Phase diagram of a frustrated Heisenberg model: From disorder to order and back again

We study the effects of bond and site disorder in the classical $J_1\text{−}{J}_2$ Heisenberg model on a square lattice in the order-by-disorder frustrated regime $2J_2>\left|J_1\right|$. Combining symmetry arguments, numerical energy minimization, and large-scale Monte Carlo simulations, we establish that the finite-temperature Ising-type transition of the clean system is destroyed in the presence of any finite concentration of impurities. We explain this finding via a random-field mechanism which generically emerges in systems where disorder locally breaks the same real-space symmetry spontaneously globally broken by the associated order parameter. We also determine that the phase replacing the clean one is a paramagnet polarized in the nematic glass order with nontrivial magnetic response. This is because disorder also induces noncollinear spin-vortex-crystal order and produces a conjugated transverse dipolar random field. As a result of these many competing effects, the associated magnetic susceptibilities are nonmonotonic functions of the temperature. As a further application of our methods, we show the generation of random axes in other frustrated magnets with broken SU(2) symmetry. We also discuss the generality of our findings and their relevance to experiments.

Figure 1

(a) The clean ($\delta J_{\alpha ,ij}=0$) classical ground state of the Hamiltonian (1) in the regime $J_1<2J_2$: two decoupled antiferromagnetic sublattices (red and blue arrows) with the polar $\varphi$ and azimuthal $\phi$ (not shown) angles parametrizing a nontrivial O(3) degeneracy. (b) The low-$T$ corrections to the free energy $\delta F$ as a function of the polar angle $\varphi$, indicating the selection of the states $\varphi =0$ or $\pi$. (c) The collinear (stripe) state $\varphi =0$ [ordering wave vector ${\mathbf{Q}}_{+}=(0,\pi )$ ]. (d) The noncollinear (spin-vortex-crystal) state $\varphi =\frac{\pi }{2}$, $\phi =0$ with staggered “handness” pattern.

Figure 2

(a) Ground-state configuration obtained by the iterative energy minimization method of the spin configurations. The strong links represent missing nearest-neighbor $J_1$ bonds ($\delta J^{\text{(v)}}=\delta J^{\text{(h)}}=-J_1$). (b) The ground-state energy (measured with respect to the clean one $E_S$) (solid line) and the perfect-stripe-state energy [$E_S-(J_1+\delta J^{\text{(v)}})$ ] (dashed line) as a function of the bond magnitude $J_1+\delta J^{\text{(v)}}$ (the magnitude of the other bond is fixed at $\delta J^{\text{(h)}}=-J_1$). Inset: the corresponding nematic $m_{\parallel }$ and SVDW $m_{\perp }$ order-parameter magnitudes. (c) Same as (a) but with far-apart defects. The system size is $L=16$ and periodic boundary conditions are considered. The value $J_2=0.55J_1$.

Figure 3

(a) The spin-rotation field on a plaquette induced by a strong ($\delta J>0$) $J_1$-bond impurity (dotted line). (Dashed arrows represent the unperturbed state.) (b) The spin-rotation field magnitude $\left|\delta \theta \right|$ plotted as a function of distance $r$ from the defect (see text for details) for the cases of a $J_1$-bond vacancy ($\delta J=-J_1$) (black squares), a single vacancy (red triangles), and a pair of nearest-neighbor vacancies (green circles). We have used $J_2=0.55J_1$ and fixed $K=0.05J_1$. We used system sizes up to $L=2^{10}$ with periodic boundary conditions. The dashed lines are power-law fits $\propto r^{-1}$ and $\propto r^{-2}$ in the range $10\gtrsim r\gtrsim {10}^2$. Deviations at large $r$ are due to finite-size effects. We have obtained similar results (not shown) for system size $L=400$, $K=0.1J_1$, and $J_2/J_1=0.55$, 0.6 and 0.8.

Figure 4

The MC results for clean system and $J_2=0.55J_1$. (a) The modulus of the nematic order parameter, (b) the specific heat, (c) the nematic susceptibility, and (d) the Binder cumulant as a function of the temperature for several system sizes. The inset shows a magnification around the crossing points signaling the location of the transition temperature $T_c\approx 0.197\left(1\right)J_1$.

