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

Numerical methods for quantum systems with infinite-dimensional space of states: two examples, two approaches

Marco Beria

Published Oct 30, 2013• 0 citations
OpenAlex

Abstract

In this thesis two rather different topics are addressed: • the application of the Truncated Conformal Spectrum Approach (TCSA) to pertur- bations of SU(2)k Wess-Zumino-Witten (WZW) models, and to Landau-Ginzburg (LG) theories; • the analysis of the role of quantum statistics in the steady flow patterns of 2d driven ideal Fermi gases in a non-integrable geometry. A common methodological question unifies the two topics: how to simulate numerically quantum systems with infinite spaces of states. This problem is typically faced either introducing a lattice in the real space or directly in the continuum, and the two addressed topics can be seen as examples of these two cases. The TCSA is aimed to tackle perturbed conformal field theories directly in the continuum. Instead, to study the ideal Fermi gas we introduce a lattice which naturally captures the features of the chosen non-integrable geometry. Chapter 1: This part of the thesis is mainly methodological and it is focused on the application of the TCSA to c = 1 conformal field theories. In particular, our interest is concerned with perturbations of WZW models with SU(2)k symmetry, which capture low energy properties of k/2-Heisenberg spin-chains. The new problem faced is how to apply the method in the presence of marginal perturbations, where UV divergences result in non-universal contributions in the simulations. A generic framework to cure this pathology is proposed, and tested in the case of the current-current deformation of SU(2)1. Among c = 1 theories, Landau-Ginzburg theories are of great interest, since their classical structure of the vacua captures the massive phases of deformation of minimal models. The most important methodological problem we faced here is that LG theories have an uncountable space of states. We show that the TCSA applies also in this case, once the target space of the LG field is properly compactified. The TCSA is thus used to very an existing conjecture on the number of stable neutral bound states. Chapter 2: Interacting quantum gases, such as the electron liquid, are known to admit a hydrodynamic description and, as for classical interacting fluids, they develop turbulence at sufficiently strong driving. On the other hand, ideal classical fluids cannot develop any turbulent flow. We ask whether this property extends to ideal quantum gases. Indeed, Fermi gases satisfy the exclusion principle that could play the role of an effective interaction, resulting in non-trivial flows. We are going to describe an ideal Fermi gas which flows form a narrow channel into a wider region. A lattice which approximates the geometry is introduced, and the system is tackled within two approaches: the micro-canonical formalism and the Lindblad equation. In the first case the system is closed, while in the second one it is coupled to external baths. Despite this difference, the observed physics turns out to be the same far from the boundaries. We show that, in specific regimes, quantum statistics induces non-trivial patterns in the vorticity, which generates local magnetic moments of a measurable intensity.

Related papers

Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-Abelian bosonization to truncated spectrum methods

A. J. A. James, Robert Konik, P. Lecheminant +2 more

We review two important non-perturbative approaches for extracting the physics of low-dimensional strongly correlated quantum systems. Firstly, we start by providing a comprehensive review of non-Abelian bosonization. This includes an introduction to the basic elements of conformal field theory as applied to systems with a current algebra, and we orient the reader by presenting a number of applications of non-Abelian bosonization to models with large symmetries. We then tie this technique into recent advances in the ability of cold atomic systems to realize complex symmetries. Secondly, we discuss truncated spectrum methods for the numerical study of systems in one and two dimensions. For one-dimensional systems we provide the reader with considerable insight into the methodology by reviewing canonical applications of the technique to the Ising model (and its variants) and the sine-Gordon model. Following this we review recent work on the development of renormalization groups, both numerical and analytical, that alleviate the effects of truncating the spectrum. Using these technologies, we consider a number of applications to one-dimensional systems: properties of carbon nanotubes, quenches in the Lieb-Liniger model, 1 + 1D quantum chromodynamics, as well as Landau-Ginzburg theories. In the final part we move our attention to consider truncated spectrum methods applied to two-dimensional systems. This involves combining truncated spectrum methods with matrix product state algorithms. We describe applications of this method to two-dimensional systems of free fermions and the quantum Ising model, including their non-equilibrium dynamics.

