The ground-state properties of the triangular antiferromagnetic Ising model with Ising constant J in a transverse field h are investigated by a real-space renormalisation group method. For h=0, the known properties (energy=-J, finite entropy, no order) are recovered. For h not=0, the ground state remains highly degenerate, no order exists and the correlation function exhibits power-law behaviour up to (h/J)c approximately=1.41, where a phase transition takes place to a non-degenerate ground state. At h/J=(h/J)c, power-law behaviour is still observed for the correlation function but with a different exponent. Just above this region a gap Delta opens ( Delta approximately ((h/J)-(h/J)c)s with s=0.8) and the coherence length of the correlation function xi diverges as xi varies as ((h/J)-(h/J)c)-v with v=0.8. This corresponds to a three-dimensional classical model where h is replaced by the temperature. In this type of system, where the ground state has a degeneracy proportional to N2/3 it is predicted that a transition will occur at a finite T c to a low-temperature phase without order but with power-law behaviour for the correlation function, as for the classical XY model in two dimensions.
The random Ising model in a transverse field in one dimension with Hamiltonian H=- Sigma ( Gamma i S i z +J i S i x S i+1 x ) is studied at T=O from a real-space renormalisation group block method which preserves duality transformations. The ground state magnetisation and the ground state energy are determined for random distributions P(J)(P( Gamma ))=N J J NJ-1 /J 0 N(J) for 0<J<J 0 and P(J)=0 for J>J O . A new critical behaviour corresponding to a new fixed point takes place in the presence of disorder. The crossover exponent describing the departure from the pure system behaviour is calculated. The second derivative of the ground state energy delta 2 E/ delta Gamma 2 which diverges logarithmically for the pure system is rounded in the presence of disorder but a sharp transition field still exists where the magnetisation goes to zero with an exponent beta about twice as large as the pure system exponent. Comparison is made with the analytical results of McCoy and Wu (1969) for the classical equivalent random-striped Ising 2D model.
A newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.
A transverse magnetic field $\mathrm{\ensuremath{\Gamma}}$ is known to induce antiferromagnetic three-sublattice order of the Ising spins ${\ensuremath{\sigma}}^{z}$ in the triangular lattice Ising antiferromagnet at low enough temperature. This low-temperature order is known to melt on heating in a two-step manner, with a power-law ordered intermediate temperature phase characterized by power-law correlations at the three-sublattice wave vector
Q: $\ensuremath{\langle}{\ensuremath{\sigma}}^{z}(\stackrel{P\vec}{R}){\ensuremath{\sigma}}^{z}(0)\ensuremath{\rangle}\ensuremath{\sim}cos(\mathbf{Q}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{R})/{|\stackrel{P\vec}{R}|}^{\ensuremath{\eta}(T)}$ with the temperature-dependent power-law exponent $\ensuremath{\eta}(T)\ensuremath{\in}(1/9,1/4)$. Here, we use a quantum cluster algorithm to study the ferromagnetic easy-axis susceptibility ${\ensuremath{\chi}}_{u}(L)$ of an $L\ifmmode\times\else\texttimes\fi{}L$ sample in this power-law ordered phase. Our numerical results are consistent with a recent prediction of a singular
L dependence ${\ensuremath{\chi}}_{u}(L)\ensuremath{\sim}{L}^{2\ensuremath{-}9\ensuremath{\eta}}$ when $\ensuremath{\eta}(T)$ is in the range
(1/9,2/9). This finite-size result implies, via standard scaling arguments, that the ferromagnetic susceptibility ${\ensuremath{\chi}}_{u}(B)$ to a uniform field
B along the easy axis is singular at intermediate temperatures in the small
B limit, ${\ensuremath{\chi}}_{u}(B)\ensuremath{\sim}{|B|}^{\ensuremath{-}\frac{4\ensuremath{-}18\ensuremath{\eta}}{4\ensuremath{-}9\ensuremath{\eta}}}$ for $\ensuremath{\eta}(T)\ensuremath{\in}(1/9,2/9)$, although there is no ferromagnetic long-range order in the low temperature state. Additionally we establish similar two-step melting behavior (via a study of the order parameter susceptibility ${\ensuremath{\chi}}_{\mathbf{Q}}$) in the case of the ferrimagnetic three-sublattice ordered phase which is stabilized by ferromagnetic next-neighbor couplings (
J2) and confirm that the ferromagnetic susceptibility obeys the predicted singular form in the associated power-law ordered phase.
