Far-from-Equilibrium Field Theory of Many-Body Quantum Spin Systems: Prethermalization and Relaxation of Spin Spiral States in Three Dimensions

We study theoretically the far-from-equilibrium relaxation dynamics of spin spiral states in the three-dimensional isotropic Heisenberg model. The investigated problem serves as an archetype for understanding quantum dynamics of isolated many-body systems in the vicinity of a spontaneously broken continuous symmetry. We present a field-theoretical formalism that systematically improves on the mean field for describing the real-time quantum dynamics of generic spin-$1/2$ systems. This is achieved by mapping spins to Majorana fermions followed by a $1/N$ expansion of the resulting two-particle-irreducible effective action. Our analysis reveals rich fluctuation-induced relaxation dynamics in the unitary evolution of spin spiral states. In particular, we find the sudden appearance of long-lived prethermalized plateaus with diverging lifetimes as the spiral winding is tuned toward the thermodynamically stable ferro- or antiferromagnetic phases. The emerging prethermalized states are characterized by different bosonic modes being thermally populated at different effective temperatures and by a hierarchical relaxation process reminiscent of glassy systems. Spin-spin correlators found by solving the nonequilibrium Bethe-Salpeter equation provide further insight into the dynamic formation of correlations, the fate of unstable collective modes, and the emergence of fluctuation-dissipation relations. Our predictions can be verified experimentally using recent realizations of spin spiral states with ultracold atoms in a quantum gas microscope [ et al., Phys. Rev. Lett. 113, 147205 (2014)].

Popular Summary

An open question is what happens when an isolated quantum system is set in motion from an initial nonequilibrium state that possesses certain orders. In classical thermodynamics, this problem is exemplified by the irreversible expansion of a gas in an isolated chamber after suddenly doubling the chamber size. Generically, one expects to observe a gradual relaxation to a thermal equilibrium state as the details of the initial state are progressively washed away in collisions. In certain cases, however, the formation of strong quantum correlations between the particles can conspire to slow down the relaxation process, resulting in a multistage dissolution of the initial information via long detours to intermediate states. In technical terms, these intermediate states are referred to as prethermal states, which are distinct from thermal equilibrium. Even though prethermalization is believed to be a rather ubiquitous phenomenon, so far it has only been experimentally observed in weakly interacting, one-dimensional systems. Unfortunately, currently available theoretical methods for describing the quantum dynamics of experimentally realizable models of strongly correlated systems in higher dimensions are inadequate. Here, we develop a new formalism using methods of quantum field theory to study the real-time dynamics of interacting quantum spins.

We focus on relaxation dynamics in the three-dimensional quantum Heisenberg model of spin-1/2 systems, one of the paradigmatic models of strong quantum correlations. Inspired by recent experiments using ultracold atoms in a quantum gas microscope, we study the evolution of spins initially prepared in helical spiral states. We find that spiral states that energetically allow spontaneous symmetry breaking upon thermalization also exhibit a pronounced two-step relaxation with long-lived prethermal states, and other states relax in a simple monotonic fashion. Curiously, the hierarchical relaxation we find here bears a strong resemblance to aging dynamics in classical glassy systems. We also show that instabilities in the system exhibit self-regulation.

Understanding the emergence of slow degrees of freedom near thermodynamic phase transitions has implications far beyond condensed matter physics and is, for example, important for the description of the early Universe. We expect that our work will contribute to establishing new connections among condensed matter, soft matter, cosmology, and atomic physics.

Figure 1

Relaxation of spin spiral states in the 3D isotropic Heisenberg model. (a) The system is prepared in a spin spiral state in the $xy$ plane with the winding $\mathbf{Q}=(Q,Q,Q)$ as tuning parameter. The figure illustrates the case $Q=\pi /2$. (b) The real-time evolution of the transverse magnetization $M_{\perp }$ for three different $Q$ as indicated in the plot. For $Q=7\pi /8$, a hierarchical relaxation process emerges with a nonthermal plateau at intermediate times. The time scale is switched to logarithmic at $tJ=5$ for better visibility. (c) A global view of the spiral dynamics. Nonthermal plateaus appear near $Q\sim 0,\pi$.

