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arXiv:2110.05124 · Journal of the Physical Society of Japan

Frustrated Ising Model on D-wave Quantum Annealing Machine

Hayun Park, Hunpyo Lee

Published Jun 2, 2022• 12 citations
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Abstract

We study the frustrated Ising model on the two-dimensional $L \times L$ square lattice with ferromagnetic (FM) nearest-neighbor and antiferromagnetic diagonal-neighbor interactions using the D-wave quantum annealing machine (D-QAM) with 5000+ qubits composed on structure of the Pegasus graph. As the former Monte Carlo and mean field results, we find the FM to stripe order phase transition, through observations of the magnetization $M$, energy, magnetic susceptibility and structure factor. We also analyze probability which occurs any $M$ at a given interaction for many quantum annealing shots to estimate the shape of objective function $f$. The only one value of $M$ with specific phase is observed in the regions far from phase transition for many quantum annealing shots, while several values of $M$ with different possibilities are appeared in the regimes of phase transition. We guess that $f$ in the regimes of phase transition retains the multi-modal structure with several local minimums, due to the strong degeneracies caused by frustrations. Finally, we discuss fail of the quantum annealing simulations, through analysis of the number of the chains, defined as the same variable with $N$-qubits, as a function of $L$.

DOI
10.7566/jpsj.91.074001

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We study the frustrated Ising model on the two-dimensional L×LL \times LL×L square lattice with ferromagnetic (FM) nearest-neighbor and antiferromagnetic diagonal-neighbor interactions using the D-wave quantum annealing machine (D-QAM) with 5000+ qubits composed on structure of the Pegasus graph. As the former Monte Carlo and mean field results, we find the FM to stripe order phase transition, through observations of the magnetization MMM, energy, magnetic susceptibility and structure factor. We also analyze probability which occurs any MMM at a given interaction for many quantum annealing shots to estimate the shape of objective function fff. The only one value of MMM with specific phase is observed in the regions far from phase transition for many quantum annealing shots, while several values of MMM with different possibilities are appeared in the regimes of phase transition. We guess that fff in the regimes of phase transition retains the multi-modal structure with several local minimums, due to the strong degeneracies caused by frustrations. Finally, we discuss fail of the quantum annealing simulations, through analysis of the number of the chains, defined as the same variable with NNN-qubits, as a function of LLL.
The Ising model with nearest-neighbor interactions on a two-dimensional (2D) square lattice is one of the simplest models for studying ferro-magnetic to para-magnetic transitions. Extensive results are available in the literature for this model, which has become a paradigm for the study of magnetic phase transitions in materials, both theoretically and numerically. After a brief review of the main results obtained with a classical computer, we show how to implement on the D- Wave quantum annealer a more complex Ising model with the addition of competing antiferromagnetic interactions between the diagonal next-to-nearest neighbors with two coupling constants J1 and J2. The dynamics of this system, owing to frustration, are richer than those of the simple Ising model and exhibit a third striped (or antiferromagnetic) phase in addition to the ferro- and para-magnetic phases. In this work, we observed all three phases on the D-Wave hardware, studied the behavior of the solution with different annealing parameters, such as the chain strength and annealing time, and showed how to identify the phase transition by varying the ratio between the ferromagnetic and antiferromagnetic couplings. The same system is studied on a classical computer, with the possibility of taking into account the temperature (fixed on D-Wave) as a free parameter and to explore the full phase diagram: some comparative conclusions with D-Wave are drawn.
The phase transitions in two-dimensional antiferromagnetic Ising model are studied on a square lattice by taking the interactions of next-nearest neighbors into account. The model is investigated on basis of the replica Monte Carlo algorithm and the histogrammic analysis of data. The diagram of the critical temperature dependence on an interaction value of next-nearest neighbors is plotted. The studied model reveals the phase transition of second order and the frustration points depending on relative interaction values and magnetic field. A effect, heat capacity splitting near the frustration points, is found.
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.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions J1<0{J}_{1}<0J1​<0 (nearest neighbor, ferromagnetic) and J2>0{J}_{2}>0J2​>0 (second neighbor, antiferromagnetic). The striped phase breaks a Z4{Z}_{4}Z4​ symmetry and is stabilized at low temperatures for g=J2/∣J1∣>1/2g={J}_{2}/|{J}_{1}|>1/2g=J2​/∣J1​∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The J1{J}_{1}J1​-J2{J}_{2}J2​ model has a line of very weak first-order phase transitions in the whole region 1/2<g<g∗1/2<g<{g}^{*}1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from g=g∗g={g}^{*}g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point g∗{g}^{*}g∗ in the J1{J}_{1}J1​-J2{J}_{2}J2​ model. The continuous transitions near g∗{g}^{*}g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of g>1/2g>1/2g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.
Quantum magnets in the MNb2O6M\mathrm{Nb_2O_6}MNb2​O6​ and BaCo2_22​V2_22​O8_88​ families realise frustrated transverse-field Ising models whose competing ferromagnetic and antiferromagnetic couplings generate a sign problem provably intractable for quantum Monte Carlo at any system size, leaving their quantum phase boundaries numerically Inaccessible. Using a D-Wave Advantage2 quantum annealer at L≤27L\leq27L≤27 (729 spins), we obtain the large-LLL critical points for this model family, measuring quantum-driven transitions at gcQPU∈{0.286, 0.210, 0.156, 0.093}{g_c^{\mathrm{QPU}}}\in\{0.286,\,0.210,\,0.156,\,0.093\}gcQPU​∈{0.286,0.210,0.156,0.093} for α∈{1.0, 0.7, 0.5, 0.3}α\in\{1.0,\,0.7,\,0.5,\,0.3\}α∈{1.0,0.7,0.5,0.3}, where the analytically exact classical threshold is gcclass(α)=2α/3{g_c^{\mathrm{class}}}(α)=2α/3gcclass​(α)=2α/3. The suppression ratio r(α)r(α)r(α) exhibits a sharp two-regime structure: the three quasi-1D geometries (α≤0.7α\leq0.7α≤0.7) are mutually consistent with a universal plateau rˉ=0.450\bar{r}=0.450rˉ=0.450 (χ2/dof=1.10χ^2/\mathrm{dof}=1.10χ2/dof=1.10, p=0.33p=0.33p=0.33), demonstrating that quantum fluctuations destroy approximately 55%55\%55% of the classical FM stability window independently of coupling anisotropy, while rrr steps down to the 2D limit above the empirical crossover scale α∗≈0.7α^*\approx0.7α∗≈0.7. Inner Binder cumulant pairs, which converge fastest to the thermodynamic limit, resolve r(1.0)≈0.412r(1.0)\approx0.412r(1.0)≈0.412 and a step Δr=0.038±0.015Δr=0.038\pm0.015Δr=0.038±0.015 from the quasi-1D plateau. A four-point linear fit r(α)=0.494−0.063 αr(α)=0.494-0.063\,αr(α)=0.494−0.063α summarises both regimes; its α→0α\to0α→0 intercept recovers the exact 1D result of Pfeuty within 1.7 standard deviations, and its slope is a lower bound on the true crossover amplitude concentrated in α∈[α∗,1]α\in[α^*,1]α∈[α∗,1]. Two sequential blind predictions, confirmed at 0.2σ0.2σ0.2σ and 0.7σ0.7σ0.7σ before each measurement, validate the crossover law. All four geometries show a direct ferromagnet-to-paramagnet transition, complete quantum ergodicity (funiq=1.000f_{\rm uniq}=1.000funiq​=1.000), and null valence-bond solid order.
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.