading to a breakdown of VMFT on the lattice) while they vanish in a continuum formulation. We give a detailed discussion of the qualitative differences of VMFT on the lattice versus the continuum. In particular, we examine the effect of having lines of zeros (lattice) versus points of zeros (continuum) for the nodes of the many-body wave function. In addition, a remarkable variational theorem is discovered for the ground-state wave function of the VMFT.

Physical review. B, Condensed matter•Mar 1, 1991•10 citations

Holographic Construction of Quantum Field Theory using Wavelets

Sukhwinder Singh, Gavin K. Brennen

Wavelets encode data at multiple resolutions, which in a wavelet description of a quantum field theory, allows for fields to carry, in addition to space-time coordinates, an extra dimension: scale. A recently introduced Exact Holographic Mapping [C.H. Lee and X.-L. Qi, Phys. Rev. B 93, 035112 (2016)] uses the Haar wavelet basis to represent the free Dirac fermionic quantum field theory (QFT) at multiple renormalization scales thereby inducing an emergent bulk geometry in one higher dimension. This construction is, in fact, generic and we show how higher families of Daubechies wavelet transforms of 1+1 dimensional scalar bosonic QFT generate a bulk description with a variable rate of renormalization flow. In the massless case, where the boundary is described by conformal field theory, the bulk correlations decay with distance consistent with an Anti-de-Sitter space (AdS3) metric whose radius of curvature depends on the wavelet family used. We propose an experimental demonstration of the bulk/boundary correspondence via a digital quantum simulation using Gaussian operations on a set of quantum harmonic oscillator modes.

arXiv:1606.05068•Jun 16, 2016•9 citations

Monte Carlo Renormalization Group for Classical Lattice Models with Quenched Disorder

Yantao Wu, Roberto Car

We extend to quenched-disordered systems the variational scheme for real-space renormalization group calculations that we recently introduced for homogeneous spin Hamiltonians. When disorder is present our approach gives access to the flow of the renormalized Hamiltonian distribution, from which one can compute the critical exponents if the correlations of the renormalized couplings retain finite range. Key to the variational approach is the bias potential found by minimizing a convex functional in statistical mechanics. This potential reduces dramatically the Monte Carlo relaxation time in large disordered systems. We demonstrate the method with applications to the two-dimensional dilute Ising model, the random transverse field quantum Ising chain, and the random field Ising in two- and three-dimensional lattices.

