High-temperature bivariate expansions have been derived for the two-spin-correlation function in a variety of classical lattice
XY (planar rotator) models in which spatially isotropic interactions among first-neighbor spins compete with spatially isotropic or anisotropic (in particular uniaxial) interactions among next-to-nearest-neighbor spins. The expansions, calculated for cubic lattices of dimensions
d=1, 2, and 3, are expressed in terms of the two variables
K1=J1/kT and
K2=J2/kT, where
J1 and
J2 are the nearest-neighbor and the next-to-nearest-neighbor exchange couplings, respectively. This paper deals in particular with the properties of the
d=3 uniaxial
XY model (ANNNXY model) for which the bivariate expansions have been computed through the 18th order, thus extending by 12 orders the results so far available and making a study of this model possible over a wide range of values of the competition parameter
R=J2/J1. Universality with respect to
R on the critical line separating the paramagnetic and the ferromagnetic phases can be verified, and at the same time the very accurate determination $\ensuremath{\gamma}=1.3177(5)$ and $\ensuremath{\nu}=0.6726(8)$ of the critical exponents of the susceptibility and of the correlation length, in the three-dimensional
XY universality class, can be achieved. For the exponents at the multicritical
(m,d,N)=(1,3,2) Lifshitz point the estimates ${\ensuremath{\gamma}}_{l}=1.535(25)$, ${\ensuremath{\nu}}_{\ensuremath{\perp}}=0.805(15)$, and ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=0.40(3)$ are obtained. Finally, the susceptibility exponent is estimated along the boundary between the disordered and the modulated phases.
A three-dimensional anisotropic sine-Gordon model, derived as the spin-wave approximation to the biaxial (
m=2) Lifshitz point problem in a uniform magnetic field, is shown to possess [in close analogy to the isotropic two-dimensional (2D) sine-Gordon theory which is well known to describe the critical behavior of the 2D
XY model], a surface of infinite-order phase transitions. This critical surface separates a phase characterized by infinite correlation length $\ensuremath{\xi}$ and power-law decay of correlations, and controlled by a stable fixed line, from one with finite $\ensuremath{\xi}$ and exponential decay. As the critical surface is approached from the latter phase, $\ensuremath{\xi}$ diverges as exp ($\ensuremath{\sigma}{t}^{\ensuremath{-}\ensuremath{\nu}}$) where $\ensuremath{\nu}=1$ is a universal number,
t measures the distance from the critical surface, and $\ensuremath{\sigma}$ is nonuniversal. On the critical surface correlations decay like ${r}^{\ensuremath{-}\ensuremath{\eta}}{(\mathrm{ln}r)}^{\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}}$, where $\ensuremath{\eta}=4$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=0.88\ensuremath{\cdots}$. Speculations on the occurrence of an infinite-order transition in liquid-crystal mixtures exhibiting nematic, smectic-
A, and smectic-
C phases are advanced.
The field-free quantum-spin-1/2 XXZ model is studied in three-dimensions using renormalization group theory. We obtain global phase diagrams and critical properties. We identify zero-temperature quantum phase transitions at isotropic points, as well as long-range order below critical temperatures at all anisotropies. We show that the excitation spectrum is gapless in XY-like, and gapped in Ising-like anisotropy regimes. Besides, internal energy density, specific heat, nearest-neighbor entanglement measures and spin-spin correlations are obtained globally at all temperatures. Our numerical results cover the global range of anisotropy parameter, for both ferromagnetic and antiferromagnetic cases.
The anisotropic triangular nearest-neighbor Ising model, with antiferromagnetic interactions, is studied. The phase diagram, as a function of temperature and field, has a complex structure. Two commensurate ordered phases are found. The transition to one is Ising-type. The second is reached from the disordered phase in one of two ways: either via an intermediate uniaxial incommensurate phase---which is reached at high temperatures by a Kosterlitz-Thouless transition and at low temperatures undergoes an incommensurate-commensurate transition---or by a single continuous transition. The latter transition is in the same universality class as the chiral three-state Potts model, with thermal exponent within 4% of that of the three-state Potts model. These two transition regimes are found to be separated by a Lifshitz point. The phase diagram and critical behavior were determined by analytical (symmetry analysis and free-fermion approximation) as well as numerical (phenomenological renormalization-group and Monte Carlo) methods.
The critical behavior of a semi-infinite
n-vector model with anisotropic ferromagnetic pair interactions on the surface and isotropic interactions everywhere else is studied with the use of renormalization-group methods for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. The effects of surface anisotropies are analyzed near the isotropic ordinary transition (where they produce corrections to scaling), near the isotropic special transition (where they are relevant perturbations), and near a new class of "anisotropic special" transitions. The latter occur if the surface exchange constants associated with
me (easy-magnetization) components take a critical value above which
me-component surface order is possible while the interaction constants of the remaining ${m}_{h}=n\ensuremath{-}{m}_{e}$ (hard-magnetization) components are weaker. Of particular interest is the anisotropic special transition with an easy axis (
me=1) because it occurs also in three dimensions, in contrast to the isotropic special one. The associated critical, crossover, and correction-to-scaling exponents are given to second order in $\ensuremath{\epsilon}$.
The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest
(J1)- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state
(T=0) and the finite-
T phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.