Conformal Minimal Models Research Articles

Quantum Simulation of the Tricritical Ising Model in Tunable Josephson Junction Ladders.

Modern hybrid superconductor-semiconductor Josephson junction arrays are a promising platform for analog quantum simulations. Their controllable and nonsinusoidal energy-phase relation opens the path to implement nontrivial interactions and study the emergence of exotic quantum phase transitions. Here, we propose the analysis of an array of hybrid Josephson junctions defining a two-leg ladder geometry for the quantum simulation of the tricritical Ising phase transition. This transition provides the paradigmatic example of minimal conformal models beyond Ising criticality and its excitations are intimately related to Fibonacci non-Abelian anyons and topological order in two dimensions. We study this superconducting system and its thermodynamic phases based on bosonization and matrix-product-state techniques. Its effective continuous description in terms of a three-frequency sine-Gordon quantum field theory suggests the presence of the targeted tricritical point and the numerical simulations confirm this picture. Our results indicate which experimental observables can be adopted in realistic devices to probe the physics and the phase transitions of the model. Additionally, our proposal provides a useful one-dimensional building block to design exotic topological order in two-dimensional scalable Josephson junction arrays.

PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{PT} $$\\end{document} breaking and RG flows between multicritical Yang-Lee fixed points

We study a novel class of Renormalization Group flows which connect multicritical versions of the two-dimensional Yang-Lee edge singularity described by the conformal minimal models M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M} $$\\end{document}(2, 2n + 3). The absence in these models of an order parameter implies that the flows towards and between Yang-Lee edge singularities are all related to the spontaneous breaking of PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{PT} $$\\end{document} symmetry and comprise a pattern of flows in the space of PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{PT} $$\\end{document} symmetric theories consistent with the c-theorem and the counting of relevant directions. Additionally, we find that while in a part of the phase diagram the domains of unbroken and broken PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{PT} $$\\end{document} symmetry are separated by critical manifolds of class M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M} $$\\end{document}(2, 2n + 3), other parts of the boundary between the two domains are not critical.

Exact solution of higher-derivative conformal theory and minimal models

I investigate the two-dimensional four-derivative conformal theory that emerges from the Nambu-Goto string after the path-integration over all fields but the metric tensor. Using the method of singular products which accounts for tremendous cancellations in perturbation theory, I show the (intelligent) one-loop approximation to give an exact solution. It is conveniently described through the minimal models where the central charge c in the Kac spectrum depends on the parameters of the four-derivative action. The relation is nonlinear so the domain of physical parameters is mapped onto c<1 thus bypassing the KPZ barrier of the Liouville action.

Multicriticality in Yang-Lee edge singularity

In this paper we study the non-unitary deformations of the two-dimensional Tricritical Ising Model obtained by coupling its two spin ℤ2 odd operators to imaginary magnetic fields. Varying the strengths of these imaginary magnetic fields and adjusting correspondingly the coupling constants of the two spin ℤ2 even fields, we establish the presence of two universality classes of infrared fixed points on the critical surface. The first class corresponds to the familiar Yang-Lee edge singularity, while the second class to its tricritical version. We argue that these two universality classes are controlled by the conformal non-unitary minimal models mathcal{M} (2, 5) and mathcal{M} (2, 7) respectively, which is supported by considerations based on PT symmetry and the corresponding extension of Zamolodchikov’s c-theorem, and also verified numerically using the truncated conformal space approach. Our results are in agreement with a previous numerical study of the lattice version of the Tricritical Ising Model [1]. We also conjecture the classes of universality corresponding to higher non-unitary multicritical points obtained by perturbing the conformal unitary models with imaginary coupling magnetic fields.

On the topological and crosscap entropies in non-oriented RCFTs

We establish a relation between the boundary and the topological entropies for the conformal minimal models in some of the simplest models of the unitary A–A series. We show that in these models the boundary entropy is a difference of topological entropies. Furthermore, we define the crosscap entropy as the analog to the boundary entropy in non-oriented theories. The crosscap entropy is defined as the logarithm of the degeneracy of the ground state due to the presence of crosscaps, and it can be expressed in terms of the crosscap coefficients. This crosscap entropy has not an explicit relation to the topological entropy as the boundary entropy. However, we propose a new quantity, \(\hat{S} = \text {ln}P_{0i}\) defined in terms of the modular transformation P between the open and closed channel of the Möbius partition function. With this quantity, the crosscap entropy can be regarded as a difference of entropies, very similar to the boundary entropy. We also compute the left-right entanglement entropy (LREE) for crosscap states, and we express it in terms of \(\hat{S}\). An explicit example of the LREE of the crosscap state in Wess–Zumino–Witten is carried out.

