Exotic critical states as fractional Fermi seas in the one-dimensional Bose gas

Abstract

Critical quantum field theories occupy a central position in modern theoretical physics for their inherent universality stemming from long-range correlations. As an example, the Tomonaga-Luttinger liquid (TLL) describes a wealth of one-dimensional quantum systems at low temperatures. Its behavior is deeply rooted in the emergence of an effective Fermi sea, leading to power-law correlations and Friedel oscillations. A promising direction to realize systems exhibiting novel universal behavior beyond TLL is through the generalization of the underlying Fermi sea. In this Letter, we show that fractional Fermi seas with reduced occupancy arise in an integrable Bose gas driven out of equilibrium by cyclic changes in interactions from repulsive to attractive. The correlation functions feature signatures of criticality incompatible with a conventional TLL, suggesting a novel critical phase. Our predictions, based on Generalized Hydrodynamics, are directly relevant to cold atoms.

Figures (8)

• Figure 1: Interaction cycles and fractional Fermi seas.--- (a) We perform slow cyclical changes of $g_\text{1D}$ crossing the TG-sTG transition. (b) Each cycle maps the initial GGE onto a GGE with reduced occupancy, realizing an analogue of a GES. (c) In particular, the initial ground state is transformed into a fractional Fermi sea.
• Figure 2: Rapidity and momentum distribution, and one-particle correlation.---Panels (a,b,c). We plot the smooth evolution of the root density $\rho(\lambda)$ versus the normalized interaction strength $\gamma\equiv 2mg_\text{1D}/(\hbar^2 n)$ in the cycle, where the vertical axis is shown on the non-linear scale $(2/\pi)\arctan(\gamma/3)+(1-\text{sign}(\gamma))+2\mathcal{W}$. Due to finite interaction strength, $\rho(\lambda)$ shows a $\gamma$- dependent concavity (more prominent at small $\gamma$). In the repulsive regime (a.1,a.2,a.3), we also plot the momentum distribution $P(p)$ as a function of $\lambda\equiv p/\hbar$, to facilitate its comparison with $\rho(\lambda)$. The statistical error is negligible on the plot scale: in the inset of panel a.2, we show in light blue the partial averages obtained by dividing the total number of samples in five groups with the same size. The small relative distance shows the convergence of the Monte Carlo sampling, see SM suppmat for details. Panels (d,e,f). Examples of $g^{(1)}(x)$ in linear (d.1,e.1,f.1) and log-log (d.2,e.2,f.2) scales, showing FO with a power-law decaying amplitude and a crossover between a short-distance (SD) and a faster long-distance (LD) power-law at a distance $\bar{x}$. We show the case of $\gamma=25$, with a prominent power-law crossover. Gray lines are data of five independent Monte Carlo simulations and the black line is their average. Deviations from the average are appreciable only in the tails of the plots in log-log scale. The apparent large-distance plateau visible in the log-log scale is due to the statistical noise of the Monte Carlo simulations suppmat.
• Figure 3: Power law exponent and FO in one-particle correlation function for $\gamma>0$--- Panel (a) shows the FO's frequency, which is $2\pi n \mathcal{W}$ at $\gamma=0$Marciniak2025 and it departs from it at finite $\gamma$, shown on the same nonlinear scale of Fig. \ref{['Fig_n2']}. Panel (b). Short-distance (SD) and long-distance (LD) power law exponents, see Fig. \ref{['Fig_n2']}(b). For details on the fit, see SM suppmat. Panel (c). Crossover distance $\bar{x}$ between the two power laws, defined as the intersection of the two power laws as in Fig. \ref{['Fig_n2']}(b). Error bars obtained by comparing the fits of five independent Monte Carlo samplings for each value of $\gamma$ and $\mathcal{W}$, and are omitted when negligible on the plot scale. We show the error bars only for the LD exponent in Panel (b) and $\bar{x}$ in Panel (a).
• Figure 4: Energy and reversibility--- We show in blue the energy $E_\text{tot}$ and in red the kinetic energy $E_\text{kin}$ during the cycle. The normalized interaction strength (lower axis) is shown on a non-linear scale (same as Fig. \ref{['Fig_n2']}). Reversibility is broken at $\gamma=0$ as manifested in both the jump in entropy in the forward cycle and a strong variation in the energies ($E_\text{tot}$ dashed blue line, $E_\text{kin}$ dashed red line), due to the formation of bound states in the reversed cycle Koch2021horvath2025suppmat.
• Figure S1: Bound state production in the reversed cycle.--- We show the bound state participation, normalized on the particle's density $n$, obtained by reversing the cycle and crossing $g_\text{1D}=0^+\to g_\text{1D}=0^-$ after having first reached the cycle's winding $\mathcal{W}$. As $\mathcal{W}$ is increased, the formation of large bound states is more unlikely.