Figure 2

The spin-2PI formalism illustrated. The spins (green arrows) precess about a fluctuating exchange field (uncertain blue arrows). The quantum fluctuations of the exchange field are mediated by real-vector bosons (wiggly lines) and are suppressed by a factor of $1/N$, permitting a systematic expansion.

Figure 3

Thermalization of the spin spiral state. (a) The effective inverse temperature of local spin $T_{\mathrm{spin}}$ and local exchange-field fluctuations $T_{\text{fluct}}$ obtained from the fluctuation-dissipation relations in the steady state. The two temperatures are in agreement for spiral windings near $Q\sim \pi /2$, supporting the true thermalization of the system. The temperatures calculated in the prethermalized plateaus $Q\sim 0,\pi$ (shaded regions) disagree with each other and generically differ from the temperature of the true thermal states that emerge at later times. Inset: The temperature $k_{B}T$ as a function of $Q$ (same data as in the main panel) displays a resonance from positive infinite temperature to negative infinite temperature at the classical duality point $Q=\pi /2$. (b) The approach of $T_{\text{fluct}}$ to the steady state (light to dark) as obtained from fluctuation-dissipation relations for $Q=\pi /4$ (left) and $Q=\pi$ (right). The steady-state temperatures are shown on the plots.

Figure 4

The evolution of spin correlations. Top panels: Growth rate of out-of-plane instable modes obtained from a linear response analysis. Bottom panels: Numerically calculated correlation function $⟨{\widehat{S}}_{\mathbf{k}}^z{\widehat{S}}_{-\mathbf{k}}^z⟩(t)=i{\chi }_{\mathbf{k}}^{zz,K}(t,t)$ as a function of the lattice wave vector $\mathbf{k}=(k,k,k)$ within the spin-2PI formalism including NLO corrections. (a) $Q=3\pi /8$, (b) $Q=\pi /2$, and (c) $Q=3\pi /4$. The inset in (c) shows the connected part of the in-plane correlations $⟨{\widehat{S}}_{\mathbf{k}}^{+}{\widehat{S}}_{-\mathbf{k}}^{-}⟩(t)$.

Figure 5

Dynamics of exchange-field fluctuations. (a) The out-of-plane (top) and in-plane (bottom) exchange-field fluctuations as a function of time and momentum $\mathbf{k}=(k,k,k)$ for $Q=\pi /4$ (left) and $Q=7\pi /8$ (right). The red lines indicate the most enhanced mode in the long-time limit; the dashed blue line in the lower right plot corresponds to the $k=Q$ in-plane mode, which initially exhibits the strongest enhancement of correlations. (b) The evolution of the late-time most enhanced mode for $Q=\pi /4,\pi /2$ (left) and $Q=7\pi /8,\pi$ (right). In cases where the system thermalizes, left column, $SU(2)$ symmetry emerges in the long-time limit, while it is broken in the prethermal case $Q=7\pi /8$ and for $Q=\pi$, right column. In the latter case, the system can exhibit true long-range order, provided its effective temperature is below the critical temperature of the equilibrium phase transition and can thus be thermal and simultaneously break $SU(2)$ symmetry.

Figure 6

Comparison between spin-2PI and semiclassical dynamics. The spin-2PI results (solid black lines) are compared to the semiclassical dynamics obtained from the dTWA (dashed blue lines). The short-time analytic result from Sec. 3a is also shown for reference (thick red lines). The long-time dynamics of the two methods are significantly different. In particular, the dTWA is not capable of describing the long-lived prethermal plateaus in contrast to the spin-2PI formalism. The scale of the time axis is switched from linear to logarithmic at $tJ=5$ for better visibility.