Metastable quantum phase transitions in a periodic onedimensional Bose gas:
II. Manybody theory
Abstract
We show that quantum solitons in the LiebLiniger Hamiltonian are precisely the yrast states. We identify such solutions with Lieb’s type II excitations from weak to strong interactions, clarifying a longstanding question of the physical meaning of this excitation branch. We demonstrate that the metastable quantum phase transition previously found in meanfield analysis of the weakly interacting LiebLiniger Hamiltonian [Phys. Rev. A 79, 063616 (2009)] extends into the medium to strongly interacting regime of a periodic onedimensional Bose gas. Our methods are exact diagonalization, finitesize Bethe ansatz, and the bosonfermion mapping in the TonksGirardeau limit.
pacs:
03.75.Hh,03.75.LmI Introduction
Exactly solvable quantum systems 94:Mattis (); 97:KY () are now within reach of experiments. This is best accomplished in highly controllable systems, such as ultracold quantum gases 02:PS (), because one has precise control over the effective dimensionality, so that one and two dimensions can be studied for a wide range of interactions, both repulsive and attractive, from the weakly to the strongly interacting, over seven orders of magnitude 09:Hulet (). Moreover, these systems are well insulated, where there is only negligible exchange of energy and particles with the environment, and thus suitable for the study of the metastable quantum phase transitions in excited states 08:CCI (), as well as groundstate quantum phases.
In this article we investigate the manybody ground and excited eigenstates of a periodic onedimensional (1D) Bose gas 63:LL () under an external rotating drive going beyond the meanfield regime. Such a geometry has been realized in experiments 05:GMMPK (); 06:AGR (); 07:M (); 07:NISTex (); 03:Dem () from the weakly interacting condensate regime to the strongly interacting TonksGirardeau (TG) limits 04:KWW (); 60:Gir (); 04:TG (). In our previous analyses we showed that the average angular momentum of weakly repulsive bosons in a onedimensional ring undergoes a quantum phase transition (QPT) in the metastable states as a function of interaction and rotation 08:KCU (). In the meanfield theory this phenomenon is intuitively understood in terms of bifurcation of stationary excitedstate energy branches of the planewave state propagating on the ring, and of localized soliton trains 09:KCU (). Each excited state has a denumerably infinite number of bifurcations from the plane wave to a state containing one or more gray or dark solitons; each such bifurcation corresponds to a QPT. Formally these QPTs are in fact “crossovers”, because the two different kinds of physical behavior, superflow and soliton, can be connected by analytic continuation. However, these QPTs have no meaning in the thermodynamic limit, where there is no BoseEinstein condensation in 1D, and as such are fundamentally restricted to the finitesize isolated systems typically found in experiments on BoseEinstein condensates. Moreover, such crossovers can appear quite sharp in experiments, so that the matter of terminology becomes a question of theory, not experiment. In metastable states of matter waves, such as soliton trains 02:ENSbs (); 02:Ricebs (), the effects of dissipation can be suppressed and the metastable condensate is observable. However, this picture does not extend into the medium to strongly interacting regime, where quantum fluctuations cause meanfield solitons to decay 09:MC (); 09:MDCC (). Two questions follow. (1) Does the QPT indicated by meanfield analysis hold for stronger interactions? (2) If so, in what way is the system characterized on either side of this QPT, given that meanfield solitons are clearly no longer eigenstates?
Our answer lies in the special class of manybody eigensolutions called yrast states 99:BP (); 99:BR (), defined as the lowestenergy solutions for fixed angular momentum. Studies of onedimensional systems relevant to our chosen model have a long history, including exactly solvable quantum systems 63:LL (); 63:L (); 89:LH2 (), decay of persistent current 67:LA (); 67:Litt (); 61:BY (); 72:PKU (); 73:Blo (), and classical solitons 80:IT (); 89:LH (); 95:Agr (). In the thermodynamic limit it has been known since Lieb that there exist two excitation branches in the system, called type I and type II excitations. While the physical meaning of type I was clarified as the particle excitation and was found to agree with the Bogoliubovtype excitation in the weakly interacting limit, the meaning of type II was elusive, described only as hole excitations 63:L (). Seventeen years after their discovery, type II hole excitations were identified as a soliton branch by analysis of the energy of a classical soliton in terms of the nonlinear Schrödinger equation 80:IT (). However, the validity range of the nonlinear Schrödinger equation is limited only within the range where the matter wave possesses offdiagonal longrange coherence.
