Introduction

Finding the best solution to complex problems is a central challenge in many fields, spanning from mathematics and computer science to statistical physics. The prototypical problem for studying lowest-energy configurations in many-body systems is likely probing the ground state of a correlated Ising model1,2,3,4,5,6,7. The annealing of a complex Ising spin system towards its optimal state can be time-consuming, and typically characterized by a relaxation time constant τ8. For a thermal (or classical) annealing, τ rapidly approaches infinity as temperature (T) decreases to 0 K, following a thermally activated Arrhenius form, τ = \({\tau }_{{{{{{{{\rm{m}}}}}}}}}\exp (\Delta E/T)\), where τm and ΔE represent the relaxation time at high temperatures and the barrier energy, respectively. At low temperatures, this classical form hinders the system from converging towards the lowest-energy configuration, resulting in reduced work efficiency. In contrast, quantum annealing (QA) with a transverse field exhibits superiority over classical annealing3,9, resulting in a much shorter τ that is temperature-independent as T → 0 K.

Various Ising models have recently been simulated using programmable superconducting QA processors4, Rydberg atoms10, and other approaches, yielding exotic many-body phenomena such as dynamical phase transitions and topological configurations5,6,11. While these simulations are of great interest as quantum simulators, ultimately, research efforts should be directed towards exploring and developing real-world materials. However, real materials are typically highly complex systems that are difficult to model due to the numerous perturbation interactions arising from structural imperfections12,13,14. Thus, experimentalists and materials scientists have been making significant efforts to search for “ultra-clean” materials that exhibit precisely solvable models, allowing for the observation of well-defined many-body phenomena15. To the best of our knowledge, there exists only one reported example of an Ising spin glass, LiHoxY1−xF4, that exhibits many-body QA phenomena9,16,17. However, the site-mixing disorder of Ho and Y introduces interaction randomness, making it challenging to create a precise microscopic model of this system. Moreover, in this compound, QA phenomena are only visible at very low temperatures, much lower than ΔE ≤ 0.54 K16, due to the weak couplings between the rare-earth magnetic moments. The triangular-lattice Ising antiferromagnet TmMgGaO4 also exhibits interaction randomness14, but no QA has been reported for this material. The frustrated spin-chain compound Ca3Co2O6 has no apparent structural disorder and was predicted to exhibit QA using a D-WAVE QA computer18, but no measurable effect of QA has been observed in this material19. Therefore, further investigation and exploration of other candidate materials for realizing QA is required.

Previous studies on α-CoV2O6 have suggested that the compound can experimentally realize the spatially anisotropic triangular lattice of ferromagnetic Ising spin chains (Fig. 1a) with no apparent structural disorder20,21. Additionally, high-quality single crystals of this compound are available for further investigation22. However, no reports on quantum effects of transverse magnetic fields have been made in α-CoV2O6. Here, we propose the use of such an “ultra-clean” magnetic material for QA studies, utilizing ultra-low-T measurements of physical properties and large-scale Monte Carlo (MC) computations, in transverse magnetic fields. Under zero applied transverse field and at temperatures below ~ 2 K, the frustrated spin system has a strong tendency to get stuck in metastable Kosterlitz-Thouless (KT) phases characterized by the appearance of topological vortices and antivortices23,24 around the domain walls. By contrast, a small transverse field, achieved by breaking the time-reversal symmetry in an applied transverse magnetic field, can profoundly enhance quantum-mechanical tunneling at low temperatures, triggering QA towards the optimum state with a short and nearly temperature-independent relaxation time.

Fig. 1: Frustrated Ising model of α-CoV2O6.
figure 1

a The crystal structure. The ferromagnetic intrachain interaction (J0) and antiferromagnetic interchain interactions (J1, J2, J3) of the original Hamiltonian are indicated. The unit cell is depicted with thin lines and the coordinate system for the spin components is established. b Simplified schematic of the four lowest-lying crystal-electric-field levels of the Co2+ (4F) ion. The ground-state Kramers doublet at zero applied transverse magnetic field (H⊥ = 0) is lifted and a small gap of Γ = E2 − E1 is opened due to the time-reversal symmetry breaking in H⊥ ≠ 0. c The decrease in the Néel transition temperature at H⊥, TN(0 T) − TN(μ0H⊥), measured by specific heat (Supplementary Fig. 2). Error bars, 1σ s.e., and the colored lines present the H⊥ dependence of Γ calculated from the single-ion Hamiltonian (see Supplementary Note 2). d Quasi-equilibrium-state temperature-longitudinal field (μ0Hz) phase diagram of α-CoV2O6 (Supplementary Fig. 4). Four phases are indicated: Stripe antiferromagnetic (AF), 1/3-plateau magnetization, fully ferromagnetic (FM), and paramagnetic (PM) phases, along with two critical longitudinal fields.

