Self-dual Critical Point Research Articles

A finite temperature study of ideal quantum gases in the presence of one dimensional quasi-periodic potential

We study the thermodynamics of ideal Bose gas as well as the transport properties of non interacting bosons and fermions in a one dimensional quasi-periodic potential, namely Aubry–André (AA) model at finite temperature. For bosons in finite size systems, the effect of quasi-periodic potential on the crossover phenomena corresponding to Bose–Einstein condensation (BEC), superfluidity and localization phenomena at finite temperatures are investigated. From the ground state number fluctuation we calculate the crossover temperature of BEC which exhibits a non monotonic behavior with the strength of AA potential and vanishes at the self-dual critical point following power law. Appropriate rescaling of the crossover temperatures reveals universal behavior which is studied for different quasi-periodicity of the AA model. Finally, we study the temperature and flux dependence of the persistent current of fermions in presence of a quasi-periodic potential to identify the localization at the Fermi energy from the decay of the current.

Critical points in coupled Potts models and critical phases in coupled loop models

We show how to couple two critical Q-state Potts models to yield a new self-dual critical point. We also present strong evidence of a dense critical phase near this critical point when the Potts models are defined in their completely packed loop representations. In the continuum limit, the new critical point is described by an SU(2) coset conformal field theory, while in this limit of the critical phase, the two loop models decouple. Using a combination of exact results and numerics, we also obtain the phase diagram in the presence of vacancies. We generalize these results to coupling two Potts models at different Q.

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.

SLE in self-dual critical [formula omitted] spin systems: CFT predictions

The Schramm–Loewner evolution (SLE) describes the continuum limit of domain walls at phase transitions in two-dimensional statistical systems. We consider here the SLE in Z ( N ) spin models at their self-dual critical point. For N = 2 and N = 3 these models correspond to the Ising and three-state Potts model. For N ⩾ 4 the critical self-dual Z ( N ) spin models are described in the continuum limit by non-minimal conformal field theories with central charge c ⩾ 1 . By studying the representations of the corresponding chiral algebra, we show that two particular operators satisfy a two level null vector condition which, for N ⩾ 4 , presents an additional term coming from the extra symmetry currents action. For N = 2 , 3 these operators correspond to the boundary conditions changing operators associated to the SLE 16 / 3 (Ising model) and to the SLE 24 / 5 and SLE 10 / 3 (three-state Potts model). We suggest a definition of the interfaces within the Z ( N ) lattice models. The scaling limit of these interfaces is expected to be described at the self-dual critical point and for N ⩾ 4 by the SLE 4 ( N + 1 ) / ( N + 2 ) and SLE 4 ( N + 2 ) / ( N + 1 ) processes.

Thermal melting of density waves on the square lattice

We present the theory of the effect of thermal fluctuations on commensurate $p\ifmmode\times\else\texttimes\fi{}p$ density wave ordering on the square lattice ($p\ensuremath{\geqslant}3$, integer). For the case in which this order is lost by a second-order transition, we argue that the adjacent state is generically an incommensurate striped state, with commensurate $p$-periodic long-range order along one direction, and incommensurate quasilong-range order along the orthogonal direction. We also present the routes by which the fully disordered high temperature state can be reached. For $p=4$, and at special commensurate densities, the $4\ifmmode\times\else\texttimes\fi{}4$ commensurate state can melt directly into the disordered state via a self-dual critical point with nonuniversal exponents.

The spin-1/2XXZ Heisenberg chain, the quantum algebra Uq[sl(2)], and duality transformations for minimal models

The finite-size scaling spectra of the spin-1/2 XXZ Heisenberg chain with toroidal boundary conditions and an even number of sites provide a projection mechanism yielding the spectra of models with a central charge c<1 including the unitary and non-unitary minimal series. Taking into account the half-integer angular momentum sectors - which correspond to chains with an odd number of sites - in many cases leads to new spinor operators appearing in the projected systems. These new sectors in the XXZ chain correspond to a new type of frustration lines in the projected minimal models. The corresponding new boundary conditions in the Hamiltonian limit are investigated for the Ising model and the 3-state Potts model and are shown to be related to duality transformations which are an additional symmetry at their self-dual critical point. By different ways of projecting systems we find models with the same central charge sharing the same operator content and modular invariant partition function which however differ in the distribution of operators into sectors and hence in the physical meaning of the operators involved. Related to the projection mechanism in the continuum there are remarkable symmetry properties of the finite XXZ chain. The observed degeneracies in the energy and momentum spectra are shown to be the consequence of intertwining relations involving U_q[sl(2)] quantum algebra transformations.

Critical exponents of the four-state Potts model

The two-dimensional four-state Potts model at finite temperature can be transformed, via the transfer matrix, into a one-dimensional quantum mechanical model at zero temperature. The duality invariant renormalisation group introduced by Fernandez-Pacheco (1979) is then employed to study the ground-state critical properties of this model. The fixed point is located at exactly the self-dual critical point K*=1. The thermal exponent is calculated to be yT=1.3219. It is in excellent agreement with the recent series value of Ditzian and Kadanoff (1979) (yT=1.33). Although it is not inconsistent with den Nijs's (1979) conjectured exact value of 3/2, the difference is nevertheless substantial.

Critical exponents of the three-state potts model

The Fernandez-Pacheco duality invariant renormalization group is applied to the hamiltonian version of the two-dimensional three-state Potts model. The fixed point is located at exactly the self-dual critical point K ∗ = 1. The thermal exponent is calculated to be y T=1.1814. This value is in excellent agreement with the low temperature series expansion result of Zwanzig and Ranshaw ( y T = 1.174) and the strong coupling expansion result of Elitzur, Pearson and Shigemitsu ( y T = 1.190). It also seems to lend strong support to den Nijs' recent conjecture that the exact value should be y T = 6/5.