The central finding of this article is that quantum solitons in the LiebLiniger Hamiltonian are precisely the yrast states, and such states are the key to the metastable QPT previously identified in the meanfield context 08:KCU (); 09:KCU (). We first show how to distill the meanfield branches and QPT, which are previously found in the meanfield theory, from the metastable yrast states. Throughout the manuscript, this is our basic fashion of discussing the metastable states. The meanfield superflowsoliton QPT found in Refs. 08:KCU (); 09:KCU () is shown to be obtained by extremizing the yrast spectra. In the weakly interacting regime, this type II excitation can indeed be called a soliton branch as shown in Ref. 80:IT () because of quantitative agreement with the GrossPitaevskii meanfield theory note1 (). We next introduce the concept of the “particle” and “hole” excitations which are clearly defined in the strongly interacting TonksGirardeau (TG) limit. We recover Lieb’s result, and the metastable condition for the type II excitation branch leads to the observable quantum phase with a nonintegral singleparticle average momentum. The type II “hole” excitation branch is made metastable (as opposed to unstable) by subjecting the gas to a rotating drive, and is observable in typical “rotatingbuckettype” experiments 73:Legg () in a manner similar to the method used to create quantized vortices. Finally we apply this concept to the regime of medium interaction strength.
This article is structured as follows. In Sec. II we introduce the LiebLiniger Hamiltonian subject to an external rotating drive. The yrast problem, and basic properties of the eigenstates are described. In Sec. III we investigate the manybody spectrum by exact diagonalization of the Hamiltonian in a truncated angularmomentum basis in the weakly interacting regime, comparing with those obtained by the meanfield theory. In Sec. IV we study the opposite limit of the interaction strength, i.e., the strongly interacting TG limit where the manybody eigenproblem can be analytically solved using the BoseFermi mapping. In Sec. V we address the intermediate regime of repulsive interaction between the weakly and strongly interacting limits via the finitesize Bethe ansatz approach. Finally, we summarize the results in Sec. VI.
Ii Formulation of the Problem
ii.1 The model and yrast states
We consider the same model as in Refs. 08:KCU (); 09:KCU (), and solve its eigenproblem beyond the meanfield and Bogoliubov theories. The Hamiltonian for periodic onedimensional bosons with a contact interaction,
(1) 
is known as the LiebLiniger Hamiltonian (LLH) 63:LL (), where is the azimuthal angle that satisfies , the number of bosonic atoms, and the effective strength of wave interatomic interaction in one dimension (1D) 98:Ols (). The length and energy units are the circumference of the ring , and with being the atomic mass, respectively. The coupling constant is measured in units of and is hence dimensionless. The purpose of this article is to elucidate the manybody properties of these bosons subjected to a rotating drive. The LLH in a rotating frame of reference with an angular frequency is given by
(2) 
where
(3) 
is the angularmomentum operator. From the singlevaluedness boundary condition of the manybody wave function 73:Legg (), one can show that solving the eigenproblem in the rest frame suffices in order to obtain solutions to the eigenproblem 09:KCU (). The eigenvalue is simply given by , which is periodic with respect to .
Throughout this article our approach is based on yrast problems 99:BR (); 99:BP (). Yrast, a Swedish term originally used in nuclear physics which can be translated as “dizziest,” refers to the lowest energy state for a given angular momentum. This approach is particularly profitable for a finite system, because all the information about physical properties in the rotating frame are embedded within the spectrum in the rest frame. Thus the physical meanings of yrast states can be extracted by the simple transformation of yrast spectra. Since the LLH commutes with the angularmomentum operator, , the yrast problem is well defined irrespective of the sign and strength of interaction, and all the yrast states are eigenstates of both the Hamiltonians and .
All the eigensolutions of Hamiltonians and are classified according to the number of atoms and total angular momentum . Let us write the set of eigenstates classified into the subspace given by parameters as where is an energy quantum number that arranges the eigenvalues for fixed in ascending order. The yrast states (the lowestenergy state under a given set of and ) are denoted as . The essential properties of the ground and lowlying excited states can be described within the yrast states. Thus we henceforth omit the quantum number from the notations for eigensolutions. There are two external parameters, the coupling constant and the external angular frequency of the rotating drive (divided by 2), written as , respectively. With the abbreviation of the quantum number and for fixed coupling constant , the eigenvalues that correspond to the yrast states are written as , where we explicitly write the parameter in the notation in order to clarify in which frame the system is. With this notation, the eigensolutions in the rest (nonrotating) frame are written as .
ii.2 Centerofmass rotation states
Due to the translational invariance of the LLH with respect to and , properties of a particular set of yrast states can be analyzed without solving the problem. We denote the set of yrast states for which total angular momentum is equal to an integral multiple of the total number of atoms , as centerofmass rotation (CMR) states. The energy of the CMR state takes the form
(4) 
where is the interaction energy and is an integer. We call the centerofmass quantum number, because it physically expresses the amount of uniform translation of the centerofmass momentum. In the GrossPitaevskii meanfield theory, is conventionally called the phase winding number; out of the meanfield regime, such terminology becomes questionable if not meaningless. In the rotating frame, the energy of the CMR state is given by
(5) 
where the change in energy associated with the frame change is involved only in the kinetic energy term, and the interaction energy is completely separated from the parameter .
