Massless Flows Research Articles

Entanglement entropy along a massless renormalisation flow: the tricritical to critical Ising crossover

We study the Rényi entanglement entropies along the massless renormalisation group flow that connects the tricritical and critical Ising field theories. Similarly to the massive integrable field theories, we derive a set of bootstrap equations, from which we can analytically calculate the twist field form factors in a recursive way. Additionally, we also obtain them as a non-trivial ‘roaming limit’ of the sinh-Gordon theory. Then the Rényi entanglement entropies are obtained as expansions in terms of the form factors of these branch point twist fields. We find that the form factor expansion of the entanglement entropy along the flow organises in two different kind of terms. Those that couple particles with the same chirality, and reproduce the entropy of the infrared Ising theory, and those that couple particles with different chirality, which provide the ultraviolet contributions. The massless flow under study possesses a global ℤ2 spin-flip symmetry. We further consider the composite twist fields associated to this group, which enter in the study of the symmetry resolution of the entanglement. We derive analytical expressions for their form factors both from the bootstrap equations and from the roaming limit of the sinh-Gordon theory.

Mingling of the infrared and ultraviolet and the “cosmological constant” for interacting QFT in 2d

We propose a proper definition of the vacuum expectation value of the stress energy tensor 〈0| Tμν| 0〉 for integrable quantum field theories in two spacetime dimensions, which is the analog of the cosmological constant in 4d. For a wide variety of models, massive or massless, we show {rho}_{textrm{vac}}=-{m}^2/2mathfrak{g} exactly, where mathfrak{g} is a generalized coupling which we compute and m is a basic mass scale. The kinds of models we consider are the massive sinh-Gordon and sine-Gordon theories and perturbations of the Yang-Lee and 3-state Potts models, pure Toverline{T} perturbations of infra-red QFT’s, and UV completions of the latter which are massless flows between UV and IR fixed points. In the massive case m is the physical mass of the lightest particle and mathfrak{g} is related to parameters in the 2-body S-matrix. In some examples ρvac = 0 due to a fractional supersymmetry. For massless cases, m can be a scale of spontaneous symmetry breaking. The “cosmological constant problem” generically arises in the free field limit mathfrak{g} → 0, thus interactions can potentially resolve the problem at least for most cases considered in this paper. We speculate on extensions of these results to 4 spacetime dimensions and propose {rho}_{textrm{vac}}={m}^4/2mathfrak{g} , however without integrability we cannot yet propose a precise manner in which to calculate mathfrak{g} . Nevertheless, based on cosmological data on ρvac, if mathfrak{g} ~ 1 then it is worth pointing out that the lightest mass particle is on the order of experimental values of proposed neutrino masses.

The staircase model: massless flows and hydrodynamics

The staircase model (SM) is a simple generalization of the sinh-Gordon model, obtained by complexifying the coupling constant. This produces a new theory with many interesting features. Chief among them is the fact that scaling functions such as Zamolodchikov’s c-function display ‘roaming’ behaviour, that is, they visit the vicinity of an infinite number of conformal fixed points, the unitary minimal models of conformal field theory. This rich structure also makes the model an interesting candidate for study using the generalized hydrodynamic approach to integrable quantum field theory (IQFT). By studying hydrodynamic quantities such as the effective velocities of quasiparticles we can develop a more physical picture of interaction in the theory, both at and away from equilibrium. Indeed, we find that in the SM the effective velocity displays monotonicity features not found in any other IQFT. These admit a natural interpretation when investigated in the context of the massless theories known as models, which provide an effective description of massless flows between consecutive unitary minimal models. Consequently, we find that the effective velocity in the SM can be reconstructed by a ‘cut and paste’ procedure involving the equivalent functions associated to the massless excitations of suitable models. We also investigate the average currents and densities of higher spin conserved quantities in the partitioning protocol.

