Emergent gapless spiral phases and conformal Lifshitz criticality in the cluster Ising model with off-diagonal interactions

TL;DR

The paper investigates a one-dimensional cluster-Ising chain with off-diagonal Gamma interactions by mapping to a free-fermion BdG problem via the Jordan–Wigner transformation. It unveils a phase diagram with gapped AFM_y and SPT phases and two gapless spiral phases related by duality, separated by four phase-transition lines: Ising and three-copy Ising conformal transitions, plus nonconformal Lifshitz transitions with z=2, and a Lifshitz multicritical point with emergent conformal symmetry. The work provides analytical and numerical evidence for the central charges $c=\tfrac{1}{2}$ and $c=\tfrac{3}{2}$ on conformal lines, a gapless phase with $c=1$, and emergent conformal symmetry at a Lifshitz multicritical point, offering a valuable reference for exotic gapless phases in exactly solvable systems. It also emphasizes the role of Kennedy–Tasaki duality in relating order parameters across dual gapless phases and suggests experimental avenues in quantum simulators for realizing such intertwined SPT and off-diagonal-interaction physics.

Abstract

We perform a comprehensive analytical study of the exotic quantum phases and phase transitions emerging from the cluster-Ising model with off-diagonal Gamma interactions. Specifically, we map out the ground-state phase diagram by analyzing both local and nonlocal order parameters, together with the energy spectra. The results reveal two pairs of gapped phases, namely and antiferromagnetic (AFM) long-range ordered phases, symmetry-protected topological (SPT) phases, as well as two distinct gapless spiral phases induced by the off-diagonal interactions, which are related by a duality transformation and are numerically confirmed through the long-distance behavior of various order parameters. Remarkably, four distinct phase transition lines emerge in the phase diagram. Two of them, which separate the distinct gapped or gapless phases, are described by the Ising and three copy Ising conformal field theories, respectively. In contrast, the remaining two transition lines, between the gapless spiral and gapped phases, belong to a nonconformal Lifshitz criticality with dynamical critical exponent $z = 2$. More importantly, the intersection of these four transition lines gives rise to a new Lifshitz multicritical point exhibiting emergent conformal symmetry, marking a fundamental departure from all previously known nonconformal Lifshitz points. This work provides a valuable reference for future investigations of exotic gapless phases and their transitions in exactly solvable many-body systems.

Figures (11)

• Figure 1: (Color Online) A schematic plot of the one-dimensional cluster-Ising model with off-diagonal Gamma interactions. The green-filled ellipsoids represent the three-spin cluster interaction $J$, the blue dotted lines denote the two-spin Ising interaction $\lambda$, and the red dashed lines indicate the off-diagonal Gamma interactions, including their strength $\Gamma$ and relative weight $\alpha$. In this work, we set $J = 1$ as the energy unit and fix $\Gamma = 0.5$.
• Figure 2: (Color Online) The global phase diagram of the cluster Ising chain with off-diagonal Gamma interactions is shown as a function of the tuning parameters ($\alpha$, $\lambda$), diagnosed by the second derivative of the ground-state energy density, $-\partial^2 e_0/\partial \lambda^{2}$. The diagram comprises four distinct regions: the cluster SPT phase, the $\text{AFM}_y$ phase, and two distinct gapless spiral phases (I and II). The phase transitions from SPT to $\text{AFM}_y$ and from Spiral I to Spiral II are conformal and described by the Ising and three-copy Ising CFTs, respectively. In contrast, the transitions between the gapless spiral phases and the gapped phases are associated with non-conformal Lifshitz criticality. However, the red star denotes a multicritical Lifshitz point featuring emergent conformal symmetry.
• Figure 3: (Color online). Long-distance behavior of order parameters in the gapped region for $\alpha=0.5$. The string order parameter $|O_{x}|$ is shown as a function of distance $r$ for various values of $\lambda$ in panel (a), while the spin correlation function $|G_{yy}|$ is plotted against $r$ in panel (b). Notably, when $\lambda<1.0$, the string order parameter displays long-range order, suggesting the presence of a cluster SPT order. Conversely, when $\lambda>1.0$, the spin correlation exhibits long-range order, indicating that the ground state features $\text{AFM}_y$ order. The simulated system size for panels (a)–(b) is $N =3000$.
• Figure 4: (Color online). Long-distance behavior of order parameters and scaling of entanglement entropy in the gapless region for $\alpha = -0.5$. The string order parameter $|O_{x}|$ is plotted as a function of distance $r$ for different values of $\lambda$ in panel (a); the spin correlation function $|G_{yy}|$ is shown as a function of $r$ in panel (b); and the vector chiral order parameter $|O_{xy}|$ is plotted versus $r$ in panel (c). The insets in panels (a)–(c) display the corresponding data on a log-log scale, confirming power-law scaling as $1/\sqrt{r}$. Panel (d) shows the entanglement entropy $S_L$ as a function of subsystem size for various $\lambda$, with the inset showing the same data on a log-log scale, indicating a logarithmic scaling $S_L \sim (1/3)\ln L$. The simulated system size for panels (a)–(d) is $N = 3000$.
• Figure 5: (Color online). Long-distance behavior of order parameters, entanglement entropy scaling, and energy spectra along the phase transition line at $\lambda = 1$ for various values of $\alpha$. Panel (a) shows the string order parameter $|O_{x}|$ as a function of distance $r$ on a log-log scale for different $\alpha$ values. Panel (b) displays the spin correlation function $|G_{yy}|$ versus $r$, also on a log-log scale. In panel (c), the entanglement entropy $S_{L}$ is plotted against the subsystem size $L$. The inset highlights the logarithmic scaling behavior: $S_L \sim (1/6)\ln L$ for $\alpha = 1, 0$ and $S_L \sim (1/2)\ln L$ for $\alpha = -1$. Panel (d) shows the energy spectrum $\varepsilon_{k}$ as a function of momentum $k$. The simulated system size for panels (a)–(d) is $N = 3000$.