For repulsive interactions , the ground state in the absence of the rotating drive is the state with zero angular momentum, . The excitation energy of the CMR states with a finite angular momentum is thus given by
(6) 
which is independent of the strength of interaction . This is natural because changing the total angular momentum by the amount is just a frame change and the interaction is isotropic. The ground state in the presence of the rotating drive is characterized by the CMR quantum number
(7) 
where denotes an integer that does not exceed .
Because of the periodicity in the eigensolutions, an eigenstate with the energy has a denumerably infinite number of counterparts and , corresponding to arbitrary values of . Solving the yrast problem for a limited range of fixed angularmomentum states, e.g., , therefore suffices to obtain all the eigensolutions. Moreover, the spectra are degenerate for the same magnitude of angular momentum, in the absence of rotating drive, while this degeneracy is resolved in the presence of rotation due to the Sagnac effect 13:Sag (). All other yrast states for out of this limited range can be obtained by shifting the total angular momentum by while keeping the internal structure of the eigenstates. This is similar to a band theory concept, as discussed in Ref. 09:KCU (), with playing the role of the Brillouin zone.
Iii Weakly Interacting Limit
In our previous studies 08:KCU (); 09:KCU (), we investigated the weakly interacting limit of the Bose gas on a rotating ring. The GrossPitaevskii equation, which corresponds to the meanfield approximation for the Hamiltonian (2), has two kinds of solutions, namely, uniform superflow and soliton train 00:CCR (). The meanfield energy diagram is characterized by the set of bifurcations of the soliton branch from the superflow branch. These bifurcations make a continuous topological crossover in the condensate wave function possible by changing . The motivation of this section is to demonstrate how in general to distill the meanfield branches from a sea of manybody eigenvalues. We argue how the meanfield soliton branch, for which average angular momentum is not quantized, emerges from the yrast spectra. The meaning of spectra related to symmetry breaking associated with the existence of soliton branch is also discussed.
iii.1 Solution of the yrast problem
To rewrite the LLH in secondquantized form, the bosonic field operator is expanded in terms of a planewave basis with the singleparticle angular momentum ,
(8) 
where the prefactor of comes from the normalization of the plane wave, and and are annihilation and creation operators which obey the standard commutation relations for bosons. Equation (8) manifestly satisfies the periodic boundary condition . The Hamiltonian (1) in second quantized form is then given by
(9)  
Note that all the angular momenta are measured in units of . The eigenstates can be expanded in terms of a Fockstate basis that represents the occupation number of each singleparticle angularmomentum state,
(10) 
These states satisfy the conservation laws
(11) 
In practice, for numerical calculations we require a cutoff angular momentum . The range of the possible total angular momenta for numerical diagonalization is hence limited to the interval of integer values . In the weakly repulsive interacting regime , which we study in this section, a cutoff of provides a quantitative agreement in energy eigenvalues note2 () with those obtained by the Bethe ansatz shown in Sec. V. Thus all the results from this section are obtained with a cutoff of .
Figure 1 shows the yrast energies , namely, the smallest eigenvalue obtained by the diagonalization of the Hamiltonian within the restricted Hilbert space for (a) and (b) for . The ratio of the meanfield interaction energy to the kinetic energy corresponding to these values of and is (a) , and (b) , respectively. The case of attractive interaction is shown just for reference, as our main interest is in the case of repulsive interactions. As the cutoff angular momentum is , yrast states (eigenstates corresponding to the eigenvalues for ) are plotted. Recall that energies depend only on the magnitude of the angular momentum: for .
The key feature of the spectrum is that there appears a prominent kink at every CMR state . As shown in Sec. II.2, the excitation energy of these states is given by . We note, however, that other states as well as the curvature of the spectrum are also important and determine the existence of another quantum phase, as we show next.
iii.2 Superflow and soliton components
In order to obtain the eigenstates of the LLH in the rotating frame, we transform the yrast spectrum according to the Legendre transformation,
(12) 
Figure 2 plots energies as a function of , where the finite number of dots, each of which is characterized by the different angular momentum in Fig. 1, become convex downward curves in Fig. 2. Each curve is thus characterized by a different total angular momentum and has a minimum at a certain value of . The degeneracy in the absence of a rotating drive is resolved for finite due to the Sagnac effect 13:Sag (), i.e., the energy difference naturally arises in the corotating, and counterrotating states with the external rotating drive.
For repulsive interactions [Fig. 2(a)], the energy corresponds to the ground state where is the groundstate CMR quantum number given by Eq. (7). The angularmomentum states with correspond to the CMR states, and the center of the parabola is located at at which the CMR state becomes the ground state. On the other hand, for attractive interactions [Fig. 2 (b)] the CMR state is not always a ground state, and is partially substituted for by the nonintegral average angularmomentum states.
