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Evaluation of correlation functions in integrable quantum field theories

James Brian Silk

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

The aim of this thesis is to explore correlation functions in two dimensional quantum field theories in two distinct ways. In part I a new method for calculating the differential equations parametrising the correlation functions of twist fields associated with the U (1) symmetry of the Dirac model is presented. While developing this method a new family of descendent twist fields are identified and their form factors calculated. This provides a novel way of calculating the vacuum expectation values of the primary twist fields and is shown to be entirely consistent with known results. The method of calculating the correlation functions of twist fields provides a parametrisation of several other correlation functions for various quantum states. Since this method relies on the Ward identities found in a double copy model it is hoped to have wider applications in other free fermion models. Part II concerns the truncated conformal space approach which has been developed to approximate perturbed conformal field theories. In this part the theory underpinning the approach is discussed and a working algorithm is developed for both bulk and boundary perturbed minimal models. The energy levels, mass gaps and one point functions of various models are computed using the truncated conformal space approach and are shown to be in good agreement with previous calculations. A possible method for using this approach to approximate two point functions in perturbed conformal field theories is discussed.

Related papers

Correlation functions of twist fields from Ward identities in the massive Dirac theory

Benjamin Doyon, James Silk

We derive non-linear differential equations for correlation functions of U (1) twist fields in the two-dimensional massive Dirac theory. Primary U (1) twist fields correspond to exponential fields in the sine-Gordon model at the free-fermion point, and it is well-known that their vacuum two-point functions are determined by integrable differential equations. We extend part of this result to more general quantum states (pure or mixed) and to certain descendents, showing that some two-point functions are determined by the sinh-Gordon differential equations whenever there is translation and parity invariance, and the density matrix is the exponential of a bilinear expression in fermions. We use methods involving Ward identities associated to the copy-rotation symmetry in a model with two independent, anti-commuting copies. Such methods were used in the context of the thermally perturbed Ising quantum field theory model. We show that they are applicable to the Dirac theory as well, and we suggest that they are likely to have a much wider applicability to free fermion models in general. Finally, we note that our form-factor study of descendents twist fields combined with a CFT analysis provides a new way of evaluating vacuum expectation values of primary U (1) twist fields: by deriving and solving a recursion relation.

arXiv:1103.2328•Journal of Physics A Mathematical and Theoretical•Jun 17, 2011•5 citations

More general correlation functions of twist fields from Ward identities in the massive Dirac theory

James Silk

Following on from previous work we derive the nonlinear differential equations of more general correlators of U (1) twist fields in two-dimensional massive Dirac theory. Using the conserved charges of the double-copy model equations parametrizing the correlators of twist fields with arbitrary twist parameter are found. This method also gives a parametrization of the correlation functions of general, fermionic, descendent twist fields. The equations parametrizing correlators of primary twist fields are compared to those of the literature and evidence is presented to confirm that these equations represent the correct parametrization.

arXiv:1112.3537•Journal of Physics A Mathematical and Theoretical•Apr 26, 2012•1 citation

Higher particle form factors of branch point twist fields in integrable quantum field theories

Olalla A. Castro-Alvaredo, Emanuele Levi

In this paper we compute higher particle form factors of branch point twist fields. These fields were first described in the context of massive (1+1)-dimensional integrable quantum field theories and their correlation functions are related to the bi-partite entanglement entropy. We find analytic expressions for some form factors and check those expressions for consistency, mainly by evaluating the conformal dimension of the corresponding twist field in the underlying conformal field theory. We find that solutions to the form factor equations are not unique so various techniques need to be used to identify those corresponding to the branch point twist field we are interested in. The models for which we carry out our study are characterized by staircase patterns of various physical quantities as functions of the energy scale. As the latter is varied, the β-function associated with these theories comes close to vanishing at several points between the deep infrared and deep ultraviolet regimes. In other words, renormalization group flows approach the vicinity of various critical points before ultimately reaching the ultraviolet fixed point. This feature provides an optimal way of checking the consistency of higher particle form factor solutions, as the changes on the conformal dimension of the twist field at various energy scales can only be accounted for by considering higher particle form factor contributions to the expansion of certain correlation functions.

