New trends in quantum integrability: Recent experiments with ultracold atoms

The Hamiltonian of the Lieb-Liniger gas of $N$ particles with periodic boundary conditions in a length $L$ is given by Lieb-Liniger:1963

where $m$ is the mass of the bosons, $g_{\rm 1D}$ is the coupling constant which is determined by the the 1D scattering length $g_{\rm 1D}=-2\hbar^{2}/ma_{\rm 1D}$. The scattering length is given by $a_{1D}=\left(-a_{\perp}^{2}/2a_{s}\right)\left[1-C\left(a_{s}/a_{\perp}\right)\right]$ Olshanii_PRL_1998 ; Dunjko_PRL_2001 ; Olshanii_PRL_2003 . In the above equation, the canonical quantum Bose fields $\hat{\psi}(x)$ satisfying the following commutation relations

Solving the Schrödinger equation ${H}\varPsi(x)=E\varPsi(x)$ reduces to the eigenvalue problem of the many-particles Hamiltonian

Here the effective coupling constant $c=\frac{g_{\rm 1D}}{2}$.

Following the Bethe ansatz Bethe:1931 the wave function is divided into $N!$ domains according to the positions of the particles $\varTheta({\cal Q}):x_{{\cal Q}_{1}}<x_{{\cal Q}_{2}}<\cdots<x_{{\cal Q}_{N}}$, where $\cal Q$ is the permutation of number set $\{1,2\cdots,N\}$. The wave function can be written as $\varPsi(\boldsymbol{x})=\sum_{\cal Q}\varTheta({\cal Q})\psi_{\cal Q}(\boldsymbol{x})$. Considering the symmetry of bosonic statistics, all the $\psi$ in different domain ${\cal Q}$ should be the same, i.e., $\psi_{\cal Q}=\psi_{{\cal I}}$, where we use ${{\cal I}}$ to denote the elements of the identity permutation, ${{\cal I}}=\{1,2,\cdots,N\}$. Lieb and Liniger Lieb:1968 wrote the wave function for the model (2) as the superposition of $N!$ plane waves

where ${\cal P}$ denotes the permutation of number set $\{1,2\cdots,N\}$ with the pseudo-momenta $k_{i}$s carried by the particles under periodic boundary conditions. They satisfy a set of Bethe ansatz equations, called the Lieb–Liniger equations

Here $j=1,\ldots,N$. For a given set of quasi-momenta $\left\{k_{j}\right\}$, the total momentum and the energy of the system are obtained by

The fundamental physics of the model (2) are essentially determined by the wave function (3) and the Lieb–Liniger equations (4), see a review Jiang:2015 .

In 1969 C N Yang and C P Yang Yang-Yang:1969 published their seminal work on the thermodynamics of the Lieb-Liniger Bose gas. The thermodynamics of the Lieb-Liniger gas is determined from the minimization conditions of the Gibbs free energy in terms of the roots of the Bethe ansatz equations (4). In the thermodynamic limit, the Bethe ansatz equations (4) become

where $\rho(k)$ and $\rho^{h}(k)$ are respectively the particle and hole density distribution functions at finite temperatures. The Gibbs free energy per unit length is given by $G/L=E/L-\mu n-Ts$ with the relation to the free energy $F=G+\mu N$. Here $\mu$ is the chemical potential, $n$ is the linear density, and $s$ is the entropy. The minimization condition $\frac{\delta G}{L}=0$ with respect to particle density $\rho$ leads to the Yang-Yang thermodynamic Bethe ansatz (TBA) equation Yang-Yang:1969

which determines the thermodynamics of the system in the whole temperature regime through the Gibbs free energy per length, i.e.,

Thus other thermodynamic quantities can be obtained through thermodynamic relations, for example the particle density, entropy density, compressibility, specific heat are given by

In the above equation $\partial_{x}$ denotes the derivative respect to the potentials. Exact Bethe ansatz solution provides an analytical way to access to quantum critical behaviour of the system at finite temperatures Guan:2011 ; Jiang:2015 .

