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:1509.07414 · Nuclear Physics B

Matrix product approximations to conformal field theories

Robert König, Volkher B. Scholz

Published Apr 11, 2017• 6 citations
arXivHTMLPDF

Abstract

We establish rigorous error bounds for approximating correlation functions of conformal field theories (CFTs) by certain finite-dimensional tensor networks. For chiral CFTs, the approximation takes the form of a matrix product state. For full CFTs consisting of a chiral and an anti-chiral part, the approximation is given by a finitely correlated state. We show that the bond dimension scales polynomially in the inverse of the approximation error and sub-exponentially in inverse of the minimal distance between insertion points. We illustrate our findings using Wess–Zumino–Witten models, and show that there is a one-to-one correspondence between group-covariant MPS and our approximation.

DOI
10.1016/j.nuclphysb.2017.04.006

Related papers

Matrix Product Approximations to Multipoint Functions in Two-Dimensional Conformal Field Theory

Robert König, Volkher B. Scholz

Matrix product states (MPSs) illustrate the suitability of tensor networks for the description of interacting many-body systems: ground states of gapped 1D systems are approximable by MPSs, as shown by Hastings [M. B. Hastings, J. Stat. Mech. (2007) P08024]. By contrast, whether MPSs and more general tensor networks can accurately reproduce correlations in critical quantum systems or quantum field theories has not been established rigorously. Ample evidence exists: entropic considerations provide restrictions on the form of suitable ansatz states, and numerical studies show that certain tensor networks can indeed approximate the associated correlation functions. Here, we provide a complete positive answer to this question in the case of MPSs and 2D conformal field theory: we give quantitative estimates for the approximation error when approximating correlation functions by MPSs. Our work is constructive and yields an explicit MPS, thus providing both suitable initial values and a rigorous justification of variational methods.

arXiv:1601.00470•Physical Review Letters•Sep 14, 2016•7 citations

Conformal Field Theory from Lattice Fermions

Tobias J. Osborne, Alexander Stottmeister

Abstract We provide a rigorous lattice approximation of conformal field theories given in terms of lattice fermions in 1+1-dimensions, focussing on free fermion models and Wess–Zumino–Witten models. To this end, we utilize a recently introduced operator-algebraic framework for Wilson–Kadanoff renormalization. In this setting, we prove the convergence of the approximation of the Virasoro generators by the Koo–Saleur formula. From this, we deduce the convergence of lattice approximations of conformal correlation functions to their continuum limit. In addition, we show how these results lead to explicit error estimates pertaining to the quantum simulation of conformal field theories.

arXiv:2107.13834•Communications in Mathematical Physics•Nov 21, 2022•13 citations

Fractional Quantum Hall Matrix Product States For Interacting Conformal Field Theories

Benoit Estienne, Nicolas Regnault, Bogdan Andrei Bernevig

Using truncated conformal field theory (CFT), we present the formalism necessary to obtain exact matrix product state (MPS) representations for any fractional quantum hall model state which can be written as an expectation value of primary fields in some conformal field theory. The auxiliary bond space is the Hilbert space of the descendants of primary fields and matrix elements can be calculated using well-known CFT commutation relations. We obtain a site-independent MPS representation on the thin annulus and cylinder, while a site-dependent representation is possible on the disk and sphere geometries. We discuss the complications that arise due to the presence of null vectors in the theory's Hilbert space, and present an array of procedures which optimize the numerical calculation of matrix elements. By truncating the Hilbert space of the CFT, we then build an approximate MPS representation for Laughlin, Moore-Read, Read-Rezayi, Gaffnian, minimal $M(3,r+2)$ models, and a series of superconformal minimal models. We show how to obtain, in the MPS representation, the Orbital, Particle, and Real Space entanglement spectrum in both the finite and infinite systems.

