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 $\ensuremath{\alpha}$ of the interactions. The phase boundary for the ferromagnet is obtained as a function of $\ensuremath{\alpha}$. For antiferromagnetic interactions, there is strong indication that the transverse field stabilizes a clock ordered phase with sublattice magnetization $(M,\ensuremath{-}\frac{M}{2},\ensuremath{-}\frac{M}{2})$ 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.
Working within the Stochastic Series Expansion (SSE) framework, we construct efficient quantum cluster algorithms for transverse field Ising antiferromagnets on the pyrochlore lattice and the planar pyrochlore lattice, for the fully frustrated square lattice Ising model in a transverse field (dual to the 2+1 dimensional odd Ising gauge theory), and for a transverse field Ising model with multi-spin interactions on the square lattice, which is dual to a 2+1 dimensional even Ising gauge theory (and reduces to the two dimensional quantum loop model in a certain limit). Our cluster algorithms use a microcanonical update procedure that generalizes and exploits the notion of "pre-marked motifs" introduced earlier in the context of a quantum cluster algorithm for triangular lattice transverse field Ising antiferromagnets. We demonstrate that the resulting algorithms are significantly more efficient than the standard link percolation based quantum cluster approach. We also introduce a new canonical update scheme that leads to a further improvement in measurement of some observables arising from its ability to make one-dimensional clusters in the "imaginary time" direction. Finally, we demonstrate that refinements in the choice of premarking strategies can lead to additional improvements in the efficiency of the microcanonical updates. As a first example of the physics that can be studied using these algorithmic developments, we obtain evidence for a power-law ordered intermediate-temperature phase associated with the two-step melting of long-range order in the fully frustrated square lattice transverse field Ising model.
The antiferromagnetic Ising model on the pyrochlore lattice exhibits a quantum phase transition in an applied transverse field from the low-field quantum spin-ice phase to the high-field polarized regime. Recent field-theoretical analysis and series expansion results indicate this to be a discontinuous, first-order transition. Here, we explore this transition using quantum Monte Carlo simulations in order to assess this scenario and study the thermodynamic properties in the vicinity of the quantum phase transition. For this purpose, we also consider several variants of extended cluster-update schemes for the transverse-field Ising antiferromagnet on frustrated lattices and compare their performance to the conventional bond-based algorithm for the transverse-field Ising model on the pyrochlore lattice.
The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series expansion quantum Monte Carlo, on both the one-dimensional linear chain and the two-dimensional square lattice. We extract the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$ as a function of the decay exponent of the long-range interactions. For ferromagnetic Ising interactions, we resolve the limiting regimes known from field theory, ranging from the nearest-neighbor Ising to the long-range Gaussian universality classes, as well as the intermediate regime with continuously varying critical exponents. In the long-range Gaussian regime, we treat the effect of dangerous irrelevant variables on finite-size scaling forms. For antiferromagnetic and therefore competing Ising interactions, the stochastic series expansion algorithm displays growing autocorrelation times leading to a reduced performance. Nevertheless, our results are consistent with the nearest-neighbor Ising universality for all investigated interaction ranges both on the linear chain and the square lattice.
A transverse magnetic field $\mathrm{\ensuremath{\Gamma}}$ is known to induce antiferromagnetic three-sublattice order of the Ising spins ${\ensuremath{\sigma}}^{z}$ in the triangular lattice Ising antiferromagnet at low enough temperature. This low-temperature order is known to melt on heating in a two-step manner, with a power-law ordered intermediate temperature phase characterized by power-law correlations at the three-sublattice wave vector
Q: $\ensuremath{\langle}{\ensuremath{\sigma}}^{z}(\stackrel{P\vec}{R}){\ensuremath{\sigma}}^{z}(0)\ensuremath{\rangle}\ensuremath{\sim}cos(\mathbf{Q}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{R})/{|\stackrel{P\vec}{R}|}^{\ensuremath{\eta}(T)}$ with the temperature-dependent power-law exponent $\ensuremath{\eta}(T)\ensuremath{\in}(1/9,1/4)$. Here, we use a quantum cluster algorithm to study the ferromagnetic easy-axis susceptibility ${\ensuremath{\chi}}_{u}(L)$ of an $L\ifmmode\times\else\texttimes\fi{}L$ sample in this power-law ordered phase. Our numerical results are consistent with a recent prediction of a singular
L dependence ${\ensuremath{\chi}}_{u}(L)\ensuremath{\sim}{L}^{2\ensuremath{-}9\ensuremath{\eta}}$ when $\ensuremath{\eta}(T)$ is in the range
(1/9,2/9). This finite-size result implies, via standard scaling arguments, that the ferromagnetic susceptibility ${\ensuremath{\chi}}_{u}(B)$ to a uniform field
B along the easy axis is singular at intermediate temperatures in the small
B limit, ${\ensuremath{\chi}}_{u}(B)\ensuremath{\sim}{|B|}^{\ensuremath{-}\frac{4\ensuremath{-}18\ensuremath{\eta}}{4\ensuremath{-}9\ensuremath{\eta}}}$ for $\ensuremath{\eta}(T)\ensuremath{\in}(1/9,2/9)$, although there is no ferromagnetic long-range order in the low temperature state. Additionally we establish similar two-step melting behavior (via a study of the order parameter susceptibility ${\ensuremath{\chi}}_{\mathbf{Q}}$) in the case of the ferrimagnetic three-sublattice ordered phase which is stabilized by ferromagnetic next-neighbor couplings (
J2) and confirm that the ferromagnetic susceptibility obeys the predicted singular form in the associated power-law ordered phase.
We report on a systematic study of two-dimensional, periodic, frustrated Ising models with quantum dynamics introduced via a transverse magnetic field. The systems studied are the triangular and kagom\'e lattice antiferromagnets, fully frustrated models on the square and hexagonal (honeycomb) lattices, a planar analog of the pyrochlore antiferromagnet, a pentagonal lattice antiferromagnet, as well as two quasi-one-dimensional lattices that have considerable pedagogical value. All of these exhibit a macroscopic degeneracy at
T=0 in the absence of the transverse field, which enters as a singular perturbation. We analyze these systems with a combination of a variational method at weak fields, a perturbative Landau-Ginzburg-Wilson approach from large fields, as well as quantum Monte Carlo simulations utilizing a cluster algorithm. Our results include instances of quantum order arising from classical criticality (triangular lattice) or classical disorder (pentagonal and probably hexagonal) as well as notable instances of quantum disorder arising from classical disorder (kagom\'e). We also discuss the effect of finite temperature, as well as the interplay between longitudinal and transverse fields---in the kagom\'e problem the latter gives rise to a nontrivial phase diagram with bond-ordered and bond-critical phases in addition to the disordered phase. We also note connections to quantum-dimer models and thereby to the physics of Heisenberg antiferromagnets in short-ranged resonating valence-bond phases that have been invoked in discussions of high-temperature superconductivity.