Quantum Annealing Simulation of Out-of-Equilibrium Magnetization in a Spin-Chain Compound

Geometrically frustrated spin-chain compounds such as ${\mathrm{Ca}}_3{\mathrm{Co}}_2{\mathrm{O}}_6$ exhibit extremely slow relaxation under a changing magnetic field. Consequently, both low-temperature laboratory experiments and Monte Carlo simulations have shown peculiar out-of-equilibrium magnetization curves, which arise from trapping in metastable configurations. In this work, we simulate this phenomenon in a superconducting quantum annealing processor, allowing us to probe the impact of quantum fluctuations on both the equilibrium and dynamics of the system. Increasing the quantum fluctuations with a transverse field reduces the impact of metastable traps in out-of-equilibrium samples and aids the development of three-sublattice ferrimagnetic (up-up-down) long-range order with magnetization 1/3. At equilibrium, we identify a finite-temperature shoulder in the 1/3-to-saturated phase transition, promoted by quantum fluctuations but with an entropic origin. This work demonstrates the viability of dynamical as well as equilibrium studies of frustrated magnetism using large-scale programmable quantum systems and is therefore an important step toward programmable simulation of dynamics in materials using quantum hardware.

Popular Summary

Many compounds exhibit geometric frustration, wherein local sets of interactions are impossible to satisfy simultaneously. This frustration can have many consequences, such as spin-liquid behavior at low temperatures, magnetization plateaus, and extensive ground-state degeneracy. In the presence of quantum fluctuations, this degeneracy can lead to a variety of exotic phases.

In this work, we simulate a geometrically frustrated Ising spin system realized in a lattice of superconducting flux qubits, using a D-Wave quantum annealing processor. Ferromagnetic spin chains are coupled in a triangular antiferromagnetic arrangement. We simulate the system under both a transverse field, which induces quantum fluctuations, and a sweeping longitudinal field, which affects the magnetization. The results are in qualitative agreement with ${\mathrm{Ca}}_3{\mathrm{Co}}_2{\mathrm{O}}_6$, a spin-chain antiferromagnet that exhibits slow relaxation under a changing longitudinal field. We observe two important consequences of the transverse field. First, quantum fluctuations hasten the hysteretic relaxation. Second, at equilibrium, a new magnetization shoulder is identified, which we attribute to a combination of quantum and entropic effects. These experiments, which have close ties to materials such as ${\mathrm{Ca}\mathrm{Co}}_2{\mathrm{O}}_6$ and ${\mathrm{Tm}\mathrm{Mg}\mathrm{Ga}\mathrm{O}}_4$, represent important progress in quantum simulation, toward spin Hamiltonians with real-world applications.

Figure 1

A geometrically frustrated square-octagonal lattice viewed as a spin-chain antiferromagnet. The red and blue lines indicate antiferromagnetic and ferromagnetic Ising exchange with couplings $J_1$ and $J_0=-1.8J_1$, respectively, where ferromagnetic bonds form four-spin chains. As in ${\mathrm{Ca}}_3{\mathrm{Co}}_2{\mathrm{O}}_6$, the strong FM chains interact via weaker AFM couplings in a triangular geometry.

Figure 2

The thermodynamic phase diagram of $\mathcal{H}$ for different transverse fields $\mathrm{\Gamma }$. For fixed value of $\mathrm{\Gamma }/J_1$, the system has a FIM phase separated by a PM phase; the phase-transition locations are estimated using Monte Carlo and QA simulations. For $\mathrm{\Gamma }>0$, $B=0$, there are two BKT transitions delimiting a critical phase (BKT) and an OBD phase [14, 27, 29].

Figure 3

Equilibrium magnetization and susceptibility observations: (a),(c) low tunneling, $\mathrm{\Gamma }/J_1=0.51$; (b),(d) high tunneling, $\mathrm{\Gamma }/J_1=1.06$. The susceptibility is computed as $dM/d(B/B_{\mathrm{MAX}})$. In both cases, $J_1\approx 46$ mK. Increasing temperature corresponds to broadening of the transverse-field-induced magnetization shoulder, consistent with an entropic origin.

