Metastable quantum phase transitions in a periodic one-dimensional Bose gas. II. Many-body theory

We show that quantum solitons in the Lieb-Liniger Hamiltonian are precisely the yrast states. We identify such solutions with Lieb’s type II excitations from weak to strong interactions, clarifying a long-standing question of the physical meaning of this excitation branch. We demonstrate that the metastable quantum phase transition previously found in mean-field analysis of the weakly interacting Lieb-Liniger Hamiltonian [Phys. Rev. A 79, 063616 (2009)] extends into the medium- to strongly interacting regime of a periodic one-dimensional Bose gas. Our methods are exact diagonalization, finite-size Bethe ansatz, and the boson-fermion mapping in the Tonks-Girardeau limit.

Figure 1

Yrast energy eigenstates for (a) repulsive and (b) attractive interaction for $N=40$ bosons on the ring, obtained by diagonalization of Hamiltonian (9) with the cutoff angular momentum $\left|l_c\right|=2$. The spectrum has kinks where $L$ is an integral multiple of $N$ and is symmetric with respect to $L=0$, i.e., $E_{N,-L}(0)=E_{N,L}(0)$. In the large $N$ limit, the number of points increases and the discrete yrast energies approach a continuous curve while the curvature and kink points remain unchanged.

Figure 2

Energy eigenvalues of the Hamiltonian (2) obtained by the Legendre transformations (12) of yrast eigenstates in Fig. 1. Each curve is distinguished by a different total angular momentum $L$. The lower panels are enlargements of the upper panels near one of the swallowtail regions. The dashed curves show a comparison to the Gross-Pitaevskii superflow (blue) and soliton (red) branches.

Figure 3

Energy eigenvalues of the Hamiltonian (2) that satisfy the extremization condition ${\mathit{dE}}_{L,N}(\Omega )/\mathit{dL}=0$ for $N=40$ and (a) $g_{1\mathrm{D}}=0.025\pi$ and (b) $g_{1\mathrm{D}}=-0.02\pi$. Each thin curve is characterized by an integral average angular momenta, $L/N=J\in \{0,\pm 1,\dots \}$, while each thick point by different nonintegral angular momentum $L/N\ne J$ so that each thick curve smoothly connects two thin curves with different angular-momentum states. The energy of the former group agrees with the mean-field energy of the uniform superflow state, while the latter is well approximated by the mean-field energy of solitons.

Figure 4

Average angular momentum $L/N$ that gives ${\mathit{dE}}_{L,N}(\Omega )/\mathit{dL}=0$ for (a) repulsive and (b) attractive interactions with $N=40$. Solid curves plot the single-particle angular momentum obtained by the mean-field theory.

Figure 5

Low-lying yrast ground and excited states of free fermions. (I) Particle excitations where the angular momentum of a particle increases while a hole is positioned at $l_F+J$. (II) Hole excitations where a particle is placed at the lowest unoccupied state and the angular momentum of a hole decreases. When $L$ is an integral multiple of $N$, only the center-of-mass angular momentum is shifted from the ground-state configuration. The lower panel shows type I and type II excitation energies as a function of $L/N$ for $N=11$ free fermions.

Figure 6

Realizable yrast eigenstates in the strongly interacting TG limit for $N=11$. All the energies are plotted relative to the ground-state energy of $E_{N,L=0}(0)$. The thin curves plot the energy of the states with angular momentum $L=\mathit{JN}$ and $L=(J+1)N$, while the bold curve is the type II excitation energy for $\mathit{JN}<L⩽(J+1)N$. The angular momentum decreases from $(J+1)N$ to $\mathit{JN}$ according to Eq. (23) along the bold curve.

Figure 7

Comparison of exact yrast eigenenergies of Eq. (36) obtained from the Bethe ansatz, yrast states obtained from the diagonalization of Hamiltonian (9) for $N=10$, and the plane-wave energy branches for $J\in \{0,1,2\}$ given by the mean-field theory. For Bethe ansatz and diagonalization results, the ground ($L=0$), and $2N$ yrast states $\left|L\right|\in \{1,\dots ,2N\}$ are plotted on a log scale. The inset enlarges the region $1⩽g_{1\mathrm{D}}⩽5$ on a linear scale.

Figure 8

(a) Eigenvalues $E_{N,L}$ of the Lieb-Liniger Hamiltonian (1) obtained by the finite-size Bethe ansatz as a function of interaction $g_{1\mathrm{D}}$ for $L=0$ (ground state) and $L\in \{\pm 1,\dots ,\pm 2N\}$ (excited states), where $E_L=E_{-L}$. The horizontal dashed line corresponds to the energy of free fermions (15) with the same number of atoms. (b) Excitation energies $(E_{N,L}-E_{N,0})/N$ of yrast states as a function of angular momentum $\left|L\right|/N$ for fixed strengths of interaction. A, B, C, D, and E correspond to the vertical line in (a). At $L=\mathit{JN}\in \{0,\pm N,\pm 2N,\dots \}$ the energy is that of a CMR state (relative to the unboosted ground state), therefore independent of $g_{1\mathrm{D}}$, and is given by $J^2$, even in the thermodynamic limit.

Figure 9

Eigenvalues of the Hamiltonian (2) that satisfy the extremization condition $\partial E_{L,N}/\partial L=0$ for (a) weakly interacting mean-field regime, (b) and (c) medium-interaction regime, and (d) strongly interacting TG regime. The corresponding average angular momentum of the thin curves is given by integers $L/N=J$. The thick curves, whose average angular momentum is noninteger, come from the type II excitation branch.

Figure 10

(a) Two CMR states with $L=0,N$ (thin curves) and the type II branch (thick curve) that connects them. The energy is defined relative to the interaction energy $V_{\mathrm{int}}$. (b) The existence range of the type II branch with angular momenta $L\in [0,N]$. At the left (right) boundary, the angular momentum is $L=N$ ($L=0$), and $L=N/2$ at $\Omega =0.5$ (vertical dashed line), decreasing linearly with respect to $\Omega$ in the shaded area. (c) Enlargement of (b) in the weakly interacting regime. The solid curve is the phase boundary given by the Bogoliubov theory. (d) Difference between the phase boundaries given by the Bethe ansatz and the Bogoliubov theory. The Bogoliubov phase boundary overestimates the extent of the shaded area.