Harmonic measure and winding of random conformal paths: A Coulomb gas perspective

We consider random conformally invariant paths in the complex plane (SLEs). Using the Coulomb gas method in conformal field theory, we rederive the mixed multifractal exponents associated with both the harmonic measure and winding (rotation or monodromy) near such critical curves, previously obtained by quantum gravity methods. The results also extend to the general cases of harmonic measure moments and winding of multiple paths in a star configuration.     

Exact multifractal spectra for arbitrary laplacian random walks

Iterated conformal mappings are used to obtain exact multifractal spectra of the harmonic measure for arbitrary Laplacian random walks in two dimensions. Separate spectra are found to describe scaling of the growth measure in time, of the measure near the growth tip, and of the measure away from the growth tip. The spectra away from the tip coincide with those of conformally invariant equilibrium systems with arbitrary central charge c < or = 1, with c related to the particular walk chosen, while the scaling in time and near the tip cannot be obtained from the equilibrium properties.     

Conformal field theories and critical phenomena

In this article we present a brief review of the conformal symmetry and the two-dimensional conformal quantum field theories. As concrete applications of the conformal theories to the critical phenomena in statistical systems, we calculate the value of central charge and the anomalous scale dimensions of the Z ₂ symmetric quantum chain with boundary condition. The results are compatible with the prediction of the conformal field theories.     

Numerical study on schramm-loewner evolution in nonminimal conformal field theories

The Schramm-Loewner evolution (SLE) is a powerful tool to describe fractal interfaces in 2D critical statistical systems, yet the application of SLE is well established for statistical systems described by quantum field theories satisfying only conformal invariance, the so-called minimal conformal field theories (CFTs). We consider interfaces in Z(N) spin models at their self-dual critical point for N = 4 and N = 5. These lattice models are described in the continuum limit by nonminimal CFTs where the role of a ZN symmetry, in addition to the conformal one, should be taken into account. We provide numerical results on the fractal dimension of the interfaces which are SLE candidates for nonminimal CFTs. Our results are in excellent agreement with some recent theoretical predictions.     

Nonlocal current algebra and N = 2 superconformal field theory in two dimensions

We present a new construction of the unitary highest weight representations of the N = 2 superconformal algebras in two dimensions. This construction is based on the non-local current in the Z_N conformal theory and a free scalar field. It provides a physical realization of all the unitary N = 2 superconformal field theories by critical systems. The correlation functions of the theories can be calculated through this construction.     

Ashkin-teller criticality and pseudo-first-order behavior in a frustrated Ising model on the square lattice

We study the challenging thermal phase transition to stripe order in the frustrated square-lattice Ising model with couplings J(1) < 0 (nearest-neighbor, ferromagnetic) and J(2) > 0 (second-neighbor, antiferromagnetic) for g = J(2)/|J(1| > 1/2. Using Monte Carlo simulations and known analytical results, we demonstrate Ashkin-Teller criticality for g ≥ g*; i.e., the critical exponents vary continuously between those of the 4-state Potts model at g = g* and the Ising model for g → ∞. Thus, stripe transitions offer a route to realizing a related class of conformal field theories with conformal charge c = 1 and varying exponents. The transition is first order for g < g* = 0.67 ± 0.01, much lower than previously believed, and exhibits pseudo-first-order behavior for |g* ≤ g 相似文献   

Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum

The entanglement entropy of a pure quantum state of a bipartite system A union or logical sumB is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. In one dimension, the entanglement of critical ground states diverges logarithmically in the subsystem size, with a universal coefficient that for conformally invariant critical points is related to the central charge of the conformal field theory. We find that the entanglement entropy of a standard class of z=2 conformal quantum critical points in two spatial dimensions, in addition to a nonuniversal "area law" contribution linear in the size of the AB boundary, generically has a universal logarithmically divergent correction, which is completely determined by the geometry of the partition and by the central charge of the field theory that describes the critical wave function.     