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Results

Spin Hamiltonian

In α-CoV2O6, the 28 electronic states of Co2+ (4F) linearly superpose into 14 doublets under the crystal electric field and spin-orbit coupling, preserving the time-reversal symmetry as described by the Kramers theorem. The lowest-lying doublet (\(\left\vert {E}_{1}\right\rangle\) and \(\left\vert {E}_{2}\right\rangle\)) is well-separated from the first excited doublet with an energy gap of E3 − E1 ~ 140 K, indicating the effective spin-1/2 dipole moment of Co2+ with Ising anisotropy (g factors of gx ~ gy ~ 0 and gz ~ 11, see Supplementary Note 1 and Supplementary Fig. 1) at low temperatures. Under nonzero transverse magnetic field, the time-reversal symmetry is broken, which results in the slight splitting of the ground-state doublet with an inner gap of Γ = E2 − E1 (Fig. 1b, c). Hence, transverse-field terms, \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{TF}}}}}}}}}=-{\varGamma }{\sum }_{i}{S}_{i}^{x}\)14,25, are induced in the Ising spin Hamiltonian of α-CoV2O6 (Supplementary Note 2 and Supplementary Fig. 3), which no longer commutes with \({S}_{i}^{z}\). The transverse-field terms cause the quantum tunneling between states with \({S}_{i}^{z}=\pm 1/2\), compete with the Ising couplings (Fig. 1a), and thus suppress the antiferromagnetic ordering temperature of TN (Fig. 1c). In a linear relationship between TN and Γ as Γ approaches 0 K26, the single-ion Hamiltonian shows good agreement with experimental results, as depicted in Fig. 1c.

The exchange couplings beyond fourth-nearest neighbors are neglected in α-CoV2O6, based on the previously reported density functional theory calculation21. The original three-dimensional (3D) Hamiltonian is formulated with the transverse field per spin,

$${{{{{{{\mathcal{H}}}}}}}}= {J}_{0}{\sum}_{\langle i,{i}_{0}\rangle }{S}_{i}^{z}{S}_{{i}_{0}}^{z}+{J}_{1}{\sum}_{\langle i,{i}_{1}\rangle }{S}_{i}^{z}{S}_{{i}_{1}}^{z}+{J}_{2}{\sum}_{\langle i,{i}_{2}\rangle }{S}_{i}^{z}{S}_{{i}_{2}}^{z}+{J}_{3}{\sum}_{\langle i,{i}_{3}\rangle }{S}_{i}^{z}{S}_{{i}_{3}}^{z}\\ -{\mu }_{0}{\mu }_{{{{{{{{\rm{B}}}}}}}}}{H}^{z}{g}^{z}{\sum}_{i}{S}_{i}^{z}-{\varGamma }{\sum}_{i}{S}_{i}^{x}.$$
(1)

After fitting the quasi-equilibrium-state thermodynamic data measured above 1.9 K at Γ = 0 K, we have refined the strengths of the couplings (Supplementary Note 3). As a result, we obtained J0 = − 30.73 K, J1 = 3.60 K, J2 = 14.21 K, and J3 = 2.55 K, with an improved goodness of fit (Supplementary Table 1, Supplementary Figs. 5, 6).

The original 3D Hamiltonian with Γ ~ 0 K (Eq. (1)) analytically yields three different lowest-energy phases separated by two critical longitudinal fields, \({\mu }_{0}{H}_{{{{{{{{\rm{c1}}}}}}}}}^{z}\) = \(\frac{2{J}_{1}+{J}_{2}-4{J}_{3}}{{\mu }_{{{{{{{{\rm{B}}}}}}}}}{g}^{z}}\) (~ 1.5 T) and \({\mu }_{0}{H}_{{{{{{{{\rm{c2}}}}}}}}}^{z}\) = \(\frac{2{J}_{1}+{J}_{2}+2{J}_{3}}{{\mu }_{{{{{{{{\rm{B}}}}}}}}}{g}^{z}}\) (~ 3.6 T), as shown in Fig. 1d. At ∣H z∣ < \({H}_{{{{{{{{\rm{c1}}}}}}}}}^{z}\), the ground phase is a stripe antiferromagnetic state with zero longitudinal magnetization M z = 0 μB/Co, at \({H}_{{{{{{{{\rm{c1}}}}}}}}}^{z}\,\) < ∣H z∣ < \({H}_{{{{{{{{\rm{c2}}}}}}}}}^{z}\) the system enters a “up-up-down” state with ∣M z∣ = gz/6 ~ 1.8 μB/Co, whereas at ∣H z∣ > \({H}_{{{{{{{{\rm{c2}}}}}}}}}^{z}\) all the Ising spins are fully polarized with ∣M z∣ = g z/2 ~ 5.5 μB/Co. These lowest-energy configurations have been verified by neutron diffraction measurements20.

Superiority of quantum annealing over thermal annealing

All three lowest-energy configurations of α-CoV2O6 have very different longitudinal magnetization values, and thus we measured the dc magnetization to assess the convergence of the annealing process. For comparison, at each low temperature the system was initialized with the same fully-inversely-polarized state in μ0H z = − 4.2 T, and then the external longitudinal μ0H z was quickly raised to 2 T with a constant ramp rate μ0dHz/dt = 10 mT/s. As μ0H z reaches 2 T, we started measuring the time dependence of M z.

