^{†}Corresponding author. Email: xwe105@wipm.ac.cn
^{*}Project supported by the National Basic Research Program of China (Grant No. 2012CB922101) and the National Natural Science Foundation of China (Grant Nos. 11374331 and 11304357).
This article presents an elementary introduction on various aspects of the prototypical integrable model the Lieb–Liniger Bose gas ranging from the cooperative to the collective features of manybody phenomena. In 1963, Lieb and Liniger first solved this quantum field theory manybody problem using Bethe’s hypothesis, i.e., a particular form of wavefunction introduced by Bethe in solving the onedimensional Heisenberg model in 1931. Despite the Lieb–Liniger model is arguably the simplest exactly solvable model, it exhibits rich quantum manybody physics in terms of the aspects of mathematical integrability and physical universality. Moreover, the Yang–Yang grand canonical ensemble description for the model provides us with a deep understanding of quantum statistics, thermodynamics, and quantum critical phenomena at the manybody physical level. Recently, such fundamental physics of this exactly solved model has been attracting growing interest in experiments. Since 2004, there have been more than 20 experimental papers that reported novel observations of different physical aspects of the Lieb–Liniger model in the laboratory. So far the observed results are in excellent agreement with results obtained using the analysis of this simplest exactly solved model. Those experimental observations reveal the unique beauty of integrability.
Mathematical principles play significant roles in understanding quantum physics. The concept “ exact integrability” originally came from the early study of the classical dynamical systems which were described by some differential equations. Usually, the solutions of those differential equations are tantamount to the determination of enough integration constants, i.e., the integrals of motions. In this sense, classical integrability is synonymous with exact solution. However, conceptual understanding of quantum integrability should trace back to Hans Bethe’ s seminal work to obtain the energy eigenstates of the onedimensional Heisenberg spin chain with the nearest interaction in 1931.^{[1]} He proposed a special form of the wavefunction — superposition of all possible permutations of plane waves in a ring of size L, namely,
where N is the number of down spins and
The Bethe’ s ansatz appeared to be escaped from physicists’ attention that time. It was only 30 years later that in 1963 Lieb and Liniger^{[2]} first solved the onedimensional (1D) manybody problem of deltafunction interacting bosons by Bethe’ s hypothesis. The exact solution for the deltafunction interacting Bose gas was given in terms of the wave numbers k_{i} (i = 1, … , N) satisfying a set of Bethe ansatz equations, called the Lieb– Liniger equations. The spectrum was given by summing up all
Towards a deeper understanding of the physics of the Lieb– Liniger gas, a significant next step was made by Yang C N and Yang C P on the thermodynamics of this manybody problem in 1969.^{[3]} They for the first time present a grand canonical description of the model in equilibrium. In fact, there are many microscopic states for an equilibrium state of the system at finite temperatures. The minimization of Gibbs free energy gives rise to the socalled Yang– Yang equation that determines the true physical state in an analytical way. In the grand canonical ensemble, the total number of particles can be associated with chemical potential μ . The temperature is associated with the entropy S, counting thermal disorder. This canonical Yang– Yang approach marks a significant step to the exact solutions of finite temperature manybody physics. It shows the subtlety of vacuum fluctuation, interaction effect, excitation modes, criticality, quantum statistics, thermalization, dynamics, correlations, and Luttinger liquid.
The name Bethe’ s hypothesis was coined by Yang C N and Yang C P in the study of the Heisenberg spin chain. The Bethe ansatz is now well accepted as a synonym of quantum integrability. From solving eigenvalue problem of spin1/2 deltafunction interacting Fermi gas, Yang^{[4]} found that the manybody scattering matrix can be reduced to a product of many twobody scattering matrices, i.e., a necessary condition for solvability of the 1D manybody systems. The twobody scattering matrix satisfies a certain intertwined relation, called Yang– Baxter equation. This seminal work has inspired a great deal of developments in physics and mathematics. The Yang– Baxter relation was independently shown by Baxter as the conditions for commuting transfer matrices in twodimensional statistical mechanics.^{[5, 6]} For such exactly solved models, the energy eigenspectrum of the model Hamiltonian can be obtained exactly in terms of the Bethe ansatz equations, from which physical properties can be derived via mathematical analysis. The Lieb– Liniger Bose gas^{[2]} and Yang– Gaudin model^{[4, 7]} are both notable Bethe ansatz integrable models.
Yang– Baxter solvable models have flourished into majority in physics since last 70’ s. Later it turned out that the Yang– Baxter integrability plays an important role in physical and mathematical studies.^{[8– 15]} The Bethe ansatz approach has also found success in the realm of condensed matter physics, such as Kondo impurity problems, BCS pairing models, strongly correlated electron systems and spin ladders, cold atoms, quantum optics, quantum statistical mechanics, etc. The Yang– Baxter equation has led to significant developments in mathematics, such as 2D conformal field theory, quantum croups, knot theory, 2D statistical problems, lattice loop models, random walks, etc. Recent research showed that there exists a remarkable connection between conformal field theory and Yang– Baxter integrability of 2D lattice models. Remarkably, conformal field theory has led to the theory of vertex operator algebras, modular tensor categories, and algebraic topology in connection to new states of matter with topological order, such as fractional quantum Hall effect, topological insulators, etc. In this elementary introduction to the exactly solvable Lieb– Liniger model, we will discuss the rigorousness of mathematical integrability and the novelty of quantum manybody effects that comprise the beautiful cold world of manybody systems in the lab. The content of this article involves understanding the fundamental manybody physics through the model of Lieb– Liniger Bose gas.
