arXiv subjects

Benjamin Doyon

Publications and source records attributed to Benjamin Doyon.

124 records · Page 7Linked to original sources

arXiv2009

Conformal loop ensembles and the stress-energy tensor. II. Construction of the stress-energy tensor

This is the second part of a work aimed at constructing the stress-energy tensor of conformal field theory (CFT) as a local "object" in conformal loop ensembles (CLE). This work lies in the wider context of re-constructing quantum field theory from mathematically well-defined ensembles of random objects. In the present paper, based on results of the first part, we identify the stress-energy tensor in the dilute regime of CLE. This is done by deriving both its conformal Ward identities for single insertion in CLE probability functions, and its properties under conformal transformations involving the Schwarzian derivative. We also give the one-point function of the stress-energy tensor in terms of a notion of partition function, and we show that this agrees with standard CFT arguments. The construction is in the same spirit as that found in the context of SLE(8/3) by the author, Riva and Cardy (2006), which had to do with the case of zero central charge. The present construction generalises this to all central charges between 0 and 1, including all minimal models. This generalisation is non-trivial: the application of these ideas to the CLE context requires the introduction of a renormalised probability, and the derivation of the transformation properties and of the one-point function do not have counterparts in the SLE context.

math-ph↗

arXiv2009

Bi-partite entanglement entropy in massive 1+1-dimensional quantum field theories

This manuscript is a review of the main results obtained in a series of papers involving the present authors and their collaborator J.L. Cardy over the last two years. In our work we have developed and applied a new approach for the computation of the bi-partite entanglement entropy in massive 1+1-dimensional quantum field theories. In most of our work we have considered these theories to be also integrable. Our approach combines two main ingredients: the "replica trick" and form factors for integrable models and more generally for massive quantum field theory. Our basic idea for combining fruitfully these two ingredients is that of the branch-point twist field. By the replica trick we obtained an alternative way of expressing the entanglement entropy as a function of the correlation functions of branch-point twist fields. On the other hand, a generalisation of the form factor program has allowed us to study, and in integrable cases to obtain exact expressions for, form factors of such twist fields. By the usual decomposition of correlation functions in an infinite series involving form factors, we obtained exact results for the infrared behaviours of the bi-partite entanglement entropy, and studied both its infrared and ultraviolet behaviours for different kinds of models: with and without boundaries and backscattering, at and out of integrability.

hep-th↗

arXiv2009

Conformal loop ensembles and the stress-energy tensor. I. Fundamental notions of CLE

This is the first part of a work aimed at constructing the stress-energy tensor of conformal field theory as a local "object" in conformal loop ensembles (CLE). This work lies in the wider context of re-constructing quantum field theory from mathematically well-defined ensembles of random objects. The goal of the present paper is two-fold. First, we provide an introduction to CLE, a mathematical theory for random loops in simply connected domains with properties of conformal invariance, developed recently by Sheffield and Werner. It is expected to be related to CFT models with central charges between 0 and 1 (including all minimal models). Second, we further develop the theory by deriving results that will be crucial for the construction of the stress-energy tensor. We introduce the notions of support and continuity of CLE events, about which we prove basic but important theorems. We then propose natural definitions of CLE probability functions on the Riemann sphere and on doubly connected domains. Under some natural assumptions, we prove conformal invariance and other non-trivial theorems related to these constructions. We only use the defining properties of CLE as well as some basic results about the CLE measure. Although this paper is guided by the construction of the stress-energy tensor, we believe that the theorems proved and techniques used are of interest in the wider context of CLE. The actual construction will be presented in the second part of this work.

math-ph↗

arXiv2008

Bi-partite entanglement entropy in massive two-dimensional quantum field theory

Recently, Cardy, Castro Alvaredo and the author obtained the first exponential correction to saturation of the bi-partite entanglement entropy at large region length, in massive two-dimensional integrable quantum field theory. It only depends on the particle content of the model, and not on the way particles scatter. Based on general analyticity arguments for form factors, we propose that this result is universal, and holds for any massive two-dimensional model (also out of integrability). We suggest a link of this result with counting pair creations far in the past.

hep-th↗

arXiv2008

Bi-partite entanglement entropy in massive QFT with a boundary: the Ising model

In this paper we give an exact infinite-series expression for the bi-partite entanglement entropy of the quantum Ising model both with a boundary magnetic field and in infinite volume. This generalizes and extends previous results involving the present authors for the bi-partite entanglement entropy of integrable quantum field theories, which exploited the generalization of the form factor program to branch-point twist fields. In the boundary case, we isolate in a universal way the part of the entanglement entropy which is related to the boundary entropy introduced by Affleck and Ludwig, and explain how this relation should hold in more general QFT models. We provide several consistency checks for the validity of our form factor results, notably, the identification of the leading ultraviolet behaviour both of the entanglement entropy and of the two-point function of twist fields in the bulk theory, to a great degree of precision by including up to 500 form factor contributions.