In zero transverse magnetic field Hx ~ 0, as shown in Fig. 2a, for instance, the longitudinal magnetization keeps at low values (< 0.2 μB/Co), without showing any tendency towards the one-third plateau value of ~ gz/6 at least up to 15 hours, indicating that the system gets deeply stuck in metastable states, below ~ 1 K. We also performed the similar experiments with Hx ~ 0 at various temperatures. Above ~ 1 K, Mz slowly relaxes to a value that monotonically decreases from ~ 1.7 μB/Co (at 2.8 K) to 0.3 μB/Co (at 1.3 K), within our 5-hour experimental time window. In sharp contrast, when a transverse magnetic field is applied, H x ~ H z, M z rapidly relaxes to ~ 1.4 μB/Co with a time constant of τ ~ 10 s as shown in Fig. 2c, suggesting successful annealing through enhanced quantum fluctuations (Fig. 2e). Moreover, we repeated the experiments with Hx ~ Hz at various temperatures down to 80 mK, which confirm that the system quickly anneals to high-magnetization states with long-time Mz ranging from ~ 1.4 μB/Co (at low temperatures) to ~ 1.8 μB/Co (at high temperatures), as soon as μ0Hz reaches 2 T from −4.2 T. Alternately, one can anneal the system thermally by increasing the temperature (Fig. 2d). When the temperature is raised to 1.85 K from 0.5 K, Mz relaxes to a high value of ~ 1.2 μB/Co (Fig. 2b), following a stretching-exponent behavior27, with a time constant of τ = 2,550(30) s much longer than the quantum counterpart. The measured stretching exponents \({\beta }^{{\prime} }\) range from ~ 0.5 to ~ 0.9 at T ≤ 2 K (refer to the inset of Supplementary Fig. 14d), which are roughly consistent with the MC calculations. The relaxation processes of independent samples have varying durations, resulting in a substantial distribution of relaxation times and thus exhibiting a pronounced stretched-exponential relaxation behavior8.

Fig. 2: Quantum annealing versus thermal annealing.
figure 2

a–c The temporal variation of magnetization measured on α-CoV2O6. We define t = 0 as the moment when the field component along the z-axis (μ0H z) reaches 2 T from −4.2 T (where all spins are fully-inversely-polarized), at a temperature of 0.5 K and with a constant ramp rate of μ0dH z/dt = 10 mT/s. As shown in (a), under zero transverse field the system is frozen into metastable states with small longitudinal magnetization of M z ~ 0.1 μB/Co at low temperatures. By raising the temperature T or applying a transverse magnetic field H x, the system can relax towards the equilibrium state with M z = g z/6 ~ 1.8 μB/Co through thermal or quantum annealing, respectively. The green lines in (b) and (c) represent the stretched-exponential fits, \({M}^{z}({t}^{{\prime} })\) = \(({M}_{0}-{M}_{\infty }) \exp [-{({t}^{{\prime} }/\tau )}^{\beta ^{\prime} }]+{M}_{\infty }\), where τ is the relaxation time, \({\beta }^{{\prime} }\) is the stretching exponent, M0 and M∞ are the initial and final magnetization, and \({t}^{{\prime} }\) = t − t0 with t0 presenting the start time of the fit. d, e Sketches of thermal (by increasing T) and quantum (by applying H x) annealing. f, h M z-H z hysteresis loops measured at 0.5 and 1.8 K with Hx ~ 0, Hz. g, i The loop widths of the curves in (f, h).

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We also measured the isothermal magnetic hysteresis loops at 0.5 and 1.8 K as shown in Fig. 2f, h, respectively. At longitudinal magnetic fields μ0∣H z∣ > \({\mu }_{0}{H}_{{{{{{{{\rm{c1}}}}}}}}}^{z}\,\) ~ 1.5 T, the transverse magnetic field clearly suppresses the hysteresis loop (Fig. 2g, i). In the stripe antiferromagnetic phase (∣H z∣ < \({H}_{{{{{{{{\rm{c1}}}}}}}}}^{z}\)), the internal transverse magnetic fields contributed by the ordered dipole moments of Co are calculated as \({\mu }_{0}{H}_{{{{{{{{\rm{in}}}}}}}}}^{x}\) ~ ± 0.2 T on the spin-up and -down sites, respectively. The observation of narrowed loop widths in zero external transverse field at ∣H z∣ < \({H}_{{{{{{{{\rm{c1}}}}}}}}}^{z}\) may be attributed to these internal dipole fields.

At a low temperature of 1.8 K, the reduce of annealing time τ by transverse fields is most profound under the longitudinal field of μ0H z = 2 T: τ = 3,378 s at H x ~ 0 is progressively suppressed to 89 s at H x ~ H z and 66 s at H x ~ 1.7H z, respectively (Fig. 3a). At this longitudinal-field strength, slightly above the critical field \({\mu }_{0}{H}_{{{{{{{{\rm{c1}}}}}}}}}^{z}\,\) ~ 1.5 T, the Zeeman interaction drives the frustrated spin system close to the critical point. This configuration results in the highest concentrations of domain walls and topological defects along with the longest relaxation time at μ0H z = 2 T, according to MC simulations (Supplementary Figs. 7, 8). Consequently, QA is most clearly seen at μ0H z = 2 T when a transverse field is applied. We focused on the relaxation measurements on α-CoV2O6 at μ0Hz = 2 T. At H x ~ 0, the thermally activated Arrhenius behavior of τ is clearly observed down to the low temperatures (where τ → ∞, see Fig. 3b), τ−1 = \({\tau }_{{{{{{{{\rm{m}}}}}}}}}^{-1}\exp (-\Delta E/T)\). Here, \({\tau }_{{{{{{{{\rm{m}}}}}}}}}^{-1}\) = 1.5 ± 0.1 Hz, and the fitted barrier energy ΔE(μ0Hz = 2 T) = 15.3 ± 0.2 K, comparable with the strengths of the interchain couplings of α-CoV2O6, is significantly larger than the barrier energy reported in LiHoxY1−xF4 (≤ 0.54 K)16. In sharp contrast, the relaxation rates measured at the transverse magnetic field of μ0Hx ~ 2 T are nearly temperature-independent at low temperatures, τ−1 ~ 0.1 Hz, suggesting a clear QA effect.

