Excitation spectrum of bosons in a finite one-dimensional circular waveguide via the Bethe ansatz

The exactly solvable Lieb-Liniger model of interacting bosons in one dimension has attracted renewed interest as current experiments with ultracold atoms begin to probe this regime. Here we numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to 20 particles for attractive interactions. We discuss the features of the solutions, and how they deviate from the well-known string solutions [Thacker, Rev. Mod. Phys. 53, 253 (1981)] at finite densities. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Finally we compare our results to those of exact diagonalization of the many-body Hamiltonian in a truncated basis. We also present excited state solutions and the excitation spectrum for the repulsive one-dimensional Bose gas on a ring.

Figure 1

(Color online) Behavior of the quasimomenta for $N=6$ particles as a function of the interaction strength $c<0$, for eight low-lying states, ordered via their energy at $c=0$. The single-particle momentum states (and their energies) at $c=0$ are (a) {0,0,0,0,0,0} $\left(E=0\right)$, (b) {1,0,0,0,0,0} $\left(E=1\right)$, (c) {1,1,0,0,0,0} $\left(E=2\right)$, (d) $\left\{1,-1,0,0,0,0\right\}$ $\left(E=2\right)$, (e) {1,1,1,0,0,0} $\left(E=3\right)$, (f) $\left\{1,1,-1,0,0,0\right\}$ $\left(E=3\right)$, (g) {1,1,1,1,0,0} $\left(E=4\right)$, and (h) $\left\{1,1,1,-1,0,0\right\}$ $\left(E=4\right)$. All states shown have a total angular momentum that is either zero [(a) and (d)] or positive [(b), (c), (e), (f), (g), and (h)]—the latter set have a degenerate counterpart with a negative total angular momentum which can be found easily via the mapping $k_i\to -k_i$ for all $i$. The states plotted all have $\left|k_i\right|\le 1$ for all $i$—states with quasimomenta such as {2,0,0,0,0,0} at $c=0$, [which are degenerate with (g) and (h)] give rise to higher-energy states for $c<0$. On each graph there are six distinct lines for each particle; however, some of these overlap for large regions. To aid visualization the color of the lines indicates how many quasimomenta are overlapping. Blue indicates one root, green two, magenta three, cyan four, and red six.

Figure 2

(Color online) Quasimomenta of the first excited state for $N=20$ particles as a function of the interaction strength $c$. (a) and (b) show the real and imaginary parts, respectively, of the quasimomenta. The (blue) solid lines are the roots calculated using the Bethe ansatz, whereas the (red) dashed lines show the string solutions given by Eqs. (13), which give the asymptotic behavior of the quasimomenta in the attractive limit. The two solutions are in good agreement above the critical interaction strength, validating the essential properties of the bound states described in Refs. [12, 27, 50, 51, 52]. The inset shows the region where the quasimomenta bifurcate, causing numerical difficulties.

Figure 3

(Color online) Quasimomenta of one of the second excited states for $N=20$ particles as a function of the interaction strengnth $c$. (a) and (b) show the real and imaginary parts, respectively, of the quasimomenta. The (blue) solid lines are the roots calculated using the Bethe ansatz, whereas the (red) dashed lines show the string solutions given by Eqs. (16) with $M=1$. The inset shows where the quasimomenta bifurcate for this case.

Figure 4

(Color online) Quasimomenta distribution for $N=20$ particles in the complex plane for different values of interaction strength $c$ for the excited state with quasimomenta distribution $\left\{-1,-1,0,\dots ,0,1,1,1\right\}$ at $c=0$. (a) $\left|c\right|={10}^{-6}$. The quasimomenta are close to the single-particle momenta for the ideal gas. (b) $\left|c\right|=C_0\approx 0.079$. Near the mean-field critical point there exists a two-particle bound state, a three-particle bound state, and a 15-particle bound state. (c) $\left|c\right|=1.9$. The interaction strength is well past the critical point and the three-particle bound state has collapsed in with the 15-particle bound state; however, there is still a two-particle bound state.