Figure 5

The finite-size scaling replot of (a) the Binder cumulant and (b) the nematic susceptibility (right) data of Fig. 4. The lines are simple quartic best fits representing the corresponding scaling functions.

Figure 6

The MC results for the case of $2\%$ dilution of horizontal $J_1$ bonds where $J_2=0.55J_1$. (b) The specific heat, (a) the nematic order parameter, (c) the corresponding susceptibility, and (d) the Binder cumulant as a function of the temperature for several system sizes.

Figure 7

A typical density plot of the local nematic order parameter $\langle m_{\parallel ,i}\rangle$ [see Eq. (3)] for a lattice of size $L=80$, temperature $T=0.18J_1$, and $J_2=0.55J_1$. The light dots (top) and light ticks (bottom) show the position of the missing sites ($1\%$ on average) and vertical bonds ($0.5\%$ on average), respectively. Notice that the site impurities always appear in pairs of horizontal nearest-neighbor sites locally favoring the nematic ${\mathbf{Q}}_{-}$ state. For these particular disorder realizations, the nematic order parameter averages $\langle m_{\parallel }\rangle =-0.486\left(1\right)$ (site dilution) and $-0.521\left(1\right)$ (bond dilution).

Figure 8

The MC results when $2\%$ of $J_1$ bonds are isotropically diluted and $J_2=0.55J_1$. The (b) specific heat, the (d) nematic Binder cumulant, the (a) nematic and (e) SVDW order parameters, and their corresponding susceptibilities [(c) and (f), respectively], all as a function of the temperature and for several system sizes. [We verified that $m_{\perp }^{y,z}$ (${\chi }_{\perp }^{y,z}$) is statistically identical to $m_{\perp }^x$ (${\chi }_{\perp }^x$).]

Figure 9

The Edwards-Anderson nematic and SVDW order parameters and the corresponding susceptibilities as a function of the temperature for the same parameters as in Fig. 8.

Figure 10

The conventional and the Edwards-Anderson nematic and SVDW order parameters and the corresponding susceptibilities as a function of the temperature for $10\%$ of isotropically dilution of $J_1$ bonds. [We verified that $m_{\perp }^{y,z}$ (${\chi }_{\perp }^{y,z}$) is statistically identical to $m_{\perp }^x$ (${\chi }_{\perp }^x$).]

Figure 11

(Left panels) The nematic order-parameter normalized histogram $P\left(m_{\parallel }\right)$ for different system sizes and temperatures and (right panels) the density of the local nematic order $\langle m_{\parallel ,i}\rangle$ for a typical disorder realization. Here, the vertical (light) and horizontal (dark) ticks represent the missing $J_1$ bonds (2% on average).

Figure 12

Critical temperature $T_c$ vs the dilution concentration of $J_2$ bonds $x$. The dashed line is the prediction given by mean field theory $T_c=0.62\sqrt{(2(1-x)J_2-J_1)J_1}$. Here, $J_2=0.55J_1$ and we have studied systems of sizes up to $L=80$. The MC critical temperature was estimated from the crossings of the Binder cumulant. The solid line is just a guide for the eyes.

Figure 13

The $T=0$ disorder average of the $z$ component of the SVDW order parameter (5) as a function of the system size $L$ for various disorder strength $\mathrm{\Delta }$ (see text). The spin configuration is on the $xy$ plane in the weak disorder limit $\mathrm{\Delta }\le \mathrm{\Delta }\approx 0.2$ and the SVDW order is finite. For stronger disorder $\mathrm{\Delta }>{\mathrm{\Delta }}_c$, the spin configuration is no longer coplanar and $\left[m_{\perp }^z\right]$ vanishes as $L\to \infty$.

Figure 14

The sketch of the six nearest-neighbor sites in a pyrochlore lattice. The local $z$ axes are also shown as red arrows. For clarity, the local $x$ and $y$ axes are not shown.