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

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, etc., 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, etc., 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.

Figure 1

Illustration of the YC(6) lattice. Upper panel: $\mathrm{YC}\left(6\right)$ embedded in 3D to illustrate the periodic boundary conditions. Lower panel: illustration as a cutout of the 2D triangular lattice with periodic boundary conditions denoted with dashed lines. The vectors ${\mathit{e}}_1$ and ${\mathit{e}}_2$ are the unit vectors which span the whole triangular cylinder.

Figure 2

Illustration of the maximally flippable state of $\mathrm{YC}\left(4\right)$ in the quantum-dimer model. Flippable plaquettes are shown as red and green hexagons. The original triangular cylinder is shown in the background together with a spin configuration resulting in the displayed dimer configuration. The periodic boundary is reflected in lighter gray.

Figure 3

Illustration of the maximally flippable state of $\mathrm{YC}\left(6\right)$ in the quantum-dimer model. Flippable plaquettes are shown as red and green hexagons. The original triangular cylinder is shown in the background together with a spin configuration resulting in the displayed dimer configuration. The periodic boundary is reflected in lighter gray.

Figure 4

Bare series of the ground-state energy per site $e_0^{\mathrm{stripe}}$ for orthogonal stripes as a function of $h/J$ for $\alpha =6$ for the $\mathrm{YC}\left(4\right)$ cylinder. The bare series is converged up to $h/J\approx 0.6$. The behavior of other stripe configurations as well as other $\alpha$ values is similar. The calculated phase transition into the clock order, visualized by the black vertical line, takes place at approximately $h/J=0.17$.

Figure 5

Illustration of orthogonal-, zigzag-, and plain-stripe states in real space on a $\mathrm{YC}\left(4\right)$ lattice. Red and blue circles denote spins pointing in opposite directions. The infinite extension of the cylinder is in horizontal direction.

Figure 6

Energy per site $e_0^{\mathrm{stripe}}$ for the considered stripe patterns evaluated on the $\mathrm{YC}\left(4\right)$ lattice for $N=4\times 1000$ (upper panel) and on the $\mathrm{YC}\left(6\right)$ lattice for $N=6\times 1000$ (lower panel) with periodic boundary conditions. The dotted vertical line in the upper panel indicates the first-order phase transition ${\alpha }_{\mathrm{c}}$ between zigzag ($\alpha <{\alpha }_{\mathrm{c}}$) and orthogonal ($\alpha >{\alpha }_{\mathrm{c}}$) stripes for the $\mathrm{YC}\left(4\right)$ cylinder. For the $\mathrm{YC}\left(6\right)$, orthogonal stripes are realized for all $\alpha$.

Figure 7

Ground-state phase diagram calculated for the $\mathrm{YC}\left(6\right)$ cylinder using series expansions. The red line is determined by the closing of the one-QP gap ${\mathrm{\Delta }}_{1\mathrm{QP}}$ at momentum ${(2\pi /3,-2\pi /3)}^T$ with the standard deviation of the Padé approximants up to 10th order. The blue line is determined by the energy intersection between the clock-order energy $e_0^{\mathrm{clock}}$ and the stripe energies in sixth order. The question mark indicates the region of the phase diagram where the used methods break down.

Figure 8

Exemplary depiction of the phase transition points for $\alpha =6$ between the $x$-polarized phase and the clock-ordered phase for the $\mathrm{YC}\left(6\right)$ cylinder. The figure displays the one-QP gap ${\mathrm{\Delta }}_{1\mathrm{QP}}$ at momentum ${(2\pi /3,-2\pi /3)}^T$ as a function of $\lambda =h/2J$ using the order-9 and -10 Padé extrapolants with no poles before the gap closing. Depicted Padé extrapolants are (4,5), (5,4), (5,5), (6,4), and (4,6). The critical point is extracted as ${\lambda }_{\mathrm{c}}=0.360\left(15\right)$.

Figure 9

Exemplary depiction of the phase-transition points for $\alpha =6$ between the orthogonal-stripe phase and the clock-ordered phase for the $\mathrm{YC}\left(6\right)$ cylinder. The figure shows the ground-state energies of the stripe and clock-order phases as a function of $h/J$. Red crosses denote calculated points in the parameter space of $h$ and $\alpha =6$ for the clock-order energy, the blue line denotes the orthogonal-stripe energy, the yellow line denotes the plain-stripe energy, and the violet line represents the zigzag-stripe energy. The crossing between the red symbols and the blue line indicates the first-order phase transition.

Figure 10

Ground-state phase diagram calculated for the $\mathrm{YC}\left(4\right)$ cylinder using series expansions. The red line is determined by the closing of the one-QP gap with the standard deviation of the Padé approximants up to order 10. The blue line is determined by the energy intersection between the clock-order energy $e_0^{\mathrm{clock}}$ and the orthogonal ($\perp$) stripe energy in sixth order. The violet line denotes the intersection between the energy of the quantum dimer model in third order and the zigzag stripe energy in sixth order. The almost vertical black line represents the first-order phase transition line between the zigzag and orthogonal stripes, which is determined by comparing the ground-state energies of both stripe phases. The question mark indicates the region of the phase diagram where the used methods break down.