Figure 5

(Color online) Quasimomenta distribution for $N=20$ particles in the complex plane for different values of interaction strength $c$ for the excited state with quasimomenta distribution $\left\{-1,0,\dots ,0,1,1,1,1,1\right\}$. (a) $\left|c\right|={10}^{-6}$. The quasimomenta are close to the single-particle momenta for the ideal gas. (b) $\left|c\right|=C_0\approx 0.079$. Near the mean-field critical point, there exists a free particle, a five-particle bound state, and a14-particle bound state. (c) $\left|c\right|=1.9$. The interaction strength is well past the critical point and the five-particle bound state has combined with the 14-particle bound state, leaving a single free particle.

Figure 6

(Color online) Excitation spectrum of an $\left(N=6\right)$-particle system obtained via the Bethe ansatz. The solid (red) lines show the formation of a single bound state. The (green) dashed lines show the formation of two bound states. The (blue) dot-dashed lines show the formation of three separate bound states. The qualitative change in the behavior of the eigenstates at the phase transition point becomes more visible than for the two-particle case (see Fig. 9).

Figure 7

(Color online) Excitation spectrum of an $\left(N=20\right)$-particle system obtained via the Bethe ansatz. The solid (red) lines show the formation of a single bound state. The (green) dashed lines show the formation of two bound states. The (blue) dot-dashed lines show the formation of three separate bound states.

Figure 8

Bogoliubov excitation spectrum of the 1D Bose gas on a ring with attractive interations. The comparison with Fig. 7 shows reasonable quantitative agreement to the left of the critical point for only $N=20$ particles. To the right of the critical point, there is qualitative agreement through the emergence of a Goldstone mode as well as higher-order branches of excitation.

Figure 9

(Color online) Comparison of the excitation spectrum found via the Bethe ansatz Eqs. (8) [(red) dashed lines] to that found via diagonalization of the truncated Hamiltonian [solid (blue) lines] for an $\left(N=2\right)$-particle system. The vertical dash-dotted line marks the meanfield crititical interaction strength $C_0$ derived in Appendix . Single-particle momentum state cutoff of (a) $\pm 1$, (b) $\pm 2$, and so on, up to (f), which has a singleparticle momentum cutoff of $\pm 6$.

Figure 10

(Color online) Comparison of the excitation spectrum found via the Bethe ansatz Eqs. (8) [(red) dashed lines] to that found via diagonalization of the truncated Hamiltonian [solid (blue) lines] for an $\left(N=20\right)$-particle system. (a) Truncation at $m=3$ (single-particle momentum states $0,\pm 1$). (b) Truncation at $m=5$ (single-particle momentum states $0,\pm 1,\pm 2$).

Figure 11

Quasimomenta as a function of interaction strength for $c>0$. (a) Ground state of the ideal Bose gas. (b) First excited state of the ideal Bose gas (degenerate with the state $k_j\to -k_j$). (c),(d) Degenerate second excited states of the ideal Bose gas (also degenerate with the state $k_j\to -k_j$). (e),(f) Degenerate third excited states of the ideal Bose gas (also degenerate with the state $k_j\to -k_j$). (g),(h),(i) Degenerate fourth excited states of the ideal Bose gas (also degenerate with the state $k_j\to -k_j$). The insets in (a)–(i) show the behavior close to $c=0$.

Figure 12

Quasimomenta as a function of interaction strength for an $\left(N=50\right)$-particle repulsive gas.

Figure 13

(Color online) Excitation spectrum of an $\left(N=20\right)$-particle gas with repulsive interactions $c>0$ [solid (black) lines]. The (red) dashed lines were obtained by truncating the Hilbert space down to five single-particle momentum states. The inset shows the region of $0<c<1$ where the truncated diagonalization is accurate. The dotted lines shows the ground state energy of a free Fermi gas, and the so-called Tonks Girardeau ground state asymptotes to this value.