Antiferromagnetic order without recourse to staggered fields

In the theory of antiferromagnetism, the staggered field—an external magnetic field that alternates in sign on atomic length scales—is used to select the classical Néel state from a quantum magnet but justification is missing. This work examines, within the decoherence framework, whether repeated local measurement can replace a staggered field. Accordingly, the conditions under which local decoherence can be considered a continuous measurement are studied. The dynamics of a small magnetic system is analyzed to illustrate that local decoherence can lead to (symmetry-broken) order similar to order resulting from a staggered field.

Figure 1

Schematic of models A and B. In model A (B), the central system consists of an open (periodic) chain of $N_S=6$ ($N_S=4$) spin-1/2 particles and spin $S_1$ is strongly coupled to an environment $\mathcal{E}$ (environment fragment ${\mathcal{E}}_1$) of spins via Ising coupling. In model B, the spin chain is immersed in fragment ${\mathcal{E}}_2$ that resembles a thermal reservoir (inverse temperature $\beta =50/J_S$). The number of spins in each environment (fragment) and the respective coupling strengths are indicated in the figure.

Figure 2

Pictorial representation of the initial decoherence process in model A. Different parts of the (initial) singlet state evolve to (approximately) orthogonal parts of the environment (denoted by $\mid \Uparrow \rangle$ and $\mid \Downarrow \rangle$) via spin $S_1$. The boxes represent a product state of the central system with the environment. The magnetization of each spin is indicated by the arrows, and the red (blue) color intensity illustrates the net magnetization parallel (antiparallel) to the $z$ axis.

Figure 3

Simulation results of model $\mathrm{A}$, where spin $S_1$ of a $N_S=6$ Heisenberg (open) spin chain is coupled to $N_{\mathcal{E}}=8$ environment spins via Ising coupling with random uniform interaction strength $I=20J_S$. The central system is prepared in the (singlet) ground state at $t=0$. (a) Coherence local to spin $S_1$ [dashed line, Eq. (14)] and the remainder of the subsystem [solid line, Eq. (15)], as a function of time. Coherence is with respect to the computational basis. (b) Entropy of the entire subsystem $S\left(t\right)=S\left[\rho \right(t\left)\right]$ (solid line) and that complementary to spin $S_1$: $S_{\uparrow \uparrow }\left(t\right)=S\left[{\tilde{\rho }}_{\uparrow \uparrow }\left(t\right)\right]$ (dashed line) and $S_{\downarrow \downarrow }\left(t\right)=S\left[{\tilde{\rho }}_{\downarrow \downarrow }\left(t\right)\right]$ (dotted line).

Figure 4

Nearest-neighbor correlations $\langle {\mathit{S}}_i\left(t\right)·{\mathit{S}}_{i+1}\left(t\right)\rangle$ of a $N_S=6$ antiferromagnetic open Heisenberg chain for different sites $i$. Repeated collapse (solid lines) along the $z$ direction of spin $S_1$ (performed every $\mathrm{\Delta }t={10}^{-1}$ units of time) is compared with decoherence model $\mathrm{A}$ (markers). For clarity, the correlations $\langle {\mathit{S}}_i\left(t\right)·{\mathit{S}}_{i+1}\left(t\right)\rangle$ with $i=1,2,3$ ($i=4,5$) are shown in the left (right) panel and the $t=0$ ground-state values, $\langle {\mathit{S}}_i\left(0\right)·{\mathit{S}}_{i+1}\left(0\right)\rangle \equiv {\langle {\mathit{S}}_i·{\mathit{S}}_{i+1}\rangle }_0$, have been subtracted for each $i$.

Figure 5

Simulation data of model $\mathrm{B}$: a $N_S=4$ antiferromagnetic spin ring, whereby spin $S_1$ is connected to ${\mathcal{E}}_1$ containing $N_{{\mathcal{E}}_1}=6$ spins, and the entire central system is in contact with ${\mathcal{E}}_2$, a $N_{{\mathcal{E}}_2}=12$ spin state that resembles a thermal reservoir at $\beta =50/J_S$. ${\mathcal{E}}_1$ (${\mathcal{E}}_2$) is Ising coupled to the central system with random uniform (random binary) strength $I=20J_S$ ($I^{\prime }=0.1J_S$), and without (with) intraenvironment coupling (of strength $K=0.1J_S$). The figures show the static structure factor [see Eq. (22)], whereby the component $a$ is indicated in the panels. (a) Simulation data of the static structure factor evaluated at the antiferromagnetic reciprocal lattice vector (i.e., $\kappa =\pi$) as a function of time. (b) The values of the structure factor in the ground state of the Heisenberg Hamiltonian without [with] an additional staggered field is denoted by $K_0^{aa}\left(\kappa \right)$ [$K_{\mathrm{st}}^{aa}\left(\kappa \right)$] and indicated by square [star] markers. $K_{\infty }^{aa}\left(\kappa \right)$ (filled circles) denotes the simulation data evaluated at $t={10}^3$.

Figure 6

Loss of phase coherence in model $\mathrm{B}$. The maximum off-diagonal component [see Eq. (23)] of the density matrix ${\tilde{\rho }}_{\uparrow \uparrow }$ (${\tilde{\rho }}_{\downarrow \downarrow }$) is evaluated in the basis diagonalizing ${H_{\mathrm{eff}}^{\prime }}^{+}$ (${H_{\mathrm{eff}}^{\prime }}^{-}$).