arXiv:1311.2936•Nov 12, 2013•9 citations

Finite-size and finite bond dimension effects of tensor network renormalization

Atsushi Ueda, Masaki Oshikawa

We propose a general procedure for extracting the running coupling constants of the underlying field theory of a given classical statistical model on a two-dimensional lattice, combining tensor network renormalization (TNR) and the finite-size scaling theory of conformal field theory. By tracking the coupling constants at each scale, we are able to visualize the renormalization group flow and demonstrate it with the classical Ising and three-state Potts models. Furthermore, utilizing this methodology, we reveal the limitations due to finite bond dimension $D$ on TNR applied to critical systems. We find that a finite correlation length is imposed by the finite bond dimension in TNR, and it can be attributed to an emergent relevant perturbation that respects the symmetries of the system. The correlation length shows the same power-law dependence on $D$ as the ``finite entanglement scaling'' of the matrix product states.

arXiv:2302.06632•Physical review. B./Physical review. B•Jul 14, 2023•30 citations

Conformal data from finite entanglement scaling

Vid Stojevic, Jutho Haegeman, Ian P. McCulloch +2 more

In this paper, we apply the formalism of translation invariant (continuous) matrix product states in the thermodynamic limit to $(1+1)$-dimensional critical models. Finite bond dimension bounds the entanglement entropy and introduces an effective finite correlation length, so that the state is perturbed away from criticality. The assumption that the scaling hypothesis holds for this kind of perturbation is known in the literature as finite entanglement scaling. We provide further evidence for the validity of finite entanglement scaling and based on this formulate a scaling algorithm to estimate the central charge and critical exponents of the conformally invariant field theories describing the critical models under investigation. The algorithm is applied to three exemplary models; the cMPS version to the nonrelativistic Lieb-Liniger model and the relativistic massless boson, and MPS version to the one-dimensional quantum Ising model at the critical point. Another new aspect to our approach is that we directly use the (c)MPS induced correlation length rather than the bond dimension as scaling parameter. This choice is motivated by several theoretical arguments as well as by the remarkable accuracy of our results.

arXiv:1401.7654•Physical Review B•Jan 20, 2015•72 citations

On the origin of finite entanglement scaling

Luke Hodgkiss, Laurens Lootens, Atsushi Ueda +2 more

The concept of finite entanglement scaling forms one of the pillars on which the tensor network ecosystem is built. In this paper, we resolve the open problem of determining the actual perturbations induced by matrix product state approximations of critical systems, and we demonstrate that these can be quite different than the ones predicted by conformal field theory. To that aim, we develop a sparse linear solver to calculate the forward and backward derivatives of 2-dimensional tensor networks with respect to their defining parameters in an implicit way. This algorithm is of independent interest as it provides a primitive for the variational optimization of projected entangled pair states that circumvents the instabilities plaguing traditional automatic differentiation methods.