The transformation of the yrast spectrum according to Eq. (12) tells us that the eigensolutions of the Hamiltonian in the rotating frame have an extremely high density of states around due to the crossing of many eigenvalues. These regions are enlarged in the lower panels of Fig. 2 for both signs of , where we also plot the energies of the stationary states given by the meanfield theory. Swallowtails were previously found within the meanfield theory to occur past the phase transition boundary between the uniform superflow and brokensymmetry soliton states 09:KCU (). In the microscopic quantum theory, the highdensity region also forms an upward/downward swallowtailshape domain for repulsive/attractive interactions, and the region is almost filled by various energy eigenvalues of various angularmomentum states crossing each other. The domain with the highdensity swallowtail shape looks as if it is enclosed by the two kinds of stationary branches predicted by the meanfield theory.
Although all the angularmomentum states shown in Fig. 2 are eigenvalues of the Hamiltonian (2), not all states are realized in practice. One example is vortex formation in a scalar condensate under rotation. Solving the yrast problem in two dimensions results in all the angularmomentum states, including the rest condensate (), offaxis vortex (), a centered vortex (), and vortex lattices (). In experiment, however, one drives the system with a specific angular frequency. In such a situation, there exists a small distortion in the trap, which “selects” a metastable angularmomentum state with respect to the variation in the angular momentum of the condensate. As a result, in reality one does not observe a stationary offcentered vortex except as a transient state.
The same argument applies to our case. In the presence of any kind of noise, such as an infinitesimal distortion of the trapping potential, quantum measurement of the matter wave, or whatever else breaks the translation symmetry of the ring trap, the realizable stationary state or metastable stationary state is determined by extremization with respect to variations in angular momentum. In order to find the metastable states we impose the condition
(13) 
with and being fixed.
Figure 3 plots energy eigenvalues that satisfy the condition (13) as a function of ; and Fig. 4 shows the corresponding angular momentum. These figures are quite similar to those given by meanfield theory, i.e., by imposing the stationary condition (13) for the manifold of eigenvalues we identify the meanfield stationary branches. The resultant branches are classified into two kinds according to the value of the angular momentum and have physical meanings as follows:
Superflow: Due to the kink in the yrast spectrum at in Fig. 1, these CMR states always satisfy the condition (13). In particular, for the weakly interacting regime the CMR states can be specifically called uniform superflow states, of which energies are given by Eq. (5) and which correspond to the thin parabolic curves in Fig. 3. The energy of superflow states agrees very well with the planewave energy in the meanfield theory 09:KCU ().
Soliton Components: Other kinds of metastable angularmomentum states appear that connect distinct superflow states as a function of , as shown by the thick curves in Fig. 3. The corresponding angular momentum divided by is nonintegral (see Fig. 4), but it approaches integral values at both ends of this branch. These branches are equivalent to the maximum/minimum envelope of the highdensity swallowtail domain for repulsive/attractive interactions, and can be approximated by soliton energies given by the GrossPitaevskii meanfield theory.
However, we should not call the thick curve a soliton branch in a rigorous sense, because each point of this branch in Fig. 3 is the eigenvalue of the Hamiltonian and thus still possesses translational symmetry, unlike solitons. Instead we should call all the angularmomentum states inside the swallow tail in Fig. 2 the soliton components, because in the presence of infinitesimal noise these states do form a brokensymmetry state, which we denote . Soliton solutions of a GrossPitaevskii equation can be interpreted in terms of the eigensolutions of the manybody Hamiltonian as a state where the several eigenvalues in the swallowtail region are collectively superimposed. The delocalization of a meanfieldlike soliton in weakly interacting theories, as demonstrated by Dziarmaga et al. 03:DKS (), is a dynamical demonstration of this idea.
The energy associated with this superposition does not change significantly because the energy required to make it is on the order of . As a result, the energy of the brokensymmetry soliton state is also well approximated by the thick curve in Fig. 3. In the presence of an infinitesimal symmetrybreaking potential, the angular momentum is no longer a good quantum number. However, the expectation value of the angular momentum agrees well with that of the solitons obtained by meanfield theory, and thus behaves like that shown in Fig. 4 08:KCU (). With all these caveats in mind, we briefly say the branch drawn by thick curve in 4 is the quantum soliton branch in the weakly interacting regime.
We also calculate the second derivative with respect to in order to check whether the metastable angularmomentum state is a local maximum or minimum. For repulsive interactions the superflow state with a CM quantum number is indeed the ground state because the second derivative is positive at that point, while the thick points are local maxima with respect to , since the second derivative is negative. For attractive interactions, the ground state is either a plane wave or a bright soliton in but the soliton becomes the sole ground state for . Consistently, the thick curve in Fig. 3 (b) becomes the global minimum.