•Jan 1, 2011•13 citations

Higher particle form factors of branch point twist fields in integrable quantum field theories

Olalla A Castro-Alvaredo, Emanuele Levi

In this paper we compute higher particle form factors of branch point twist fields. These fields were first described in the context of massive (1+1)-dimensional integrable quantum field theories and their correlation functions are related to the bi-partite entanglement entropy. We find analytic expressions for some form factors and check those expressions for consistency, mainly by evaluating the conformal dimension of the corresponding twist field in the underlying conformal field theory. We find that solutions to the form factor equations are not unique so various techniques need to be used to identify those corresponding to the branch point twist field we are interested in. The models for which we carry out our study are characterized by staircase patterns of various physical quantities as functions of the energy scale. As the latter is varied, the β-function associated with these theories comes close to vanishing at several points between the deep infrared and deep ultraviolet regimes. In other words, renormalization group flows approach the vicinity of various critical points before ultimately reaching the ultraviolet fixed point. This feature provides an optimal way of checking the consistency of higher particle form factor solutions, as the changes on the conformal dimension of the twist field at various energy scales can only be accounted for by considering higher particle form factor contributions to the expansion of certain correlation functions.

arXiv:1103.2069•Journal of Physics A Mathematical and Theoretical•May 16, 2011•16 citations

Finite volume form factors and correlation functions at finite temperature

Balázs Pozsgay

In this thesis we investigate finite size effects in 1+1 dimensional integrable QFT. In particular we consider matrix elements of local operators (finite volume form factors) and vacuum expectation values and correlation functions at finite temperature. In the first part of the thesis we give a complete description of the finite volume form factors in terms of the infinite volume form factors (solutions of the bootstrap program) and the S-matrix of the theory. The calculations are correct to all orders in the inverse of the volume, only exponentially decaying (residual) finite size effects are neglected. We also consider matrix elements with disconnected pieces and determine the general rule for evaluating such contributions in a finite volume. The analytic results are tested against numerical data obtained by the truncated conformal space approach in the Lee-Yang model and the Ising model in a magnetic field. In a separate section we also evaluate the leading exponential correction (the $μ$-term) associated to multi-particle energies and matrix elements. In the second part of the thesis we show that finite volume factors can be used to derive a systematic low-temperature expansion for correlation functions at finite temperature. In the case of vacuum expectation values the series is worked out up to the third non-trivial order and a complete agreement with the LeClair-Mussardo formula is observed. A preliminary treatment of the two-point function is also given by considering the first nontrivial contributions.

arXiv:0907.4306•Jul 24, 2009•15 citations

Survey on the Conformal Field Theories in two dimensions and Minimal Models

Atlas Çelik

The fundamentals of the conformal symmetry transformations in two dimensions are introduced. From the adapted \textit{Ward identities under conformal constraints}, the \textit{operator product expansion} (OPE) is defined for free fields and the energy momentum tensor (EM). After demonstrating how the energy momentum tensor transforms under conformal invariance, the notion of the \textit{central charge c} is defined. Conformal fields in two dimensions are \textit{radially} quantized on the cylinder within the framework of the second quantization of quantum field theory and are then mapped onto complex plane via a conformal mapping. The \textit{radial ordering} of conformal fields is adopted. The \textit{Virasoro algebra} is derived from the identification of the \textit{conformal generators}, which follow from relations between equal-time commutators and contour integrals of conformal fields. The radial quantization is applied to the free fermionic and bosonic fields, where for the latter the \textit{Vertex operator} is defined and specified. The Fock space of the free boson is constructed through the \textit{operator state correspondence}. Various boundary conditions are applied to free fields, and their effects on the ground-state energy density are detailed, leading to the notion of the \textit{Casimir energy}. The OPEs are regularized for the \textit{interacting} conformal fields, namely the E-M tensor, and the related definition of \textit{normal ordering} is revisited. Utilization of the Virasoro algebra led first to the \textit{descendant states} and subsequently to the \textit{conformal families}. The OPEs are generalized for the conformal families, and it is demonstrated how the three-point function coefficients may be obtained in terms of the two-point functions. The same procedure is employed generically through the definition of the \textit{conformal blocks}. The \textit{conformal Bootstrap method} is introduced to calculate the coefficients of $n$ point functions. The Hilbert space of the conformal fields in two dimensions is formally constructed through the \textit{representation theory of Virasoro algebra}. Accordingly, the reducible and irreducible \textit{Verma modules} are specified. Due to the unitarity requirement of the Hilbert space, the famous \textit{Kac determinant} is introduced to obtain constraints on the inner products of the Verma modules, whence on the \textit{conformal weights} and the central charge. The decoupling of the \textit{null states} from rest of the their conformal families is used to obtain \textit{fusion rules} of the families. The \textit{Kac table} for the constrained conformal weights is constructed, and its analysis under the requirement of unitarity leads to the identification of the \textit{minimal models}. The correspondence between the discrete statistical systems at criticality and the minimal models is studied, where special attention is given on the \textit{Ising model}.