Topological order and conformal quantum critical points

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary two-dimensional free boson, the two-dimensional Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously varying critical exponents separating phases with long-range order from a deconfined topologically ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-Abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang–Mills gauge theory with a Chern–Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.     

Gravity duals for nonrelativistic conformal field theories

We attempt to generalize the anti-de Sitter/conformal field theory correspondence to nonrelativistic conformal field theories which are invariant under Galilean transformations. Such systems govern ultracold atoms at unitarity, nucleon scattering in some channels, and, more generally, a family of universality classes of quantum critical behavior. We construct a family of metrics which realize these symmetries as isometries. They are solutions of gravity with a negative cosmological constant coupled to pressureless dust. We discuss realizations of the dust, which include a bulk superconductor. We develop the holographic dictionary and find two-point correlators of the correct form. A strange aspect of the correspondence is that the bulk geometry has two extra noncompact dimensions.     

Multivalued fields on the complex plane and conformal field theories

In this paper a class of conformal field theories with nonabelian and discrete group of symmetry is investigated. These theories are realized in terms of free scalar fields starting from the simpleb-c systems and scalar fields on algebraic curves. The Knizhnik-Zamolodchikov equations for the conformal blocks can be explicitly solved. Besides the fact that one obtains in this way an entire class of theories in which the operators obey nonstandard statistics, these systems are interesting in exploring the connection between statistics and curved space-times, at least in the two dimensional case.Work supported by the Consiglio Nazionale Ricerche, P.le A. Moro 7, Roma, Italy     

Conformal field theories and modular invariant partition functions for parafermion field theories

Parafermion conformal field theories with D_2N discrete symmetry are examined in detail. The structure of field space of parafermion field theories is studied with the help of a projection operator G. Characters of the representations of the twist sector of parafermion algebra and projected characters are given. A new class of modular invariant partition functions, therefore conformal field theories, for parafermion theories are found. We argue that the principal theories correspond to the generic critical SOS models of Andrew, Baxter and Forrest.     

Conformal invariance,the KdV equation and coadjoint orbits of the Virasoro algebra

We use the transformation properties of the energy-momentum tensor of 2-dimensional conformal field theories to study the symplectic geometry of DiffS¹/S¹ together with the other coadjoint orbits of the Virasoro algebra in the space of quadratic differentials. Also, we examine orbits of the N = 1 and N = 2 super-Virasoro algebras and investigate the connections with nonlinear differential equations of KdV type. The importance that the dual of the Virasoro algebra may have in the geometric formulation of string theory is discussed as well.     

The Large-N Limit of Superconformal Field Theories and Supergravity

We show that the large-N limits of certainconformal field theories in various dimensions includein their Hilbert space a sector describing supergravityon the product of anti-de Sitter spacetimes, spheres, and other compact manifolds. This is shown bytaking some branes in the full M/string theory and thentaking a low-energy limit where the field theory on thebrane decouples from the bulk. We observe that, in this limit, we can still trust thenear-horizon geometry for large N. The enhancedsupersymmetries of the near-horizon geometry correspondto the extra supersymmetry generators present in thesuperconformal group (as opposed to just the super-Poincaregroup). The 't Hooft limit of 3 + 1 N = 4 super-Yang–Mills at the conformal pointis shown to contain strings: they are IIB strings. Weconjecture that compactifications of M/string theory on various anti-de Sitterspacetimes is dual to various conformal field theories.This leads to a new proposal for a definition ofM-theory which could be extended to include fivenoncompact dimensions.     

