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Physical Review B

Excited states of the saddle point equation of the Landau-Ginzburg-Wilson Hamiltonian with random temperature

Xintian Wu

Published May 26, 2009• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.79.184208

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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 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 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.
The effects of quenched disorder on the critical properties of itinerant quantum antiferromagnets and ferromagnets are considered. Particular attention is paid to locally ordered spatial regions that are formed in the presence of quenched disorder even when the bulk system is still in the paramagnetic phase. These rare regions or local moments are reflected in the existence of spatially inhomogeneous saddle points of the Landau-Ginzburg-Wilson functional. We derive an effective theory that takes into account small fluctuations around all of these saddle points. The resulting free energy functional contains a new term in addition to those obtained within the conventional perturbative approach, and it comprises what would be considered nonperturbative effects within the latter. A renormalization group analysis shows that in the case of antiferromagnets, the previously found critical fixed point is unstable with respect to this new term, and that no stable critical fixed point exists at one-loop order. This is contrasted with the case of itinerant ferromagnets, where we find that the previously found critical behavior is unaffected by the rare regions due to an effective long-ranged interaction between the order parameter fluctuations.
Abstract In this paper, we review recent results on sample-to-sample fluctuations in a critical Ising model with quenched random ferromagnetic couplings. In particular, in terms of the renormalized replica Ginzburg–Landau Hamiltonian in dimensions D < 4 an explicit expression for the probability distribution function (PDF) of the critical free energy fluctuations is derived. Next, using known fixed-point values for the renormalized coupling parameters the universal curve for such PDF in the dimension D = 3 is obtained. For the specific case of the two-dimensional Ising model, using replica calculations in the renormalization group framework, we derive explicit expressions for the PDF of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energy is Gaussian, and its typical sample-to-sample fluctuations as well as its average value scale with the system size L like ~ Llnln(L). In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a δ-peak in the thermodynamic limit L → ∞.
The phase transition in the q -state Potts model with homogeneous ferromagnetic couplings is strongly first order for large q, while it is rounded in the presence of quenched disorder. Here we study this phenomenon on different two-dimensional lattices by using the fact that the partition function of the model is dominated by a single diagram of the high-temperature expansion, which is calculated by an efficient combinatorial optimization algorithm. For a given finite sample with discrete randomness the free energy is a piecewise linear function of the temperature, which is rounded after averaging, however, the discontinuity of the internal energy at the transition point (i.e., the latent heat) stays finite even in the thermodynamic limit. For a continuous disorder, instead, the latent heat vanishes. At the phase transition point the dominant diagram percolates and the total magnetic moment is related to the size of the percolating cluster. Its fractal dimension is found d(f) = ( 5 + square root of 5)/4 and it is independent of the type of the lattice and the form of disorder. We argue that the critical behavior is exclusively determined by disorder and the corresponding fixed point is the isotropic version of the so-called infinite randomness fixed point, which is realized in random quantum spin chains. From this mapping we conjecture the values of the critical exponents as beta=2- d(f), beta(s) =1/2, and nu=1.