The paper is organized as follows. In Section 2, we present a rigorous derivation of the Bethe ansatz for the Lieb– Liniger Bose gas. In this section we show how a field theory problem reduces to a quantummechanical manybody systems following the Lieb– Liniger’ s derivation. In Section 3, the properties of the ground state are discussed, including groundstate energy, excitations, cooperative and collective features, Luttinger parameter, etc. In Section 4, we introduce the Yang– Yang grand canonical approach to the finite temperature physics of the Lieb– Linger gas. In this section, we demonstrate how the Yang– Yang equation encode the subtle Bose– Einstein statistics, Fermi– Dirac statistics, and Boltzmann statistics. In particular, we give an insightful understanding of quantum criticality. In Section 5, we briefly review some of recent experimental measurements related to the Lieb– Liniger Bose gas from which one can conceive the beauty of the integrability.
We start with introduction of canonical quantum Bose fields
The Hamiltonian of the 1D single component bosonic quantum gas of N particles in a 1D box with length L is given by^{[2]}
where m is the mass of the bosons, g_{1D} is the coupling constant which is determined by the 1D scattering length g_{1D} = − 2ħ ^{2}/ma_{1D}. The scattering length is given by
In order to process Lieb and Liniger’ s solution, we first define the vacuum state in the Fock space as
Considering
are commutative with the Hamiltonian (1), i.e., [Ĥ , N̂ ] = 0 and [Ĥ , P̂ ] = 0. They are among the conserved quantities of this model. The eigenfunction of the Nparticle state  Ψ 〉 for the operators Ĥ , N̂ , and P̂ is given by
where x = {x_{1}, x_{2}, … , x_{N}} and
Here x_{j} is the coordinate position of the jth particle. For the bosons, the first quantized wavefunction Ψ is symmetric with respect to any exchange of two particles in space x, namely,
In the following discussion, we set ħ = 2m = 1 and c = mg_{1D}/ħ ^{2}. After some algebra, one can find that the eigenvalue problem of the Schrö dinger equation Ĥ  Ψ 〉 = E Ψ 〉 in Nparticle sector reduces to the quantummechanical manybody problem which is described by the Schrö dinger equation HΨ (x) = EΨ (x) with the first quantized form of the Hamiltonian
This manybody Hamiltonian describes N bosons with δ function interaction in one dimension, called the Lieb– Liniger model. This is a physical realistic model in quantum degenerate gases with swave scattering potential. In the dilute quantum gases, when the average distance between particles is much larger than the scattering length, the swave scattering between two particles at x_{ξ } and x_{η } have the following short distance behavior:^{[19]}
where Ψ (x) is the relative wavefunction of the two particles and x is the relative distance between the two particles, i.e., x = x_{η } – x_{ξ }. In the above equation, the prime denotes the derivative with respect to x.
In the model (6), the interaction only occurs when two particles contact with each other. Following the Bethe ansatz, ^{[1]} we can divide the wavefunction into N! domains according to the positions of the particles Θ (𝒬 ) : x_{𝒬 1} < x_{𝒬 2} < … < x_{𝒬 N}, where 𝒬 is the permutation of number set {1, 2, … , N}. The wavefunction can be written as Ψ (x) = ∑ _{𝒬 }Θ (𝒬 ) ψ _{𝒬 }(x). Considering the symmetry of bosonic statistics, all the ψ in different domain 𝒬 should be the same, i.e., ψ _{𝒬 } = ψ _{1}, where we denote the unitary element of the permutation group as 1 = {1, 2, … , N}. Lieb and Linger wrote the wavefunction for the model (6) as the superposition of N! plane waves^{[2]}
where k’ s are the pseudomomenta carried by the particles under a periodic boundary condition.
Indeed, after solving Schrö dinger equation HΨ (x) = EΨ (x), we can obtain the same swave scattering boundary condition (7) that provides the twobody scattering relation among the coefficients
where
When c ≠ 0, all the quasimomenta are different. If there are two equal pseudo momenta k_{a} = k_{b}, we can prove that the wavefunction ψ _{1} = 0.
In general, equation (9) gives the twobody scattering matrix Ŝ ,
then the system is integrable. This relation was independently found by Baxter^{[5, 6]} in studying twodimensional statistical models. Nowadays, equation (11) is called Yang– Baxter equation. The Yang– Baxter equation guarantees that the multibody scattering process can be factorized as the product of many twobody scattering processes. This factorization reveals the nature of the integrability, i.e., no diffraction in outgoing waves. For the Lieb– Linger model (6), Ŝ matrix is a scalar function so that the scattering matrix satisfies the Yang– Baxter equation trivially. In the scattering process from 1 to
With the help of Eq. (12), the eigen wavefunction of the system is given by
Submitting the periodic boundary conditions Ψ (… , x_{ξ } = 0, … ) = e^{iα }Ψ (… , x_{ξ } = L, … ) into the wavefunction (13), we can find that the pseudo momenta k_{l} satisfies the following Bethe ansatz equations (BAE):
which are called the Lieb– Liniger equations. When α = π , the wavefunction is antiperiodic; while when α = 0, it is periodic. In the following discussion, we only consider the periodic boundary conditions.