Fig. 3: Annealing time constants.
figure 3

a The relaxation time τ of α-CoV2O6 measured with different applied transverse fields (H x) with respective to the longitudinal field (H z), H x ~ 0, 0.3H z, 0.6H z, H z, 1.7Hz, respectively. b Inverse-temperature dependence of the relaxation rate τ−1. The blue line represents the fit to the experimental data with H x ~ 0 by the Arrhenius function τ−1 = \({\tau }_{{{{{{{{\rm{m}}}}}}}}}^{-1}\exp (-\Delta E/T)\) below 2.8 K, whereas the red line shows a constant fit to the data with μ0H x ~ 2 T below 1.1 K. The raw data of magnetization relaxations, used in extracting τ, are shown in Supplementary Fig. 14. Error bars, 1σ s.e.

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By conducting systematic measurements of magnetic hysteresis loops and relaxations, we observed a distinct classical-quantum crossover behavior induced by the transverse magnetic field below ~ 4 K. This behavior is illustrated in Fig. 3, and further details can be found in Supplementary Note 6 and Supplementary Fig. 17. The temperature-dependent relaxation rate peaks at T ~ 3-4 K (1/T ~ 0.27-0.34 K−1, see Fig. 3b), indicating a maximum in spin fluctuations. This is likely due to a crossover in spin dynamics driven by thermal fluctuations with rising temperature. Notably, this observed crossover is reminiscent of similar findings previously reported in LiHoxY1−xF416.

Quantum Monte Carlo simulations

The above observations demonstrate the occurrence of QA in α-CoV2O6 and provide experimental evidence supporting its superiority over thermal annealing, as previously proposed in theoretical work on the Ising spin glass3. Although we have presented the processes of quantum and thermal annealing towards the lowest-energy configuration in experiments, it remains unclear whether the microscopic model of Eq. (1) can capture the observed QA effect of Γ. To investigate this, we have conducted stochastic series expansion (SSE) quantum Monte Carlo (QMC) simulations at low temperatures28,29. The microscopic model (Eq. (1)) has shown good agreement with both the quasi-equilibrium-state and dynamic properties of α-CoV2O6 at Γ = 0 K (see Supplementary Figs. 8, 10). We further use it to study the quasi-equilibrium and out-of-equilibrium properties of the system at Γ > 0 K.

Compared with the quantum Ising spin-glass systems30, in α-CoV2O6 achieving critical suppression of the ground-state magnetic ordering necessitates high transverse fields of Γc ~ 36 and 28 K (comparable to ∣J0∣) at μ0Hz = 0 and 2 T, respectively, as estimated using the SSE method based on the original 3D Hamiltonian (Fig. 4a, b). This would require excessively high transverse magnetic fields of > 50 T, leading to significant stress within the single crystal and potential sample destruction due to the strong Ising magnetic anisotropy31. The primary focus of this work is to investigate the QA effect in a tiny transverse field, Γ ≪ Γc.

Fig. 4: Simulations of annealing processes.
figure 4

a, b Transverse field dependence of staggered magnetization \({m}_{{{{{{{{\rm{s}}}}}}}}}^{z}=\langle | {\sum }_{j}{S}_{j}^{z}{(-1)}^{{j}_{a}+{j}_{c}}| \rangle /(NS)\) and 3\(\langle | {\sum }_{j}{S}_{j}^{z} \exp [i2\pi ({j}_{a}+ {j}_{c})/3]| \rangle /(2NS)\) calculated at μ0Hz = 0 and 2 T, respectively. Here, N is the site number, S = 1/2, and ja and jc are site indexes along the a- and c-axis, respectively. The critical transverse fields Γc are marked. c Density matrix ratios, W↓/W↑. Here, \({W}_{\uparrow }=\langle \uparrow | \exp (-\beta {{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}})| \uparrow \rangle\) and \({W}_{\downarrow }=\langle \downarrow | \exp (-\beta {{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}})| \downarrow \rangle\) are exactly calculated using the single-site molecular-field Hamiltonian \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}}\) = − hSz − ΓSx. d, e, g, h Monte Carlo step (MCS) dependence of the energy per site (E) calculated at T = 1, 1.8 K in μ0Hz = 0, 2 T. f, i MCS dependence of longitudinal magnetization calculated at μ0H z = 2 T. The original spin Hamiltonian of α-CoV2O6 is used in (a, b, d–i), and the datasets in (d–i) are averaged over 64 independent samples. To mimic the experimental procedures, the simulations (d–i) initiated with random states at μ0H z = − 4.2 T. We gradually raised μ0Hz until reaching the target field and then simulated the relaxation processes in subsequent MCS. All the SSE-QMC calculations (a, b, d–i) were performed on a 6 × 6 × 12 (432-site) cluster with periodic boundary conditions.