Työväentutkimus Vuosikirja•May 20, 2026•0 citations
The aim of this thesis is to explore correlation functions in two dimensional quantum field theories in two distinct ways. In part I a new method for calculating the differential equations parametrising the correlation functions of twist fields associated with the U (1) symmetry of the Dirac model is presented. While developing this method a new family of descendent twist fields are identified and their form factors calculated. This provides a novel way of calculating the vacuum expectation values of the primary twist fields and is shown to be entirely consistent with known results. The method of calculating the correlation functions of twist fields provides a parametrisation of several other correlation functions for various quantum states. Since this method relies on the Ward identities found in a double copy model it is hoped to have wider applications in other free fermion models. Part II concerns the truncated conformal space approach which has been developed to approximate perturbed conformal field theories. In this part the theory underpinning the approach is discussed and a working algorithm is developed for both bulk and boundary perturbed minimal models. The energy levels, mass gaps and one point functions of various models are computed using the truncated conformal space approach and are shown to be in good agreement with previous calculations. A possible method for using this approach to approximate two point functions in perturbed conformal field theories is discussed.
We derive non-linear differential equations for correlation functions of U (1) twist fields in the two-dimensional massive Dirac theory. Primary U (1) twist fields correspond to exponential fields in the sine-Gordon model at the free-fermion point, and it is well-known that their vacuum two-point functions are determined by integrable differential equations. We extend part of this result to more general quantum states (pure or mixed) and to certain descendents, showing that some two-point functions are determined by the sinh-Gordon differential equations whenever there is translation and parity invariance, and the density matrix is the exponential of a bilinear expression in fermions. We use methods involving Ward identities associated to the copy-rotation symmetry in a model with two independent, anti-commuting copies. Such methods were used in the context of the thermally perturbed Ising quantum field theory model. We show that they are applicable to the Dirac theory as well, and we suggest that they are likely to have a much wider applicability to free fermion models in general. Finally, we note that our form-factor study of descendents twist fields combined with a CFT analysis provides a new way of evaluating vacuum expectation values of primary U (1) twist fields: by deriving and solving a recursion relation.
Following on from previous work we derive the nonlinear differential equations of more general correlators of U (1) twist fields in two-dimensional massive Dirac theory. Using the conserved charges of the double-copy model equations parametrizing the correlators of twist fields with arbitrary twist parameter are found. This method also gives a parametrization of the correlation functions of general, fermionic, descendent twist fields. The equations parametrizing correlators of primary twist fields are compared to those of the literature and evidence is presented to confirm that these equations represent the correct parametrization.
In this paper we compute higher particle form factors of branch point twist fields. These fields were first described in the context of massive (1+1)-dimensional integrable quantum field theories and their correlation functions are related to the bi-partite entanglement entropy. We find analytic expressions for some form factors and check those expressions for consistency, mainly by evaluating the conformal dimension of the corresponding twist field in the underlying conformal field theory. We find that solutions to the form factor equations are not unique so various techniques need to be used to identify those corresponding to the branch point twist field we are interested in. The models for which we carry out our study are characterized by staircase patterns of various physical quantities as functions of the energy scale. As the latter is varied, the β-function associated with these theories comes close to vanishing at several points between the deep infrared and deep ultraviolet regimes. In other words, renormalization group flows approach the vicinity of various critical points before ultimately reaching the ultraviolet fixed point. This feature provides an optimal way of checking the consistency of higher particle form factor solutions, as the changes on the conformal dimension of the twist field at various energy scales can only be accounted for by considering higher particle form factor contributions to the expansion of certain correlation functions.