Since partial number N̂ and momentum P̂ are conversed quantities of the Lieb– Liniger model, the Hamiltonian together with N̂ and P̂ can be simultaneously diagonalized. For the eigenstate (13), the corresponding particle number 〈 N̂ 〉 = N. For a given set of quasimomenta {k_{j}}, the total momentum and the energy of the system are obtained as
The solutions to the BAE (14) provide complete spectra of the Lieb– Linger model. The physical solutions to the Bethe ansatz equations require that all the pseudo momenta are distinct to each other. The BAE (14) can be written in the form of phase shift function θ (k) as
where {I_{i}} are the quantum numbers of pseudo momenta. If N is odd, these quantum numbers are integers, whereas they are half odd integers when N is even. For a given set of quantum numbers {I_{i}}, there is a unique set of real values {k_{i}} for c > 0. These quantum numbers are independent of coupling constant c. The total momentum can be expressed as
For the ground state, P = 0, where all quasimomenta {k_{i}} are located in an interval (– Q, Q). Here Q is the cutoff. For the ground state, the quantum numbers are given by
In the thermodynamic limit, i.e., N, L → ∞ and the particle density n = N/L is a constant, the BAE (16) can be written in the integral form
where ρ (k) is the density distribution function of the quasimomenta defined by the particle numbers in a small interval of (k, k + Δ k), i.e.,
Here the cutoff Q is the “ Fermi point” of the pseudo momenta. It is determined by the particle density
For the ground state, the energy depends on the dimensionless scale γ = Lc/N. Let us make a scaling transformation
we find the groundstate energy per particle,
with
with the cutoff condition
For the ground state, the competition of kinetic energy and interaction energy is represented by the dimensionless coupling strength γ . When γ = 0, the system is free bosons, and all the particles are condensed at the zero momentum state; while if there is a small coupling constant, all the k’ s are distinct. At the limit γ = ∞ , the strong repulsion makes the quasimomentum distribution the same as the one of the free fermions.
It is very insightful to examine the physics of the model in weak coupling limit. In this case the Bethe ansatz roots comprise the semicircle law. Gaudin firstly noticed such a kind of distribution, ^{[20]} followed by several groups.^{[21, 22]} The Fredholm equation (23) is known as the Love equation for the problem of the circular disk condenser. By using Hutson’ s method, Gaudin found the density distribution function and energy density
This result coincides with the perturbative calculation by using the Bogoliubov method.
This problem can also be solved from the original BAE (14). In the weak coupling limit, i.e., Lc ≪ 1, the quasimomenta k_{j} are proportional to the square root of c and c/(k_{j} – k_{l}) is a small value. Up to the second order of c, the BAE (14) is expanded as^{[22]}
where
The solution of this differential equation is a Hermite polynomial and q_{j} is the root of the corresponding polynomial of degree N, i.e., H_{N}(q) = 0. We set the order of q_{j} as q_{j} < q_{j+ 1}, then we have
where the cutoff
In the strong repulsion limit, the gas is known as the Tonks– Girardeau (TG) gas. In realistic experiment with cold atoms, it is practicable to observe the quantum degenerate gas with the strong coupling regime.^{[23, 24]} In the regime γ → ∞ , the Bose– Fermi mapping method^{[18]} can map out the groundstate properties of the Bose gas through the wavefunction of the noninteracting fermions. For finitely strong interaction, γ ≫ 1, we can obtain the groundstate pseudo momenta from the Bethe asnatz equations (14)^{[25, 26]}
where I_{j} take the groundstate quantum numbers
This asymptotic result fits well with the numerical result obtained by solving the integral BAE (19) (see Fig. 2(a)). The groundstate energy (29) can also be obtained from the integral BAE (19) by strong coupling expansion method.^{[26, 28]} We see that for γ → ∞ the leading order of the groundstate energy reduces to that of the free fermions e_{f} = π ^{2}n^{3}/3. One can also calculate other important properties such as compressibility, sound velocity, and Luttinger parameter, once we know the dimensionless function e_{0}(γ ).
Very recently, Ristivojevic obtained highprecision groundstate distribution function for the strong coupling regime^{[29]}
The groundstate energy per unit length and particle density are given by
where λ = (γ + 2)/π − 4π /(3γ ^{2}) + 16π /(3γ ^{3}). The result e_{0} = E/(Ln^{3}) is plotted in Fig. 2.
As discussed in previous section, the eigenstates of the model are described by the quantum numbers {I_{j}}. We define I_{j} as a function of the psudomomenta, i.e., I(k_{j}) = I_{j}, thus we have
Usually we define the occupied Bethe ansatz roots as particles, while the unoccupied roots as holes. For a hole quasimomentum k_{h}, the quantum number I_{h} = I(k_{h}) ∈ ℤ + (N + 1)/2. We demonstrate the quantum numbers I_{j} for the ground state and excited states in Figs. 3(a)– 3(d). For the ground state, there is no hole below the quasiFermi momentum, i.e.,  k < Q, whereas the quantum numbers I’ s for the holes site outside the interval [I_{1}, I_{N}] (see Fig. 3(a)).
