Spin Seebeck Effect of Triangular-lattice Spin Supersolid

TL;DR

The paper develops a thermal tensor-network framework to quantify the spin Seebeck effect (SSE) in triangular-lattice quantum antiferromagnets hosting spin supersolid phases, linking the SSE to the local dynamical susceptibility through $\tilde{I}_S = \int d\omega\, k^2(\beta\omega)\, \mathrm{Im}[\chi^{-+}_{\mathrm{loc}}(\omega)]$ and showing how the observable current depends on low-energy spin dynamics. It demonstrates two sign reversals in the 1D Heisenberg chain and, in the 2D spin-supersolid NBCP, a persistent negative spin current at low $T$ mediated by Goldstone modes, with a universal $T^{d/z}$ scaling at polarization quantum critical points. The work establishes SSE currents as sensitive probes of spin-supersolid states and maps phase boundaries, providing a route to ultralow-temperature spin caloritronics and guiding experimental studies on NBCP and related materials. It also highlights the limitations of linear spin-wave theory in capturing supersolid SSE features, underscoring the necessity of thermal tensor-network methods for strongly correlated frustrated magnets.

Abstract

Using thermal tensor-network approach, we investigate the spin Seebeck effect (SSE) of the triangular-lattice quantum antiferromagnet hosting spin supersolid phase. We focus on the low-temperature scaling behaviors of the normalized spin current across the interface. For the 1D Heisenberg chain, we find a negative spinon spin in the bulk current with algebraic temperature scaling; at low fields, boundary effects induce a second sign reversal at lower temperatures. These benchmark results are consistent with field-theoretical analysis. On the triangular lattice, spin frustration dramatically enhances the low-temperature SSE, with distinct spin-current signatures -- particularly the sign reversal and characteristic temperature dependence -- distinguishing different spin states. Remarkably, we discover a persistent, negative spin current in the spin supersolid phase, which saturates to a non-zero value in the low-temperature limit and can be ascribed to the Goldstone-mode-mediated spin supercurrents. Moreover, a universal scaling $T^{d/z}$ is found at the U(1)-symmetric polarization quantum critical points. These distinct quantum spin transport traits provide sensitive spin current probes for spin supersolid states in quantum magnets such as Na$_2$BaCo(PO$_4$)$_2$. Furthermore, our results also establish spin supersolids as a tunable quantum platform for spin caloritronics in the ultralow-temperature regime.

Figures (13)

• Figure 1: (a) Longitudinal SSE setup: the quantum magnet (triangular lattice, with temperature $T_s$) and metal substrate ($T_m$) maintain a temperature difference $\delta T = T_s - T_m$. The resulting spin current $I_S$ flows across the magnet-metal interface along the $x$-axis, parallel to the thermal gradient $-\nabla T$. Red(blue) arrow represents the positive(negative) current. A perpendicular magnetic field $B$ is applied along the $z$-axis, and the spin current is measured by the voltage $V$ along the $y$-axis through the inverse spin Hall effect in the metal substrate. (b) The spin current $\tilde{I}_2$ is efficiently computed by contracting the density matrix operator $\rho(\beta/2)$ with its Hermitian conjugate. $A$ and $A^\dagger$ are rank-4 tensors and $\mathcal{O}_j$ and $S_j^+$ are the inserted operators.
• Figure 2: Benchmarks on normalized spin current in 1D Heisenberg chain ($L=128$, $D=500$). (a) The simulated spin current $\tilde{I}_2$, where the black dotted line marks the sign reversal, and the white dotted line locates the maximum of $\tilde{I}_2$ under a fixed field. The red dot labels the QCP at $B_c=2$, separating the TLL and polarized (PL) phases. (b) presents the temperature dependence of spin currents calculated through real-time dynamics ($\tilde{I}_{S}$, with $t_{\rm max}=40$, $D=500$ and $j=64$) and imaginary-time correlations ($\tilde{I}_2$, $j\in[33,96]$ for $B=1, B_c$ and $j\in[1,2]$ for $B=0.1$). At low temperatures, we observe algebraic spin current $\tilde{I}_{S,2} \sim T^{\alpha}$ with $\alpha \simeq 1.59(2)$ for $B=1$ (TLL phase) and $\tilde{I}_{S,2} \sim \sqrt{T}$ at $B=B_c$ (QCP). Given the undetermined prefactors in simulated spin currents, we shift the $\tilde{I}_{S}$ data to align with the low-temperature $\tilde{I}_2$. The slight shift of the sign-reversal temperature is ascribed to their different kernel functions, namely, $k(\beta\omega)$ versus $k^2(\beta\omega)$. For a small magnetic field $B=0.1$, the boundary contributions from first two sites exhibit an additional sign reversal at a lower temperature.
• Figure 3: (a) Simulated spin current $\tilde{I}_2$ of TLAF model (6$\times$18 cylinder, $D=3000$) for the compound NBCP Gao2022Super. Three QCPs $B_{c1,2,3}$ (red dots) separate supersolid-Y (SSY), up-up-down (UUD), supersolid-V (SSV), and the PL phases. Dashed lines indicate schematic phase boundaries of SSY and SSV determined from the sign reversal in $\tilde{I}_2$. (b) Isothermal $\tilde{I}_2$ cuts reveal three QCPs at $B_{c1} \simeq 0.4$ T, $B_{c2} \simeq 1.1$ T, and $B_{c3} \simeq 1.75$ T (vertical gray lines), with prominent peaks or dips. (c) $\tilde{I}_2 \sim T^{d/z}$ with $d/z=1$ (black dashed line) at QCPs ($B_{c1,2,3}$). The exponentially decaying $\tilde{I}_2$ within the UUD phase ($B=0.6~{\rm T}$) is also plotted as a comparison.
• Figure 4: (a) The simulated $\tilde{I}_2$ results in the SSY phase, where the data are well converged with $D=5000$. (b) and (c) present the momentum-resolved spin current $\tilde{I}_k$ at two temperatures. The gray dashed line shows the boundary of the 1st Brillouin zone. The red dots mark the involved momentum points in the calculations of $\tilde{I}_{\rm G}$. The black dots label the $\Gamma$ and K points. (a) shows the spin supercurrent under a 0.05 T field (see inset); results for a different field in SSY phase can be found in the Appendix.
• Figure 5: The simulated spin current $\tilde{I}_2$ is shown both at the outermost site ($j=1$) and averaged over several adjacent edge sites ($j \in [1, l]$) on a spin-1/2 chain with length $L=256$. The bulk results computed on an infinite chain are also included. A magnetic field $B=0.1$ is applied and bond dimension $D=500$ is kept. The two arrows indicate the higher sign-reversal temperature $T_{\rm R1}$ and the lower temperature $T_{\rm R2}$, respectively.