Variational approach to open quantum systems with long-range competing interactions

Abstract

Competition between short- and long-range interactions underpins many emergent phenomena in nature. Despite rapid progress in their experimental control, computational methods capable of accurately simulating open quantum many-body systems with complex long-ranged interactions at scale remain scarce. Here, we address this limitation by introducing an efficient and scalable approach to dissipative quantum lattices in one and two dimensions, combining matrix product operators and time-dependent variational Monte Carlo. We showcase the versatility, effectiveness, and unique methodological advantages of our algorithm by simulating the non-equilibrium dynamics and steady states of spin-$\frac{1}{2}$ lattices with competing algebraically-decaying interactions for as many as $N=200$ sites, revealing the emergence of spatially-modulated magnetic order far from equilibrium. This approach offers promising prospects for advancing our understanding of the complex non-equilibrium properties of a diverse variety of experimentally-realizable quantum systems with long-ranged interactions, including Rydberg atoms, ultracold dipolar molecules, and trapped ions.

Figures (6)

• Figure 1: Model of interacting spins on a square lattice, undergoing drive $h$ and dissipation $\gamma$ from contact with the external environment. The spins interact non-locally, as indicated with the example nearest-neighbor and next-nearest-neighbor couplings $J_1$ and $J_2$.
• Figure 2: Illustration of the Dirac-Frenkel variational principle applied to the Lindblad master equation \ref{['eq: lindblad master equation']}. Given a set of variational parameter values $\boldsymbol{a}$, spanning a variational submanifold $\pazocal{M}$ of the projective Hilbert space, the exact time-evolution vector $\pazocal{L}\rho(\boldsymbol{a})$, which generally lies outside of $\pazocal{M}$, is orthogonally projected onto the tangent space $T_{\rho(\boldsymbol{a})}\pazocal{M}$ of $\pazocal{M}$ at $\rho(\boldsymbol{a})$ by means of the projection operator $\pazocal{P}_{T_{\rho(\boldsymbol{a})}\pazocal{M}}$. The blue path $\rho(\boldsymbol{a}(t))$ on $\pazocal{M}$, resultant from a succession of projections, is the best approximation to the exact dynamics PhysRevLett.107.070601. The red path $\rho(\tilde{\boldsymbol{a}}(t))$ is a stochastic solution to the variational equations of motion.
• Figure 3: Dynamics of the bulk (a) magnetizations, (b) nereast-neighbor, and (c) next-nearest-neighbor spin-spin correlation functions for the dissipative anisotropic antiferromagnetic Heisenberg spin chain with $(J_x, J_y, J_z)=(-1.0,-0.9,-1.2)$ and $N=200$ sites at $\gamma=-h=1$, starting from the product state $\expval{\sigma^x}=1$. Solid and darker dashed lines are respectively t-VMC+MPO and t-MPS results, each with $\chi=20$. (d) Long-time relaxation dynamics of the difference of bulk spin magnetization and corresponding exact steady-state expectation value for a dissipative transverse-field Ising chain with $N=10$, $J_z=-0.5$, and $\gamma=-h=1$. Compared are t-VMC+MPO and t-MPS dynamics (for multiple time-step sizes), each with $\chi=20$.
• Figure 4: Dynamics of (a) magnetizations and (b) nearest-neighbor correlation functions for a dissipative spin chain with $N=200$ sites and competing long-ranged dipolar Ising interactions ($J_1=-1/2$, $\alpha_1=3$ and $J_2=1$, $\alpha_2=6$), starting from the product state $\langle \sigma^y \rangle=-1$ at $\gamma = -2h = 1$. (c)-(d) as in (a)-(b) for a square lattice with $N=4\times4$ sites ($J_1=-1/4$ and $J_2=1/2$). Darker dashed lines are corresponding exact dynamics obtained for a reduced number of $N=10$ and $N=3\times 3$ sites, respectively. Lighter solid lines in (c-d) represent variational results for $N=3\times 3$ sites.
• Figure 5: (a) Steady-state structure factor phase diagrams for a dissipative spin chain with long-range dipolar $(\alpha_2 = 3)$ Ising interactions as function of the interaction strength $J_2$ for $N=20$. (b) Steady-state spin-spin correlation functions as function of separation distance for $N=30$. (c)-(d) As in (a)-(b) with addition of competing antiferromagnetic Coulomb Ising interaction ($\alpha_1 = 1, J_1=-1.5$). In (c), the reduction of $S_{zz}(q=2\pi/N)$ with increasing system sizes from $N=20$ to $N=50$ is shown in lighter shades of red. To ensure extensivity with system size, the interaction strengths were renormalized by the Kac normalization factor. Inset in (d): reduction over time of the paramagnetic order parameter $\langle \sigma^x (t) \rangle$ for $J_1=-J_2=0.5$ and $N=50$ as compared to mean-field prediction $\langle \sigma^x (t) \rangle_\text{MF} = 1$.