arXiv:1703.08421•Reports on Progress in Physics•Oct 9, 2017•107 citations

On asymptotic behaviour in truncated conformal space approach

Anatoly Konechny, Dermot McAteer

Abstract The Truncated conformal space approach (TCSA) is a numerical technique for finding finite size spectrum of Hamiltonians in quantum field theory described as perturbations of conformal field theories. The truncation errors of the method have been systematically studied near the UV fixed point (when the characteristic energy related to the coupling is less than the truncation cutoff) where a good theoretical understanding has been achieved. However numerically the method demonstrated a good agreement with other methods for much larger values of the coupling when the RG flow approaches a new fixed point in the infrared. In the present paper we investigate this regime for a number of boundary RG flows testing the leading exponent and truncation errors. We also study the flows beyond the first fixed point which have been observed numerically but yet lack a theoretical understanding. We show that while in some models such flows approximate reversed physical RG flows, in other models the spectrum approaches a stable regime that does not correspond to any local boundary condition. Furthermore we find that in general the flows beyond the first fixed point are very sensitive to modifications of the truncation scheme.

arXiv:1904.09616•Journal of High Energy Physics•Sep 1, 2019•4 citations

Investigations in Two-Dimensional Quantum Field Theory by the Bootstrap and TCSA Methods

Gábor Zsolt Tóth

This thesis contains three main parts, which are largely independent. In the first part we deal with the boundary bootstrap in supersymmetric factorized scattering theory. We give a description of supersymmetry in the case when the space is a half-line and present rules for the determination of the representations in which higher level boundary bound states transform, and for the determination of the supersymmetric one-particle reflection matrix factors for the higher level boundary bound states. These rules apply under the condition that the bulk particles transform in the kink or in the boson-fermion representation. Examples for the application of these rules to specific models are also given. In the second part we investigate the problem whether the TCSA spectrum can be approximated by the spectrum of the original Hamiltonian operator in which the coefficients of the terms are suitably changed. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. Another truncation method that preserves the solvability of the model is also considered. The results of perturbative and numerical calculations show that the above approximation is possible and that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method. In the third part we investigate the phase structure of the two- and three-frequency sine-Gordon models using the TCSA. In the case of the three-frequency model the tricritical point, several points of the critical line and a few points of the line of first order transition are found.

arXiv:0707.0015•Jun 30, 2007•4 citations

Deux essais sur les theories quantiques des champs conformes et fortement couplees en d > 2 dimensions

Matthijs Hogervorst

This thesis investigates two aspects of Conformal Field Theories (CFTs) in D dimensions. Its first part is devoted to conformal blocks, special functions that arise in the partial wave expansion of CFT four-point functions. We prove that these conformal blocks admit an expansion in terms of polar coordinates and show that the expansion coefficients are determined by recursion relations. Conformal blocks are naturally defined on the complex plane: we study their restriction to the real line, and show that they obey a fourth-order differential equation there. This ODE can be used to efficiently compute conformal blocks and their derivatives in general D. Several applications to the conformal bootstrap program are mentioned.The second half of this thesis investigates RG flows that are defined by perturbing a CFT by a number of relevant operators. We study such flows using the Truncated Conformal Space Approach (TCSA) of Yurov and Zamolodchikov, a numerical method that allows for controlled computations in strongly coupled QFTs. Two different RG flows are considered: the free scalar field deformed by a mass term, and phi^4 theory. The former is used as a benchmark, in order to compare numerical TCSA results to exact predictions. TCSA results for phi^4 theory display spontaneous Z_2 symmetry breaking at strong coupling: we study the spectrum of this theory both in the Z_2-broken and preserved phase, and we compare the critical exponents governing the phase transition to known values. In a separate chapter, we show how truncation errors can be reduced by adding suitable counterterms to the bare TCSA action, following earlier work in two dimensions.