Figure 4

Out-of-equilibrium experimental results obtained from hysteresis simulation. (a) Annealing protocols for simulating hysteresis. To take an observation at field $B_f$ from initial field $B_0$ of either zero or a saturating field $B_{\mathrm{MAX}}=2J_1$, the system is annealed [via $\mathrm{\Gamma }(t)$ and $J_1(t)$] under $B_0$; $B(t)$ is swept at the end of the protocol from $B_0$ to $B_f$ at a rate of $dt/d(B/B_{\mathrm{MAX}})=1\mu \mathrm{s}$, $10\mu \mathrm{s}$, or $100\mu \mathrm{s}$ before $J(t)$ and $\mathrm{\Gamma }(t)$ are rapidly quenched prior to state readout [$B_f=1$, $\frac{3}{2}$, and 2 shown for $dt/d(B/B_{\mathrm{MAX}})=100\mu \mathrm{s}$]. (b) Magnetization curves. Shown are measurements at $T=10$ mK for increasing field ($B_0=0$, left) and decreasing field ($B_0=2$, right) with a low and high transverse field (top and bottom, respectively). The black curves show equilibrium estimates (Appendix pp1). (c) The FIM long-range order $m_{\mathrm{FIM}}$, which is saturated in the threefold degenerate ground state of the 1/3 plateau.

Figure 5

Experimental output samples. The darkness of the markers indicates magnetization of the four-qubit chain; the darkness of the plaquettes indicates plaquette magnetization. The plaquette hue indicates the pseudospin phase; the 1/3 plateau has three degenerate FIM ground states with varying phase. The lattice is periodic at the top (bottom) and open on the left (right). Six of 450 sites are empty due to inoperable qubits. (a),(b) Equilibrium samples at $H=1.48$; the transverse field modulates long-range order: (a) at $\mathrm{\Gamma }/J_1=0.51$, the system adopts long-range FIM order and is dominated by plaquettes of a single phase, with $m_{\mathrm{FIM}}$ large; (b) with $\mathrm{\Gamma }/J_1=1.06$, the system is characterized by competing FIM domains, with saturated plaquettes at the interfaces. (c),(d) Out-of-equilibrium samples, with $H$ decreasing from 2 to 1 over $0.5\mu \mathrm{s}$: (c) with $\mathrm{\Gamma }/J_1=0.51$, the system is chaotic, with many small FIM domains—the interfaces of these domains host trapped excitations, many in the form of broken four-qubit chains with magnetization zero; (d) with $\mathrm{\Gamma }/J_1=1.06$, the system forms large FIM domains during the sweep of $H$—the interfaces between these domains host plaquettes of magnetization $+1$ and $-1/3$.

Figure 6

Reverse-annealing protocols for equilibrium estimates. For estimating the equilibrium statistics, we perform “quantum evolution Monte Carlo,” forming a chain of reverse anneals. Each reverse anneal starts from a classical state with $\mathrm{\Gamma }=0$, then increases $\mathrm{\Gamma }$, dwells, and decreases $\mathrm{\Gamma }$ to 0, arriving at a new classical state. In the easy case (left), the protocol involves a simple dwell. In the slow-relaxing case (right), $\mathrm{\Gamma }$ is attenuated from a higher value before the dwell. Each chain repeats the reverse anneal 100 times, producing 100 classical output samples that ideally represent projections of the system to the ${\sigma }^z$ basis. In both cases, $B(t)/J_1$ is held constant [e.g., $B(t)/J_1=1$ as shown here], in contrast to the out-of-equilibrium protocols shown in Fig. 4.

Figure 7

The local entropy of the observed states. The ground state at $\mathrm{\Gamma }=0$, $T=0$, $H=\frac{3}{2}$ has extensive entropy, consisting of triangular plaquettes with one or zero down-spins, which is lifted by the transverse field. The local entropy, which we define as the number of neighboring states differing by a swap of a single up-down spin chain pair, becomes increasingly favorable with increasing $\mathrm{\Gamma }$ and $T$. The error bars indicate the standard deviation of the estimates for individual 50-sample QA programmings.

Figure 8

The finite-size scaling of the magnetization that maximizes the entropy within the subspace $\mathcal{S}$ of low-energy states.

Figure 9

Out-of-equilibrium QMC simulations. The simulations are performed on a fully periodic lattice with no site vacancies, with a longitudinal field swept from ${10}^3$ to ${10}^7$ Monte Carlo sweeps. Sweeps are performed either one spin at a time (top) or four spins at a time (bottom). Compare with Fig. 4.