Quantum Monte Carlo study of long-range transverse-field Ising models on the triangular lattice

Motivated by recent experiments with a Penning ion trap quantum simulator, we perform numerically exact Stochastic Series Expansion quantum Monte Carlo simulations of long-range transverse-field Ising models on a triangular lattice for different decay powers $\alpha$ of the interactions. The phase boundary for the ferromagnet is obtained as a function of $\alpha$. For antiferromagnetic interactions, there is strong indication that the transverse field stabilizes a clock ordered phase with sublattice magnetization $(M,-\frac{M}{2},-\frac{M}{2})$ with unsaturated $M<1$ in a process known as “order by disorder” similar to the nearest-neighbor antiferromagnet on the triangular lattice. Connecting the known limiting cases of nearest-neighbor and infinite-range interactions, a semiquantitative phase diagram is obtained. Magnetization curves for the ferromagnet for experimentally relevant system sizes and with open boundary conditions are presented.

Figure 1

Mean-field phase boundary (red line) of the ferromagnetic Ising model on the triangular lattice and in a transverse field as given by Eq. (8). The dotted lines are for successively larger hexagonally shaped systems of radius $R=10,100,1000$, and 2000.

Figure 2

Mean-field phase boundary of the antiferromagnetic Ising model on the triangular lattice and in a transverse field as given by Eq. (8).

Figure 3

Comparison between Lanczos exact diagonalization and SSE QMC for a hexagonal system of 19 spins with AFM interactions. Shown are the fluctuations of the modulus of the complex XY order parameter $\langle m^2\rangle$. In the region of interest around $\mathrm{\Gamma }\approx 1$, the results from both algorithms agree nicely. For some values of $\mathrm{\Gamma }$, the mean fluctuation of the magnetization could not be determined due to technical issues. Those points lie around $\langle m^2\rangle \approx 0.17$ and are meaningless. For comparison, the critical value in the AFM mean-field theory for $\alpha =2.5$ is indicated by an arrow at ${\mathrm{\Gamma }}_c^{\text{MF}}(\alpha =2.5)=2.1758$.

Figure 4

Squared magnetization per site for $\alpha =3.0$. For a mean-field transition, $\beta =0.5$ and $\langle m_z^2\rangle$ should vanish linearly in the vicinity of the critical field.

Figure 5

Binder cumulants for $\alpha =3.0$.

Figure 6

Extrapolation of the crossing points of the Binder cumulant for $\alpha =3.0$. $N_R$ is the number of lattice sites for a system with radius $R$; $C_{N_R}=C(U_{N_R},U_{N_{R-1}})$ denotes the value of the Binder cumulant $U_{N_R}$ for system size $N_R$ at the crossing point with the Binder cumulant for the successively smaller system size $N_{R-1}$. The range of system sizes is $R=3,4,...,13$.

Figure 7

Data collapse for $\alpha =3.0$ with mean-field critical exponents $\beta =1/2$ and $\nu =1$. The linear extent of the system is measured in terms of the radius $R$ of the simulation cell. ${\mathrm{\Gamma }}_r=(\mathrm{\Gamma }-{\mathrm{\Gamma }}_c)/{\mathrm{\Gamma }}_c$ denotes the reduced field. The best data collapse is achieved for ${\mathrm{\Gamma }}_c\approx (9.6\pm 0.1)J$.

Figure 8

Squared magnetization per site for $\alpha =2.5,2.0$, and 1.5. The error bars are smaller than the symbol size.

Figure 9

Structure factor for $R=8,N=241$ spins, and $\alpha =3.0$ for different values of the transverse field $\mathrm{\Gamma }$ below $(\mathrm{\Gamma }=0.8J)$ and above $(\mathrm{\Gamma }=1.2J,1.4J)$ the phase transition, i.e., in the clock ordered and the fully $x\text{-polarized}$ phase, respectively.

Figure 10

Structure factor $S\left(\mathbf{Q}\right)$ at ${\mathbf{Q}}_{+}=(4\pi /3,0)$ for $\alpha =3$.

Figure 11

Squared sublattice magnetizations for decay exponent $\alpha =3.0$ for $R=3, N=37$ spins (upper panel); $R=4, N=61$ spins (middle panel); and $R=7, N=183$ spins (lower panel).

Figure 12

Rescaled structure factor $S\left(\mathbf{Q}\right)/N$ at ${\mathbf{Q}}_{+}=(4\pi /3,0)$ around the critical field ${\mathrm{\Gamma }}_c=1.05$ for $\alpha =3$.

Figure 13

Bootstrap analysis for decay exponent $\alpha =3$. The main panel shows the structure factor $S\left(\mathbf{Q}\right)/N$ at ${\mathbf{Q}}_{+}=(4\pi /3,0)$, extrapolated to the thermodynamic limit. The extrapolation vs $1/N$ with a third-order polynomial is shown in the inset for different transverse fields $\mathrm{\Gamma }$.

Figure 14

Binder cumulants $U_{N_R}$ for different system sizes; $\alpha =3.0$. (Inset) Extrapolation of their crossing points $C_{N_R}=C(U_{N_R},U_{N_{R-1}})$, which are indicated with horizontal error bars in the main graph, vs inverse system size.

Figure 15

Semiquantitative phase diagram for the long-range transverse-field Ising AFM on the triangular lattice. FP: fully $x\text{-polarized}$ phase, CO: clock-ordered phase, and class: region dominated by classical ground states.

Figure 16

Modulus $m$ of the complex $XY$ order parameter $m{e}^{i\theta }\equiv (m_A+m_B{e}^{i4\pi /3}+m_C{e}^{-i4\pi /3})/\mathcal{C}$ with $\mathcal{C}=3/2$ computed with Lanczos exact diagonalization for a hexagon of 19 spins.

Figure 17

Simulation cell with open boundary conditions with radius $R=8$.

Figure 18

Histograms for the average energy per spin and the modulus squared ${\left|m\right|}^2$ of the complex XY order parameter for $(\alpha =3,\mathrm{\Gamma }=0.7)$ and different radii $R$ of the simulation cell. (a) $\mathrm{R}={3,\left|m\right|}^2$, (b) $\mathrm{R}=3$, energy, (c) $\mathrm{R}={4,\left|m\right|}^2$, (d) $\mathrm{R}=4$, energy, (e) $\mathrm{R}={5,\left|m\right|}^2$, (f) $\mathrm{R}=5$, energy, (g) $\mathrm{R}={6,\left|m\right|}^2$, (h) $\mathrm{R}=6$, energy, (i) $\mathrm{R}={7,\left|m\right|}^2$, and (j) $\mathrm{R}=7$, energy.

Figure 19

Data collapse for $\alpha =3$ with mean-field exponents and ${\mathrm{\Gamma }}_c=1.15$ (upper panel) and data collapse with 3D XY exponents and ${\mathrm{\Gamma }}_c=1.05$ (lower panel). ${\mathrm{\Gamma }}_r\equiv (\mathrm{\Gamma }-{\mathrm{\Gamma }}_c)/{\mathrm{\Gamma }}_c$ is the reduced field.