1D models of Bethe ansatz solvable quantum gases have become an important subject of experiment, such as two-component spinor fermions and bosons as well as multi-component fermions. Models of this kind have rich internal degrees of freedom and high symmetries. In 1967, C N Yang Yang:1967 solved the 1D delta-function interacting Fermi gas wit the discovery of quantum integrability, i.e. the necessary condition for the Bethe ansatz solvability. This quantum integrability condition was referred to the factorization condition–the scattering matrix of a quantum many-body system can be factorized into a product of many two-body scattering matrices. At the same time, M Gaudin also rigorously derived the Bethe ansatz equations for the spin-$1/2$ Fermi gas for spin balance case Gaudin:1967 . Therefore, the 1D spin-$1/2$ Fermi gas is now called the Yang-Gaudin model. On the other hand, in 1972, R J Baxter Baxter:1972 independently found that a similar factorization relation also occurred as the conditions for commuting transfer matrices in 2D vertex models in statistical mechanics. Such a factorization condition is now known as the Yang-Baxter equation and becomes a significant key condition to quantum integrability.

The many-body Hamiltonian for the Yang-Gaudin model is given by Yang:1967 ; Gaudin:1967

describes $N$ fermions of the same mass $m$ with two internal spin states confined to a length $L$ interacting via a $\delta$-function potential. In the above equation, prime denotes the exclusion of the interaction between two atoms with the same spin. Here we denoted the numbers of fermions in the two hyperfine levels $|\uparrow\rangle$ and $|\downarrow\rangle$ as $N_{\uparrow}$ and $N_{\downarrow}$, respectively. The total number of fermions and the magnetization were denoted as $N=N_{\uparrow}+N_{\downarrow}$ and $M^{z}=(N_{\uparrow}-N_{\downarrow})/2$, respectively. The interaction strength is given by $g_{1D}=\hbar^{2}c/m$ with $c=-2/a_{1D}$, where $a_{1D}$ is the effective 1D scattering length, see the discussion on the Hamiltonian of the Lieb-Liniger gas. For convenience, we also define a dimensionless interaction strength $\gamma=c/n$. Here the linear density is defined by $n=N/L$; meanwhile, $c>0$ for a repulsive interaction and $c<0$ for an attractive interaction, see review Guan:2013 .

C N Yang generalized Bethe’s wave function to the following many-body wave function of the spin-$1/2$ Fermi gas

for the domain $0<x_{Q1}<x_{Q2}<\ldots<x_{QN}<L$. Where $\{k_{i}\}$ denote a set of unequal wave numbers and $\sigma_{i}$ with $i=1,\ldots,N$ indicate the spin coordinates. Both $P$ and $Q$ are permutations of indices $\{1,2,\ldots,N\}$, i.e. $P=\left\{P_{1},\ldots,P_{N}\right\}$ and $Q=\left\{Q_{1},\ldots,Q_{N}\right\}$. The sum runs over all $N!$ permutations $P$ and the coefficients of the exponentials are column vectors with each of the $N!$ components representing a permutation $Q$. For determining the wave function associated with the irreducible representations of the permutation group $S_{N}$ and the irreducible representation of the Young tableau for different up- and down-spin fermions, C N Yang found a key condition for solvability, i.e., the two-body scattering matrix acting on three linear tensor spaces $V_{1}\otimes V_{2}\otimes V_{3}$

satisfies the following cubic equation

which has been known as the Yang-Baxter equation Yang:1967 ; Baxter:1972 . It is worth mentioning that an experimental simulation of the Yang-Baxter equation itself through the linear quantum optics Zhang:2013 and the Nuclear Magnetic Resonance interferometric setup Vind:2016 has been performed. In the above equations, we denoted the operator $T_{ij}=-P_{ij}$, here $P_{ij}$ is the permutation operator. The Yang-Baxter equation has found a profound legacy in both mathematics and physics, see review 1D-Hubbard ; Korepin ; Sutherland-book ; Takahashi-b ; Wang-book ; Guan:2013 .

Solving the eigenvalue problem of $N$ interacting particles (10) in a periodic box of length $L$ with the periodic boundary conditions, C N Yang obtained the Bethe ansatz equations for the Fermi gas Yang:1967

with $j=1,2,\cdots,N$ and $\alpha=1,2,\cdots,M$. Here $M$ is the number of atoms with down-spins. The energy eigenspectrum is given in terms of the quasimomenta $\left\{k_{i}\right\}$ of the fermions via $E=\frac{\hbar^{2}}{2m}\sum_{j=1}^{N}k_{j}^{2}$. In the above equations (14) and (15), $\lambda_{\alpha}$ denoted spin rapidities. All quasimomenta $\left\{k_{i}\right\}$ are distinct and uniquely determine the wave function of model Eq.(11).

