Generalized mean-field description of entanglement in dimerized spin systems

We discuss a generalized self-consistent mean-field (MF) treatment, based on the selection of an arbitrary subset of operators for representing the system density matrix, and its application to the problem of entanglement evaluation in composite quantum systems. As a specific example, we examine in detail a pair MF approach to the ground state (GS) of dimerized spin-$1/2$ systems with anisotropic ferromagnetic-type $XY$ and $XYZ$ couplings in a transverse field, including chains and arrays with first neighbor and also longer range couplings. The approach is fully analytic and able to capture the main features of the GS of these systems, in contrast with the conventional single-spin MF. Its phase diagram differs significantly from that of the latter, exhibiting $\left(S_z\right)$ parity breaking just in a finite field window if the coupling between pairs is sufficiently weak, together with a fully dimerized phase below this window and a partially aligned phase above it. It is then shown that through symmetry restoration, the approach is able to correctly predict not only the concurrence of a pair, but also its entanglement with the rest of the chain, which shows a pronounced peak in the parity breaking window. Perturbative corrections allow to reproduce more subtle observables like the entanglement between weakly coupled spins and the low lying energy spectrum. All predictions are tested against exact results for finite systems.

Figure 1

Schematic plot of the dimerized cyclic chain.

Figure 2

Phase diagram of the dimerized spin-$1/2$ chain according to the conventional (top panel) and pair (bottom panel) mean-field approaches, for $J_x>0$ and $J_y=J_x/2$. The corresponding states for the unit cell are indicated. While in the conventional MF the $S_z$ parity breaking phase arises below a critical field $B_c^{\alpha }$, in the pair MF it occurs within a field window $B_{c1}^{\alpha }<B<B_{c2}^{\alpha }$ if $\alpha <{\alpha }_c$ [Eq. (33)]. For $B<B_{c1}^{\alpha }$, a dimerized state with maximally entangled pairs is preferred. The dashed lines denote the factorizing field $B_s^{\alpha }$ where both MF approaches coincide and are exact.

Figure 3

Exact and approximate results for the GS entanglement entropy of a strongly coupled spin pair (top), a single spin (center), and a weakly coupled neighboring pair (bottom), with the rest of the chain, for $\alpha =0.1$ and $J_y/J_x=1/2$, as a function of the (scaled) magnetic field. MF denotes the conventional single-spin MF treatment (17, 18, 19, 20, 21), while GMF the pair MF approach (23)–(26), both with symmetry restoration [Eq. (13)], and GMF+P the perturbed pair MF approach (9)–(11).

Figure 4

The concurrence of a strongly (top) and weakly (center) coupled pair of neighboring spins, as a function of the scaled magnetic field for the chain of Fig. 5, according to exact and approximate results. The bottom panel depicts the concurrence of a strongly coupled pair as a function of the weak coupling parameter $\alpha$, at fixed field $B=0.3J_x$. The standard MF result vanishes in all panels.

Figure 5

Top: The difference $\Delta E_0/n=(E_0^{\mathrm{app}}-E_0^{\mathrm{ex}})/n$ between the approximate and exact GS energies per pair, according to conventional and pair MF approaches, for the chain of Fig. 4. The inset depicts for reference the corresponding intensive GS energies. (Bottom) The first excitation energies of a small chain with 8 spins with the same parameters, according to pair MF and exact results. The inset depicts a blow up of the exact first excitation energy in the parity breaking region.

Figure 6

The dimerized systems corresponding to Hamiltonians (41) (left) and (43) (right).

Figure 7

Results for the spin ladder and lattice of Fig. 6 [Eqs. (41) and (43)]. The entanglement of strongly coupled pairs with the rest of the system $S\left({\rho }_{12}\right)$ (top), and their concurrence $C\left({\rho }_{12}\right)$ (center), are plotted for increasing fields for a common value $\alpha =0.2$ [Eqs. (42, 43, 44)]. Results for both systems are very close and almost coincident with those for the cyclic chain of Fig. 1, also depicted, in agreement with the common pair MF prediction (GMF). The bottom panel depicts the concurrence for increasing values of the total coupling parameter $\alpha$, for two fixed values of the field.

Figure 8

The angle $\theta$ of the pair MF approach for the $XYZ$ Hamiltonian (45), as a function of the transverse field for different values of $J_z/J_x$. The dimerized phase corresponds to $\theta =\pi$, the partially aligned phase to $\theta =0$ and the parity breaking phase to $0<\theta <\pi$. We have set $J_y/J_x=1/2$ and $\alpha =0.1$. A positive (negative) $J_z$ in (45) unfavors (favors) the dimerized phase, which will exist for $J_z<J_y-2\alpha J_x$ [Eq. (50)].

Figure 9

GS results for the $XYZ$ chain of Eq. (45). The entanglement entropy $S\left({\rho }_{12}\right)$ (top) of strongly coupled pairs with the rest of the chain and their concurrence $C\left({\rho }_{12}\right)$ (bottom) are plotted for increasing fields at $\alpha =0.1$ for $J_z=\pm 0.2J_x$. Exact results (solid lines) are again in agreement with those of the pair MF (GMF, dashed lines), which predicts a peak of $S\left({\rho }_{12}\right)$ in a displaced (with respect to that for $J_z=0)$ parity breaking sector, and a lower (higher) critical field for the dimerized phase if $J_z>0(J_z<0)$. The concurrence vanishes at the factorizing field (46).