To sum up this section, we obtained the yrast states of the LLH by diagonalization of the Hamiltonian in the weakly interacting regime. Among these eigenvalues of the Hamiltonian in the rotating frame, we distilled the metastable branches from the variety of yrast spectra by imposing an extremization condition. Two kinds of metastable branches, superflow and quantum soliton, were found, consistent with meanfield theory. The region where the different angularmomentum states in the quantum theory densely cross indeed agrees with the soliton regime predicted by the meanfield and Bogoliubov theories. The phrase “quantum flesh sewn onto classical bones” has been used elsewhere davisED2004 (); bokulich2008 () as a visual metaphor, perhaps inspired by xray images, to describe this accord. As we show later, the simple method shown in this section for obtaining metastable states is applicable to other regimes; these two branches continuously exist over a wide range of interaction, from the weakly interacting regime all the way to the strongly interacting TG gas.
Iv TonksGirardeau Limit
In the previous section, we studied the weakly interacting limit of the LLH in the rotating frame to demonstrate how to obtain the meanfieldlike stationary states. In the opposite limit of the strongly interacting TG regime, where the bosons are impenetrable and hence behave like spinless fermions, the eigenproblem can be calculated via the BoseFermi mapping. In this section we thus solve the yrast eigenproblem of free spinless fermions. In particular, we introduce the particle and hole excitations, which are well defined in the fermionized gas, and show that these excitations are related to the meanfield stationary states in the opposite weakly interacting limit.
iv.1 BoseFermi mapping
The BoseFermi mapping theorem 60:Gir () states that the eigenvalues of impenetrable bosons are identical to those of spinless free fermions of the same form of Hamiltonian, and the eigenfunctions of bosons are generally written in terms of those of free fermions as
(14) 
where is a unit antisymmetric function that takes the value or depending on the order of coordinates. This theorem holds for all the eigensolutions, and hence significantly simplifies our eigenproblem. The detailed properties of the TG gas are reviewed in Ref. 02:GW ().
We first calculate the ground and excitedstate energies of free fermions without taking the thermodynamic limit. For simplicity of notation we show the analytic expression only for an odd total number of particles. For an even number of particles the periodic boundary condition must be taken as antisymmetric.
The ground state of (odd) free fermions is obtained by the occupation of the lowest angularmomentum states from to [see in Fig. 5], where is the Fermi momentum. The groundstate energy is thus
(15) 
We note that the dependence of Eq. (15) is the same as that of the bound state for attractive interactions, , except for the prefactors 64:MG ().
iv.2 Particle and hole excitations
We next consider the lowlying excitations. Lieb has shown 63:L () that excitation of the repulsively interacting Bose gas in the thermodynamic limit has two branches. The first branch is called type I and was shown to be in agreement with the Bogoliubov spectrum of plane waves in the weakly interacting regime. The second branch is called type II, and this was supposed to be absent in the Bogoliubov spectrum. Intuitively, the type I and II branches correspond to the particle and hole excitations, respectively.
We reconsider these branches in the context of yrast states. For the excited state with total angular momentum , there exist two kinds of excitations, type I (particle excitations) and II (hole excitations), as originally named by Lieb. To obtain these excitations one uses the following procedure.
(I) Remove a particle at the Fermi momentum and place it at the momentum . For free fermions, there is no energylevel reconstruction in an particle system associated with removal or addition of a particle. The energy of the type I excited state is thus obtained as
(16) 
Relative to the ground state the energy is
(17) 
There is no limitation on the singleparticle angular momentum for this excitation. Such excitations are doubly degenerate for , for .
This type of excitation has an infinite set for the different CMR states . The particle excitation of the state is achieved by removing a particle at and replacing it at . The resulting excitation energy is given by
(18) 
(II) Starting from the ground state, remove a particle (create a hole) at the momentum and place the particle at , where . It is clear from Fig. 5 that the hole with this kind of lowlying excitation energy be created only within the range . The energy of this excited state is given by
(19) 
and the excitation energy is thus
(20) 
At , the particle configuration in the angularmomentum space is the same as in the ground state, provided that the centerofmass angular momentum is shifted. Starting from the state , we can consider the same kind of hole excitation, where we now place a particle at the lowestunoccupied angular momentum , and place a hole at . In this way, the type II excitation is extended for states. This is a hole excitation of angular momentum with a particle fixed at , starting from the yrast state . Since the excitation energy of the CMR state is , the type II hole excitation energy is
(21)  
where .
iv.3 Metastable hole excitation under rotation
Next we consider the excitations under rotation, i.e., rotate all the yrast spectra according to . The energy of the type II excited state for measured relative to is given by
(22) 
In a similar manner to the previous section, we look for the metastable angularmomentum states by imposing the extremization condition . By inspection CMR states are (either ground or excited) metastable states. This extremization condition gives another metastable angular momentum,
(23) 
and the corresponding energy,
(24) 
As a function of this is a parabolic curve the minimum of which is located at with energy , as shown in Fig. 6. Let us next compare this curve with those of two CMR states and . These CMR energies intersect at with the energy . The minimum of (24) is thus higher than the value by at .