•Jun 29, 2015•2 citations

Z(_N)-symmetric field theories and the thermodynamic Bethe ansatz

Kevin E. Thompson

This thesis is concerned with perturbed conformal field theory, the thermodynamic Bethe ansatz technique and applications to statistical mechanics. In particular, the phase space of two dimensional Z(_N)-symmetric statistical models is examined using these techniques. The aim of the first two chapters is to review some general material concerning statistical mechanics, perturbed conformal field theory, integrable two-dimensional quantum field theory and the thermodynamic Bethe ansatz (TBA) technique. In the third chapter Z(_N)-symmetric statistical theories are discussed and the known features of the phase space of such models are surveyed. The field content of the conformal models in this space (called parafermionic models) is investigated in order to determine which perturbations can be used to investigate the phase space. In the fourth and fifth chapters TBA equations are proposed to describe massless and massive renormalisation flows from the Z(_N)-symmetric conformal theories under self-dual Z(_N)-symmetric perturbations. According to the sign of the perturbation parameter the infrared limits are shown to be either conformal c = 1 or massive theories. The ground state energies of these models can be discovered in all perturbative regimes via the TBA method and the results agree with perturbation theory in ultraviolet and infrared limits. Results from detailed studies of the N = 5, 6..10 models are presented throughout. It is also deduced that the parafermionic models lie exactly at the bifurcation point of the first-order transition region into the Kosterlitz-Thouless region of the Z(_N)-symmetric phase space. The sixth and seventh chapters deal solely with massive perturbations. In chapter six, results from the TBA equations are used to deduce the mass spectrum and the vacuum structure of the underlying scattering theory. In chapter seven, proposals for the massive S-matrices are made. For N odd the mass spectra proposed by the TBA method and that predicted by the S-matrix approach (using the minimality principle) differ. It is suggested therefore, that the N odd S-matrices contain zeroes in the physical strip, violating the minimality principle.

•Jan 1, 1996•0 citations

Perturbed conformal field theory, nonlinear integral equations and spectral problems

Clare Dunning

This thesis is concerned with various aspects of perturbed conformal field theory and the methods used to calculate finite-size effects of integrable quantum field theories. Nonlinear integral equations are the main tools to find the exact ground-state energy of a quantum field theory. The thermodyamic Bethe ansatz (TBA) equations are a set of examples and are known for a large number of models. However, it is also an interesting question to find exact equations describing the excited states of integrable models. The first part of this thesis uses analytical continuation in a continuous parameter to find TBA like equations describing the spin-zero excited states of the sine-Gordon model at coupling β(^2) = 16π/3. Comparisons are then made with a further type of nonlinear integral equation which also predicts the excited state energies. Relations between the two types of equation are studied using a set of functional relations recently introduced in integrable quantum field theory. A relevant perturbation of a conformal field theory results in either a massive quantum field theory such as the sine-Gordon model, or a different massless conformal field theory. The second part of this thesis investigates flows between conformal field theories using a nonlinear integral equation. New families of flows are found which exhibit a rather unexpected behaviour. The final part of this thesis begins with a review of a connection between integrable quantum field theory and properties of certain ordinary differential equations of second- and third-order. The connection is based on functional relations which appear on both sides of the correspondence; for the second-order case these are exactly the functional relations mentioned above. The results are extended to include a correspondence between n(^th) order differential equations and Bethe ansatz system of SU(n) type. A set of nonlinear integral equations are derived to check the results.

