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Iranian Journal of Physics Research

FLOW EQUATIONS APPROACH TO ONE DIMENSIONAL QUANTUM ISING MODEL

Azin Kiani, Farhad Shahbazi, S. A. Jafari

Published Mar 15, 2011• 0 citations
OpenAlex

Abstract

One dimensional quantum Ising model with nearest neighbor interaction in transverse magnetic field is one of the simplest spin models which undergo quantum phase transition. This model has been precisely solved using different methods. In this paper, we solve this model in uniform magnetic field -Jgσxn precisely using a new method called Continuous Unitary Transformations (CUT) or flow equations, and derive its expectation values <σxn>, <σzn> and <σxnσxn+j>. Then, we apply this method on one dimensional quantum Ising model in staggered magnetic field (-1)nJgσxn. Results show that both models have the same critical properties as expected, and it was also found that spontaneous symmetry breaking cannot be derived from CUT.

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One dimensional quantum Ising model with nearest neighbor interaction in transverse magnetic field is one of the simplest spin models which undergo quantum phase transition. This model has been precisely solved using different methods. In this paper, we solve this model in uniform magnetic field -Jgσxn precisely using a new method called Continuous Unitary Transformations (CUT) or flow equations, and derive its expectation values <σxn>, <σzn> and <σxnσxn+j>. Then, we apply this method on one dimensional quantum Ising model in staggered magnetic field (-1)nJgσxn. Results show that both models have the same critical properties as expected, and it was also found that spontaneous symmetry breaking cannot be derived from CUT.
In this work we present a semi-classical transfer matrix method to study one-dimensional modified quantum transverse-field Ising model. Both average magnetization per spin along the z-direction 〈σz〉 and along the x-direction 〈σx〉 are calculated in this picture. It is predicted that the t > 1 case is the corresponding effective ferromagnetic phase [Formula: see text] at sufficiently high Hzwhereas t < 1 case is an effective paramagnetic phase [Formula: see text]. Consequently, it is concluded that t = Hz/K = 1, here Hzis the transverse field along the z-axis and K is the coupling strength, which corresponds to quantum ferromagnetic-paramagnetic phase transition. In other words, the same result of the sophisticated quantum field theoretical calculation is predicted unexpectedly in the naive semiclassical picture.
We apply a finite-size scaling approach to the one-dimensional spin-1/21/21/2 Ising model with nearest- and next-nearest-neighbor interactions in the presence of a transverse magnetic field. By using the scaling behavior of the energy gap we are able not only to determine the ferromagnetic-paramagnetic transition line at zero absolute temperature but also to compute the corresponding critical exponent. A comparison with other approaches is made, and aside for small discrepancies likely to occur at the thermodynamic limit, we believe that the present results are quite close to the unknown exact ones.
The one-dimensional transverse field Ising model is solved by continuous unitary transformations in the high-field limit. A high accuracy is reached due to the closure of the relevant algebra of operators, which we call string operators. The closure is related to the possibility to map the model by Jordan-Wigner transformation to noninteracting fermions, but it is proven without referring to this mapping. The effective model derived by the continuous unitary transformations is used to compute the contributions of one, two, and three elementary excitations to the diagonal dynamic structure factors. The three-particle contributions have, so far, not been addressed analytically, except close to the quantum critical point.
The critical properties of the one-dimensional spin-1/2 transverse Ising model in the presence of a longitudinal magnetic field were studied by the quantum fidelity method. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins. The maximum of the fidelity susceptibility was used to locate the various phase boundaries present in the system. The type of dominant spin ordering for each phase was identified by examining the corresponding ground-state eigenvector. For a given antiferromagnetic nearest-neighbor interaction J_{2}, we calculated the fidelity susceptibility as a function of the transverse field B_{x} and the longitudinal field B_{z}. The phase diagram in the (B_{x},B_{z})-plane shows three phases. These findings are in contrast with the published literature that claims that the system has only two phases. For B_{x}<1, we observed an antiferromagnetic phase for small values of B_{z} and a paramagnetic phase for large values of B_{z}. For B_{x}>1 and low B_{z}, we found a disordered phase that undergoes a second-order phase transition to a paramagnetic phase for large values of B_{z}.
We use a recently developed quantum simulation approach to study the properties of a three-dimensional Ising model consisting of S = 1/2 quantum spins localized at the sites of a simple cubic lattice. We assume nearest-neighbor interaction between spins with an exchange interaction that can be either ferromagnetic or antiferromagnetic. It is found that the computational method quickly converges towards the expected equilibrium spin configurations. The resulting spontaneous magnetization curves corresponding to the two types of magnetic interactions under consideration were found to be almost identical to the ones obtained via quantum mean field theory at all temperatures. The derived total energies, total free energies, magnetic entropies and specific heats per mole of spins show no sizeable differences from known theoretical values. Furthermore, the results of the simulations for two different 3D Ising systems containing 4×4×4 and 20×20×20 spins localized at the sites of a simple cubic lattice were found to be almost identical to each other. This finding suggests that the self-consistent algorithm approach of the current simulation method allows one to obtain the physical bulk properties of a large magnetic system by relying on simulations of a much smaller spin system sample. Therefore, the method presently considered appears to be not only very accurate as gauged by comparison to mean field theory, but also able to greatly increase the speed of simulations.
We study the asymptotic scaling of the entanglement of a block of spins for the ground state of the one-dimensional quantum Ising model with transverse field. When the field is sufficiently strong, the entanglement grows at most logarithmically in the number of spins. The proof utilises a transformation to a model of classical probability called the continuum random-cluster model, and is based on a property of the latter model termed ratio weak-mixing. Our proof applies equally to a large class of disordered interactions. 1 The quantum Ising model The quantum Ising model in a transverse magnetic field is one of the most famous examples of exactly solvable one-dimensional quantum models. The solution was first given by Pfeuty in [26], based on earlier works by Lieb, Schultz, and Mattis [18] and by McCoy [21]. The diagonalisation of the Hamiltonian and the determination of the energy eigenstates is based on methods developed by Jordan and Wigner [16] in the theory of second quantisation of fermion fields, and by Bogoliubov [7] in the theory of superconductivity. This model exhibits a second-order phase transition in the ground state when the temperature of the system is zero. The existence of the phase transition and