Linked cluster expansions for open quantum systems on a lattice

We propose a generalization of the linked-cluster expansions to study driven-dissipative quantum lattice models, directly accessing the thermodynamic limit of the system. Our method leads to the evaluation of the desired extensive property onto small connected clusters of a given size and topology. We first test this approach on the isotropic spin-1/2 Hamiltonian in two dimensions, where each spin is coupled to an independent environment that induces incoherent spin flips. Then we apply it to the study of an anisotropic model displaying a dissipative phase transition from a magnetically ordered to a disordered phase. By means of a Padé analysis on the series expansions for the average magnetization, we provide a viable route to locate the phase transition and to extrapolate the critical exponent for the magnetic susceptibility.

Figure 1

Steady-state average magnetization along the $z$ direction for the isotropic Heisenberg model, evaluated by means of the NLCE (bare sum) at different orders $R$ in the cluster size, as a function of $1/\alpha$. The arrows indicate the values of ${\alpha }^{★}$ at which each curve at the $R\mathrm{th}$ order ($R<8$) starts deviating significantly from the highest accuracy curve ($R=8$; thick black line) that we have.

Figure 2

Steady-state average $z$-magnetization as a function of $\alpha$, after implementing two resummation techniques on the bare data at order $R=8$ (black curve, same as in Fig. 1): Wynn's algorithm (colored curves in the upper panel) and Euler transformation (colored curves in the lower panel). The symbols denote the results of QT simulations for a finite system using periodic boundary conditions, with a $4\times 4$ square plaquette (red circles) and a $4\times 3$ plaquette (black circles) constructed from the previous one after removing the four sites at the corners. Note that quite counterintuitively, in the case of Wynn's algorithm, ${\epsilon }_4^{\left(4\right)}$ seems to converge for larger values of $\alpha$, with respect to ${\epsilon }_2^{\left(6\right)}$.

Figure 3

Angularly averaged magnetic susceptibility to an external field in the $x\text{−}y$ plane, as a function of $\alpha =J_y-0.9$. The continuous curves denote Euler resummed data, to the best achievable expansion, of the bare NLCE results up to the order $R=8$. The dashed lines are the results of the bare expansions. Symbols are the results from the corner-space renormalization method, taken from Ref. [39]. The dotted area highlights the region of alpha $0\le \alpha \le 0.02$, for which the NLCE converges.

Figure 4

Logarithmic derivative of ${\chi }_{\mathrm{av}}$ as a function of $\alpha$. The black line is obtained from Euler resummed data to the order $R=8$ (blue line of Fig. 3). The red and blue lines are the results of the Padé analysis with different $\left[3\right|3]$ and $\left[3\right|4]$ approximants, respectively.

Figure 5

Position of the critical point ${\alpha }_c$ (top panel) and value of the critical exponent $\gamma$ (bottom panel) as a function of the upper boundary of the fitting region ${\alpha }_{\mathrm{fin}}$.