arXiv:1810.09579•Physical Review Letters•Nov 3, 2020•2 citations
We investigate the description of statistical field theories using Daubechies' orthonormal compact wavelets on a lattice. A simple variational approach is used to extend mean field theory and make predictions for the fluctuation strengths of wavelet coefficients and thus for the correlation function. The results are compared to Monte Carlo simulations. We find that wavelets provide a reasonable description of critical phenomena with only a small number of variational parameters. This lets us hope for an implementation of the renormalization group in wavelet space.
In this thesis, it has been studied and repeated C. Best and A. Schafer’s work on the application of wavelet to the Gaussian model and the Landau-Ginzburg model. The fields of the Gaussian model are expanded in wavelet representation and with the fluctuation strengths of wavelet coefficients as variational parameter, the free energy of the model in Gaussian ensemble is minimized analytically. We then numerically recalculate the fluctuation strengths of wavelet coefficients and thus the correlation function. The result confirms that wavelets provide a reasonable description of the critical phenomena with only a small number of variational parameters, which is the same as the application of wavelet in other fields. In Landau-Ginzburg model, by analytic approximation, it is shown how the phase-transition is represented in wavelet space and how critical exponents can be computed from the wavelet expansion.
In the theoretical study of phase transitions and critical phenomena, renormalization groups (RG) provide a course-grain representation of a many particle system and aid in the determination of the critical exponents associated with phase transitions. Monte Carlo and Molecular Dynamic simulations are often the simulation tools used for verification. However, the rich structure and multiscale properties of wavelets have found limited use in these applications. To promote exploration here, we apply and investigate wavelets to the solution of the simplest of molecular models, the Ising model and evaluate the most fundamental of objects in statistical mechanics, the partition function, now based on a wavelet-modifed Hamiltonian. As in we iteratively compute the waveletbased coarse grain 2-D finite-size Ising model. Thermodynamic properties are calculated to predict, given the finite size effect, potential for phase transition. Exact calculations are confirmed using Monte Carlo simulations. Based on this investigation and corresponding insight gained, we propose a wavelet-based investigation in three open areas: Wavelet Bases, Critical Exponents and Phase Transitions, Renormalization Groups and Continuous Wavelets.
We introduce a variational method to solve continuum quantum models with discrete tensor network techniques. The method leverages wavelet matrix product states (wMPS): matrix product states built on top of sufficiently regular (N≥6N\geq 6N≥6) Daubechies scaling functions. These states live in the continuum field theory Fock space, have finite energy density, and can be optimized with standard algorithms, without restriction to free theories. Further, exploiting the multi-resolution analysis built into wavelets, and its quantum circuit description, we can iteratively refine wMPS to obtain accurate approximations at arbitrarily fine length-scales. We showcase the efficiency of the method on the Lieb-Liniger model, computing energy density and correlation functions.
We examine the validity of vector-mean-field theory (VMFT) for statistical transmutation on large lattices with a high density of particles per site (1/2 and 1/4). We take as a difficult test case the representation of hard-core bosons as fermions plus attached flux tubes. We use a variational Monte Carlo method to test the variational properties of the mean-field ground-state wave function against the predictions of the VMFT. We find a discrepancy of order 1 in the thermodynamic limit. This leads us to postulate that a better formulation on a lattice may be that of a renormalized vector-mean-field theory. We show that the renormalization coefficients can be understood by an analysis of the phase fluctuations (whose magnitude we estimate) of the long-range gauge interaction. These phase fluctuations are of order \ensuremath{\pi} on the lattice (thus leading to a breakdown of VMFT on the lattice) while they vanish in a continuum formulation. We give a detailed discussion of the qualitative differences of VMFT on the lattice versus the continuum. In particular, we examine the effect of having lines of zeros (lattice) versus points of zeros (continuum) for the nodes of the many-body wave function. In addition, a remarkable variational theorem is discovered for the ground-state wave function of the VMFT.
Wavelets encode data at multiple resolutions, which in a wavelet description of a quantum field theory, allows for fields to carry, in addition to space-time coordinates, an extra dimension: scale. A recently introduced Exact Holographic Mapping [C.H. Lee and X.-L. Qi, Phys. Rev. B 93, 035112 (2016)] uses the Haar wavelet basis to represent the free Dirac fermionic quantum field theory (QFT) at multiple renormalization scales thereby inducing an emergent bulk geometry in one higher dimension. This construction is, in fact, generic and we show how higher families of Daubechies wavelet transforms of 1+1 dimensional scalar bosonic QFT generate a bulk description with a variable rate of renormalization flow. In the massless case, where the boundary is described by conformal field theory, the bulk correlations decay with distance consistent with an Anti-de-Sitter space (AdS3) metric whose radius of curvature depends on the wavelet family used. We propose an experimental demonstration of the bulk/boundary correspondence via a digital quantum simulation using Gaussian operations on a set of quantum harmonic oscillator modes.
We extend to quenched-disordered systems the variational scheme for real-space renormalization group calculations that we recently introduced for homogeneous spin Hamiltonians. When disorder is present our approach gives access to the flow of the renormalized Hamiltonian distribution, from which one can compute the critical exponents if the correlations of the renormalized couplings retain finite range. Key to the variational approach is the bias potential found by minimizing a convex functional in statistical mechanics. This potential reduces dramatically the Monte Carlo relaxation time in large disordered systems. We demonstrate the method with applications to the two-dimensional dilute Ising model, the random transverse field quantum Ising chain, and the random field Ising in two- and three-dimensional lattices.