Reentrance of disorder in the anisotropic shuriken Ising model

Frustration is often a key ingredient for reentrance mechanisms. Here we study the frustrated anisotropic shuriken Ising model, where it is possible to extend the notion of reentrance between disordered phases, i.e., in absence of phase transitions. By tuning the anisotropy of the lattice, we open a window in the phase diagram where magnetic disorder prevails down to zero temperature, in a classical analogy with a quantum critical point. In this region, the competition between multiple disordered ground states gives rise to a double crossover where both the low- and high-temperature regimes are less correlated than the intervening classical spin liquid. This reentrance of disorder is characterized by an entropy plateau and a multistep Curie law crossover. Our theory is developed based on Monte Carlo simulations, analytical Husimi-tree calculations and an exact decoration-iteration transformation. Its relevance to experiments, in particular, artificial lattices, is discussed.

Figure 1

The shuriken lattice with six sites per unit cell and two sublattices $A$ and $B$. Interactions between $A$ sites (red square plaquettes) are described with coupling constant $J_{AA}$, while interactions between $A$ and $B$ sites (black octagonal plaquettes) are described with $J_{AB}$. By convention, we chose the triangles to be equilateral.

Figure 2

Phase diagram of the Ising model on the anisotropic shuriken lattice. (a) The circles (triangles) correspond to phase transitions (crossovers), obtained by Monte Carlo simulations (Husimi-tree calculations) [see Appendix pp3 for further details]. As a function of the coupling ratio $x=\frac{J_{AB}}{J_{AA}}$, the model supports a long-range ordered ferromagnet (FM) [see (b)], a long-range ordered ferrimagnet (FiM) [see (c)], a binary paramagnet (BPM) [see (d)], and two classical spin liquids $\left({\mathrm{SL}}_{1,2}\right)$. The BPM illustrated in (d) is made of antiferromagnetically ordered square plaquettes, decoupled from each other and from the intermediate spins sitting on the $B$ sublattice. For $\left|x\right|\gtrsim 1$, on cooling, the system undergoes an evolution from “gas $\stackrel{\mathrm{crossover}}{\to }$ liquid $\stackrel{\mathrm{transition}}{\to }$ solid.” As for $\left|x\right|\lesssim 1$, it provides a remarkable example of reentrance from “gas $\stackrel{\mathrm{crossover}}{\to }$ liquid $\stackrel{\mathrm{crossover}}{\to }$ gas.” The spin liquids $\left(x=\pm 1\right)$ can also be seen as classical analogues of quantum critical points: they sit at the zero-temperature frontiers between extended regions of order and disorder, resulting in a persistence of the spin-liquid physics at finite temperature (the blue regions). However, please note that the spin-spin correlations are not “critical” in the sense that they do not decay algebraically.

Figure 3

Multiple crossovers between the paramagnetic, spin-liquid, and binary regimes as observed in the energy $E$, specific heat $C_h$, entropy $S$ and reduced magnetic susceptibility $\chi T$. The models correspond to (a) $x=\pm 1$, (b) $\pm 0.9$, and (c) 0. There is no phase transition for this set of parameters, which is why the Husimi tree calculations (lines) perfectly match the Monte Carlo simulations (circles) for all temperatures. The double crossover is present for $x=\pm 0.9$, with the low-temperature regime being the same as for $x=0$, as confirmed by its entropy and susceptibility. The entropy is obtained by integration of $C_h/T$, setting $S(T\to +\infty )=ln2$. The vertical dashed lines represent estimates of the crossover temperatures determined by the local specific-heat maxima in Husimi-tree calculations. The temperature axis is on a logarithmic scale. All quantities are given per number of spins and the Boltzmann constant $k_B$ is set to 1. Details on the Husimi-tree calculations and Monte Carlo simulations are given in the appendices.

Figure 4

Reduced susceptibility $\chi T$ with coupling ratios of $x=\pm 1,\pm 0.99,\pm 0.9$ and 0, obtained from Husimi-tree calculations (solid lines) and Monte Carlo simulations (circles). The Curie-law crossover of classical spin liquids is standard, i.e., $\chi T$ is monotonic, for $x=\pm 1$ and 0, and takes a multistep behavior for intermediate values of $x$, due to the double crossover. The characteristic values of the entropy and reduced susceptibility are given in Table 1. The temperature axis is on a logarithmic scale. Details on the Husimi-tree calculations and Monte Carlo simulations are given in the appendices.

Figure 5

Behavior of the coupling constants of the effective checkerboard model as a function of temperature [Eq. (12)] for $x=-0.9$ (top) and $-1$ (bottom). (Top) All couplings vanish at both high and low temperatures with an intermediate regime at $T\sim 1$ where the effective interactions are stronger. The intermediate regime corresponds to the spin liquid region of the phase diagram Fig. 2, with the high- and low-temperature regimes corresponding to the paramagnet and binary paramagnet respectively. (Bottom) For all couplings ${\mathcal{J}}_i,\beta {\mathcal{J}}_i$ vanishes at high temperature and tends to a finite constant of magnitude $|\beta {\mathcal{J}}_i\left(T\right)|\ll 1$ at low temperature. The short-range correlated, spin-liquid regime, thus extends all the way down to $T=0$.