Hydrodynamics of massless integrable RG flows and a non-equilibrium c-theorem

We study Euler scale hydrodynamics of massless integrable quantum field theories interpolating between two non-trivial renormalisation group fixed points after inhomogeneous quantum quenches. Using a partitioning protocol with left and right initial thermal states and the recently developed framework of generalised hydrodynamics, we focus on current and density profiles for the energy and momentum as a function of ξ = x/t, where both x and t are sent to infinity. Studying the first few members of the An and Dn massless flows we carry out a systematic treatment of these series and generalise our results to other unitary massless models.In our analysis we find that the profiles exhibit extended plateaux and that non-trivial bounds exist for the energy and momentum densities and currents in the non-equilibrium stationary state, i.e. when ξ = 0. To quantify the magnitude of currents and densities, dynamical central charges are defined and it is shown that the dynamical central charge for the energy current satisfies a certain monotonicity property. We discuss the connection of the Landauer-Büttiker formalism of transport with our results and show that this picture can account for some of the bounds for the currents and for the monotonicity of the dynamical central charge. These properties are shown to be present not only in massless flows but also in the massive sinh-Gordon model suggesting their general validity and the correctness of the Landauer-Büttiker interpretation of transport in integrable field theories. Our results thus imply the existence of a non-equilibrium c-theorem as well, at least in integrable models. Finally we also study the interesting low energy behaviour of the A2 model that corresponds to the massless flow from the tricritical to the critical Ising field theory.

Backward Finite-Time Lyapunov Exponents in Inertial Flows.

Inertial particles are finite-sized objects that are carried by fluid flows and in contrast to massless tracer particles they are subject to inertia effects. In unsteady flows, the dynamics of tracer particles have been extensively studied by the extraction of Lagrangian coherent structures (LCS), such as hyperbolic LCS as ridges of the Finite-Time Lyapunov Exponent (FTLE). The extension of the rich LCS framework to inertial particles is currently a hot topic in the CFD literature and is actively under research. Recently, backward FTLE on tracer particles has been shown to correlate with the preferential particle settling of small inertial particles. For larger particles, inertial trajectories may deviate strongly from (massless) tracer trajectories, and thus for a better agreement, backward FTLE should be computed on inertial trajectories directly. Inertial backward integration, however, has not been possible until the recent introduction of the influence curve concept, which - given an observation and an initial velocity - allows to recover all sources of inertial particles as tangent curves of a derived vector field. In this paper, we show that FTLE on the influence curve vector field is in agreement with preferential particle settling and more importantly it is not only valid for small (near-tracer) particles. We further generalize the influence curve concept to general equations of motion in unsteady spatio-velocity phase spaces, which enables backward integration with more general equations of motion. Applying the influence curve concept to tracer particles in the spatio-velocity domain emits streaklines in massless flows as tangent curves of the influence curve vector field. We demonstrate the correlation between inertial backward FTLE and the preferential particle settling in a number of unsteady vector fields.

Roaming form factors for the tricritical to critical Ising flow

We study the massless flows described by the staircase model introduced by Al.B. Zamolodchikov through the analytic continuation of the sinh-Gordon S-matrix, focusing on the renormalisation group flow from the tricritical to the critical Ising model. We show that the properly defined roaming limits of certain sinh-Gordon form factors are identical to the form factors of the order and disorder operators for the massless flow. As a by-product, we also construct form factors for a semi-local field in the sinh-Gordon model, which can be associated with the twist field in the ultraviolet limiting free massless bosonic theory.

Massless renormalization group flow in [formula omitted] perturbed conformal field theory

We investigate the infrared properties of SU(N)k conformal field theory perturbed by its adjoint primary field in 1+1 dimensions. The latter field theory is shown to govern the low-energy properties of various SU(N) spin chain problems. In particular, using a mapping onto k-leg SU(N) spin ladder, a massless renormalization group flow to SU(N)1 criticality is predicted when N and k have no common divisor. The latter result extends the well-known massless flow between SU(2)k and SU(2)1 Wess–Zumino–Novikov–Witten theories when k is odd in connection to the Haldane's conjecture on SU(2) Heisenberg spin chains. A direct approach is presented in the simplest N=3 and k=2 case to investigate the existence of this massless flow.