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The transverse field is expected to induce quantum-mechanical tunneling between states with \({S}_{i}^{z}=\pm 1/2\), thereby causing QA. To quantify this more precisely, we estimate the quantum effects of the tiny transverse field using a single-site Hamiltonian in the molecular-field approximation, \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}}\) = − hSz − ΓSx, where h ~ ∣J0∣ represents the mean field. Through exact diagonalization of \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}}\), we calculate the density elements, \({W}_{\uparrow }=\langle \uparrow | \exp (-\beta {{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}})| \uparrow \rangle,{W}_{\downarrow }=\langle \downarrow | \exp (-\beta {{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}})| \downarrow \rangle\), and the matrix ratios W↓/W↑ as shown in Fig. 4c, where β ≡ 1/(kBT). When applying a small transverse field (Γ ≤ 0.1 K), the probability of flipping the spin, as indicated by W↓/W↑, can be significantly enhanced at T ≤ 1.8 K. Therefore, Γ facilitates quantum-mechanical tunneling between different states, likely inducing the QA effects observed at low temperatures in α-CoV2O6. It is noteworthy that the simplified molecular-field model cannot directly account for the observed very slow spin dynamics and, consequently, the many-body QA effects. To investigate these effects more precisely, we employ SSE-QMC simulations for small transverse fields using the original Hamiltonian of α-CoV2O6, Eq. (1) (Fig. 4d–i).

When Γ raises to 0.1 K, for example by applying μ0Hy ~ 10 T on α-CoV2O6, the energy (per site) is significantly reduced (see Fig. 4d, e) and the longitudinal magnetization at μ0Hz = 2 T is profoundly increased (see Fig. 4f) after the same large MC steps (MCS). The equilibrium-state energy and magnetization decreases induced by Γ, estimated using \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}}\), are entirely negligible, ∣ΔEΓ∣ ~ \(\sqrt{{\varGamma }^{{{{{{{2}}}}}}}+{h}^{{{{{{{2}}}}}}}}\)/2 − h/2 ≤ 8 × 10−5 K and ∣\(\Delta {M}_{\varGamma }^{z}\)∣ ≤ 3 × 10−5μB/Co, at Γ ≤ 0.1 K. Clearly, the observed energy decrease of  > 0.2 K caused by Γ (≤ 0.1 K) at large MCS (see Fig. 4d, e) cannot be attributed to the above equilibrium-state energy decrease. The observed increase in the longitudinal magnetization caused by Γ (≤ 0.1 K) at large MCS (see Fig. 4f) also contradicts the equilibrium-state magnetization decrease. Furthermore, the energies calculated at various transverse fields (Γ ≤ 0.1 K) align with each other at small MCS, and the distinct energy decrease caused by Γ becomes apparent only at larger MCS and lower temperatures, in the QMC simulations (Fig. 4d, e). Therefore, these simulations for both energies and longitudinal magnetization unequivocally demonstrate QA effects caused by the small transverse field (0.02 ≤ Γ ≤ 0.1 K, ≪ Γc) at low temperatures (~ 1 K), providing a qualitative description of the previously mentioned experimental observations. Moreover, our heat transport measurements, which also exhibit clear QA effects around 1 K, were conducted at a transverse field of up to 0.07 K (Fig. 5), significantly smaller than Γc ~ 28 K, and more than three times larger than 0.02 K.

Fig. 5: Heat transport in transverse fields.
figure 5

a Zero-field thermal conductivity of two α-CoV2O6 crystals measured with the heat flow along the b- and a-axis, κb and κa, respectively. The dashed lines depict the power-law fits. b, c κb and κa measured under various transverse magnetic fields of μ0Hy. The dashed lines in (b) and (c) show the κ ∝ T3 behaviors of the zero-field data. d, e Temperature dependence of κa(0 T)/κa(μ0Hy) − 1. In (e), the zero-field data measured earlier is used (Supplementary Note 5), and the red and blue lines display the Arrhenius and constant fits above ~ 0.7 K. f Transverse field (Γ) dependence of κa(0 T)/κa(μ0Hy) − 1.

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At T = 1 K, the ratios W↓/W↑ at Γ = 3.5 mK and 0.02 K are estimated to be ~ 3 × 10−9 and 1 × 10−7, respectively, markedly larger than W↓/W↑ ~ 5 × 10−14 at Γ = 0 K (Fig. 4c). To observe clear QA effects of Γ, a minimal number of MCS roughly proportional to ~ W↑/W↓ is required, resulting in W↑/W↓ ~ 3 × 108 (Γ = 3.5 mK) and 9 × 106 (Γ = 0.02 K). Although W↓/W↑ is also significantly enhanced by Γ = 3.5 mK, much larger MCS are needed to observe clear QA effects compared to Γ = 0.02 K. At Γ = 0.02 K, QA effects are evident only after ~ 3 × 104 SSE MCS ( × 100) (Fig. 4e, f), necessitating more than ~ 1 × 106 SSE MCS (×100) at Γ = 3.5 mK. Therefore, observing clear QA effects at 0 < Γ < 0.02 K remains extremely challenging due to the increased computational cost associated with larger MCS.

