Phase diagram of the quantum Ising model with long-range interactions on an infinite-cylinder triangular lattice

Abstract

Obtaining quantitative ground-state behavior for geometrically-frustrated quantum magnets with long-range interactions is challenging for numerical methods. Here, we demonstrate that the ground states of these systems on two-dimensional lattices can be efficiently obtained using state-of-the-art translation-invariant variants of matrix product states and density-matrix renormalization-group algorithms. We use these methods to calculate the fully-quantitative ground-state phase diagram of the long-range interacting triangular Ising model with a transverse field on six-leg infinite-length cylinders and scrutinize the properties of the detected phases. We compare these results with those of the corresponding nearest neighbor model. Our results suggest that, for such long-range Hamiltonians, the long-range quantum fluctuations always lead to long-range correlations, where correlators exhibit power-law decays instead of the conventional exponential drops observed for short-range correlated gapped phases. Our results are relevant for comparisons with recent ion-trap quantum simulator experiments that demonstrate highly-controllable long-range spin couplings for several hundred ions.

Quantum criticality of the transverse-field Ising model with long-range interactions on triangular-lattice cylinders

“…This gap-closing we can track up to decay exponents α = 1.5. The calculated phase-transition points for α 2.4 are within error bars in good agreement with the numerical findings by Saadatmand et al 22 . Specifically, for the NNTFIM α → ∞, the pCUT high-field calculation yields a gap closing at a transverse field h = 1.54 (7) J, which has to be compared to h = 1.5(1) J determined numerically by investigating the order parameter for the clock order 22 .…”

“…We find that cylinders with circumference 4, 8, 12, et cetera display two distinct striped phases as a function of the long-range interaction, while cylinders with circumference 6, 10, 14, and so on always realize the same stripe structure for all long-range Ising interactions, which is, however, distinct from the one found numerically in Ref. 22. Next we focus on the cylinder with minimal circumference of arXiv:1907.10693v1 [cond-mat.str-el] 24 Jul 2019 both families, namely of length 4 and 6, and study the ground-state phase diagram of the full LRTFIM.…”

“…In contrast, we (slightly) overestimate e clock 0 for large α due to the finite cluster extension and the approximate treatment of the field-induced quantum fluctuations. It is therefore plausible that for α 2.4 no clock order is present anymore in the phase diagram as suggested by Saadatmand et al 22 and there is a direct phase transition between the x-polarized and the orthogonal-stripe phase. Another scenario is the presence of an intermediate (gapless) phase as we discuss in Sect.…”

“…However, the situation for more slowly decaying Ising interactions is far less understood. This lead Saadatmand et al 22 to investigate the LRTFIM on a quasi-one-dimensional triangular cylinder lattice with circumference six by infinite-size density matrix renormalization group (iDMRG) calculations. Interestingly, apart from a similar type of clock order as well as a trivial polarized phase, a symmetry-broken columnar stripe phase is present in the ground-state phase diagram.…”

Review on novel methods for lattice gauge theories

“…The efficiency of these algorithms makes it possible to optimize MPS with very large bond dimension, and has helped DMRG-like methods becoming one of the most powerful tools for the numerical investigation of condensed matter problems. For instance, MPS have been successfully applied to describe states that violate the area law, including critical systems, long range interactions and even two-dimensional problems [255][256][257][258].…”