hep-th↗

arXiv2007

Integral equations and large-time asymptotics for finite-temperature Ising chain correlation functions

This work concerns the dynamical two-point spin correlation functions of the transverse Ising quantum chain at finite (non-zero) temperature, in the universal region near the quantum critical point. They are correlation functions of twist fields in the massive Majorana fermion quantum field theory. At finite temperature, these are known to satisfy a set of integrable partial differential equations, including the sinh-Gordon equation. We apply the classical inverse scattering method to study them, finding that the ``initial scattering data'' corresponding to the correlation functions are simply related to the one-particle finite-temperature form factors calculated recently by one of the authors. The set of linear integral equations (Gelfand-Levitan-Marchenko equations) associated to the inverse scattering problem then gives, in principle, the two-point functions at all space and time separations, and all temperatures. From them, we evaluate the large-time asymptotic expansion ``near the light cone'', in the region where the difference between the space and time separations is of the order of the correlation length.

math-ph↗

arXiv2007

Twisted modules for vertex operator algebras

This contribution is mainly based on joint papers with Lepowsky and Milas, and some parts of these papers are reproduced here. These papers further extended works by Lepowsky and by Milas. Following our joint papers, I explain the general principles of twisted modules for vertex operator algebras in their powerful formulation using formal series, and derive general relations satisfied by twisted and untwisted vertex operators. Using these, I prove new "equivalence" and "construction" theorems, identifying a set of sufficient conditions in order to have a twisted module for a vertex operator algebra, and a simple way of constructing the twisted vertex operator map. This essentially combines our general relations for twisted modules with ideas of Li (1996), who had obtained similar construction theorems using different relations. Then, I show how to apply these theorems in order to construct twisted modules for the Heisenberg vertex operator algebra. I obtain in a new way the explicit twisted vertex operator map, and in particular give a new derivation and expression for the formal operator $Δ_x$ constructed some time ago by Frenkel, Lepowsky and Meurman. Finally, I reproduce parts of our joint papers. The untwisted relations in the Heisenberg vertex operator algebra are employed to explain properties of a certain central extension of a Lie algebra of differential operators on the circle, in relation to the Riemann Zeta function at negative integers. A family of representations for this algebra are constructed from twisted modules for the vertex operator algebra, and are related to the Bernoulli polynomials at rational values.

math.QA↗

arXiv2007

New method for studying steady states in quantum impurity problems: The interacting resonant level model

We develop a new perturbative method for studying any steady states of quantum impurities, in or out of equilibrium. We show that steady-state averages are completely fixed by basic properties of the steady-state (Hershfield's) density matrix along with dynamical "impurity conditions". This gives the full perturbative expansion without Feynman diagrams (matrix products instead are used), and "re-sums" into an equilibrium average that may lend itself to numerical procedures. We calculate the universal current in the interacting resonant level model (IRLM) at finite bias V to first order in Coulomb repulsion U for all V and temperatures. We find that the bias, like the temperature, cuts off low-energy processes. In the IRLM, this implies a power-law decay of the current at large V (also recently observed by Boulat and Saleur at some finite value of U).

cond-mat.str-el↗

arXiv2007

Calogero-Sutherland eigenfunctions with mixed boundary conditions and conformal field theory correlators

We construct certain eigenfunctions of the Calogero-Sutherland hamiltonian for particles on a circle, with mixed boundary conditions. That is, the behavior of the eigenfunction, as neighbouring particles collide, depend on the pair of colliding particles. This behavior is generically a linear combination of two types of power laws, depending on the statistics of the particles involved. For fixed ratio of each type at each pair of neighboring particles, there is an eigenfunction, the ground state, with lowest energy, and there is a discrete set of eigenstates and eigenvalues, the excited states and the energies above this ground state. We find the ground state and special excited states along with their energies in a certain class of mixed boundary conditions, interpreted as having pairs of neighboring bosons and other particles being fermions. These particular eigenfunctions are characterised by the fact that they are in direct correspondence with correlation functions in boundary conformal field theory. We expect that they have applications to measures on certain configurations of curves in the statistical O(n) loop model. The derivation, although completely independent from results of conformal field theory, uses ideas from the "Coulomb gas" formulation.

