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Journal of Physics A Mathematical and General

Invalidity of the replica trick for a two-dimensional fermion model

K. Ziegler

Published Oct 21, 1986• 4 citations
Publisher

Abstract

Non-interacting fermions in a random potential are considered on a square lattice. Such a model was previously proposed as a description of the quenched thermodynamics of a disordered Ising model. The author compares in a perturbative approach the replica trick and a formal functional integral representation and find that they yield different results.

DOI
10.1088/0305-4470/19/15/015

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Fluctuation effects in the Ising model with reduced interaction and quenched disorder

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The effect of quenched random fields and local perturbations of critical temperature on the critical behavior at phase transitions is studied within the framework of an exactly solvable model that takes into account interaction of fluctuations with equal and opposite momenta. Using the replica method the dimensional reduction by 2 for systems with finite-range interaction and quenched random fields is explicitly shown. For interaction of the infinite range the model demonstrates the mean-field critical asymptotics independently of dimensionality or the presence of random fields. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matter•Apr 1, 1996•11 citations
Non-interacting fermions in a random potential are considered on a square lattice. Such a model was previously proposed as a description of the quenched thermodynamics of a disordered Ising model. The author compares in a perturbative approach the replica trick and a formal functional integral representation and find that they yield different results.
The notion of the integral over the anticommuting Grassmann variables is applied to analyze the fermionic structure of the 2D Ising model with quenched site dilution. In the NNN-replica scheme, the model is explicitly reformulated as a theory of interacting fermions on a lattice. For weak dilution, the continuum-limit approximation implies the log-log singularity in the specific heat near TcT_cTc​.
The partition function of the 2D Ising model with random nearest-neighbour coupling is expressed in the dual lattice made of square plaquettes. The dual model is solved in the mean field and in different types of Bethe-Peierls approximations, using the replica method. The application of methods of mean-field type to Ising models allows one to obtain very accurate approximations of the thermodynamic quantities. However, in the presence of quenched disorder this approach is difficult to implement. In this paper, we show that rather good results can be obtained after performing a duality transformation of the Ising model with random nearest-neighbour coupling that assumes the values Jij D1 with equal probability. The model is thus defined on a dual lattice where the spin variables are attached to the square plaquettes. The advantage is that the quadratic term of the dual Hamiltonian has constant coefficients instead of random ones. It is therefore possible to use the standard methods of the mean field to estimate the quenched free energy. Our results can be generalized to higher dimensions, although the approximations become rougher, because the number of plaquette spins over the number of interaction links increases with the dimensionality (1). In particular, we obtain an extremely good estimate of the ground- state energy of the random Ising model, by applying the Bethe-Peierls approximation where part of the short-range order is taken into account. In section 1, we introduce the dual lattice made of elementary square plaquettes. On this lattice the partition function can be expressed as a function of the inverse temperature e D 1 ln tanh.1=T/ whereT D 1 is the temperature of the original lattice. In section 2, we apply the mean-field approximation to the dual model, using the replica method. In section 3, we introduce the Bethe-Peierls approximation. This allows us to obtain a very precise estimate of the ground-state energy of the two-dimensional Ising model with random coupling. In section 4, we show that it is possible to improve the Bethe-Peierls approximation by introducing an interaction between different replicas. In section 5, the reader will find some remarks and conclusions.
We apply a new anticommuting path integral technique to clarify the fermionic structure of the 2D Ising model with quenched site dilution. In the N-replica scheme, the model is explicitly reformulated as a theory of interacting fermions on a lattice. An unusual feature is that the leading term of interaction in the exact lattice theory is of order 2N in fermions, where N is the number of replicas. The continuum-limit approximation near Tc for weak dilution produces, however, an effective four-fermion interaction. In particular, this implies the doubled-logarithmic singularity in the specific heat near Tc for weak site dilution. The exact value of the initial slope is also obtained.
Disorder is ubiquitous in real materials and influences the physical properties like the conductivity to varying degrees. \nIf electron-electron interactions are strong, theoretical and numerical treatment of these systems becomes challenging. In this thesis a numerical approach is developed to address these systems, \ntreating both interactions and disorder on equal footing. \nThe approach is based on the dual fermion approach for interacting systems developed by Rubtsov et al. \nTerletska et al. applied the ideas of the dual fermion approach to disordered non-interacting systems. In this approach, the replica trick is used to integrate out the disorder in favor of an effective electron-electron interaction. \nWe extended the approach from Terletska et al. to treat disordered interacting systems. \n \nDual Fermions allow to take into account non-local fluctuations by means of a perturbative expansion around an impurity problem. The impurity reference system is determined self-consistently, analogously to the dynamical mean-field \ntheory. \nThe perturbative expansion is expected to yield good results for small and large values of interaction strength and disorder. A priori, it is not clear what to expect for intermediate values, \nbut experience shows that oftentimes good results are obtained for this region. \n \nAn advantage of the dual fermion approach is that there is no sign-problem for a single orbital model if quantum Monte Carlo is used to solve the interacting reference system. \nAdditionally, perturbation theory is usually numerically much cheaper than fully solving an interacting lattice or cluster problem. \nThus, the dual fermion approach allows to address regions of parameter space that are not accessible to lattice quantum Monte Carlo calculations or cluster extension of dynamical mean-field theory. \nCluster extensions of the dynamical mean-field theory are for example the dynamical cluster approximation or the cellular dynamical mean-field theory. \n \nThe new approach is benchmarked with respect to the Anderson-Falicov-Kimball model. The Anderson-Hubbard model is the main application. \nFor both models, the algorithm is tested in one dimension for the single particle Green function and compared to the dynamical cluster approximation to estimate the quality of the results. \nThe Anderson-Falicov-Kimball model is studied further in two dimensions and the phase diagram on the disorder-interaction plane is calculated from both the single particle Green function and the dc conductivity. \n \nFor the Anderson-Hubbard model, the antiferromagnetic phase transition, the Mott transition and the Anderson transition are investigated in three dimensions. The dynamical mean-field theory results are \nqualitatively confirmed for the most part. Quantitative corrections are in agreement with dynamical cluster approximation results, thus confirming the validity of the results. \nWe found a qualitative difference between the dual fermion approach and dynamical mean-field theory for the temperature dependence of the hysteresis of the double occupancy. The dual fermion result \nshows a decreasing lower critical interaction strength for decreasing temperature, which is in contrast with the dynamical mean-field result that shows an increase of the lower critical interaction strength. \nThis behavior happens for small temperatures that are not accessible using for example the dynamical cluster approximation, thus highlighting the usefulness of the dual fermion approach. \nWe found this behavior for all values of the disorder that we considered.
The effect of quenched random fields and local perturbations of critical temperature on the critical behavior at phase transitions is studied within the framework of an exactly solvable model that takes into account interaction of fluctuations with equal and opposite momenta. Using the replica method the dimensional reduction by 2 for systems with finite-range interaction and quenched random fields is explicitly shown. For interaction of the infinite range the model demonstrates the mean-field critical asymptotics independently of dimensionality or the presence of random fields. \textcopyright{} 1996 The American Physical Society.