Pattern Description of Quantum Phase Transitions in the Transverse Antiferromagnetic Ising Model with a Longitudinal Field

TL;DR

This work develops a pattern-based framework to map the quantum phase diagram of the 1D transverse antiferromagnetic Ising model under a uniform longitudinal field. By diagonalizing the Hamiltonian in operator space, the authors identify three key patterns, $\\lambda_1$, $\\lambda_9$, and $\\lambda_{16}$, that encode ferromagnetic-like, antiferromagnetic-like, and large-$J$ antiferromagnetic behavior, respectively; the occupancy of these patterns tracks the ground-state transitions as $J$ and $h$ vary. They demonstrate two QPTs/crossovers: an initial transition driven by competition among patterns and a broad crossover to a fully ordered AF phase at large $J$ (roughly $J \gtrsim h/2$), with the latter not involving symmetry breaking. The pattern picture remains robust across system sizes and offers experimentally accessible predictions for quantum simulators such as optical lattices, trapped ions, and Rydberg-atom arrays.

Abstract

Despite of simplicity of the transverse antiferromagnetic Ising model with a uniform longitudinal field, its phases and involved quntum phase transitions (QPTs) are nontrivial in comparison to its ferromagnetic counterpart. For example, what is the nature of the mixed-order in such a model and does there exist a disorder phase? Here we use a pattern picture to explore the competitions between the antiferromagnetic Ising interaction, the transverse and longitudinal fields and uncover what kind of pattern takes responsibility of these three competing energy scales, thus determine the possible phases and their QPTs or crossovers. Our results not only unveil rich physics of this paradigmatic model, but also further stimulate quantum simulation by using current available experimental platforms.

Figures (5)

• Figure 1: The ground state and first eight excited states energies of the transverse antiferromagnetic Ising model in the absence (a) and presence (b, c, d) of longitudinal fields. The circles for the ground and first excited states are obtained by numerical ED, confirming the validity of the present pattern picture.
• Figure 2: The patterns and their relative phases obtained by the first diagonalization, marked by the single-body operators $\hat{A}_n = \sum_{i=1}^L \left[u_{n,3i-2} \hat{\sigma}^x_i+u_{n,3i-1} (i\hat{\sigma}^y_i) + u_{n,3i}\hat{\sigma}^z_i\right]$ with $(\pm,\pm,\pm)$ denoting the signs of $(u_{n,3i-2},u_{n,3i-1},u_{n,3i})$ for the 1D tranverse antiferromagnetic Ising model with $L=8$ under PBC. All patterns are divided into three groups marked by dashed red, dashed green, and dashed blue frames, respectively. For each pattern, a phase factor $e^{i\pi}$ is free, not affecting the relative signs within and between patterns. Marked patterns' energy is zero, dividing all patterns into positive and negative ones.
• Figure 3: (a) The eigenvalues and their eigenfunctions [(b1)-(b8)] of patterns as functions of $J$ and $h = 5.0$ is fixed. It is noted that $\lambda_n = - \lambda_{3L-n+1}$ and $u_{n, m} = - u_{3L-n+1, m}$, and $m$ denotes the spin components. Thus the eigenfunctions from $u_{14,m}$ to $u_{24,m}$ are not shown.
• Figure 4: (a1)-(d1) The ground state energies (thick solid lines) and their pattern components (thin colored solid lines) as functions of $J$ for $h = 0.0, 1.0, 2.5$ and $5.0$, respectively. The right-upper insets in corresponding plots give an enlarged view of the pattern components. (a2)-(d2) The second derivatives of the energy components of patterns.
• Figure 5: Comparison of patterns' occupancy histograms of the ground state for different longitudinal fields $h = 0.0$[(a1)-(a10)], $1.0$[(b1)-(b10)], $2.5$[(c1)-(c10)], and $5.0$[(d1)-(d10)] at different antiferromagnetic Ising interactions $J$'s.