Integrable and Conformal Boundary Conditions for \widehat{s\ell}(2) A–D–E Lattice Models and Unitary Minimal Conformal Field Theories

Integrable boundary conditions are studied for critical A–D–E and general graph-based lattice models of statistical mechanics. In particular, using techniques associated with the Temperley–Lieb algebra and fusion, a set of boundary Boltzmann weights which satisfies the boundary Yang–Baxter equation is obtained for each boundary condition. When appropriately specialized, these boundary weights, each of which depends on three spins, decompose into more natural two-spin edge weights. The specialized boundary conditions for the A–D–E cases are naturally in one-to-one correspondence with the conformal boundary conditions of $\widehat{s\ell }$ (2) unitary minimal conformal field theories. Supported by this and further evidence, we conclude that, in the continuum scaling limit, the integrable boundary conditions provide realizations of the complete set of conformal boundary conditions in the corresponding field theories.     

Nonabelions in the fractional quantum hall effect

Applications of conformal field theory to the theory of fractional quantum Hall systems are discussed. In particular, Laughlin's wave function and its cousins are interpreted as conformal blocks in certain rational conformal field theories. Using this point of view a hamiltonian is constructed for electrons for which the ground state is known exactly and whose quasihole excitations have nonabelian statistics; we term these objects “nonabelions”. It is argued that universality classes of fractional quantum Hall systems can be characterized by the quantum numbers and statistics of their excitations. The relation between the order parameter in the fractional quantum Hall effect and the chiral algebra in rational conformal field theory is stressed, and new order parameters for several states are given.     

Exact g-function flow between conformal field theories

Exact equations are proposed to describe g-function flows in integrable boundary quantum field theories which interpolate between different conformal field theories in their ultraviolet and infrared limits, extending previous work where purely massive flows were treated. The approach is illustrated with flows between the tricritical and critical Ising models, but the method is not restricted to these cases and should be of use in unravelling general patterns of integrable boundary flows between pairs of conformal field theories.     

Emergent Spacetime from Modular Motives

The program of constructing spacetime geometry from string theoretic modular forms is extended to Calabi-Yau varieties of dimensions three and four, as well as higher rank motives. Modular forms on the worldsheet can be constructed from the geometry of spacetime by computing the L?Cfunctions associated to omega motives of Calabi-Yau varieties, generated by their holomorphic n?forms via Galois representations. The modular forms that emerge in this way are related to characters of the underlying rational conformal field theory. The converse problem of constructing space from string theory proceeds in the class of diagonal theories by determining the motives associated to modular forms in the category of pure motives with complex multiplication. The emerging picture suggests that the L?Cfunction can be viewed as defining a map between the geometric category of motives and the category of conformal field theories on the worldsheet.     

c-Theorem for disordered systems

We find an analog of Zamolodchikov's c-theorem for disordered two-dimensional non-interacting systems in their supersymmetric field theory representation. We show that the energy momentum tensor of such field theories must be a part of a supermultiplet, and that a new parameter b can be introduced with the help of that multiplet. b flows along the renormalization group trajectories much like the central charge for unitary two-dimensional field theories. While it has not been established if this flow is irreversible, that is, if b always flows down to lower values, it does so for all the cases worked out so far. b gives a new way to label different conformal field theories for disordered systems whose central charge is always 0. b turns out to be related to the central extension of a certain algebra, a generalization of the Virasoro algebra, which we show may be present at the critical points of these theories. b is also related to the finite size corrections of the physical free energy of disordered systems. We discuss possible applications by computing b for two-dimensional Dirac fermions with random gauge potential, in other words, for U(1∣1) Kac-Moody algebra.     

Modular invariance and the spectrum of two dimensional conformal field theories

Modular invariance has recently emerged as a powerful tool in conformal field theory. In conjunction with the representation theory of infinite dimensional Lie algebras, the study of modular invariance gave the spectrum of several families of theories. These include the minimal conformal models (Cardy and others), WZW theories which describe string propagation on group manifolds (Gepner and Witten) and parafermionic field theories (Gepner and Qiu). The minimal conformal models models were shown to be a product of two SU(2) WZW theories (Gepner). These results represent a step towards a complete classification of conformal field theories, an important goal both for the study of critical phenomena and string theory.