In the thermodynamic limit, the pseudo momentum distribution function ρ _{0}(k) for the ground state is determined by Eq. (19) with the cutoff of Q. The groundstate energy and the particle density can be regarded as the function of the cutoff Q, i.e., e(Q) and n(Q). The changes over the configuration of quantum numbers for the ground state give rise to excited states. Figure 3(d) shows such an excitation where the particle with a quasimomentum k_{h} below the pseudo Fermi point is excited outside the Fermi point with a new quasimomentum k_{e}.
For a singleparticle excitation, we decompose the pseudo momentum density into two parts,
Here we choose a state which satisfies Eq. (19) with cutoff Q as a reference state, i.e.,
The addition of the excited particle leads to a collective rearrangement of the distribution of the pseudo momenta of the N − 1 particles. We denote the difference of the pseudo momentum distribution functions between the excited and the reference state
The addition of the particle also leads to a shift of the cutoff over the psuadoFermi point of the ground state of Nparticle, i.e., Δ Q = Q – Q_{G}. Here, Q_{G} is the cutoff for the ground state with the particle number N = Ln_{G}(Q_{G}), and Q is that for the excited state with
where μ is the chemical potential, and
For convenience, we introduce a useful relation between the dressed energy and density. Assuming that two functions f (k) and g (k) satisfy the following equations:
where f_{0}(k) and g_{0}(k) are the driving terms of these integral equations. Then we have the useful relation
By using Eq. (40), from Eqs. (38) and (39), we thus prove that the dressed energy is nothing but the excitation energy of a particle,
On the other hand, the hole excitations impose an additional condition, namely, the maximum momentum is nπ . Similarly, one can prove that the excited energy reads
We plot the dressed energies for different coupling strengths in Fig. 1(b). In order to see clearly excitation spectra, we need to calculate the total momentum using Eq. (17). For both the particle– hole excitation and the TypeII hole excitation, the total particle numbers of particles do not change, as shown in Figs. 3(d) and 4(b). Therefore, the total momenta of the excited states are given by
For the lowenergy excitations, k_{e} − Q is a very small value. Therefore, the lowlying behavior can be described by a linear dispersion relation
where v_{s} is the sound velocity of the system. In the above equation, ± corresponds to the excitations at left/right Fermi point, respectively. Nevertheless, for  k > Q, the relation (44) is the dispersion relation for particle– hole excitations. This linear spectra uniquely determine the universal Luttinger liquid behaviour at the low temperatures. The essential feature resulted from the dispersion relation (44) is such that the system exhibits the conformal invariant in low energy sector. We will further discuss this universal nature of the 1D manybody physics. For the strong coupling, we can obtain the sound velocity from the groundstate energy through the relation
In Figs. 2(a) and 2(b), we present the analytical and numerical results for the dimensionless energy ē (γ ) = e/n^{3} and v_{s}/c, respectively.
In regard of the strong interacting bosons in 1D, the super Tonks– Girardeau gas is particularly interesting. It describes a gaslike phase of the attractive Bose gas which was first proposed in a system of attractive hard rods by Astrakharchik et al.^{[30]} Batchelor and his coworkers showed its existence of such a novel state in the Lieb– Liniger model with a strong attraction.^{[31]} Due to the large kinetic energy inherited from the repulsive Tonks– Girardeau gas, the hardcore behavior of the particles with Fermilike pressure prevents the collapse of the super TG phase after the switch of interactions from repulsive to attractive interactions.^{[32– 35]} In fact, the energy can be continuous in the limits c → ± ∞ . From the Bethe ansatz equations (14), near c → ± ∞ , the compressibility is given by
However, for the Lieb– Liniger gas with weak repulsive interaction (0 < c ≪ 1), compressibility is given by
In contrast, the energy for the gaslike phase (super Tonks– Girardeau gas) in weak attractive interaction limit (− 1 ≪ c < 0) is given by
Thus, the compressibility of the super Tonks– Giraread gas with weak attraction is given by
Such different forms of compressibility reveal an important insight into the root patterns of the quasimomenta. We now show the subtlety of the Bethe ansatz roots for the super Tonks– Girardeau state^{[32]} in Fig. 5.
The 1D integrable system gives rise to the power law behavior of long distance or long time asymptotics of correlation functions for the ground state. The effective Hamiltonian can be approximately described by the conformal Hamiltonian which can be written in terms of the generators of the underlying Virasoro algebra with the central charge C = 1. The lowlying excitations present the phonon dispersion Δ E(p) = v_{s}p in the longwavelength limit. In this limit, all particles participate in the excitations and form a collective motion of bosons which is called the Luttinger liquid.^{[35]} The Lieb– Liniger field theory Hamiltonian can be rewritten as an effective Hamiltonian in longwavelength limit, which essentially describes the lowenergy physics of the Lieb– Linger Bose gas
where the canonical momenta π conjugate to the phase ϕ obeying the standard Bose commutation relations [ϕ (x), π (y)] = iδ (x − y). ∂ _{x}ϕ is proportional to the density fluctuations. In this effective Hamiltonian, v_{s}/K fixes the energy for the change of density. In this approach, the density variation in space is viewed as a superposition of harmonic waves.