Entanglement entropy from corner transfer matrix in Forrester–Baxter non-unitary RSOS models

Using a corner transfer matrix approach, we compute the bipartite entanglement Rényi entropy in the off-critical perturbations of non-unitary conformal minimal models realised by lattice spin chains Hamiltonians related to the Forrester–Baxter RSOS models (Bianchini et al 2015 J. Stat. Mech. P03010) in regime III. This allows to show on a set of explicit examples that the Rényi entropies for non-unitary theories rescale near criticality as the logarithm of the correlation length with a coefficient proportional to the effective central charge. This complements a similar result, recently established for the size rescaling at the critical point (Bianchini et al 2015 J. Phys. A: Math. Theor. 48 04FT01), showing the expected agreement of the two behaviours. We also compute the first subleading unusual correction to the scaling behaviour, showing that it is expressible in terms of expansions of various fractional powers of the correlation length, related to the differences between the conformal dimensions of fields in the theory and the minimal conformal dimension. Finally, a few observations on the limit leading to the off-critical logarithmic minimal models of Pearce and Seaton (2012 J. Stat. Mech. P09014) are put forward.

Form factors of descendant operators: reduction to perturbed M (2, 2s + 1) models

In the framework of the algebraic approach to form factors in two-dimensional integrable models of quantum field theory we consider the reduction of the sine-Gordon model to the $\Phi_{13}$-perturbation of minimal conformal models of the $M(2,2s+1)$ series. We find in an algebraic form the condition of compatibility of local operators with the reduction. We propose a construction that make it possible to obtain reduction compatible local operators in terms of screening currents. As an application we obtain exact multiparticle form factors for the compatible with the reduction conserved currents $T_{\pm2k}$, $\Theta_{\pm(2k-2)}$, which correspond to the spin $\pm(2k-1)$ integrals of motion, for any positive integer~$k$. Furthermore, we obtain all form factors of the operators $T_{2k}T_{-2l}$, which generalize the famous $T\bar T$ operator. The construction is analytic in the $s$ parameter and, therefore, makes sense in the sine-Gordon theory.

Three loop analysis of the criticalO(N)models in6−εdimensions

We continue the study, initiated in arXiv:1404.1094, of the $O(N)$ symmetric theory of $N+1$ massless scalar fields in $6-\epsilon$ dimensions. This theory has cubic interaction terms $\frac{1}{2}g_1 \sigma (\phi^i)^2 + \frac{1}{6}g_2 \sigma^3$. We calculate the 3-loop beta functions for the two couplings and use them to determine certain operator scaling dimensions at the IR stable fixed point up to order $\epsilon^3$. We also use the beta functions to determine the corrections to the critical value of $N$ below which there is no fixed point at real couplings. The result suggests a very significant reduction in the critical value as the dimension is decreased to $5$. We also study the theory with $N=1$, which has a $Z_2$ symmetry under $\phi\rightarrow -\phi$. We show that it possesses an IR stable fixed point at imaginary couplings which can be reached by flow from a nearby fixed point describing a pair of $N=0$ theories. We calculate certain operator scaling dimensions at the IR fixed point of the $N=1$ theory and suggest that, upon continuation to two dimensions, it describes a non-unitary conformal minimal model.

Critical excitation spectrum of a quantum chain with a local three-spin coupling

Using the phenomenological renormalization group (PRG), we evaluate the low-energy excitation spectrum along the critical line of a quantum spin chain having a local interaction between three Ising spins and longitudinal and transverse magnetic fields, i.e., a Turban model. The low-energy excitation spectrum found with the PRG agrees with the spectrum predicted for the (D(4),A(4)) conformal minimal model under a nontrivial correspondence between translations at the critical line and discrete lattice translations. Under this correspondence, the measurements confirm a prediction that the critical line of this quantum spin chain and the critical point of the two-dimensional three-state Potts model are in the same universality class.

Energy level distribution of perturbed conformal field theories

We study the energy level spacing of perturbed conformal minimal models infinite volume, considering perturbations of such models that are massive but notnecessarily integrable. We compute their spectrum using a renormalization groupimproved truncated conformal spectrum approach. With this method we are able tostudy systems where more than 40 000 states are kept and where we determine theenergies of the lowest several thousand eigenstates with high accuracy. We find, asexpected, that the level spacing statistics of integrable perturbed minimal modelsare Poissonian while the statistics of non-integrable perturbations are GOE-like.However, by varying the system size (and so controlling the positioning of the theorybetween its IR and UV limits) one can induce crossovers between the two statisticaldistributions.

Effective potentials and kink spectra in non-integrable perturbed conformal field theories

We analyze the evolution of the effective potential and the particle spectrum of two-parameter families of non-integrable quantum field theories. These theories are defined by deformations of conformal minimal models by using the operators Φ1,3, Φ1,2 and Φ2,1. This present work extends the analysis, previously done for the universality classes of Ising/tricritical Ising/RSOS models, to all minimal models. We establish the symmetry and the duality properties of the various models also identifying the limiting theories that emerge when m → ∞.

Boundary States and Correlation Functions of Tricritical Ising Model from Coulomb-Gas Formalism

We consider the minimal conformal model describing the tricritical Ising model on the disk and on the upper half plane. Using the coulomb-gas formalism we determine its consistents boundary states as well as its one-point and two-point correlation functions.

