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European Journal of Physics

A mean field approach to the Ising chain in a transverse magnetic field

Carlos Osácar, Amalio F. Pacheco

Published Apr 12, 2017• 1 citation
Publisher

Abstract

Abstract We evaluate a mean field method to describe the properties of the ground state of the Ising chain in a transverse magnetic field. Specifically, a method of the Bethe–Peierls type is used by solving spin blocks with a self-consistency condition at the borders. The computations include the critical point for the phase transition, exponent of magnetisation and energy density. All results are obtained using basic quantum mechanics at an undergraduate level. The advantages and the limitations of the approach are emphasised.

DOI
10.1088/1361-6404/aa6cd4

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Abstract We evaluate a mean field method to describe the properties of the ground state of the Ising chain in a transverse magnetic field. Specifically, a method of the Bethe–Peierls type is used by solving spin blocks with a self-consistency condition at the borders. The computations include the critical point for the phase transition, exponent of magnetisation and energy density. All results are obtained using basic quantum mechanics at an undergraduate level. The advantages and the limitations of the approach are emphasised.
The random Ising model in a transverse field in one dimension with Hamiltonian H=- Sigma ( Gamma i S i z +J i S i x S i+1 x ) is studied at T=O from a real-space renormalisation group block method which preserves duality transformations. The ground state magnetisation and the ground state energy are determined for random distributions P(J)(P( Gamma ))=N J J NJ-1 /J 0 N(J) for 0<J<J 0 and P(J)=0 for J>J O . A new critical behaviour corresponding to a new fixed point takes place in the presence of disorder. The crossover exponent describing the departure from the pure system behaviour is calculated. The second derivative of the ground state energy delta 2 E/ delta Gamma 2 which diverges logarithmically for the pure system is rounded in the presence of disorder but a sharp transition field still exists where the magnetisation goes to zero with an exponent beta about twice as large as the pure system exponent. Comparison is made with the analytical results of McCoy and Wu (1969) for the classical equivalent random-striped Ising 2D model.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
The ground state phase diagram of the alternating spin-1/2 chains with anisotropic ferromagnetic coupling under the influence of a symmetry breaking transverse magnetic field is studied. We have used the exact diagonalization technique. In the limit where the antiferromagnetic coupling is dominant, we have identified two Ising-type quantum phase transitions. We have calculated two critical fields hc1h_{c_{1}}hc1​​ and hc2h_{c_{2}}hc2​​, corresponding to the transition between different magnetic phases of the system. It is found that the intermediate state ($h_{c_{1}}
In this set of notes, a complete, pedagogical tutorial for applying mean field theory to the two-dimensional Ising model is presented. Beginning with the motivation and basis for mean field theory, we formally derive the Bogoliubov inequality and discuss mean field theory itself. We proceed with the use of mean field theory to determine a magnetisation function, and the results of the derivation are interpreted graphically, physically, and mathematically. We give a new interpretation of the self-consistency condition in terms of intersecting surfaces and constrained solution sets. We also include some more general comments on the thermodynamics of the phase transition. We end by evaluating symmetry considerations in magnetisation, and some more subtle features of the Ising model. Together, a self-contained overview of the mean field Ising model is given, with some novel presentation of important results.
Abstract In undergraduate statistical mechanics courses the Ising model always plays an important role because it is the simplest non-trivial model used to describe magnetic systems. The one-dimensional model is easily solved analytically, while the two-dimensional one can be solved exactly by the Onsager solution. For this reason, numerical simulations are usually used to solve the two-dimensional model. Keeping in mind that the two-dimensional model is the platform for studying phase transitions, it is usually an exercise in computational undergraduate courses because its numerical solution is relatively simple to implement and its critical exponents are perfectly known. The purpose of this article is to present a detailed numerical study of the second-order phase transition in the two-dimensional Ising model at an undergraduate level, allowing readers not only to compare the mean-field solution, the exact solution and the numerical one through a complete study of the order parameter, the correlation function and finite-size scaling, but to present the techniques, along with hints and tips, for solving it themselves. We present the elementary theory of phase transitions and explain how to implement Markov chain Monte Carlo simulations and perform them for different lattice sizes with periodic boundary conditions. Energy, magnetization, specific heat, magnetic susceptibility and the correlation function are calculated and the critical exponents determined by finite-size scaling techniques. The importance of the correlation length as the relevant parameter in phase transitions is emphasized.