quant-pharXiv:2607.15124•Jul 17, 2026•0 citations
We establish rigorous error bounds for approximating correlation functions of conformal field theories (CFTs) by certain finite-dimensional tensor networks. For chiral CFTs, the approximation takes the form of a matrix product state. For full CFTs consisting of a chiral and an anti-chiral part, the approximation is given by a finitely correlated state. We show that the bond dimension scales polynomially in the inverse of the approximation error and sub-exponentially in inverse of the minimal distance between insertion points. We illustrate our findings using Wess–Zumino–Witten models, and show that there is a one-to-one correspondence between group-covariant MPS and our approximation.
Matrix product states (MPSs) illustrate the suitability of tensor networks for the description of interacting many-body systems: ground states of gapped 1D systems are approximable by MPSs, as shown by Hastings [M. B. Hastings, J. Stat. Mech. (2007) P08024]. By contrast, whether MPSs and more general tensor networks can accurately reproduce correlations in critical quantum systems or quantum field theories has not been established rigorously. Ample evidence exists: entropic considerations provide restrictions on the form of suitable ansatz states, and numerical studies show that certain tensor networks can indeed approximate the associated correlation functions. Here, we provide a complete positive answer to this question in the case of MPSs and 2D conformal field theory: we give quantitative estimates for the approximation error when approximating correlation functions by MPSs. Our work is constructive and yields an explicit MPS, thus providing both suitable initial values and a rigorous justification of variational methods.
Abstract We provide a rigorous lattice approximation of conformal field theories given in terms of lattice fermions in 1+1-dimensions, focussing on free fermion models and Wess–Zumino–Witten models. To this end, we utilize a recently introduced operator-algebraic framework for Wilson–Kadanoff renormalization. In this setting, we prove the convergence of the approximation of the Virasoro generators by the Koo–Saleur formula. From this, we deduce the convergence of lattice approximations of conformal correlation functions to their continuum limit. In addition, we show how these results lead to explicit error estimates pertaining to the quantum simulation of conformal field theories.
Using truncated conformal field theory (CFT), we present the formalism necessary to obtain exact matrix product state (MPS) representations for any fractional quantum hall model state which can be written as an expectation value of primary fields in some conformal field theory. The auxiliary bond space is the Hilbert space of the descendants of primary fields and matrix elements can be calculated using well-known CFT commutation relations. We obtain a site-independent MPS representation on the thin annulus and cylinder, while a site-dependent representation is possible on the disk and sphere geometries. We discuss the complications that arise due to the presence of null vectors in the theory's Hilbert space, and present an array of procedures which optimize the numerical calculation of matrix elements. By truncating the Hilbert space of the CFT, we then build an approximate MPS representation for Laughlin, Moore-Read, Read-Rezayi, Gaffnian, minimal M(3,r+2)M(3,r+2)M(3,r+2) models, and a series of superconformal minimal models. We show how to obtain, in the MPS representation, the Orbital, Particle, and Real Space entanglement spectrum in both the finite and infinite systems.
We propose a general procedure for extracting the running coupling constants of the underlying field theory of a given classical statistical model on a two-dimensional lattice, combining tensor network renormalization (TNR) and the finite-size scaling theory of conformal field theory. By tracking the coupling constants at each scale, we are able to visualize the renormalization group flow and demonstrate it with the classical Ising and three-state Potts models. Furthermore, utilizing this methodology, we reveal the limitations due to finite bond dimension DDD on TNR applied to critical systems. We find that a finite correlation length is imposed by the finite bond dimension in TNR, and it can be attributed to an emergent relevant perturbation that respects the symmetries of the system. The correlation length shows the same power-law dependence on DDD as the ``finite entanglement scaling'' of the matrix product states.
In this paper, we apply the formalism of translation invariant (continuous) matrix product states in the thermodynamic limit to (1+1)(1+1)(1+1)-dimensional critical models. Finite bond dimension bounds the entanglement entropy and introduces an effective finite correlation length, so that the state is perturbed away from criticality. The assumption that the scaling hypothesis holds for this kind of perturbation is known in the literature as finite entanglement scaling. We provide further evidence for the validity of finite entanglement scaling and based on this formulate a scaling algorithm to estimate the central charge and critical exponents of the conformally invariant field theories describing the critical models under investigation. The algorithm is applied to three exemplary models; the cMPS version to the nonrelativistic Lieb-Liniger model and the relativistic massless boson, and MPS version to the one-dimensional quantum Ising model at the critical point. Another new aspect to our approach is that we directly use the (c)MPS induced correlation length rather than the bond dimension as scaling parameter. This choice is motivated by several theoretical arguments as well as by the remarkable accuracy of our results.
The concept of finite entanglement scaling forms one of the pillars on which the tensor network ecosystem is built. In this paper, we resolve the open problem of determining the actual perturbations induced by matrix product state approximations of critical systems, and we demonstrate that these can be quite different than the ones predicted by conformal field theory. To that aim, we develop a sparse linear solver to calculate the forward and backward derivatives of 2-dimensional tensor networks with respect to their defining parameters in an implicit way. This algorithm is of independent interest as it provides a primitive for the variational optimization of projected entangled pair states that circumvents the instabilities plaguing traditional automatic differentiation methods.