Figure 6

Correlation lengths in the effective checkerboard model, calculated from Eq. (15), for $x=-0.9$ and $-1$. The correlation length is calculated to leading order in a perturbative expansion of the effective model in powers of $\beta {\mathcal{J}}_i$. Such an expansion is reasonable for $\left|x\right|\le 1$ since $\beta {\mathcal{J}}_i\ll 1$ for all $T$ (see Fig. 5). For $x=-0.9$, the behavior of the correlation length is nonmonotonic. The correlation length is maximal in the spin liquid regime but correlations remain short ranged at all temperatures. In the binary paramagnet regime, the correlation length vanishes linearly at low temperature. For $x=-1$, the correlation length enters a plateau at $T\sim 1$, and short range correlations remain down to $T=0$.

Figure 7

Spin-spin correlations in the vicinity of the spin liquid phases for $x=-1.05$ [(a) and (b)], $-1$ [(c) and (d)], and $-0.9$ [(e) and (f)], obtained from Monte Carlo simulations. The temperatures considered are $T=0.01$ $\left(♦\right)$, $1\left(\blacksquare \right)$, and 891.$25\left(•\right)$. Because of the anisotropy of the lattice, we want to separate the correlation functions that start on $A$ sites [(a), (c), and (e)] and $B$ sites [(b), (d), and (f)]. The radial distance is given in units of the unit-cell length. The agglomeration of data points around $C\sim 2\times {10}^{-5}$ is due to finite size effects. The blue data point at $\rho \approx 0.4$ and $C\approx 0.001$ is due to the fact that the paramagnetic simulations were performed at high but not infinite temperatures. The $y$ axis is on a logarithmic scale.

Figure 8

Static structure factors of the anisotropic shuriken lattice for (a) $x=-1$, (b) 0, and (c) 1 at zero temperature, obtained from Monte Carlo simulations. For $x=\pm 1$, the scattering is strongly inhomogeneous (as opposed to a standard paramagnet) and nondivergent (i.e., without long-range order), confirming the spin liquid nature of these phases. The similarity between the structure factors for $x=+1$ and $-1$ comes from the symmetry between models with positive and negative $x$ [Eq. (3)]. For (b) $x=0$, the black background underlines the absence of correlations in the binary paramagnet beyond the size of the superspins (square plaquettes). The finite size of the superspins is responsible for the finite extension of the dots of scattering. In order to restore ergodicity, a local update flipping the four spins of square plaquettes was used in the simulations. A video showing the temperature dependence of the static structure factor for $x=0.9$ is available in Ref. [55].

Figure 9

Distribution and intensity of Bragg peaks in reciprocal space $\left(q_x,q_y\right)$, as would be measured in an x-ray diffraction experiment on the shuriken lattice. Its analytical formula is given in Eq. (A5), where we set $a=1$. The first Brillouin zone corresponds to $-\pi <q_x,q_y<+\pi$. While the location of these peaks is periodic, their intensities are not, because of the irrational relative positions of sites within the unit cell [Eq. (A5)]. The apparent sixfold symmetry is deceptive; it is only fourfold, as expected for the shuriken lattice.

Figure 10

Finite size effects on the specific heat for $x=-3$ (top) and $-1.05$ (bottom). (Insets) The transition temperature is scaled as a function of $1/L$, where $L$ is the linear system size. In the thermodynamic limit, we find $T_c=2.788\left(5\right)$ for $x=-3$ and $T_c=0.0714\left(5\right)$ for $x=-1.05$

Figure 11

Finite size scaling of the magnetization $\left|M\right|$ [(a) and (c)] and susceptibility $\chi$ [(b) and (d)] for $x=-3$.

Figure 12

Finite size scaling of the magnetization $\left|M\right|$ [(a) and (c)] and susceptibility $\chi$ [(b) and (d)] for $x=-1.05$.

Figure 13

Absence of finite size effects in the double crossover region for $x=0.9$, as seen in Monte Carlo simulations of the specific heat (left) and reduced susceptibility (right). The system sizes used here are substantially bigger than the correlation length for all temperatures [Figs. 6 and 7], which is why the numerical results sit on top of each others for all system size.

Figure 14

Husimi tree made of shurikenlike building blocks. Here the first three layers of the Husimi tree are shown, with one shuriken in the central layer 0, four shurikens in layer 1, twelve shurikens in layer 2, and so on. Two successive layers are connected by sites on the $B$ sublattice. Closed loops of size 8 and beyond do not exist. The red part is a branch starting on a site of the zeroth layer.

Figure 15

The checkerboard lattice formed by the set of $B$ spins on the shuriken lattice.

Figure 16

A path (in red) between two spins on the checkerboard lattice containing only nearest-neighbor ${\mathcal{J}}_1$ bonds. The correlation function between two such spins in the disordered regime is calculated in Appendix pp5.