Nature of the deconfining phase transition in the (2+1)-dimensional SU(N) Georgi–Glashow model

Abstract The nature of the deconfining phase transition in the ( 2 + 1 )-dimensional SU( N ) Georgi–Glashow model is investigated. Within the dimensional-reduction hypothesis, the properties of the transition are described by a two-dimensional vectorial Coulomb gas models of electric and magnetic charges. The resulting critical properties are governed by a generalized SU( N ) sine-Gordon model with self-dual symmetry. We show that this model displays a massless flow to an infrared fixed point which corresponds to the Z N parafermions conformal field theory. This result, in turn, supports the conjecture of Kogan, Tekin, and Kovner that the deconfining transition in the ( 2 + 1 )-dimensional SU( N ) Georgi–Glashow model belongs to the Z N universality class.

Form factors in the massless coset models [formula omitted]: Part II

Massless flows from the coset model su ( 2 ) k + 1 ⊗ su ( 2 ) k / su ( 2 ) 2 k + 1 to the minimal model M k + 2 are studied from the viewpoint of form factors. These flows include in particular the flow from the tricritical Ising model to the Ising model. By analogy with the magnetization operator in the flow TIM → IM , we construct all form factors of an operator that flows to Φ 1 , 2 in the IR. We make a numerical estimation of the difference of conformal weights between the UV and the IR thanks to the Δ-sum rule; the results are consistent with the conformal weight of the operator Φ 2 , 2 in the UV. By analogy with the energy operator in the flow TIM → IM, we construct all form factors of an operator that flows to Φ 2 , 1 . We propose to identify the operator in the UV with σ 1 Φ 1 , 2 .

Form factors in the massless coset models [formula omitted]: Part I

Massless flows between the coset model su ( 2 ) k + 1 ⊗ su ( 2 ) k / su ( 2 ) 2 k + 1 and the minimal model M k + 2 are studied from the viewpoint of form factors. These flows include, in particular, the flow between the tricritical Ising model and the Ising model. Form factors of the trace operator with an arbitrary number of particles are constructed, and numerical checks on the central charge are performed with four-particle contribution. Large discrepancies with respect to the exact results are observed in most cases.

Excited TBA equations II: massless flow from tricritical to critical Ising model

We consider the massless tricritical Ising model M(4,5) perturbed by the thermal operator ϕ 1,3 in a cylindrical geometry and apply integrable boundary conditions, labelled by the Kac labels ( r, s), that are natural off-critical perturbations of known conformal boundary conditions. We derive massless thermodynamic Bethe ansatz (TBA) equations for all excitations by solving, in the continuum scaling limit, the TBA functional equation satisfied by the double-row transfer matrices of the A 4 lattice model of Andrews, Baxter and Forrester (ABF) in Regime IV. The resulting TBA equations describe the massless renormalization group flow from the tricritical to critical Ising model. As in the massive case of Part I, the excitations are completely classified in terms of ( m , n ) systems but the string content changes by one of three mechanisms along the flow. Using generalized q-Vandermonde identities, we show that this leads to a flow from tricritical to critical Ising characters. The excited TBA equations are solved numerically to follow the continuous flows from the UV to the IR conformal fixed points.

Crossover behavior of theJ1−J2model in a staggered magnetic field

The ground states of the $S=\frac{1}{2}$ Heisenberg chain with the nearest-neighbor and the next-nearest-neighbor antiferromagnetic couplings are numerically investigated in a staggered magnetic field. While the staggered magnetic field may induce the N\'eel-type excitation gap, and it is characterized by the Gaussian fixed point in the spin-fluid region, the crossover to the behavior controlled by the Ising fixed point is expected to be observed for the spontaneously dimerized state at finite field. Treating a low-lying excitation gap by the phenomenological renormalization-group method, we numerically determine the massless flow connecting the Gaussian and Ising fixed points. Further, to check the criticalities, we perform the finite-size-scaling analysis of the excitation gap.