While the present microscopic model of Eq. (1) is precise enough to account for the quasi-equilibrium-state thermodynamic properties (Supplementary Fig. 10), it may lack high precision in simulating the slow spin dynamics observed in α-CoV2O6, especially at smaller transverse fields (0 < Γ < 0.02 K, see Fig. 4). This discrepancy may arise from a low concentration of structural defects and internal transverse magnetic fields contributed by the dipole moments of Co, which are unavoidable in the real material and complexities not considered in our calculations. At 1.8 K, the quantum tunneling capacity of Γ (indicated by the increase in W↓/W↑ due to Γ) is significantly reduced compared to that at 1 K (Fig. 4c). Consequently, the QA effects of Γ at 1.8 K are expected to be notably weaker (Fig. 4g–i). Moreover, to mitigate the variability in the calculated MCS dependence of energy and magnetization, we averaged over 64 independent samples (despite the associated high computational cost). However, achieving complete elimination of this variability remains challenging in the MC simulations, especially at higher temperatures around 1.8 K, where thermal fluctuations come into play (Fig. 4g–i). Therefore, the main focus of this work is to present experimental findings regarding the QA effects of the small transverse field (Γ ≤ 0.1 K) in α-CoV2O6, and the numerical results only seek to provide a qualitative interpretation of the annealing effects of the transverse field at low temperatures (Fig. 4d–f).

Our large-scale classical MC simulations at Γ = 0 K demonstrate that α-CoV2O6 exhibits long-lived metastable states at low temperatures, commonly in the form of KT phases containing topological vortices, antivortices, and vortex-antivortex pairs (Supplementary Fig. 7)6,18,23,24. Moreover, we did not observe significant evidence supporting the presence of quenched structural defects that restrict the motion of domain walls. Instead, the vortices, antivortices, and vortex-antivortex pairs emerge around the domain walls, and are expected to decrease with increasing transverse field as indicated by changes in energies and longitudinal magnetization (Fig. 4d–f). Therefore, we suggest that these topological defects may play a crucial role in confining domain walls and increasing lifetimes of metastable states. However, it remains uncertain whether the applied transverse field directly affects the dynamics of domain walls or if it influences them indirectly through vortex dynamics, calling for further investigations.

Heat transport measurements

The spin fluctuations around domain walls and topological defects can scatter carriers such as electrons32 and phonons33,34,35,36, and their properties may be studied through transport measurements. As α-CoV2O6 is electrically insulating with a room temperature resistance larger than 20 MΩ, we performed heat transport measurements under transverse magnetic fields of μ0Hy up to 8 T at μ0Hz = 0 T (Fig. 5, Supplementary Figs. 15, 16). At μ0Hz = 0 T, the QA effects of transverse field are also evident in the SSE-QMC simulations at low temperatures (Fig. 4d). Below the Néel temperature of TN ~ 14 K, the magnetic specific heat of α-CoV2O6 exhibits a gaped behavior with a large spin gap of ~ 35 K (Supplementary Fig. 9), indicating that the density of states of magnetic excitations is extremely low. Thereby, the main carriers in the low-T thermal conductivity (κ) measurements should be phonons.

In the temperature range from ~ 1 to 2.3 K, the zero-field thermal conductivity can be fitted by a power law, κ = \(\gamma {T}^{{\alpha }^{{\prime} }}\) (Fig. 5a). For the measurements with heat flow along the b and a axes, we got \({\alpha }^{{\prime} }\) = 2.96(1) and 2.98(1), γ = 0.514(5) and 0.477(3) WK−4m−1 for sample nos. 1 and 2, respectively, consistent with the dominance of phonons. In α-CoV2O6, we expect the ideal phonon thermal conductivity κp ~ \(\frac{n{C}_{{{{{{{{\rm{p}}}}}}}}}{\lambda }_{{{{{{{{\rm{pp}}}}}}}}}{\bar{v}}_{{{{{{{{\rm{p}}}}}}}}}}{3{N}_{{{{{{{{\rm{A}}}}}}}}}}\,\) ~ γT3 below 2.3 K, where Cp ~ \(\frac{12{\pi }^{4}R{T}^{3}}{5{\Theta }_{{{{{{{{\rm{D}}}}}}}}}^{3}}\) is the phonon specific heat with the Debye temperature ΘD ~ 160 K estimated from fitting the experimental specific heat of nonmagnetic α-ZnV2O6, λpp represents the mean free path, \({\bar{v}}_{{{{{{{{\rm{p}}}}}}}}}\) ~ \(\frac{{k}_{{{{{{{{\rm{B}}}}}}}}}{\Theta }_{{{{{{{{\rm{D}}}}}}}}}}{\hslash }\) (6π2n)−1/3 (~ 3140 m/s) is the average phonon velocity, and n ~ 5 × 1027 m−3 the density of unit cells. Using the measured values of γ, we obtained λpp = \({\bar{v}}_{{{{{{{{\rm{p}}}}}}}}}{\tau }_{{{{{{{{\rm{pp}}}}}}}}}\,\) ~ 0.12 mm (\({\tau }_{{{{{{{{\rm{pp}}}}}}}}}^{-1}\,\) ~ 30 MHz represents the pure phonon scattering rate including crystal boundary contributions), comparable to the smallest dimension of the crystal (~ 0.15 mm).

