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Physical review. B./Physical review. B

Saddle point solution of the Landau-Ginzburg-Wilson Hamiltonian for the McCoy-Wu Ising model

Xintian Wu

Published Jun 20, 2018• 1 citation
Publisher

Abstract

The Landau-Ginzburg-Wilson Hamiltonian for the McCoy-Wu strip-random Ising model on a two-dimensional lattice is studied. In the Landau-Ginzburg-Wilson Hamiltonian, the random temperature only depends on one of the coordinates ($x$), the horizontal coordinate. The difference (rather than differential) saddle point equation is solved numerically. In the critical regime, there exist locally ordered regions, where the saddle point solutions are not zero. Due to the translational invariance in the vertical direction, the locally ordered regions are striplike in the vertical direction. In the excited state solutions, there exist horizontal domain walls across the locally ordered regions. It is found that these horizontal domain walls are stabilized by free-energy barriers due to the discreteness. The stability conditions for such domain walls are studied. Usually, the domain wall across multiple locally ordered regions can be stable. These multiple locally ordered regions form an elementary cluster. The criterion for multiple locally ordered regions forming an elementary cluster is obtained. At the saddle point level, we get an effective Hamiltonian in which the elementary clusters behave as Ising spins and are coupled with their neighbors. For the random temperatures with narrow uniform probability distributions, the distribution of elementary cluster size, the horizontal and vertical bonds between elementary clusters are calculated numerically. The vertical bonds are extremely strong and the horizontal bonds are extremely weak. The distribution of vertical bonds is exponential. There is an exponentially small probability of elementary clusters with extremely large size. These elementary clusters with extremely large size will cause the susceptibility divergent in a range of temperature. The random temperature with binary probability distributions is also studied. The qualitative properties are the same. The Griffiths-McCoy singularity in the saddle point solutions does not depend on the distribution of random temperature.

DOI
10.1103/physrevb.97.224422

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The Landau-Ginzburg-Wilson Hamiltonian for the McCoy-Wu strip-random Ising model on a two-dimensional lattice is studied. In the Landau-Ginzburg-Wilson Hamiltonian, the random temperature only depends on one of the coordinates (xxx), the horizontal coordinate. The difference (rather than differential) saddle point equation is solved numerically. In the critical regime, there exist locally ordered regions, where the saddle point solutions are not zero. Due to the translational invariance in the vertical direction, the locally ordered regions are striplike in the vertical direction. In the excited state solutions, there exist horizontal domain walls across the locally ordered regions. It is found that these horizontal domain walls are stabilized by free-energy barriers due to the discreteness. The stability conditions for such domain walls are studied. Usually, the domain wall across multiple locally ordered regions can be stable. These multiple locally ordered regions form an elementary cluster. The criterion for multiple locally ordered regions forming an elementary cluster is obtained. At the saddle point level, we get an effective Hamiltonian in which the elementary clusters behave as Ising spins and are coupled with their neighbors. For the random temperatures with narrow uniform probability distributions, the distribution of elementary cluster size, the horizontal and vertical bonds between elementary clusters are calculated numerically. The vertical bonds are extremely strong and the horizontal bonds are extremely weak. The distribution of vertical bonds is exponential. There is an exponentially small probability of elementary clusters with extremely large size. These elementary clusters with extremely large size will cause the susceptibility divergent in a range of temperature. The random temperature with binary probability distributions is also studied. The qualitative properties are the same. The Griffiths-McCoy singularity in the saddle point solutions does not depend on the distribution of random temperature.
The phase transition in quenched disordered systems is studied on the level of the saddle point solution. Two-dimensional saddle point equation of the Landau-Ginzburg-Wilson Hamiltonian with random temperature is numerically solved for the excited states. It is shown that the excited-state solutions can be described by the domain walls. The length of domain wall and the free energy increase due to the domain wall are calculated. On the level of the saddle point solution the partition function can be mapped to an Ising model approximately. The coupling between Ising spin is estimated. The phase transition is discussed according to this Ising model. It is found that there are two classes of phase transition: inhomogeneous and homogeneous. Different from the pure system, for the phase transition in disordered systems there are fluctuations on the level of saddle point solution.
The Landau-Ginzburg-Wilson Hamiltonian with random temperature for the phase transition in disordered systems from the Griffiths phase to ferromagnetic phase is reexamined. From the saddle point solutions, especially the excited state solutions, it is shown that the system self-organizes into blocks coupled with their neighbors like superspins, which are emergent variables. Taking the fluctuation around these saddle point solutions into account, we get an effective Hamiltonian, including the emergent superspins of the blocks, the fluctuation around the saddle point solutions, and their couplings. Applying Stratonovich-Hubbard transformation to the part of superspins, we get a Landau-Ginzburg-Wilson Hamiltonian for the blocks. From the saddle point equations for the blocks, we can get the second generation blocks, of which sizes are much larger than the first generation blocks. Repeating this procedure again and again, we get many generations of blocks to describe the asymptotic behavior. If a field is applied, the effective field on the superspins is multiplied greatly and proportional to the block size. For a very small field, the effective field on the higher generation superspins can be so strong to cause the superspins polarizaed radically. This can explain the extra large critical isotherm exponent discovered in the experiments. The phase space of reduced temperature vs. field is divided into many layers , in which different generation blocks dominate the critical behavior. The sizes of the different generation emergent blocks are new relevant length scales. This can explain a lot of puzzles in the experiments and the Monte Carlo simulation.
We study a d=2 Ising model with vertical coupling K>0 and random horizontal coupling KH{K}_{H}KH​(KH{K}_{H}KH​:K1{K}_{1}K1​\ensuremath{\le}KH{K}_{H}KH​,\ensuremath{\le}K2{K}_{2}K2​), which is the same within a row. This generalizes the McCoy-Wu model since it allows for frustration. By mapping it into a random-field model in d=1, we show that the free energy F has infinitely differentiable singularities at K∗{K}^{\mathrm{*}}K∗=K2{K}_{2}K2​ (Griffith's singularity in temperature); K∗{K}^{\mathrm{*}}K∗=〈K〉, where 〈K〉 is the average horizontal bond (infinite correlation length and \ensuremath{\nu}=1), and K∗{K}^{\mathrm{*}}K∗=K1{K}_{1}K1​ (Griffith's singularity, if ${K}_{1}$0, its dual version if not). We show for the last case that F(t)\ensuremath{\approxeq}${t}^{1/t}$.
We study a d=2 Ising model where the veritcal bonds are fixed and ferromagnetic and the horizontal bonds can vary randomly in sign and in magnitude (within some limits) but are same within each now. The model therefore generalizes that of McCoy and Wu since it allows for the interesting case of frustration. We use the transfer matrix to map our problem to a collection of random field d=1 problems about which a lot is known. We find generally three transitions: a Griffiths transition, its dual version, and one with infinite correlation length and index \ensuremath{\nu}=1. In all cases the free energy has infinitely differentiable singularities. In addition there are some zero-temperature transitions.