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

## TL;DR

This paper explores the quantum phase diagram of the long-range transverse-field Ising model on triangular-lattice cylinders, revealing various ordered phases, phase transitions, and potential gapless states influenced by long-range interactions.

## Contribution

It provides a detailed analysis of the ground-state phases and phase transitions in the long-range Ising model on quasi-1D triangular lattices, including the emergence of stripe, clock, and gapless phases.

## Key findings

- Stripe-ordered phases emerge from degenerate classical ground states.
- Cylinders with certain circumferences show phase transitions between stripe structures.
- Potential intermediate gapless phase with algebraic correlations and Kosterlitz-Thouless transitions.

## Abstract

To gain a better understanding of the interplay between frustrated long-range interactions and zero-temperature quantum fluctuations, we investigate the ground-state phase diagram of the transverse-field Ising model with algebraically-decaying long-range Ising interactions on quasi one-dimensional infinite-cylinder triangular lattices. Technically, we apply various approaches including low- and high-field series expansions. For the classical long-range Ising model, we investigate cylinders with an arbitrary even circumference. We show the occurrence of gapped stripe-ordered phases emerging out of the infinitely-degenerate nearest-neighbor Ising ground-state space on the two-dimensional triangular lattice. Further, while cylinders with circumferences $6$, $10$, $14$ et cetera are always in the same stripe phase for any decay exponent of the long-range Ising interaction, the family of cylinders with circumferences $4$, $8$, $12$ et cetera displays a phase transition between two different types of stripe structures. For the full long-range transverse-field Ising model, we concentrate on cylinders with circumference four and six. The ground-state phase diagram consists of several quantum phases in both cases including an $x$-polarized phase, stripe-ordered phases, and clock-ordered phases which emerge from an order-by-disorder scenario already present in the nearest-neighbor model. In addition, the generic presence of a potential intermediate gapless phase with algebraic correlations and associated Kosterlitz-Thouless transitions is discussed for both cylinders.