The thermal conductivity measured along the a axis (κa) decreases significantly with increasing transverse magnetic field (μ0Hy) as shown in Fig. 5c. In comparison, the thermal conductivity along the b axis (κb) exhibits only a weak decrease with increasing μ0Hy (Fig. 5b). These observations are in line with the MC simulations that the spin system has much more domain walls and topological defects along the frustrated triangular plane (i.e., the ac plane) compared to those along the spin chain (i.e., the b axis), at low temperatures (Supplementary Fig. 7). Hence, the suppression of κa by μ0Hy is likely related to the movements of domain walls and topological defects. Due to the significant difference in scale between the microscale distances between domain walls (or topological defects) and the macroscale wavelengths of acoustic phonons, phonon scattering by relatively static domain walls and topological defects is negligible in the zero-field heat transport at low temperatures (Fig. 5a). The transverse magnetic field is believed to induce the deconfinement of domain walls and topological defects by flipping nearby Ising spins, which in turn can increase the phonon-spin scattering rate (\({\tau }_{{{{{{{{\rm{ps}}}}}}}}}^{-1}\)) and suppress κa (when \({\tau }_{{{{{{{{\rm{ps}}}}}}}}}^{-1}\) gets comparable to \({\tau }_{{{{{{{{\rm{pp}}}}}}}}}^{-1}\)). To simplify the analysis, we used the latest measurement of zero-field thermal conductivity, κa(0 T), to approximate the ideal phonon thermal conductivity κp along the a axis, even though there are internal transverse magnetic fields present (approximately ± 0.2 T, as mentioned above) and slight derivations from the T3 law below ~ 1 K (Fig. 5c). We obtained κa(0 T)/κa(μ0Hy) − 1 ~ (λpp − λp)/λp = τpp/τps, where λp = \({\bar{v}}_{{{{{{{{\rm{p}}}}}}}}}{\tau }_{{{{{{{{\rm{p}}}}}}}}}\) is the mean free path at μ0Hy and \({\tau }_{{{{{{{{\rm{p}}}}}}}}}^{-1}\) = \({\tau }_{{{{{{{{\rm{pp}}}}}}}}}^{-1}\)+\({\tau }_{{{{{{{{\rm{ps}}}}}}}}}^{-1}\) is the total scattering rate33. The ratio of κa(0 T)/κa(μ0Hy) − 1 approximately reflects \({\tau }_{{{{{{{{\rm{ps}}}}}}}}}^{-1}\) induced by μ0Hy in α-CoV2O6, given the negligible temperature and magnetic field dependencies of τpp (and \({\bar{v}}_{{{{{{{{\rm{p}}}}}}}}}\)) below 2.3 K.

If \({\tau }_{{{{{{{{\rm{ps}}}}}}}}}^{-1}\) is thermally activated, it is expected to follow the Arrhenius form \({\tau }_{{{{{{{{\rm{ps}}}}}}}}}^{-1}\,\) ∝ \(\exp (-\Delta E/T)\). We observed this behavior and obtained a barrier energy of ΔE(μ0Hz = 0 T) ~ 4.3 ± 0.3 K by analyzing two thermal conductivity measurements taken during different run sequences (run #1 and #2) at 0.7 < T < 2.3 K and 0 T in α-CoV2O6 (see Fig. 5e). In contrast, in a transverse magnetic field of μ0Hy ≥ 1 T, \({\tau }_{{{{{{{{\rm{ps}}}}}}}}}^{-1}\), as measured by κa(0 T)/κa(μ0Hy) − 1, is nearly independent of temperature down to ~ 0.7 K (Fig. 5d, e), indicating quantum tunneling of magnetic domain walls. Above ~ 0.7 K, the QA effect of the transverse field almost saturates at Γ ~ 0.07 K, as shown in Fig. 5f. On the one hand, the applied transverse field facilitates annealing, reduces domain walls, and thus increases the thermal conductivity by decreasing \({\tau }_{{{{{{{{\rm{ps}}}}}}}}}^{-1}\) at low temperatures. On the other hand, as aforementioned, the transverse field causes quantum fluctuations between states with \({S}_{i}^{z}=\pm 1/2\) resulting in the motion of magnetic domain walls, increases \({\tau }_{{{{{{{{\rm{ps}}}}}}}}}^{-1}\), and decreases the thermal conductivity. The transverse field’s positive and negative effects likely compete, resulting in the observed oscillation of κa(0 T)/κa(μ0Hy) − 1 with temperature, below ~ 0.7 K (Fig. 5d). Further investigation is required to fully understand the intriguing behaviors of κa observed at low temperatures (Fig. 5c–f).

The thermal conductivity (κa) measured at the same transverse magnetic field and temperature exhibits a weak increase in run #2 compared to run #1, following a long-term (> 3 days) annealing in μ0Hy. This observation potentially suggests a decrease in the concentrations of domain walls and topological defects along the triangular plane, after the prolonged annealing process. However, the suppression of κa by μ0Hy remains largely repeatable in run #2, suggesting that the domain walls and topological defects do not completely disappear even after the long-term annealing in μ0Hy (up to 8 T) between temperatures of 0.05 and 2.3 K.