For example, the leading order of oneparticle correlation 〈 ψ ^{† }(x)ψ (0)〉 ∼ 1/x^{1/2K} is uniquely determined by the Luttinger parameter K. The Luttinger parameter is defined by the ratio of sound velocity to stiffness, namely,
where v_{s} is the sound velocity and v_{N} is the stiffness and defined as
In Eq. (51), the second expression of the Luttiger parameter is used for numerical calculation with the help of Eq. (22).
Using the asymptotic expansion result of the groundstate energy (24) for weak counting and (29) for the strong counting regimes, we find the asymptotic forms of the Luttinger parameter K for the two limits
In Fig. 6, we show that these asymptotic forms of the Luttinger parameters provide a very accurate expression throughout the whole parameter space. The correlation functions can be calculated by the conformal field theory.^{[11, 36– 39]}
Usually, the longdistance or longtime asymptotics of correlation functions of the 1D critical systems can be calculated by using the canonical field theory (CFT).^{[11]} From the CFT, the twopoint correlation function for primary fields with the conformal dimensions Δ ^{± } is given by^{[40]}
where τ is the Euclidean time, v is the velocity of light, G_{O}(x, t) = 〈 G Ô ^{† }(x, t)Ô (0, 0) G〉 is the correlator for the field operators Ô (0, 0) and Ô ^{† }(x, t). Equation (55) involves the contributions of the excited states which are characterized by numbers Δ D, N^{± }, and Δ N. Here N^{− } (or N^{+ }) describes the elementary particle– hole excitations by moving atoms close to the left (or right) pseudo Fermi point outside the Fermi sea with adding N^{− } (or N^{+ }) holes below the left (or right) Fermi point. Δ N is the change of particle number and it characterizes the elementary excitations by adding (or removing) particles over the ground state. N^{± } and Δ N are not enough to describe all elementary excitations. It is necessary to introduce the quantum number 2Δ D to denote the particle number difference between the right and leftgoing particles. From the analysis of finitesize corrections of the BAE, the total momentum and excited energy of the lowlying excitations in terms of these quantum numbers N^{± }, Δ N, and D are given by
where Z = 2π ρ (Q) is the dressed charge at the pseudo Fermi point for the ground state. From the conformal field theory, the excited energy and momentum are given by
By comparison between the two results obtained from finitesize corrections (56), (57), and CFT (58), (59), the conformal dimensions are analytically obtained as a function of N^{± }, Δ N, Δ D, and the dressed charge Z as well, namely,
It turns out that only the lowenergy excitations determine the long distance or time asymptotic behavior of the correlation functions. Given that the operator
We find that the Luttinger parameter K = Z^{2} by comparing with the result of the Luttinger theory,
In 1969, Yang C N and Yang C P presented a grand canonical ensemble to describe finitetemperature thermodynamics for the Lieb– Linger model.^{[3]} The Yang– Yang method has led to significant developments in quantum integrable systems.^{[12, 13, 41]} This approach allows one to access full finite temperature physics of the models in terms of the thermodynamic Bethe ansatz (TBA) equations.^{[12]} In the grand canonical ensemble, we usually convert the TBA equations in terms of the dimensionless chemical potential μ ̃ = μ /c^{2} and the dimensionless temperature T̃ = T/c^{2} with the interaction strength c. It is also convenient to use the degenerate temperature as an energy unit, i.e., T_{d} = ħ ^{2}n^{2}/2m.^{[42, 43]} It is very insightful to discuss the critical phenomena of the models in terms of the dimensionless units. We first discuss the Yang– Yang grand canonical ensemble below.
For the ground state, the set of quantum numbers (18) provides the lowest energy. However, at finite temperatures, any thermal equilibrium state involves many microscopic eigenstates. As discussed in previous section, these eigenstates are characterized by different quantum numbers {I_{j}}, see Eq. (16). In the thermodynamic limit, I(k) is a monotonic function of pseudo momenta k.^{[11]} We define dI(k)/(Ldk) = ρ (k) + ρ _{h}(k), where ρ _{h} is the density of the holes. From Eq. (16), we can obtain the integral BAE for arbitrary eigenstate as
Here we should notice that integral interval can extend to the whole real axis, i.e., particles can occupy any real quasimomentum.
In order to understand the equilibrium states of the model, it is essential to introduce the entropy. In a small interval dk, the number of total vacancies is L[ρ (k) + ρ _{h}(k)]dk with a number of Lρ (k)dk particles and a number of Lρ _{h}(k)dk holes. These particles and holes give rise to microscopic states
In the thermodynamic limit, [L(ρ (k) + ρ _{h}(k))dk] ≫ 1 and dk → 0, with the help of Stirling’ s formula, the entropy in this small interval is given by
where η (k) = ρ _{h}(k)/ρ (k). The total entropy is S = ∫ dS. It can be understood from this procedure that the entropy involves the disorder of mixing the particles and holes in the pseudo momentum space. For regions with zero ρ (k) or zero ρ _{h}(k), no disorder occurs, i.e., dS = 0. Therefore, the entropy for the ground state is zero. It is worth noting that the above discussion is valid only in the equilibrium state.
