First-Order Excited-State Quantum Phase Transition in the Transverse Ising Model with a Longitudinal Field

TL;DR

This work investigates first-order excited-state quantum phase transitions in the 1D transverse Ising model with a longitudinal field, contrasting them with the well-known second-order ground-state QPT in the absence of the field. Using a two-step pattern-diagonalization approach, the authors decompose the Hamiltonian into pattern contributions and track how competing patterns reorganize as the longitudinal field grows, revealing a dramatic first-order QPT in the first excited state around $J \approx h$. The analysis shows that two or more patterns become metastable and compete, in contrast to the single dominant pattern seen in the second-order case, and reports clear signatures in pattern occupancies and energy components. The results provide a practical diagnostic framework for excited-state QPTs in non-integrable systems and point to experimental platforms in trapped ions, superconducting circuits, and optical lattices for validation.

Abstract

The investigation of the first-order quantum phase transition (QPT) is far from clarity in comparison to that of the second-order or continuous QPT, in which the order parameter and associated broken symmetry can be clearly identified and at the same time the concepts of universality class and critical scaling can be characterized by critical exponents. Here we present a compared study of these two kinds of QPT in the transverse Ising model. In the absence of a longitudinal field, the ground state of the model exhibits a second-order QPT from paramagnetic phase to ferromagnetic one, which is smeared out once the longitudinal field is applied. Surprisingly, the first excited state involves a first-order QPT as the longitudinal field increases, which has not been reported in literature. Within the framework of a pattern picture we clearly identify the difference between these two kinds of QPT: for the continuous QPT only the pattern flavoring ferromagnetic phase is always dominant over the others, and on the contrary, there exist at least two competitive patterns in the first-order QPT, which is further indicated by patterns' occupancies calculated by pattern projections on the ground and first excited states wavefunctions. Our result has not only a fundamental significance in the understandings of the nature of QPTs, but also a practical interest in quantum simulations used to test the present finding.

Figures (5)

• Figure 1: The ground state and first five excited states energies of the 1D transverse Ising model in the absence (a) and presence (b, c, d) of longitudinal fields. The insets in (b), (c) and (d) show enlarged views of the crossovers of excited states.
• 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 Ising model with $L=8$ under PBC. All patterns are divided into three groups marked by the dashed red, green, and blue frames, respectively. For each pattern, a phase factor $e^{i\pi}$ is free, not affecting the relative signs within and between patterns.
• Figure 3: (a) The eigenvalues of patterns and [(b1)-(b8)] their eigenfunctions as functions of $J$ and $h = 1.0$ is taken. The patterns satisfy with $\lambda_n = - \lambda_{3L-n+1}$ and $u_{n, m} = - u_{3L-n+1, m}$, where $m$ denotes the spin components. Thus the eigenfunctions from $u_{14,m}$ to $u_{24,m}$ are not shown.
• Figure 4: (a1) $\&$ (b1) The ground state and first excited state energies as functions of $J$ [thick black lines (calculated by patterns) and circles (numerical ED)]. The left-lower insets in (a1) and (b1) denote the first derivatives of them. The right-upper insets in (a1) and (b1) are enlarged views of the energy components of patterns. (a2) $\&$ (b2) The second derivatives of the energy components of patterns. Here $h=1.0$ is taken.
• Figure 5: Histograms of patterns' occupancy of the ground state [(a1)-(a9)] and the first excited state [(b1)-(b9), grouped by dashed red frame] of the system with $h=1$ for different Ising interactions $J = 0.0, 0.4, 0.8, 0.9, 1.0, 1.1, 1.2, 1.5$, and $2.0$, which correspond successively from [(a1),(b1)] to [(a9),(b9)], respectively.