Discussion

At μ0Hz = 2 T, it is noteworthy that a small transverse field of Γ ~ 3.5 mK is unable to fully anneal the entire magnetic crystal of α-CoV2O6 to the lowest-energy configuration at low temperatures, as indicated by the slight deviation of long-time Mz from the lowest-energy value of gz/6 (Fig. 2c). The QMC simulation shows that the metastable states can have extremely long lifetimes, lasting up to > 400,000 SSE MCS (×100, see Fig. 4). The presence of vortex-antivortex pairs may be a key factor contributing to these long lifetimes (Supplementary Fig. 7f). Under nonzero transverse field, the QA effects manifest primarily through a significant reduction in the annealing time constant (Fig. 3) and a profound increase in Mz at long times (Fig. 2c), compared to those observed under zero transverse magnetic field (Fig. 2a), at the same low temperature. If one wishes for the system to converge completely to the optimum state, it seems to be necessary to apply a stronger Γ and to wait longer. For instance, at 1.8 K, the long-time Mz measured at μ0Hx ~ 3.4 T is closer to gz/6 than at μ0Hx ~ 2 T (Supplementary Fig. 14). However, even with increased waiting times up to 5 hours, Mz did not show further convergence towards gz/6 at μ0Hx ~ 2 T and lower temperatures. Furthermore, under transverse fields up to Γ ~ 0.07 K, κa(μ0Hy) is significantly lower than κa(0 T) at least within 0.7–2.3 K (Fig. 5), indicating an abundance of domain walls along the frustrated triangular plane even after the long-term annealing (see above). In the QMC simulations, achieving complete convergence to the lowest-energy configuration of the 432-site cluster is also extremely difficult at low temperatures. In most independent samples, increasing Γ up to 0.1 K (corresponding to e.g. μ0Hy ~ 10 T) still did not result in complete convergence to the lowest-energy configuration after ~ 300,000 SSE MCS (×100, see Fig. 4).

There are at least three possible reasons for the incomplete QA observed in α-CoV2O6 at low temperatures. First, metastable KT phases are predicted to be widespread in frustrated Ising models on triangular lattices6,29,37,38. The presence of vortices, antivortices, and vortex-antivortex pairs may confine domain walls, and prolong the annealing times in the extremely long-time range as the system approaches its lowest-energy configuration. Our many-body MC simulations, conducted at T ≥ 1 K for up to large MCS, have effectively captured this incomplete annealing phenomenon (Fig. 4). It can be expected that annealing this frustrated spin system will become even more challenging at larger scales and low temperatures. Second, the transverse field Γ can weaken the lowest-energy “up-up-down” order, leading to a decrease in longitudinal magnetization Mz. However, at Γ ≤ 0.1 K this reduction of Mz is estimated to be less than ~ 2 × 10−4μB/Co (Fig. 4b) ~ ∣\(\Delta {M}_{\varGamma }^{z}\)∣ as estimated using \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}}\) (see above), which is much smaller than the experimental value of gz/6 − Mz(t ≫ 10 s) ~ 0.4 μB/Co (see Fig. 2c). In this study, the transverse field of Γ ≤ 0.1 K is considered too weak to significantly alter the lowest-energy configurations, which cannot account for the incomplete QA observed. Finally, while we did not find strong evidence supporting the presence of structural imperfections in the single crystals of α-CoV2O6, it is still possible that the real-world compound contains a low concentration of quenched structural defects. These defects could potentially impede magnetic domain wall movement at low temperatures, making complete annealing difficult, even in the presence of transverse fields.

At low temperatures (≪ TN), (incomplete) QA towards the lowest-energy configuration is clearly observed in magnetic relaxation measurements under a tiny transverse field applied to α-CoV2O6. Experimental evidence of quantum tunneling of magnetic domain walls and topological defects is observed in both the relaxation rates of magnetization and the phonon-spin scattering rates reflected in heat transport measurements along the frustrated triangular plane. The distinctive observation of QA effects induced by the small transverse field may stimulate further experimental investigations and large-scale ab initio simulations in this frustrated magnet and other relevant materials.

Methods

In the magnetization (Mz) measurements, we tilted the single crystal to apply a transverse magnetic field with respect to the external field (H), determining the longitudinal and transverse magnetic fields for each fixed angle θ between the Ising (z) direction and H. We used five single crystals with θ ~ 0∘, 15∘, 30∘, 45∘, 60∘ to measure Mz above 1.8 K in a magnetic properties measurement system, and measured on two smaller crystals with θ ~ 0∘ and 45∘ below 1.85 K using a Faraday force magnetometer in a 3He–4He dilution refrigerator39,40 (Supplementary Fig. 13). The low-T thermal conductivity measurements were conducted using a standard four-wire method in the dilution refrigerator33. We conducted unbiased SSE-QMC calculations, utilizing both local diagonal and Wolff cluster updates28,29, to simulate the model on finite lattice sizes. Each SSE MCS involves one local diagonal update (attempting to either remove or insert a diagonal operator and traversing through the operator strings) along with a set number of Wolff cluster updates (sufficient to averagely visit all operators in the string). Due to frustration combined with a low transverse field or high longitudinal field, the SSE Wolff cluster algorithm is less efficient at low temperatures. To make the relaxation processes clear, we initiated all the simulations with random states at μ0Hz = − 4.2 T. Classical MC simulations indicate that the spin system should quickly relax to the fully inversely polarized state after only 3-4 MCS at μ0Hz ~ − 4 T, please refer to Supplementary Fig. 7h. For more details, please refer to the Supplementary Information.