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Journal of Physics Condensed Matter

The re-entrant phase diagram of the generalized Harper equation

Jukka A. Ketoja, Indubala I. Satija

Published Feb 3, 1997• 9 citations
Publisher

Abstract

We study the phase diagram of the tight-binding model for an electron on an anisotropic square lattice with a four-dimensional parameter space defined by two nearest-neighbour and two next-nearest-neighbour couplings. Using a renormalization scheme, we show that the inequality of the two next-nearest-neighbour couplings destroys the fat critical regime found in the isotropic case above the bicritical line and replaces it with another re-entrant extended phase. The scaling properties of the model are those of the corresponding tight-binding models on the nearest-neighbour square and triangular lattices. The triangular universality class also describes the quantum Ising chain in a transverse field with the only exception being the conformally invariant state of the Ising model which has no analogue in the triangular-lattice case.

DOI
10.1088/0953-8984/9/5/016

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We study the phase diagram of the tight-binding model for an electron on an anisotropic square lattice with a four-dimensional parameter space defined by two nearest-neighbour and two next-nearest-neighbour couplings. Using a renormalization scheme, we show that the inequality of the two next-nearest-neighbour couplings destroys the fat critical regime found in the isotropic case above the bicritical line and replaces it with another re-entrant extended phase. The scaling properties of the model are those of the corresponding tight-binding models on the nearest-neighbour square and triangular lattices. The triangular universality class also describes the quantum Ising chain in a transverse field with the only exception being the conformally invariant state of the Ising model which has no analogue in the triangular-lattice case.
We study the phase diagram of an electron on a square lattice in a transverse magnetic field. Using a renormalization scheme, we show that the inequality of the two next-nearest neighbor couplings destroys the fat critical regime above the bicritical line and replaces it with another reentrant extended phase. Furthermore, the universal strange attractor describing self-similar fluctuations of the localized phase is replaced by a new fixed point. The reentrant extended phase and the new fixed point of the localized phase belong to the universality class of an anisotropic triangular lattice.
The tight binding model for an electron on an anisotropic triangular lattice in a uniform magnetic field is studied using a decimation scheme. The model exhibits a transition from critical to localized phase and the phase diagram is described in terms of three nontrivial renormalization fixed points for the band edges. These subcritical, critical, and supercritical universality classes also describe the corresponding states of the quantum Ising chain in a modulating transverse field. The only exception is the conformally invariant point of the Ising model which has no analog in the electron problem.
Using the parallel tempering algorithm and graphics processing unit accelerated techniques, we have performed large-scale Monte Carlo simulations of the Ising model on a square lattice with antiferromagnetic (repulsive) nearest-neighbor and next-nearest-neighbor interactions of the same strength and subject to a uniform magnetic field. Both transitions from the (2x1) and row-shifted (2x2) ordered phases to the paramagnetic phase are continuous. From our data analysis, re-entrance behavior of the (2x1) critical line and a bicritical point which separates the two ordered phases at T=0 are confirmed. Based on the critical exponents we obtained along the phase boundary, Suzuki's weak universality seems to hold.
A renormalization scheme is developed to study an anisotropic quantum XY spin chain in a quasiperiodic transverse field. The critical phase of the quasi-particle excitations of the model with fractal wave functions exists in a finite parameter interval and is sandwiched between the extended and localized phases. The scaling properties of the critical phase fall into four distinct universality classes referred as spectral, subconformal, conformal and Harper. The spectral and conformal classes respectively describe the onsets of extended to critical and critical to localized transitions while the subconformal class describes the part of the phase diagram sandwiched between the conformal and spectral transitions. The Harper universality class describes the isotropic limit of the XY model. A decimation scheme is developed to compute the infinite sets of universal scaling ratios characterizing the wave functions in the four universality classes. The renormalization flow equations exhibit a limit cycle at the band center and at the band edges providing a new method for determining these energies with extremely high precision.
The authors study the Ising model on the triangular lattice with nearest-neighbor couplings K(nn) , next-nearest-neighbor couplings K(nnn) >0 , and a magnetic field H . This work is done by means of finite-size scaling of numerical results of transfer matrix calculations, and Monte Carlo simulations. We determine the phase diagram and confirm the character of the critical manifolds. The emphasis of this work is on the antiferromagnetic case K(nn) <0 , but we also explore the ferromagnetic regime K(nn) >/=0 for H=0 . For K(nn) <0 and H=0 we locate a critical phase presumably covering the whole range -infinity< K(nn) <0 . For K(nn) <0 , H not equal 0 we locate a plane of phase transitions containing a line of tricritical three-state Potts transitions. In the limit H-->infinity this line leads to a tricritical model of hard hexagons with an attractive next-nearest-neighbor potential.