hep-th↗

arXiv2007

Universal aspects of non-equilibrium currents in a quantum dot

We study the electric current in the non-equilibrium Kondo model at zero magnetic field, using real-time perturbation theory in the Schwinger-Keldysh formulation. We show that the perturbative coefficients to all orders have a finite limit at large switch-on time (t_0 to minus infinity), and we give a prescription for general operators to give finite coefficients in this limit. We explain how this is related to the fact that the leads play the role of thermal baths and allow relaxation to occur and the steady state to form. This proves perturbatively that a steady state is reached in the Schwinger-Keldysh formulation, and specifies which operators correspond to quantities that have a well-defined value in the steady state. Then, we show that the steady state can be described by a special type of density matrix (related to Hershfield's conjecture for the particular example of the non-equilibrium Kondo model.) In the second part of the paper we perform a renormalization-group analysis of the perturbative series. We give a general argument that strongly suggests that the perturbative series of any average in the steady state satisfies the equilibrium Callan-Symanzik equations, and show in detail how it works to one-loop order for the electric current operator inside any average. We finally compute to two loops order the average of the electric current in the steady state, and perform a renormalization-group improvement. From this, we give a universal prescription, valid in the perturbative regime, for comparing the effect of the electric current to that of the temperature on the ``Kondo cloud''.

cond-mat.str-el↗

arXiv2007

Finite-Temperature Form Factors: a Review

We review the concept of finite-temperature form factor that was introduced recently by the author in the context of the Majorana theory. Finite-temperature form factors can be used to obtain spectral decompositions of finite-temperature correlation functions in a way that mimics the form-factor expansion of the zero temperature case. We develop the concept in the general factorised scattering set-up of integrable quantum field theory, list certain expected properties and present the full construction in the case of the massive Majorana theory, including how it can be applied to the calculation of correlation functions in the quantum Ising model. In particular, we include the ''twisted construction'', which was not developed before and which is essential for the application to the quantum Ising model.

hep-th↗

arXiv2006

Twisted vertex operators and Bernoulli polynomials

Using general principles in the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. We develop new identities and principles in the theory of vertex operator algebras and their twisted modules, and explain the construction by applying general results, including an identity that we call "modified weak associativity", to the Heisenberg vertex operator algebra. This paper gives proofs and further explanations of results announced earlier. It is a generalization to twisted vertex operators of work announced by the second author some time ago, and includes as a special case the proof of the main results of that work.

math.QA↗

arXiv2005

Finite-temperature form factors in the free Majorana theory

We study the large distance expansion of correlation functions in the free massive Majorana theory at finite temperature, alias the Ising field theory at zero magnetic field on a cylinder. We develop a method that mimics the spectral decomposition, or form factor expansion, of zero-temperature correlation functions, introducing the concept of "finite-temperature form factors". Our techniques are different from those of previous attempts in this subject. We show that an appropriate analytical continuation of finite-temperature form factors gives form factors in the quantization scheme on the circle. We show that finite-temperature form factor expansions are able to reproduce expansions in form factors on the circle. We calculate finite-temperature form factors of non-interacting fields (fields that are local with respect to the fundamental fermion field). We observe that they are given by a mixing of their zero-temperature form factors and of those of other fields of lower scaling dimension. We then calculate finite-temperature form factors of order and disorder fields. For this purpose, we derive the Riemann-Hilbert problem that completely specifies the set of finite-temperature form factors of general twist fields (order and disorder fields and their descendants). This Riemann-Hilbert problem is different from the zero-temperature one, and so are its solutions. Our results agree with the known form factors on the circle of order and disorder fields.

hep-th↗

arXiv2004

Ising Field Theory on a Pseudosphere

We show how the symmetries of the Ising field theory on a pseudosphere can be exploited to derive the form factors of the spin fields as well as the non-linear differential equations satisfied by the corresponding two-point correlation functions. The latter are studied in detail and, in particular, we present a solution to the so-called connection problem relating two of the singular points of the associated Painleve VI equation. A brief discussion of the thermodynamic properties is also presented.

hep-th↗

arXiv2004

Two-point correlation functions of scaling fields in the Dirac theory on the Poincare disk

A result from Palmer, Beatty and Tracy suggests that the two-point function of certain spinless scaling fields in a free Dirac theory on the Poincare disk can be described in terms of Painleve VI transcendents. We complete and verify this description by fixing the integration constants in the Painleve VI transcendent describing the two-point function, and by calculating directly in a Dirac theory on the Poincare disk the long distance expansion of this two-point function and the relative normalization of its long and short distance asymptotics. The long distance expansion is obtained by developing the curved-space analogue of a form factor expansion, and the relative normalization is obtained by calculating the one-point function of the scaling fields in question. The long distance expansion in fact provides part of the solution to the connection problem associated with the Painleve VI equation involved. Calculations are done using the formalism of angular quantization.

hep-th↗

arXiv2003

Twisted modules for vertex operator algebras and Bernoulli polynomials

Using general principles of the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. This is explained through results in the general theory of vertex operator algebras, including a new identity, which we call ``modified weak associativity.'' This paper is an announcement. The detailed proofs will appear elsewhere.

math.QA↗