Another important point is the emergence of certain critical angular frequencies where the metastable type II branch disappears and merges into the CMR branch:
(25) 
The stable angular momentum approaches at , and at , respectively, and the corresponding energy coincides with the energy of CMR states.
This is reminiscent of the uniform superflow to dark soliton transition in the weakly interacting limit, where there exists a critical angular frequency at which the soliton branch bifurcates from the superflow branch. Thus a continuous crossover between these topologically distinct states exists in the TG limit, as well as the weakly interacting limit. Naturally, we can associate the hole excitations in the TG limit with the soliton branch in the weakly interacting limit.
V FiniteSize Bethe Ansatz
We studied metastable states drawn from the yrast spectrum in the limits of the weakly and strongly interacting TG regimes by the extremization condition of the eigenvalues and by introducing the particle and hole excitations. The goal of this section is (i) to properly interpolate these limits via the finitesize Bethe ansatz approach with the hint that the physical properties of the LLH with repulsive interaction are continuous and (ii) to show that two distinct phases exist over the entire range of repulsive interactions. We also vindicate our hypothesis that the soliton branch in the weakly interacting regime are “hole” excitations of the quasimomenta in general. The ground and excitedstate energies in the thermodynamic limit were given by the integral equation as a continuous limit of the Bethe equations 63:LL (); 63:L (). Recently, the spectrum of LLH was obtained 06:CB () by treating the inverse of the TG parameter, which is infinite at the TG regime, as the expansion parameter, and its analytical interpolation was given 09:CB ().
v.1 Bethe equations for quasimomenta
We first present how to derive the eigensolutions of the LLH by the Bethe ansatz 97:KY (). The deltafunction interaction imposes the conditions
(26) 
Equation (V.1) is rewritten by the interchange of the subscripts as
(27) 
where is the manybody wave function. The periodic boundary condition is expressed as . The entire coordinate space is expressed as where is given by a permutation of . The next procedure for solving the problem is to restrict the original coordinate space to the ordered space and solve the Hamiltonian within the ordered space, say, . The LLH and the condition Eq. (27) yield
(28) 
and
(29) 
respectively, where refers to the manybody wave function in the ordered coordinate space. The periodic boundary condition and its derivative in the region are given by
(30)  
The Schrödinger equation (28) describes free particles. All eigenstates and spectra can therefore be represented formally by those of free particles. The eigenfunction in the region can be written as
(32) 
where are called quasimomenta (or quasiangular momenta). This is the basic idea of the Bethe ansatz: one writes down the manybody wave function in terms of a symmetrized superposition of plane waves with quasimomenta, which implicitly includes all the effects of interactions. This wave function is a superposition of plane waves with distinct quasimomenta , means permutations of quasimomentum indices, and is the coefficient of superposition of plane waves with a different configuration of quasimomenta .
Substituting the wave function (32) into the conditions (29) and (30), we obtain the equations that determine the values of quasimomenta :
(33) 
where
(34) 
is the twobody phase shift.
Note that the quasiangular momenta do not have a physical meaning per se; however, the sum of quasiangular momenta does have a physical meaning as the total angular momentum,
(35) 
Energy is also given in terms of as
(36) 
where the units of quasiangular momentum and energy are the same as the previous sections, and , respectively. From the boundary condition we obtain simultaneous nonlinear equations (Bethe equations),
(37) 
which determine the set of values for each atom . Since all the quasimomenta are known to be real and continuous for positive , Eq. (37) can be separated into real and imaginary parts, both of which are found to give the same set of solutions. Therefore it is sufficient to solve only the real part of the set of equations.
v.2 Weakly interacting regime
We numerically solve the real part of the Bethe equations (37) for each set of energy levels characterized by the different total angular momenta. The numerical solution of Eqs. (37) is highly sensitive to the initial set of trial values of . If this initial set is sufficiently close to a solution for a target angularmomentum state, the set of solutions can be correctly obtained. In contrast, if the initial set is closer to another angularmomentum state, the total angular momentum given by Eq. (35) reveals undesired jumps, deviating from the target angular momentum. In such a case we again start from another initial set of trial values of quasimomenta.
For simplicity we start with consideration of the trivial noninteracting limit where all the quasiangular momenta , and hence the total angular momentum , are zero for the ground state. The energy of the first excited state corresponds to the degenerate yrast levels , where only one of the quasiangular momenta has the value or and the remaining quasiangular momenta are zero. The second excited state has total angular momentum , that is, two of the quasiangular momenta take the value . Higher excited states can be obtained in a similar way. Starting from these initial conditions, solutions can be obtained from the noninteracting to the strongly interacting regime by gradually increasing the value of .
