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arXiv:cond-mat/9912131 · Physical review. B, Condensed matter

Incommensurate structures studied by a modified density-matrix renormalization-group method

Andrej Gendiar, A. Šurda

Published Aug 1, 2000• 8 citations
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Abstract

A modified density-matrix renormalization-group (DMRG) method is introduced and applied to classical two-dimensional models: the anisotropic triangular nearest-neighbor Ising model and the anisotropic triangular next-nearest-neighbor Ising model. Phase diagrams of both models have complex structures and exhibit incommensurate phases. It was found that the incommensurate phase completely separates the disordered phase from one of the commensurate phases, i.e., the nonexistence of the Lifshitz point is conjectured in phase diagrams of both models with the DMRG method. It is shown that the DMRG method is also an effective-field approximation, but in spite of that it yields the correct value of the domain wall density critical exponent.

DOI
10.1103/physrevb.62.3960

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A modified density-matrix renormalization-group (DMRG) method is introduced and applied to classical two-dimensional models: the anisotropic triangular nearest-neighbor Ising model and the anisotropic triangular next-nearest-neighbor Ising model. Phase diagrams of both models have complex structures and exhibit incommensurate phases. It was found that the incommensurate phase completely separates the disordered phase from one of the commensurate phases, i.e., the nonexistence of the Lifshitz point is conjectured in phase diagrams of both models with the DMRG method. It is shown that the DMRG method is also an effective-field approximation, but in spite of that it yields the correct value of the domain wall density critical exponent.
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 classical two-dimensional anisotropic triangular nearest-neighbour Ising (ATNNI) model is studied by the density matrix renormalization group (DMRG) technique when periodic boundary conditions are imposed. Applying the finite-size scaling to the DMRG results, a commensurate-disordered (CD) phase transition line as well as temperature and magnetic critical exponents are calculated. We conclude that the CD phase transition in the ATNNI model belongs to the same universality class as the ordered-disordered phase transition of the Ising model.
We investigate the thermodynamic and critical properties of an interacting-domain-wall model which is derived from the triangular-lattice antiferromagnetic Ising model with the anisotropic nearest- and next-nearest-neighbor interactions. The model is equivalent to the general five-vertex model. Diagonalizing the transfer matrix exactly by the Bethe Ansatz method, we obtain the phase diagram displaying the commensurate and incommensurate (IC) phases separated by the Pokrovksy-Talapov transitions. The phase diagram exhibits commensurate phases where the domain-wall density q is locked at the values of 0, 1/2, and 1. The IC phase is a critical state described by the Gaussian fixed point. The effective Gaussian coupling constant is obtained analytically and numerically for the IC phase using the finite-size-scaling predictions of the conformal-field theory. It takes the value 1/2 in the noninteracting limit and also at the boundaries of q=0 or 1 phase and the value 2 at the boundary of q=1/2 phase, while it varies smoothly throughout the IC region.
We investigate the interacting-domain-wall model derived from the triangular-lattice antiferromagnetic Ising model with two next-nearest-neighbor interactions. The system has commensurate phases with a domain-wall density of q=2/3 as well as that of q=0 when the interaction is repulsive. The q=2/3 commensurate phase is separated from the incommensurate phase through the Kosterlitz-Thouless (KT) transition. The critical interaction strength and the nature of the KT phase transition are studied by the Monte Carlo simulations and numerical transfer-matrix calculations. For strongly attractive interaction, the system undergoes a first-order phase transition from the q=0 commensurate phase to the incommensurate phase with q\ensuremath{\ne}0. The incommensurate phase is a critical phase which is in the Gaussian model universality class. The effective Gaussian coupling constant is calculated as a function of interaction parameters from the finite-size scaling of the transfer-matrix spectra.
We numerically study the Kitaev honeycomb model with the additional XX Ising interaction between the nearest and the next nearest neighbors (Kitaev–Ising- J1{{J}_{1}}J1​ - J2{{J}_{2}}J2​ model), by using the density matrix renormalization group method. Such additional interaction correspond to the nearest and diagonal interactions on the square lattice. Phase diagram of the bare Kitaev model consist of low entangled commensurate magnetic phases and entangled Kitaev spin liquid. Anisotropic Ising interaction allows the entangled quantum paramagnetic phases in the phase diagram, which in the absence of the magnetic field previously was predicted for more complex type of interaction. We study the scaling law of the entanglement entropy and the bond dimension of the matrix product state with the size of the system. In addition, we propose an optimization algorithm to prevent density matrix renormalization group (DMRG) from getting stuck in the low-entangled phases.