Universal distribution of random matrix eigenvalues near the ‘birth of a cut’transition

We study the eigenvalue distribution of a random matrix, at a transition where a newconnected component of the eigenvalue density support appears away from other connectedcomponents. Unlike previously studied critical points, which correspond to rational singularitiesρ(x) ∼ xp/q classified by conformal minimal models and integrable hierarchies, this transition showslogarithmic and non-analytical behaviours. There is no critical exponent; instead, the power ofN changes in a sawtooth behaviour.

Paths, Virasoro characters and fermionic expressions

Baxter's corner transfer matrix method enables the one-point functions of certain 2D statistical models to be expressed as generating functions of weighted paths. In the case of the Forrester-Baxter models, the generating functions are Virasoro characters associated with the conformal minimal models M(p, p′). By combinatorially manipulating the paths, we obtain a quasiparticle description of the ensemble of paths. This directly leads to fermionic expressions for the Virasoro characters, and thence to interesting q-series identities.

Finite-Size scaling at Yang-Lee singularities of 2D discrete spin models

We numerically study the Hamiltonian limits of the two dimensional (2D) Ising and 3-state Potts models in complex magneticfields. From the Phenomenological Renormalization Group, we find the criticalfield values associated with the Yang-Lee singularity (YLS) in these models. We also determine the low-lying part of the excitation spectrum at the YLS. Finally, we compare the resulting patterns of energy levels to predictions for conformal non-unitary minimal models.

CONFORMAL THEORY FOR YANG–LEE SINGULARITY OF 2D 3-STATE POTTS MODEL

We identify a conformal field theory (CFT) that describes the Yang–Lee edge singularity of the 2-dimensional (2D) 3-state Potts model. The identification is based on a comparison of finite-size scaling measurements to predictions for conformal minimal models. The comparison shows that the Yang–Lee edge singularities of the 2D 3-state Potts and the 2D Ising models are in the same universality class.

Tachyon Condensation, Open-Closed Duality, Resolvents, and Minimal Bosonic and Type 0 Strings

Type 0A string theory in the (2,4k) superconformal minimal model backgrounds and the bosonic string in the (2,2k-1) conformal minimal models, while perturbatively identical in some regimes, may be distinguished non-perturbatively using double scaled matrix models. The resolvent of an associated Schrodinger operator plays three very important interconnected roles, which we explore perturbatively and non-perturbatively. On one hand, it acts as a source for placing D-branes and fluxes into the background, while on the other, it acts as a probe of the background, its first integral yielding the effective force on a scaled eigenvalue. We study this probe at disc, torus and annulus order in perturbation theory, in order to characterize the effects of D-branes and fluxes on the matrix eigenvalues. On a third hand, the integrated resolvent forms a representation of a twisted boson in an associated conformal field theory. The entire content of the closed string theory can be expressed in terms of Virasoro constraints on the partition function, which is realized as wavefunction in a coherent state of the boson. Remarkably, the D-brane or flux background is simply prepared by acting with a vertex operator of the twisted boson. This generates a number of sharp examples of open-closed duality, both old and new. We discuss whether the twisted boson conformal field theory can usefully be thought of as another holographic dual of the non-critical string theory.

The masses of gauge fields in higher spin field theory on AdS(4)

Higher spin field theory on AdS(4) is defined by lifting the minimal conformal sigma model in three-dimensional flat space. This allows to calculate the masses from the anomalous dimensions of the currents in the sigma model. The Goldstone boson field can be identified.

Non-Perturbative String Equations for Type 0A

Well-defined non-perturbative formulations of the physics of string theories, sometimes with D-branes present, were identified over a decade ago, from a careful study of double scaled matrix models. Following recent work which recasts some of those old results in the context of type 0 string theory, a study is made of a much larger family of models, which are proposed as type 0A models of the entire superconformal minimal series coupled to gravity. This gives many further examples of important physical phenomena, including non-perturbative descriptions of transitions between D-branes and fluxes, tachyon condensation, and holography. In particular, features of a large family of non-perturbatively stable string equations are studied, and results are extracted which pertain to type 0A string theory, with D-branes and fluxes, in this large class of backgrounds. For the entire construction to work, large parts of the spectrum of the supergravitationally dressed superconformal minimal models and that of the gravitationally dressed bosonic conformal minimal models must coincide, and it is shown how this happens. The example of the super-dressed tricritical Ising model is studied in some detail.

Exact conserved quantities on the cylinder I: conformal case

The nonlinear integral equations describing the spectra of the left and right (continuous) quantum KdV equations on the cylinder are derived from integrable lattice field theories, which turn out to allow the Bethe Ansatz equations of a twisted ``spin -1/2'' chain. A very useful mapping to the more common nonlinear integral equation of the twisted continuous spin $+1/2$ chain is found. The diagonalization of the transfer matrix is performed. The vacua sector is analysed in detail detecting the primary states of the minimal conformal models and giving integral expressions for the eigenvalues of the transfer matrix. Contact with the seminal papers \cite{BLZ, BLZ2} by Bazhanov, Lukyanov and Zamolodchikov is realised. General expressions for the eigenvalues of the infinite-dimensional abelian algebra of local integrals of motion are given and explicitly calculated at the free fermion point.