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High order perturbation study of the frustrated quantum Ising chain

Amir Hajibabaei, Farhad Shahbazi

Published Dec 1, 2014• 0 citations
OpenAlex

Abstract

In this paper, using high order perturbative series expansion method, the critical exponents of the order parameter and susceptibility in transition from ferromagnetic to disordered phases for 1D quantum Ising model in transverse field, with ferromagnetic nearest neighbor and anti-ferromagnetic next to nearest neighbor interactions, are calculated. It is found that for small value of the frustrating second neighbor coupling ( k<0.4 ), the exponents are the same as those of the classical 2D Ising model (B=1/8 and Y=7/8), but a deviation from 2D classical Ising universality class is observed for larger values of frustration (0.4< K<0.5). The reason for this deviation is the “disorder line” which becomes tangent to the Ferromagnetic phase boundary line at K=0.5

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arXiv:1105.4836•Physical Review B•Nov 9, 2011•55 citations
In this paper, using high order perturbative series expansion method, the critical exponents of the order parameter and susceptibility in transition from ferromagnetic to disordered phases for 1D quantum Ising model in transverse field, with ferromagnetic nearest neighbor and anti-ferromagnetic next to nearest neighbor interactions, are calculated. It is found that for small value of the frustrating second neighbor coupling ( k<0.4 ), the exponents are the same as those of the classical 2D Ising model (B=1/8 and Y=7/8), but a deviation from 2D classical Ising universality class is observed for larger values of frustration (0.4< K<0.5). The reason for this deviation is the “disorder line” which becomes tangent to the Ferromagnetic phase boundary line at K=0.5
We apply density-matrix renormalization-group methods to study the phase diagram of the quantum axial next-nearest-neighbor Ising model in the region of low frustration where the ferromagnetic coupling is larger than the next-nearest-neighbor antiferromagnetic one. By finite-size scaling on lattices with up to 80 sites we locate precisely the transition line from the ferromagnetic phase to a paramagnetic phase without spatial modulation. We then measure and analyze the spin-spin correlation function in order to determine the disorder transition line where a modulation appears. We give strong numerical support to the conjecture that the Peschel-Emery one-dimensional line actually coincides with the disorder line. We also show that the critical exponent governing the vanishing of the modulation parameter at the disorder transition is ${\ensuremath{\beta}}_{q}=1∕2$.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
To study the ground state of an axial next-nearest-neighbor Ising chain under transverse field as a function of frustration parameter kappa and field strength Gamma, we present here two different perturbative analyses. In one, we consider the (known) ground state at kappa=0.5 and Gamma=0 as the unperturbed state and treat an increase of the field from 0 to Gamma coupled with an increase of kappa from 0.5 to 0.5+rGamma/J as perturbation. The first-order perturbation correction to eigenvalue can be calculated exactly and we could conclude that there are only two phase-transition lines emanating from the point kappa=0.5, Gamma=0. In the second perturbation scheme, we consider the number of domains of length 1 as the perturbation and obtain the zeroth-order eigenfunction for the perturbed ground state. From the longitudinal spin-spin correlation, we conclude that floating phase exists for small values of transverse field over the entire region intermediate between the ferromagnetic phase and antiphase.
We analytically investigated the phase diagrams of one-dimensional Ising and XY models. A second-order phase transition occurs in both models only with the transverse magnetic field. In this study, the fully connected ferromagnetic and anti-ferromagnetic XX interactions are applied in addition to the transverse magnetic field. The ferromagnetic XX interaction causes a first-order phase transition in both of the one-dimensional Ising and XY models. In contrast, by applying the anti-ferromagnetic one, the second-order phase transition can be avoided in the one-dimensional Ising model, and the critical amplitude becomes smaller in the one-dimensional XY model even though the second-order phase transition remains and the critical exponent does not change. Because the anti-ferromagnetic XX interaction works to destroy the order of the transverse magnetization, it begins to decrease at a larger strength of the transverse magnetic field. As a result, the susceptibility of the transverse magnetization with respect to the strength of the magnetic field becomes small. It is conjectured that the anti-ferromagnetic XX interaction is generally preferable to the ferromagnetic interaction to improve the performance of quantum annealing.
The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series expansion quantum Monte Carlo, on both the one-dimensional linear chain and the two-dimensional square lattice. We extract the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$ as a function of the decay exponent of the long-range interactions. For ferromagnetic Ising interactions, we resolve the limiting regimes known from field theory, ranging from the nearest-neighbor Ising to the long-range Gaussian universality classes, as well as the intermediate regime with continuously varying critical exponents. In the long-range Gaussian regime, we treat the effect of dangerous irrelevant variables on finite-size scaling forms. For antiferromagnetic and therefore competing Ising interactions, the stochastic series expansion algorithm displays growing autocorrelation times leading to a reduced performance. Nevertheless, our results are consistent with the nearest-neighbor Ising universality for all investigated interaction ranges both on the linear chain and the square lattice.
We analyze the phase transition of the frustrated J1{J}_{1}J1​-J2{J}_{2}J2​ Ising model with antiferromagnetic nearest- and strong next-nearest-neighbor interactions on the square lattice. Using extensive Monte Carlo simulations we show that the nature of the phase transition for $1/2<{J}_{2}/{J}_{1}\ensuremath{\lesssim}1$ is not of the weakly universal type---as commonly believed---but we conclude from the clearly doubly peaked structure of the energy histograms that the transition is of weak first order. Motivated by these results, we analyze the phase transitions via field-theoretic methods; i.e., we calculate the central charge of the underlying field theory via transfer-matrix techniques and present, furthermore, a field-theoretic discussion on the phase-transition behavior of the model. Starting from the conformally invariant fixed point of two decoupled critical Ising models (J1=0{J}_{1}=0J1​=0), we calculate the effect of the nearest-neighbor coupling term perturbatively using operator product expansions. As an effective action we obtain the Ashkin-Teller model.