Massless flows between minimal W models

We study the renormalisation group flows between minimal W models by means of a new set of nonlinear integral equations which provide access to the effective central charge of both unitary and nonunitary models. We show that the scaling function associated to the nonunitary models is a nonmonotonic function of the system size.

Behavior of boundary string field theory associated with integrable massless flow.

We put forward an idea that the boundary entropy associated with integrable massless flow of thermodynamic Bethe ansatz (TBA) is identified with tachyon action of boundary string field theory. We show that the temperature parametrizing a massless flow in the TBA formalism can be identified with tachyon energy for the classical action at least near the ultraviolet fixed point, i.e., the open string vacuum.

The scaling supersymmetric Yang–Lee model with boundary

We define the scaling supersymmetric Yang-Lee model with boundary as the (1,3) perturbation of the superconformal minimal model SM(2/8) (or equivalently, the (1,5) perturbation of the conformal minimal model M(3/8)) with a certain conformal boundary condition. We propose the corresponding boundary S matrix, which is not diagonal for general values of the boundary parameter. We argue that the model has an integral of motion corresponding to an unbroken supersymmetry, and that the proposed S matrix commutes with a similar quantity. We also show by means of a boundary TBA analysis that the proposed boundary S matrix is consistent with massless flow away from the ultraviolet conformal boundary condition.

Non-integrable aspects of the multi-frequency sine-Gordon model

We consider the two-dimensional quantum field theory of a scalar field self-interacting via two periodic terms of frequencies α and β. Looking at the theory as a perturbed sine-Gordon model, we use form factor perturbation theory to analyse the evolution of the spectrum of particle excitations. We show how, within this formalism, the non-locality of the perturbation with respect to the solitons is responsible for their confinement in the perturbed theory. The effects of the frequency ratio α/β being a rational or irrational number and the occurrence of massless flows from the gaussian to the Ising fixed point are also discussed. A generalisation of the Ashkin-Teller model and the massive Schwinger model are presented as examples of application of the formalism.

Correlation functions along a massless flow.

A nonperturbative method based on the form factor bootstrap approach is proposed for the analysis of correlation functions of two-dimensional massless integrable theories and applied to the massless flow between the tricritical Ising and the critical Ising models.

MASSLESS FLOWS I: THE SINE-GORDON AND O(n) MODELS

The massless flow between successive minimal models of conformal field theory is related to a flow within the sine-Gordon model when the coefficient of the cosine potential is imaginary. This flow is studied, partly numerically, from three different points of view. First we work out the expansion close to the Kosterlitz-Thouless point, and obtain roaming behavior, with the central charge going up and down in between the UV and IR values of c=1. Next we analytically continue the Casimir energy of the massive flow (i.e. with real cosine term). Finally we consider the lattice regularization provided by the O(n) model in which massive and massless flows correspond to high- and low-temperature phases. A detailed discussion of the case n=0 is then given using the underlying N=2 supersymmetry, which is spontaneously broken in the low-temperature phase. The “index” tr F(−1)F follows from the Painlevé III differential equation, and is shown to have simple poles in this phase. These poles are interpreted as occurring from level crossing (one-dimensional phase transitions for polymers). As an application, new exact results for the connectivity constants of polymer graphs on cylinders are obtained. These results and points of view are used in the following paper to discuss the appropriate exact S matrices and the resulting Casimir energies.

On symmetries of some massless 2D field theories

We describe a few aspects of the quantum symmetries of some massless two-dimensional field theories. We discuss their relations with recent proposals for the factorized scattering theories of the massless PCM 1 and O(3) ϑ= π sigma models. We use these symmetries to propose massless factorized S-matrices for the su(2) sigma models with topological terms at any level, alias the PCM k models, and for the su(2)-coset massless flows.