Split Fermi seas in one-dimensional Bose fluids

For the one-dimensional repulsive Bose gas (Lieb-Liniger model), we study a special class of highly excited states obtained by giving a finite momentum to subgroups of particles. These states, which correspond to “splitting” the ground-state Fermi-sea-like quantum number configuration, are zero-entropy states which display interesting properties more normally associated with ground states. Using a numerically exact method based on integrability, we study these states' excitation spectrum, density correlations, and momentum distribution functions. These correlations display power-law asymptotics and are shown to be accurately described by an effective multicomponent Tomonaga-Luttinger liquid theory whose parameters are obtained from the Bethe ansatz. The nonuniversal correlation prefactors are moreover obtained from integrability, yielding a completely parameter-free fit of the correlator asymptotics.

Figure 1

Illustration of the notations used to specify a Moses state. (top) The ground state has quantum numbers $-(N-1)/2,\cdots ,(N-1)/2$ occupied. (bottom) The Moses state specified by $\{I_{1L},I_{1R},I_{2L},I_{2R}\}$ has occupied quantum numbers in the intervals $[I_{1L},I_{1R}]$ and $[I_{2L},I_{2R}]$.

Figure 2

The single-particle-hole excitation spectrum for a Moses state at $c=\infty$ with Fermi momenta given by $\{-\frac{3}{2}\pi ,-\frac{1}{2}\pi ,\frac{1}{2}\pi ,\frac{3}{2}\pi \}$ and a description of the excitations corresponding to the different lines. Starting with a particle excitation at $I_{2R}$ and a hole in the right Fermi sea, we get line A, which corresponds to the standard Lieb type-I excitation, and line B, which corresponds to the Lieb type-II excitation. At the end of line B, when the hole is at $I_{2L}$, the hole cannot move any farther to the left. Instead, we can now have a particle in between the two Fermi seas and a hole at $I_{1R}$; this gives line D. Line H can be seen as a continuation of the Lieb type-II excitation of line B. The other dispersion lines are easily understood in a similar fashion. The inset shows the single-particle-hole excitation spectrum for the ground state at $c=\infty$. In both Moses and ground states, the single-particle-hole continuum gives the dominant support for density correlations.

Figure 3

The dynamical structure factor $S(k,\omega )$ for the Moses state for different values of the interaction strength. From top left to bottom right: $c=1,c=4,c=16,c=64$. The calculation was performed for 64 particles, 32 in each Fermi sea, and 32 holes separating the two Fermi seas, except for $c=64$, where 128 particles were used.

Figure 4

(left) Extremal rapidities ${\lambda }_{2L},{\lambda }_{2R}$ of the right Fermi sea (dashed red line) and velocities ${\tilde{v}}_{2L}/2,{\tilde{v}}_{2R}/2$ (solid blue line) as a function of $c$ (in units where $2m=1$ and $N/L=1$, so $c$ is dimensionless) calculated for a state with Fermi momenta $k_{ia}=\{-2\pi ,-\pi ,\pi ,2\pi \}$. (right) Illustration of the Moses state in the limits $c=\infty ,0$. In the $c=\infty$ limit, the extremal rapidities correspond to ${\lambda }_{ia}=k_{ia}=2\pi I_{ia}/L$. For $c=0$ the rapidities collapse onto the inner value ${\lambda }_{2L},{\lambda }_{2R}\to k_{2L}$. In the limits $c=0$ and $c=\infty$ the velocities are consistent with quadratic dispersion, i.e., ${\tilde{v}}_{ia}=2{\lambda }_{ia}$.

Figure 5

The dynamical structure factor $S(k,\omega )$ for an asymmetric Moses state for $c=16$. The calculation was performed for 64 particles with quantum numbers $\{I_{1L},I_{1R},I_{2L},I_{2R}\}=\{-41.5,-20.5,-1.5,39.5\}$.

Figure 6

Fixed momentum cuts of the dynamical structure factor $S(k,\omega )$ at (top) $k=\pi$ and (bottom) $k=2\pi$ for $c=1,4,16,64$. These graphs are obtained from the same data as in Fig. 3, showing the threshold singularities at the edges of the single-particle-hole continuum.

Figure 7

The static structure factor for $c=1$, $c=4$, $c=16$, and $c=64$ with the exact Tonks-Girardeau result as a benchmark. The data presented here are the frequency-integrated data of Fig. 3.

Figure 8

Illustration of the effect of changing the Moses state quantum number configuration on static correlation functions, using the solvable $c=\infty$ limit. The dashed lines represent a standard situation with Fermi momenta $\{-3/2\pi ,-1/2\pi ,1/2\pi ,3/2\pi \}$; the solid lines correspond to a modified configuration. From top to bottom, increasing separation between seas, decreasing separation between seas, and asymmetric seas with Fermi momenta $\{-2\pi ,-3/2\pi ,5/2\pi ,4\pi \}$. (left) The static structure factor. (right) Momentum distribution function.

Figure 9

The momentum distribution for different values of the interaction ($c=1,c=4,c=16,c=64$) and the exact Tonks-Girardeau result. Calculations were performed on a Moses state with a symmetric configuration of filled quantum numbers: two Fermi seas of 32 particles each, separated by 32 holes.

Figure 10

The density-density correlation function $S\left(x\right)$ for a Moses state for different values of the interaction obtained from the numerical data of the DSF as shown in Fig. 3.

Figure 11

Comparison of the density-density correlation obtained numerically (dashed lines) and the analytic results from the multicomponent Tomonaga-Luttinger model (solid lines). The Tomonaga-Luttinger prediction differs from the numerical data by less than $0.01$ when $x$ is larger than $6.1\%$ of the system length for $c=1$, $2.8\%$ for $c=4$, $2.3\%$ for $c=16$, and $1.4\%$ for $c=64$. Calculations were performed at unit filling with 64 particles on length $L=64$.

Figure 12

Prefactors of the Tomonaga-Luttinger correlation asymptotics as a function of $c$.

Figure 13

The one-body reduced density matrix $g_1\left(x\right)$ for a Moses state for different values of the interaction as obtained from the numerical data of the momentum distribution function in Fig. 9.

Figure 14

Comparison of the one-body reduced density matrix obtained numerically (dashed lines) with the parameter-free fits using the multicomponent Tomonaga-Luttinger model (solid lines). The difference between the numerical plot and the analytic multicomponent Tomonaga-Luttinger curve becomes less than $0.01$ for $x$ greater than $0.038L$ for $c=1$, $0.028L$ for $c=4$, $0.030L$ for $c=16$, and $c=64$.