In this paper we compute higher particle form factors of branch point twist fields. These fields were first described in the context of massive (1+1)-dimensional integrable quantum field theories and their correlation functions are related to the bi-partite entanglement entropy. We find analytic expressions for some form factors and check those expressions for consistency, mainly by evaluating the conformal dimension of the corresponding twist field in the underlying conformal field theory. We find that solutions to the form factor equations are not unique so various techniques need to be used to identify those corresponding to the branch point twist field we are interested in. The models for which we carry out our study are characterized by staircase patterns of various physical quantities as functions of the energy scale. As the latter is varied, the β-function associated with these theories comes close to vanishing at several points between the deep infrared and deep ultraviolet regimes. In other words, renormalization group flows approach the vicinity of various critical points before ultimately reaching the ultraviolet fixed point. This feature provides an optimal way of checking the consistency of higher particle form factor solutions, as the changes on the conformal dimension of the twist field at various energy scales can only be accounted for by considering higher particle form factor contributions to the expansion of certain correlation functions.
In this thesis we investigate finite size effects in 1+1 dimensional integrable QFT. In particular we consider matrix elements of local operators (finite volume form factors) and vacuum expectation values and correlation functions at finite temperature. In the first part of the thesis we give a complete description of the finite volume form factors in terms of the infinite volume form factors (solutions of the bootstrap program) and the S-matrix of the theory. The calculations are correct to all orders in the inverse of the volume, only exponentially decaying (residual) finite size effects are neglected. We also consider matrix elements with disconnected pieces and determine the general rule for evaluating such contributions in a finite volume. The analytic results are tested against numerical data obtained by the truncated conformal space approach in the Lee-Yang model and the Ising model in a magnetic field. In a separate section we also evaluate the leading exponential correction (the μμμ-term) associated to multi-particle energies and matrix elements. In the second part of the thesis we show that finite volume factors can be used to derive a systematic low-temperature expansion for correlation functions at finite temperature. In the case of vacuum expectation values the series is worked out up to the third non-trivial order and a complete agreement with the LeClair-Mussardo formula is observed. A preliminary treatment of the two-point function is also given by considering the first nontrivial contributions.
The fundamentals of the conformal symmetry transformations in two dimensions are introduced. From the adapted \textit{Ward identities under conformal constraints}, the \textit{operator product expansion} (OPE) is defined for free fields and the energy momentum tensor (EM). After demonstrating how the energy momentum tensor transforms under conformal invariance, the notion of the \textit{central charge c} is defined. Conformal fields in two dimensions are \textit{radially} quantized on the cylinder within the framework of the second quantization of quantum field theory and are then mapped onto complex plane via a conformal mapping. The \textit{radial ordering} of conformal fields is adopted. The \textit{Virasoro algebra} is derived from the identification of the \textit{conformal generators}, which follow from relations between equal-time commutators and contour integrals of conformal fields. The radial quantization is applied to the free fermionic and bosonic fields, where for the latter the \textit{Vertex operator} is defined and specified. The Fock space of the free boson is constructed through the \textit{operator state correspondence}. Various boundary conditions are applied to free fields, and their effects on the ground-state energy density are detailed, leading to the notion of the \textit{Casimir energy}. The OPEs are regularized for the \textit{interacting} conformal fields, namely the E-M tensor, and the related definition of \textit{normal ordering} is revisited. Utilization of the Virasoro algebra led first to the \textit{descendant states} and subsequently to the \textit{conformal families}. The OPEs are generalized for the conformal families, and it is demonstrated how the three-point function coefficients may be obtained in terms of the two-point functions. The same procedure is employed generically through the definition of the \textit{conformal blocks}. The \textit{conformal Bootstrap method} is introduced to calculate the coefficients of nnn point functions. The Hilbert space of the conformal fields in two dimensions is formally constructed through the \textit{representation theory of Virasoro algebra}. Accordingly, the reducible and irreducible \textit{Verma modules} are specified. Due to the unitarity requirement of the Hilbert space, the famous \textit{Kac determinant} is introduced to obtain constraints on the inner products of the Verma modules, whence on the \textit{conformal weights} and the central charge. The decoupling of the \textit{null states} from rest of the their conformal families is used to obtain \textit{fusion rules} of the families. The \textit{Kac table} for the constrained conformal weights is constructed, and its analysis under the requirement of unitarity leads to the identification of the \textit{minimal models}. The correspondence between the discrete statistical systems at criticality and the minimal models is studied, where special attention is given on the \textit{Ising model}.