The critical coarsening dynamics of the spin S =1/2, 3/2 antiferromagnetic Ising model on a triangular lattice is studied by the dynamic Monte Carlo simulation using a heat bath algorithm. The triangular antiferromagnetic Ising (TAI) model possesses an intrinsic geometrical frustration and a large degeneracy of ground state which may affect the equilibrium and non-equilibrium critical behaviors. The S =1/2 TAI has no phase transition at a finite temperature, but it was suggested that the S =3/2 TAI has the Kosterlitz–Thouless (KT)-type phase transition at a finite temperature. We employ a finite size scaling approach for the correlation function from the short-time dynamics of the S =1/2, 3/2 TAI to calculate the values of the static critical exponent η and the dynamic exponent z. For the S =1/2 TAI, our dynamic scaling analysis provides η =0.498±0.006 and z =2.278±0.020 at T =0, agreeing with the previous results. For the S =3/2 TAI, after identifying a KT-transition temperature TKT =0.51±0.01, we find the temperature-dependent η ranging from 0.301±0.008 at T =0.51 to 0.224±0.016 at T =0 along the KT-line whereas the value of z =2.20±0.06 is constant for T≤TKT. It is shown that the spin S =3/2 TAI model and the two-dimensional XY model, sharing the KT-type phase transition, exhibit similar static critical and coarsening dynamics behavior. For both the S =1/2, 3/2 TAI, the value of z (η) is compatible with (larger than) that of the Ising model at Tc and the XY model for T≤TKT in two-dimension. Our results imply that although the quasi-long-range order disappears with ηXY =0 in the two-dimensional XY model at T =0, the S =3/2 TAI still maintains it with η TAI =0.224 due to the effect of a frustration and a high degeneracy of ground state.
A zero-temperature real-space renormalization-group method for quantum systems is applied to the
S=21XY model with a
Z field with the Hamiltonian $H=\ensuremath{-}\frac{1}{2}J\ensuremath{\Sigma}{〈i, j〉}^{}[(1+\ensuremath{\gamma}){S}_{i}^{x}{S}_{j}^{x}+(1\ensuremath{-}\ensuremath{\gamma}){S}_{i}^{y}{S}_{j}^{y}]\ensuremath{-}h\ensuremath{\Sigma}{i}^{}{S}_{i}^{z}$ on quadratic, triangular, and hexagonal lattices in two dimensions (2D) and on a cubic lattice in 3D. In all the isotropic ($\ensuremath{\gamma}=0$) 2D cases the method gives no order in the ground state
∣0〉 but changes in the correlation functions at ${(\frac{h}{J})}_{c}=\frac{1}{2}{z}^{\ensuremath{'}}$ [${z}^{\ensuremath{'}}$ is the number of nearest neighbors (NN)] are observed. An argument is presented that at
(Jh)c the method becomes exact in the limit of infinite-size spin blocks. It is suggested that the correlation functions $〈0|{{S}_{0}}^{\ensuremath{\alpha}}{{S}_{R}}^{\ensuremath{\alpha}}|0〉$ ($\ensuremath{\alpha}=x, y$) for large
R have a power-law behavior $\ensuremath{\sim}{R}^{\ensuremath{-}\ensuremath{\eta}}$ for
0< ∣Jh∣<∣(Jh)c∣ with $\ensuremath{\eta}\ensuremath{\approx}1.5$ for
∣Jh∣<(Jh)c∣ and $\ensuremath{\eta}=2$ at
(Jh)c. The dynamical exponent
z is obtained to be \ensuremath{\approx} 0.3 for $|\frac{h}{J}|<\frac{1}{2}{z}^{\ensuremath{'}}$ and at
(Jh)c close to 2. An indirect method of calculating the ground-state energy is developed, and the results are compared with the known estimates. For $\ensuremath{\gamma}\ensuremath{\ne}0$ and
h=0 the $\ensuremath{\gamma}$ dependences of the order in the ground state ${\ensuremath{\mu}}^{x}(\ensuremath{\gamma})$ is calculated to be ${\ensuremath{\mu}}^{x}\ensuremath{\sim}{\ensuremath{\gamma}}^{0.21}$ for $\ensuremath{\gamma}\ensuremath{\rightarrow}0$ and
h=0. In a 3D cubic lattice ($\ensuremath{\gamma}=0$) the analysis of recursion relations indicates the existence of order, in accord with recent exact theorems.
We study the low-field ground-state (GS) properties of the antiferromagnetic transverse-field Ising model with long-range interactions (afLRTFIM) on the triangular lattice. We use the method of perturbative continuous unitary transformations (pCUT) to derive an effective model for the degenerate GS space of the antiferromagnetic nearest-neighbour (NN) Ising model on a finite system, by treating the transverse-field (TF) and the long-range interactions (LRI) as a perturbation. We determine a level-crossing between the gapped plain stripe phase at small TF and the gapped clock-ordered phase at intermediate TF at hc≅0.129 for α=6, N=36 spins in order three perturbation theory on the finite system. We discuss the qualitative layout of the quantum phase diagram of the afLRTFIM on the triangular lattice.