Giant bubbles of Fisher zeros in the quantum XY chain

Abstract

We demonstrate an alternative approach based on complex-valued inverse temperature and partition function to probe quantum phases of matter with nontrivial spectra and dynamics. It leverages thermofield dynamics (TFD) to quantitatively characterize quantum and thermal fluctuations, and exploit the correspondence between low-energy excitations and Fisher zeros. Using the quantum XY chain in an external field as a testbed, we show that the oscillatory gap behavior manifests as oscillations in the long-time dynamics of the TFD spectral form factor. We also identify giant bubbles, i.e. large-scale closed lines, of Fisher-zeros near the gapless XX limit. They provide a characteristic energy scale that seems to contradict the predictions of the low energy theory of a featureless Luttinger liquid. We identify this energy scale and relate the motion of these giant bubbles with varying external field to the transfer of spectral weight from high to low energies. The deep connection between Fisher zeros, dynamics, and excitations opens up promising avenues for understanding the unconventional gap behaviors in strongly correlated many-body systems.

Figures (4)

• Figure 1: Phase diagram of the quantum XY model under an external field $g$. $g=1$ is the critical line separating the FM phase from the disordered phase. The thick dashed line separates the oscillatory gap region from the FM region. The thin dashed line indicates where the gap vanishes for $L = 16$. $\gamma=1$ corresponds to the Ising limit, while $\gamma=0$ represents the gapless $XX$ limit.
• Figure 2: Fisher zeros of the $XY$ model without external field ($g = 0$) in the thermodynamic limit for different values of $\gamma$. As $\gamma$ decreases from the Ising limit ($\gamma = 1$) toward the $XX$ limit ($\gamma = 0$), an upward-moving open Fisher zero line crosses over with a downward-moving closed Fisher zero line. The cases $\gamma = 0.12$ and 0.11 provide the crossover example closest to the $\beta_r$-axis.
• Figure 3: Oscillatory behavior of Fisher zeros and $S$ for $\gamma = 0.6$. (a) Evolution of Fisher zero configurations in the thermodynamic limit as $g$ increases. Before reaching $g=1$, the closed Fisher zero loops exhibit non-monotonic behavior, including disappearing and reappearing at $g\lesssim 0.5$. (b) For $L=32$ and $\beta_r=10$, the oscillation frequency of $S$ (black hollow dots) agrees well with $\Delta_\infty$ (gray line). The inset shows the oscillation of $1-S$ with $\beta_i$ for two nearby values $g=0.08$ (blue pentagram) and $g=0.115$(red pentagram). (c) For $L=32$ and $\beta_r=50$, the oscillation frequency of $S$ (black hollow dots) matches $\Delta_L$ (gray line). This inset shows the long-time oscillatory behavior of $1-S$ for the two $g$ values from (b).
• Figure 4: Large-scale closed Fisher zero patterns at $\gamma=0$. (a) For different $g$s, the Fisher zeros form only closed loops. As $g$ increases, the closed zero loops exhibit a tendency to expand. (b) Near the QCP, the expansion of the large-scale closed Fisher zero loop becomes more pronounced compared to the smaller zero loops near the $\beta_i$-axis (upper-left inset). The lower-right inset shows that the inverse of the imaginary part ($1/\beta_{i0}$) of the rightmost Fisher zero on the large loop scales linearly with $1-g$. (c) The DOS diverges near $1-g$, as illustrated for $g=0$ (blue) and $g=0.7$ (red), exhibiting van Hove singularities. The left inset shows the corresponding dispersion relations $\epsilon(q)$, where the dashed slope illustrates the Luttinger-liquid velocity, which is proportional to $\sqrt{1-g^2}$. The right inset displays the dependence of $1/\beta_{i0}$ of the open zero lines on $g$ ($1/\beta_{i0}\sim\sqrt{0.99-g^2}$) at $\beta_r=100$ when a small anisotropy $\gamma=0.1$ is introduced.