Results of the Bethe ansatz are obtained by the following steps:

For , the quasimomenta are given by , , , , for and for , for example. Each set is directly calculated by solving Eqs. (37) with the target angularmomentum state from to , respectively.

The states are obtained from a transformation of the first states of (i) via , where . The quasimomenta are given by a transformation of the following form. For example, with transforms to with .

There exist degenerate spectra with the same magnitude of angular momentum for the yrast states. The corresponding quasimomenta are given by a transformation, . Although there exist a denumerably infinite number of other kinds of excitations of higher energy, these states are irrelevant for comparison with the previous results. Hence we do not use the larger quantumnumber subscript , as in previous sections.

We gradually increase from zero with the step size for , for , and for . The convergence of the numerical solutions of Eqs. (37) is confirmed by comparing both sides of the Bethe equation with the substitution of the solution . We set the tolerance factor, i.e., the difference in the left and righthand sides of Eqs. (37), to be . We also required as a secondary convergence criterion that the total angular momentum be conserved to better than in the target angular momentum. If, during the changing of , the angular momentum unexpectedly deviates from the target angular momentum, and/or some of quasiangular momenta show a jump as a function of , these errors are detectable. For the attractive case, complex solutions of Eqs. (37) appear 07:SDD (), indicating groundstate soliton formation, which we do not treat here.
In Fig. 7, we compare lowlying excited states obtained by three different theoretical methods: Bethe ansatz, diagonalization, and GP meanfield theory. We note that the concept of yrast state for the angular momenta does not exist in the meanfield theory: this theory is concerned only with the singleparticle angular momentum, which coincides with the average angular momentum in this theory. We thus plot the meanfield energy for the integral singleparticle angular momenta. We plot the first yrast spectra , , as a function of the strength of interaction for . As expected, the rigorous Bethe ansatz spectra have the lowest energy for any , the meanfield planewave branch has the highest value, and diagonalization results have values in between those obtained by the Bethe ansatz and meanfield theory. For , the Bethe ansatz and diagonalization results quantitatively agree very well, while the latter becomes larger than the exact spectra for note2 ().
v.3 Medium to strongly interacting regime
We now further investigate the yrast states via the Bethe ansatz, going beyond the weakly interacting meanfield regime. Figure 8(a) plots for over a wide range of repulsive interactions, where the horizontal dotted line shows the groundstate energy of free fermions given by Eq. (15). All the ground and excitedstate energies monotonically increase with respect to .
Note, however, that the energy does not monotonically increase with respect to the total angular momentum for a fixed strength of interaction. This is clearly shown in Fig. 8 (b), where we show the excitation energy of yrast states for several values of fixed interaction strengths (indicated as A, B, C, D, and E) as a function of . In the noninteracting limit, the the yrast spectrum is linear with respect to with nodes at . While in a weakly interacting limit (plot A), the spectrum still looks almost linear, as the interaction increases, the kinks in the yrast spectra at become more pronounced due to the large increase in the energy of the yrast state in between . For strong interactions (curve E), the system is in the TG regime, which can be confirmed by the fact that the excitation energy has the value between Eq. (21) for and .
Finally, we observe numerically that the excitation energy of the CMR state is independent of , and is given by Eq. (6), namely . This follows from the nature of the CMR state , which is just a Galilean boost of the nonrotating state; under this transformation interactions are unchanged.
In order to see how these points are transformed in the rotating frame, we again rotate the yrast spectrum according to
(38)  
The results are shown in Fig. 9 for various strengths of interaction. The thin curves are parabolas for various values of centerofmass quantum numbers . The lowest possible energy of the CMR state is thus given by at .
The thick curves plot other metastable angularmomentum states, the angular momentum of which is given by a nonintegral multiple of . The weakly interacting meanfield regime is shown in Fig. 9(a), where the type II branch that satisfies the metastable condition just starts to appear. Thus these are the energies of the quantum solitons [see also Fig. 3(a)]. As the interaction increases [Figs. 9(b) and 9(c)] the domain with the swallowtail shape enclosed by the two CMR branches, as well as the size of the type II branch, increases. In the TG limit [9(d)], the area of the swallowtail region saturates the spectra.
These behaviors are quantitatively summarized in Fig. 10(a), which shows the energy of metastable states relative to the interaction energy at each strength of interaction. The CMR branches drawn by the thin curves no longer have a dependence because of the subtraction of , while the thick curve gradually increases the domain over which it extends as increases. For simplicity we plot only two CMR branches with angular momenta , and a metastable state associated with the type II branch that smoothly connects these two CMR states.
