Dissipation-Induced Order: The $S=1/2$ Quantum Spin Chain Coupled to an Ohmic Bath

We consider an $S=1/2$ antiferromagnetic quantum Heisenberg chain where each site is coupled to an independent bosonic bath with ohmic dissipation. The coupling to the bath preserves the global SO(3) spin symmetry. Using large-scale, approximation-free quantum Monte Carlo simulations, we show that any finite coupling to the bath suffices to stabilize long-range antiferromagnetic order. This is in stark contrast to the isolated Heisenberg chain where spontaneous breaking of the SO(3) symmetry is forbidden by the Mermin-Wagner theorem. A linear spin-wave theory analysis confirms that the memory of the bath and the concomitant retarded interaction stabilize the order. For the Heisenberg chain, the ohmic bath is a marginal perturbation so that exponentially large system sizes are required to observe long-range order at small couplings. Below this length scale, our numerics is dominated by a crossover regime where spin correlations show different power-law behaviors in space and time. We discuss the experimental relevance of this crossover phenomena.

Figure 1

(a) Antiferromagnetic correlation ratio $R$ as a function of the spin-boson coupling $\alpha$ for different system sizes $L$. A finite-size scaling of the crossings ${\alpha }_c(L)$ between data pairs $\{L,2L-2\}$ is shown as an inset. (b) $R(1/L)$ for different $\alpha$. The minima of $R(1/L)$ define a crossover scale $L_c(\alpha )$ beyond which $R$ increases. $L_c$ is estimated using spline fits and is shown in (c). The crossover scale is consistent with an exponential scaling $L_c\propto \exp (\zeta /\alpha )$.

Figure 2

Real-space correlation function $C(r)$ for even $r$ at the largest available $L\in \{82,162,322,642\}$ for different $\alpha$. For $\alpha =0$, the exact asymptotic form (dashed line), $C(r)\sim (-1{)}^r(\ln r{)}^{1/2}/[(2\pi {)}^{3/2}r]$, is approached very slowly [27]. A finite-size analysis of the boundary effects near $r=L/2$ can be found in the Supplemental Material [28].

Figure 3

Finite-size dependence of the spin structure factor $S(q)$ for (a) $\alpha =0.0$ and (b) $\alpha =0.2$. (c) Finite-size scaling of the order parameter $m^2(L)=S(q=\pi )/L$ for different $\alpha$.

Figure 4

Local spin susceptibility $\chi (r=0)$ (a) for $T\to 0$ as a function of $L$ and (b) for $L\to \infty$ as a function of $T$. The dot-dashed lines indicate an $\ln L$ dependence in (a) and an $\ln \beta$ dependence in (b), whereas the dashed line in (b) corresponds to the $\chi \sim 1/T$ behavior expected for the ordered state.