The Gibbs free energy Ω is the thermodynamic potential of grand canonical ensemble for this model,
where μ is the chemical potential, and the particle number is given by N = L ∫ ρ (k) dk. In thermal equilibrium, the true physical state is determined by the conditions of minimizing the Gibbs free energy. Making a virtual change δ ρ , δ ρ _{h} in the thermal equivalent state, we take the variation of the free energy such that
Here we should notice that the variations δ ρ and δ ρ _{h} are not independent in view of the integral BAE (62). This minimization condition leads to the TBA equations in terms of the dressed energy^{[3]}
where the dressed energy is defined by ε (k) = Tln η (k). In the above equation, we also denote ε _{− }(k) = − T ln[1 + e^{− ε (k)/T}]. Using the TBA equation (64), we further obtained the grand thermodynamic potential
This serves as the equation of state (65) from which we can calculate the thermodynamics of this model at finite temperatures. We can obtain the zerotemperature and finitetemperature phase diagrams of the Lieb– Liniger model. In the zerotemperature limit, the TBA equation (64) reduces to the dressed energy equation (38) in the limit T → 0. From the standard thermodynamic relations, one can calculate the particle density n = ∂ _{μ }p _{c, T}, entropy density s = ∂ _{Tp} _{μ , c}, compressibility
In view of the grand canonical ensemble, there exists a quantum phase transition at the chemical potential μ _{c} = 0 at zero temperature. Universal thermodynamics is expected for the temperature under the quantum degenerate regime T ≲ T_{d}, where T_{d} = ħ ^{2}n^{2}/2m.^{[42, 43]} At high temperatures (HT), i.e., T ≫ T_{d}, the system behaves like a classical Boltzmann gas. For μ < μ _{c} = 0 and at low temperatures, the density is very low and the gas becomes decoherent. This phase is semiclassical (SC). Whereas for μ > μ _{c} and the temperature T <  μ − μ _{c} , it shows the Tomonaga– Luttinger liquid (TLL) phase. The quantum critical regime lies between SC and TLL for the temperature T ≫  μ − μ _{c} . We show such different regimes through the entropy and specific heat in Fig. 7.
Dynamical interaction and thermal fluctuation drive Lieb– Liniger model from one phase into another. In particular, under the degenerate temperature T_{d}, the model has three distinct phases: semiclassical, quantum critical, and the TTL critical phases. At high temperatures, there does not exhibit universal behaviour. When the temperature tends to be infinity, the system reaches the Boltzmann gas. Therefore, the Yang– Yang equation (64) encodes different quantum statistics. For example, when the coupling strength γ → 0, the system behaves as the free bosons; for the strong coupling limit, it behaves like free fermions; at high temperatures, the system becomes the Boltzmann gas. In the following, we rigorously derive such quantum statistics in an analytical way.
When the coupling strength γ turns to be zero, the integral kernel a(k) → δ (x). The dressed energy in this limit can be expressed as
from which we obtain the thermal potential per length
If γ → ∞ , the dressed energy reads
In this limit, the Bethe ansatz equation (62) naturally reduces to the form
which indicates the Fermi– Dirac statistics. Consequently, the thermal potential per unit length is given by
We further remark that equations (66)– (70) are valid for arbitrary temperature. If the system is under the quantum degeneracy, the quantum statistical interaction is important. Thus, the particles are indistinguishable. At high temperatures, the Yang– Yang equation (64) gives rise to the Maxwell– Boltzmann statistic such that the particles are distinguishable. In the weak coupling limit and hightemperature limits, it is very convenient to consider Viral expansions with the Yang– Yang equation (64), namely,
Here
where
It is remarkable to discover the universal low temperature behavior of the Lieb– Liniger model with the Bose– Einstein statistic and Fermi– Dirac statistic. The Yang– Yang equation (64) provides full physics of the Lieb– Liniger model which goes beyond that can be found by Bose– Fermi mapping.^{[18]} In fact, for strong coupling limit, i.e., γ ≫ 1, the system can be viewed as an ideal gas with the fractional statistics.^{[43]} When the coupling strength is very weak, the ground state behaves like a quasi BEC. The Bogoliubov approach is valid in the weak coupling limit
In lowenergy physics, T ≪ T_{d}, lowlying excitations form a collective motion of bosons. The linear relativistic dispersion near the Fermi points results in the TLL behavior. At finite temperatures, the TLL can be sustained in a region of T <  μ − μ _{c} in the T– μ plane (see Fig. 7). In the TLL phase, we can take the Sommerfeld expansion with the TBA equation (64). By iterations, the pressure with the leading order temperature correction is given by
which gives the free energy per unit length as the field theory prediction
with the central charge C = 1. Here we took k_{B} = 1, so that in TLL phase the specific heat is linear temperaturedependent
This is a universal signature of the TLL.
However, for the temperature beyond the crossover temperature, i.e., T > T* ∼  μ − μ _{c} , the excitations give a nonrelativistic dispersion, i.e., Δ E ∼ p^{2}. The crossover temperature T* can be also determined by the breakdown of linear temperaturedependent relation given by Eq. (75).^{[25]} The crossover is also evidenced by the correlation length.^{[44]}
For strong coupling and low temperatures, i.e., γ ≫ 1 and T̃ ≪ 1, the pressure is given by^{[25]}
where
respectively. Here f_{n} = Li_{n} (− e^{A/T}).