•Jan 1, 2000•2 citations
In this thesis two rather different topics are addressed: • the application of the Truncated Conformal Spectrum Approach (TCSA) to pertur- bations of SU(2)k Wess-Zumino-Witten (WZW) models, and to Landau-Ginzburg (LG) theories; • the analysis of the role of quantum statistics in the steady flow patterns of 2d driven ideal Fermi gases in a non-integrable geometry. A common methodological question unifies the two topics: how to simulate numerically quantum systems with infinite spaces of states. This problem is typically faced either introducing a lattice in the real space or directly in the continuum, and the two addressed topics can be seen as examples of these two cases. The TCSA is aimed to tackle perturbed conformal field theories directly in the continuum. Instead, to study the ideal Fermi gas we introduce a lattice which naturally captures the features of the chosen non-integrable geometry. Chapter 1: This part of the thesis is mainly methodological and it is focused on the application of the TCSA to c = 1 conformal field theories. In particular, our interest is concerned with perturbations of WZW models with SU(2)k symmetry, which capture low energy properties of k/2-Heisenberg spin-chains. The new problem faced is how to apply the method in the presence of marginal perturbations, where UV divergences result in non-universal contributions in the simulations. A generic framework to cure this pathology is proposed, and tested in the case of the current-current deformation of SU(2)1. Among c = 1 theories, Landau-Ginzburg theories are of great interest, since their classical structure of the vacua captures the massive phases of deformation of minimal models. The most important methodological problem we faced here is that LG theories have an uncountable space of states. We show that the TCSA applies also in this case, once the target space of the LG field is properly compactified. The TCSA is thus used to very an existing conjecture on the number of stable neutral bound states. Chapter 2: Interacting quantum gases, such as the electron liquid, are known to admit a hydrodynamic description and, as for classical interacting fluids, they develop turbulence at sufficiently strong driving. On the other hand, ideal classical fluids cannot develop any turbulent flow. We ask whether this property extends to ideal quantum gases. Indeed, Fermi gases satisfy the exclusion principle that could play the role of an effective interaction, resulting in non-trivial flows. We are going to describe an ideal Fermi gas which flows form a narrow channel into a wider region. A lattice which approximates the geometry is introduced, and the system is tackled within two approaches: the micro-canonical formalism and the Lindblad equation. In the first case the system is closed, while in the second one it is coupled to external baths. Despite this difference, the observed physics turns out to be the same far from the boundaries. We show that, in specific regimes, quantum statistics induces non-trivial patterns in the vorticity, which generates local magnetic moments of a measurable intensity.
We review two important non-perturbative approaches for extracting the physics of low-dimensional strongly correlated quantum systems. Firstly, we start by providing a comprehensive review of non-Abelian bosonization. This includes an introduction to the basic elements of conformal field theory as applied to systems with a current algebra, and we orient the reader by presenting a number of applications of non-Abelian bosonization to models with large symmetries. We then tie this technique into recent advances in the ability of cold atomic systems to realize complex symmetries. Secondly, we discuss truncated spectrum methods for the numerical study of systems in one and two dimensions. For one-dimensional systems we provide the reader with considerable insight into the methodology by reviewing canonical applications of the technique to the Ising model (and its variants) and the sine-Gordon model. Following this we review recent work on the development of renormalization groups, both numerical and analytical, that alleviate the effects of truncating the spectrum. Using these technologies, we consider a number of applications to one-dimensional systems: properties of carbon nanotubes, quenches in the Lieb-Liniger model, 1 + 1D quantum chromodynamics, as well as Landau-Ginzburg theories. In the final part we move our attention to consider truncated spectrum methods applied to two-dimensional systems. This involves combining truncated spectrum methods with matrix product state algorithms. We describe applications of this method to two-dimensional systems of free fermions and the quantum Ising model, including their non-equilibrium dynamics.
Abstract The Truncated conformal space approach (TCSA) is a numerical technique for finding finite size spectrum of Hamiltonians in quantum field theory described as perturbations of conformal field theories. The truncation errors of the method have been systematically studied near the UV fixed point (when the characteristic energy related to the coupling is less than the truncation cutoff) where a good theoretical understanding has been achieved. However numerically the method demonstrated a good agreement with other methods for much larger values of the coupling when the RG flow approaches a new fixed point in the infrared. In the present paper we investigate this regime for a number of boundary RG flows testing the leading exponent and truncation errors. We also study the flows beyond the first fixed point which have been observed numerically but yet lack a theoretical understanding. We show that while in some models such flows approximate reversed physical RG flows, in other models the spectrum approaches a stable regime that does not correspond to any local boundary condition. Furthermore we find that in general the flows beyond the first fixed point are very sensitive to modifications of the truncation scheme.