As increases, the thick curve appears to bifurcate from the CMR branch with angular momentum at a certain , and at the energy becomes minimum. As increases further, this branch smoothly merges into the CMR branch with angular momentum and eventually disappears at a certain . We therefore find that the same kind of energy bifurcation which was found in the meanfield theory persists over the full range of repulsive interactions.
Figure 10(b) plots the existence range of the metastablestate type II excitation branch. The shaded area indicates the existence of such a branch. For higher angularmomentum uniform superflow states and , the boundary can be obtained by the parallel displacement of this figure along the axis. The angular momentum on the phase boundary is given by , and the angular momentum linearly decreases as increases, just like in Fig. 4 (a). At (vertical dashed line), the value of the angular momentum is given by irrespective of the strength of interaction. At a certain value of the angular momentum eventually goes to zero, causing the metastable hole excitation branch to disappear. This behavior corresponds to the fact that in the meanfield theory the type II branch bifurcates from the planewave regime, developing nodes, and it again merges into the planewave regime with the increase of 08:KCU (); 09:KCU (). The phase boundary approaches in the strongly interacting regime.
In the lower panel of Fig. 10 the phase boundary is compared with the one obtained by Bogoliubov theory in the weakly interacting regime. Bogoliubov theory thus predicts the quantitatively correct phase boundary to the 5 level in the weakly interacting limit (for ), but it significantly overestimates the phase boundary as the interaction increases.
These results indicate that the continuous change in the topologically distinct quantum phases can be found at any strength of interaction. For larger strength of interaction, the existence range of the type II branch increases. In this existence range the angular momentum changes linearly in , and the rate of change thus decreases for larger coupling constant.
Vi Conclusions
We addressed the continuous topological change in the repulsively interacting 1D Bose gas on a ring, previously found in the GrossPitaevskii meanfield theory 09:KCU (). In the meanfield theory the GrossPitaevskii equation has two kinds of solutions: uniform superflow and the brokensymmetry soliton train as a function of interaction strength and rotation. In the weakly interacting regime, the energy diagram is characterized by the smooth bifurcation of a soliton branch from a planewave branch in the rotating frame, which is the key to the continuous change in the topology of the condensate wave function characterized as a selfinduced phase slip.
In this article we vindicated this picture starting from the manybody Hamiltonian without assuming the existence of the condensate wave function and spontaneous symmetry breaking. We solved the yrast problem of the original LiebLiniger Hamiltonian by three methods: diagonalization of the Hamiltonian in the weakly interacting regime; BoseFermi mapping in the strongly interacting TG regime; and the Bethe ansatz approach for all regimes of repulsive interaction strength.
We then obtained the eigensolutions in the rotating frame through transforming the eigenvalues according to specific values of the angular frequency of the external rotating drive . The extremization condition is imposed so that eigensolutions which are realizable in practice are extracted from a very large number of possibilities. The realizable states, namely those metastable under symmetrybreaking perturbations, reveals that two kinds of eigensolutions are physically distinguished. One is the superflow state in which angular momentum is an integral multiple of the number of atoms. The other is a quantum soliton characterized by a set of soliton components, which are also the yrast states. In the weakly interacting regime, the energy and angular momentum obtained by exact diagonalization and the Bethe ansatz agree well with those predicted by the GrossPitaevskii equation. This fact bears out the above physical meanings of metastable states. In the opposite limit, the strongly interacting TG limit was studied by the BoseFermi mapping. We introduced the concept of particle and hole excitations, which are well defined not only in the fermionized system but also for the whole interaction range in terms of the quasimomenta. The solution was similarly transformed into the rotating frame to extract the metastable states.
In between the weakly interacting meanfield and strongly interacting TG limits, we employed the Bethe ansatz approach. In order to compare with the diagonalization results, the set of Bethe equations was solved without substituting the summation with an integral, and the lowest discrete excited states were found. By the same transformation into the rotating frame, we elucidated how the energy diagram of these topologically distinct states changes as the strength of interaction increases.
Energy and angular momentum of the two kinds of topologically distinct states exist over the whole of repulsive interaction . The quantum phase diagram in the  plane for the quantum soliton with a single density notch and with angular momentum was explicitly shown. This metastable quantum phase transition is technically a crossover: it can occur only in a finite system, as expressed by our choice of units in terms of the ring circumference, and all states are connected analytically. Nevertheless, one finds a sharp change between distinct physical states which will appear as a QPT in experiments.
Acknowledgments
We thank Joachim Brand, Marvin Girardeau, Ewan Wright, and Tetsuo Deguchi for useful discussions. This work was supported by the National Science Foundation under Grant PHY0547845 as part of the NSF CAREER program (L.D.C.), a GrantinAid for Scientific Research under Grant Numbers 17071005 (M.U.) and 21710098 (R.K.), and by the Aspen Center for Physics.
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