At zero temperature, the quantum phase transition from the vacuum phase into the TLL at the critical point μ _{c} = 0 occurs in the Lieb– Liniger Bose gas. According to the renormalized group theory, universal scaling properties are expected in the critical regime at low temperatures (see Fig. 7). In 2011, Guan and Batchelor investigated the quantum criticality of the Bose gas and found that the equation of state (76) reveals the universal scaling behavior of quantum criticality in terms of the polylogarithm functions.^{[26]} It is straightforward from the equation of the state to derive the universal scaling form of the density as
where the background density n_{0} = 0. The scaling function
where κ _{0} = 0 and
Near the critical point, the specific heat divided by the temperature c̃ _{v} ≡ c_{v}/T obeys the following scaling form:
The specific heat at different temperatures has two round peaks near the critical point μ _{c} = 0. These peaks mark the crossover temperatures that distinguish the TLL and semiclassical gas phases from the quantum critical regime. This is a very robust signature for the existence of the crossover temperatures in the 1D Bose gas. This scaling law of the entropy and specific heat is shown in Fig. 9.
In the study of the interacting Bose gas in one dimension, an important property is the local twobody correlation function g_{2}. Physically speaking, this function describes the rates of inelastic collision between pairs of particles.^{[46– 48]} This quantity reflects the probability that the two particles site at the same size. It is also known as contact that strikingly captures the universality of ultracold atoms. This has been described by Tan’ s relations.^{[49– 51]} Tan’ s contact, which measures the twobody correlations at short distances in dilute systems, is a central quantity to ultracold atoms. It builds up universal relations among thermodynamic quantities such as the large momentum tail, energy, and dynamic structure factor, through the renowned Tan’ s relations, see recent developments in this research.^{[52]}
Knowing that the twobody correlation can lead to the classification of physically distinct regimes, for example, the Tonks– Girardeau regime with g_{2} → 0, the Gross– Pitaevskii regime with g_{2} = 1 and the very weak coupling or fully decoherent regime with g_{2} = 2. The twobody correlation function can be calculated for these different regimes by expression
For strong coupling regime, the local pair correlation function is obtained from the equation of state^{[47, 48, 53, 54]}
On the other hand, in one dimension the fundamental thermodynamic relation in a harmonic trap is given by^{[52]}
where ρ _{s} and C are the densities of superfluid and contact, respectively. In this relation, w = v_{s} − v_{n} is the difference between the velocity of the superfluid and normal components. Maxwell relations build the general connections between the contact and other physical quantities such as
Furthermore, we can obtain the contact through the following relations:
The Tan’ s contact of the Lieb– Liniger gas does not have the usual scaling behavior which was found for the interacting Fermi gas in Ref. [52]. This is mainly because the critical field μ _{c} = 0 which does not depend on the scattering length a_{1D} (see Fig. 10).
Over the past few decades, experimental achievements in trapping and cooling ultracold atomic gases have revealed beautiful physics of the cold quantum world. In particular, recent breakthrough experiments on trapped ultracold bosonic and fermionic atoms confined to one dimension have provided a precise understanding of significant quantum statistical and strong correlation effects in quantum manybody systems. The particles in the waveguides are tightly confined in two transverse directions and weakly confined in the axial direction. The transverse excitations are fully suppressed by the tight confinements. Thus, the atoms in these waveguides can be effectively characterised by a quasi1D system. Thus, 1D effective interaction potentials can be controlled in the whole interacting regime by the underlying 3D scattering with tight confinements in the two transverse directions.^{[16– 18]} In such a way, these 1D manybody systems ultimately relate to the integrable models of interacting bosons and fermions. It is now possible to realize effectively onedimensional quantum Bose gases, in which the interaction strength between ultracold atoms is tunable, see recent reviews.^{[37, 41]} These experiments have successfully demonstrated the anisotropic confinements of atoms to one dimension by optical waveguides, see a feature review article.^{[55]} Particularly striking examples involve the measurements of momentum distribution profiles, ^{[24, 52]} the ground state of the Tonks– Girardeau gas, ^{[25]} quantum correlations, ^{[56– 61]} Yang– Yang thermodynamics, ^{[62, 63]} the super Tonks– Girardeau gas, ^{[64]} quantum phonon fluctuations, ^{[65– 67]} elementary excitations and dark solitons, ^{[68, 69]} thermolization and quantum dynamics.^{[70– 72]} More experimental developments of the Lieb– Liniger model are listed in Table 1.^{[73]}
The early experimental studies of the Lieb– Linger gas with cold atoms were made in the laboratory by Bloch’ s group^{[24]} and Weiss’ s group.^{[25]} In particular, the observation of the ground state energy of the Tonks– Girardeau gas provides deep insights into understanding fermionization effect induced by a strong repulsive interaction, see Fig. 11. Loading the ^{87}Rb ultracold atoms into a 2D array of 1D tubes, where the atoms were kept in the lowest energy state in the two transverse directions. Thus, the systems were realized in quasi1D systems within axial harmonic traps. The essential feature of the Tonks– Girardeau gas was observed through the groundstate energy T_{1D} of such waveguided ensembles.