This thesis contains three main parts, which are largely independent. In the first part we deal with the boundary bootstrap in supersymmetric factorized scattering theory. We give a description of supersymmetry in the case when the space is a half-line and present rules for the determination of the representations in which higher level boundary bound states transform, and for the determination of the supersymmetric one-particle reflection matrix factors for the higher level boundary bound states. These rules apply under the condition that the bulk particles transform in the kink or in the boson-fermion representation. Examples for the application of these rules to specific models are also given. In the second part we investigate the problem whether the TCSA spectrum can be approximated by the spectrum of the original Hamiltonian operator in which the coefficients of the terms are suitably changed. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. Another truncation method that preserves the solvability of the model is also considered. The results of perturbative and numerical calculations show that the above approximation is possible and that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method. In the third part we investigate the phase structure of the two- and three-frequency sine-Gordon models using the TCSA. In the case of the three-frequency model the tricritical point, several points of the critical line and a few points of the line of first order transition are found.
This thesis investigates two aspects of Conformal Field Theories (CFTs) in D dimensions. Its first part is devoted to conformal blocks, special functions that arise in the partial wave expansion of CFT four-point functions. We prove that these conformal blocks admit an expansion in terms of polar coordinates and show that the expansion coefficients are determined by recursion relations. Conformal blocks are naturally defined on the complex plane: we study their restriction to the real line, and show that they obey a fourth-order differential equation there. This ODE can be used to efficiently compute conformal blocks and their derivatives in general D. Several applications to the conformal bootstrap program are mentioned.The second half of this thesis investigates RG flows that are defined by perturbing a CFT by a number of relevant operators. We study such flows using the Truncated Conformal Space Approach (TCSA) of Yurov and Zamolodchikov, a numerical method that allows for controlled computations in strongly coupled QFTs. Two different RG flows are considered: the free scalar field deformed by a mass term, and phi^4 theory. The former is used as a benchmark, in order to compare numerical TCSA results to exact predictions. TCSA results for phi^4 theory display spontaneous Z_2 symmetry breaking at strong coupling: we study the spectrum of this theory both in the Z_2-broken and preserved phase, and we compare the critical exponents governing the phase transition to known values. In a separate chapter, we show how truncation errors can be reduced by adding suitable counterterms to the bare TCSA action, following earlier work in two dimensions.
This thesis is concerned with perturbed conformal field theory, the thermodynamic Bethe ansatz technique and applications to statistical mechanics. In particular, the phase space of two dimensional Z(_N)-symmetric statistical models is examined using these techniques. The aim of the first two chapters is to review some general material concerning statistical mechanics, perturbed conformal field theory, integrable two-dimensional quantum field theory and the thermodynamic Bethe ansatz (TBA) technique. In the third chapter Z(_N)-symmetric statistical theories are discussed and the known features of the phase space of such models are surveyed. The field content of the conformal models in this space (called parafermionic models) is investigated in order to determine which perturbations can be used to investigate the phase space. In the fourth and fifth chapters TBA equations are proposed to describe massless and massive renormalisation flows from the Z(_N)-symmetric conformal theories under self-dual Z(_N)-symmetric perturbations. According to the sign of the perturbation parameter the infrared limits are shown to be either conformal c = 1 or massive theories. The ground state energies of these models can be discovered in all perturbative regimes via the TBA method and the results agree with perturbation theory in ultraviolet and infrared limits. Results from detailed studies of the N = 5, 6..10 models are presented throughout. It is also deduced that the parafermionic models lie exactly at the bifurcation point of the first-order transition region into the Kosterlitz-Thouless region of the Z(_N)-symmetric phase space. The sixth and seventh chapters deal solely with massive perturbations. In chapter six, results from the TBA equations are used to deduce the mass spectrum and the vacuum structure of the underlying scattering theory. In chapter seven, proposals for the massive S-matrices are made. For N odd the mass spectra proposed by the TBA method and that predicted by the S-matrix approach (using the minimality principle) differ. It is suggested therefore, that the N odd S-matrices contain zeroes in the physical strip, violating the minimality principle.
This thesis is concerned with various aspects of perturbed conformal field theory and the methods used to calculate finite-size effects of integrable quantum field theories. Nonlinear integral equations are the main tools to find the exact ground-state energy of a quantum field theory. The thermodyamic Bethe ansatz (TBA) equations are a set of examples and are known for a large number of models. However, it is also an interesting question to find exact equations describing the excited states of integrable models. The first part of this thesis uses analytical continuation in a continuous parameter to find TBA like equations describing the spin-zero excited states of the sine-Gordon model at coupling β(^2) = 16π/3. Comparisons are then made with a further type of nonlinear integral equation which also predicts the excited state energies. Relations between the two types of equation are studied using a set of functional relations recently introduced in integrable quantum field theory. A relevant perturbation of a conformal field theory results in either a massive quantum field theory such as the sine-Gordon model, or a different massless conformal field theory. The second part of this thesis investigates flows between conformal field theories using a nonlinear integral equation. New families of flows are found which exhibit a rather unexpected behaviour. The final part of this thesis begins with a review of a connection between integrable quantum field theory and properties of certain ordinary differential equations of second- and third-order. The connection is based on functional relations which appear on both sides of the correspondence; for the second-order case these are exactly the functional relations mentioned above. The results are extended to include a correspondence between n(^th) order differential equations and Bethe ansatz system of SU(n) type. A set of nonlinear integral equations are derived to check the results.