The experimental measurement of the metastable highly excited state — the super Tonks– Girardeau gas was achieved by Haller^{[64]} in 2009. They made a new experimental breakthrough with a stable highly excited gaslike phase in the strongly attractive regime of bosonic Cesium atoms across a confinementinduced resonance, see Fig. 12. This particular state was first predicted theoretically by Astrakharchik et al.^{[30]} Using the Monte Carlo method and by ANU group from the integrable interacting Bose gas with attractive interactions.^{[31]} This model has improved our understanding of quantum statistics and dynamical interaction effect in manybody physics. It turns out that a highly excited state of gaslike gas could be stable as the interaction is switched from strongly repulsive into strongly attractive interactions due to the the existence of Fermilike pressure.^{[32– 34]} This phenomenon has triggered much attention in theory.^{[80]}
In fact, many experiments have successfully demonstrated the confinements of atoms to one dimension by optical waveguides. Another particularly interesting example involves the measurement of photoassociation rates in onedimensional Bose gases of ^{87}Rb atoms to determine the local pair correlation function g^{(2)}(0) over a range of interaction strengths, see Fig. 13. This experiment provides a direct observation of the fermionization of bosons with increasing interaction strength. It sheds light on the phase coherence behavior.^{[29, 34, 42, 43, 81]} At zero temperature, the local pair correlation is g^{(2)}(0) ∼ 1 for the weakly interacting Bose gas and g^{(2)}(0) → 0 as the system enters into the Tonks– Girardeau regime.
As discussed in previous sections, the finitetemperature problem of the Lieb– Liniger Bose gas was solved by Yang and Yang in 1969.^{[3]} It turns out that the Yang– Yang thermodynamic equation is an elegant way to analytically access the thermodynamics, quantum fluctions, and quantum criticality. The Yang– Yang thermodynamics have been confirmed in the recent experiments through various thermodynamical properties^{[62, 63]} and quantum fluctuations.^{[65– 67]} A typical example is the measurement of the Yang– Yang thermodynamics on the atom chip, see Fig. 14.
Moreover, recent experimental simulations with ultracold atoms provide promising opportunities to test quantum dynamics of manybody systems. In particular, nonequilibrium evolution of an isolated system involves transport and quench dynamics beyond the usual thermal Gibbs mechanism where the ground state and low lying excitation play an essential role. The experimental study^{[70, 71]} of thermalization of 1D ensemble of cold atoms has led to significant developments in this field.^{[82, 83]} In these experiments, it was demonstrated that quenching the dynamics into the isolated systems can lead to nonthermal distributions if conserved laws exist. So far a generalized Gibbs ensemble is believed to present the nonthermal distributions in the isolated systems with conserved laws. The manybody density matrix is written as
in terms of conserved quantities
Recently, Langen et al.^{[76]} showed that a degenerate 1D Bose gas relaxes to a state that can be described by such a generalized Gibbs ensemble. By splitting a 1D Bose gas into two halves, they prepared a nonequilibrium system of ^{87}Rb atoms trapped in an atomic chip. They measured the local relative phase profile φ (z) between the two halves. It was shown that most of the experimentally reachable initial states evolve in time into the steady states which can be determined within a reasonable precision by far less than N Lagrange multipliers. It was particularly interesting to see that the experimental data of the reduced χ ^{2} values can be well fitted with about 10 modes although there exist a much larger number of conserved quantities in the system. This research further opens the study of the generalized Gibbs ensemble for the quantum systems out of equilibrium.
We have introduced a fundamental understanding of manybody phenomena in the Lieb– Liniger model. The exact results for various physical properties of the Lieb– Liniger model at T = 0 and at finite temperature were obtained by using the Bethe ansatz equations. In particular, we have presented a precise understanding of the excitation modes, Luttinger liquid, quantum statistics, quantum criticality, correlations, and dynamics in the context of Bethe ansatz. In fact, there have been great developments in the study of the Lieb– Liniger model in Refs. [11, 15, 37] via various methods, such as field theory methods, ^{[34, 84]} Luttinger liquid theory and bosonization, etc.^{[36, 85]} It was shown that the repulsive LiebLiniger Bose gas can be obtained as the nonrelativistic limit of the sinhGordon model.^{[86– 88]} Moreover, the study of the nonthermal distributions for the isolated systems with many conserved laws has attracted much attention. In this scenario, the generalized Gibbs ensemble^{[83, 89– 92]} has been used to study the thermalization of the isolated systems. More recently, there has been growing interest on quench dynamics in terms of the generalized Gibbs Ensemble.^{[93– 98]} This research has been becoming a new frontier in cold atoms and condensed matter physics. It turns out that the integrable systems of this kind thus provide a promising platform to advance the basic understanding of new quantum effects in manybody physics, such as fewbody problems, universal thermodynamics, universal contact, quench dynamics, and correlation functions. These studies will further place mathematical theories of exactly solvable models into the laboratory for a wide range of physical phenomena.
This brief introduction to the Bethe ansatz solvable model is based on some informal lectures delivered by the author Guan X W at the University of Science and Technology of China, and Institute for Advanced Study at Tsinghua University.
The author Guan X W thanks Murray T Batchelor, YouJin Deng, Hui Zhai, ZhenSheng Yuan, Fei Zhou, and Qi Zhou for helpful discussion. He acknowledges the Department of Physics, Chinese University of Hong Kong for their kind hospitality.
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