The fundamental physics of the model (10) can be obtained by solving the transcendental Bethe ansatz equations (14) and (15). Finding the solution of the Bethe ansatz equations (14) and (15) is extremely cumbersome. For a repulsive interaction, in the thermodynamic limit, i.e., $L,N\to\infty$, and $N/L$ is finite, the above Bethe ansatz equations can be written as the generalized Fredholm equations

The associated integration boundaries $B_{1}$, $B_{2}$ are determined by the relations

where $n_{t}$ denotes the linear density, while $n_{\downarrow}$ is the density of spin-down fermions. In the above equations we introduced the quasimomentum distribution function $\rho(k)$ and distribution function of the spin rapidity $\sigma_{1}(\lambda)$ for the ground state.

The thermodynamics of the Yang-Gaudin model with a repulsive interaction was studied by C K Lai Lai:1971 ; Lai:1973 and M Takahashi Takahashi:1971 . In the same fashion as the derivation of the thermodynamics of the Lieb-Liniger Bose gas by C. N. Yang and C. P. Yang in 1969 Yang-Yang:1969 , M Takahashi Takahashi:1971 gave the thermodynamic Bethe ansatz equations for the repulsive Yang-Gaudin model

with $n=1,\ldots,\infty$. Here $*$ denotes the convolution, i.e. $\hat{a}\ast f(x)=\int_{-\infty}^{\infty}a(x-y)f(y){\rm d}y$, $\varepsilon(k)$ and $\phi_{n}(\lambda)$ are the dressed energies for the charge and the length-$n$ spin strings, respectively, with $k$’s and $\lambda$’s being the rapidities; and the function $T_{mn}(x)$ is given by

Where we denoted $a_{m}(x)=\frac{1}{2\pi}\frac{mc}{(mc/2)^{2}+x^{2}}$. The pressure is given by

from which all the thermal and magnetic quantities, as well as other relevant physical properties, such as Wilson ratio Guan:2013b and Grüneisen parameter Peng:2019 ; Yu:2020 , can be derived according to the standard statistical relations.

For a attractive interaction, i.e. $c<0$, the root patterns of the BA equations (14) and (15) are significantly different from that of the model for $c>0$. In this regime the quasimomenta $\left\{k_{i}\right\}$ of the fermions with different spins form two-body bound states Takahashi-a ; Gu-Yang ; Guan:2007 , leading to the existence of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing state Larkin1965 ; Fulde1964 ; Yang2001 . Here we will not discuss the attractive Yang-Gaudin model Orso:2007 ; Hu:2007 ; Guan:2007 ; Liu:2008 ; Zhao:2009 ; LeeGuan:2011 ; HeJS:2009 , a more detailed study of this model can be found in the review article Guan:2013 .

It is shown that the fermionic alkaline-earth atoms display an exact $SU(w)$ spin symmetry with $w=2I+1$, where $I$ is the nuclear spin Cazalilla:2009 ; Gorshkov:2010 . For example, the experiment Taie:2010 for ¹⁷¹Yb dramatically realised the model of fermionic atoms with $SU(2)\times SU(6)$ symmetry where electron spin decouples from its nuclear spin $I=5/2$, and ⁸⁷Sr atoms have $SU(10)$ symmetry XZhang2014S . These experimental developments have opened exciting opportunities to explore a wide range many-body phenomena such as spin and orbital magnetism Cappellini2014PRL ; Scazza2014NP , Kondo spin-exchange physics Nakagawa:2015 ; Zhang2015 and the one-dimensional (1D) Tomonaga-Luttinger liquid (TLL) Pagano:2014 etc. Such fermionic systems with enlarged $SU(w)$ spin symmetry are expected to display a remarkable diversity of new quantum phases and quantum critical phenomena due to the existence of multiple charge bound states Guan:2013 ; Wu:2006 . Here we do not wish to review the developments of $SU(w)$ Fermi gases in details. We prefer to briefly introduce the integrable $SU(w)$ Fermi gases with repulsive interactions.

We introduce the Hamiltonian of $w$-component Fermi gas of $N$ particles with mass $m$ Sutherland:1968

where $\hat{S}_{z}$ is the total spin of the $z$-direction, $\hat{S}_{z}=\sum_{r=1}^{w}[-(w+1)/2+r)]N_{r}$ and $h$ is the external magnetic field. In the above equation, the prime stands for the exclusion of the interaction between two atoms with the same spin. Here $N_{r}$ is the particle number in the hyperfine state $r$. There are $w$ possible hyperfine states $|1\rangle,|2\rangle,\ldots,|w\rangle$ that the fermions can occupy. Again the interaction coupling constant is given by $g_{\mathrm{1D}}=-2\hbar^{2}/ma_{\mathrm{1D}}$, and $c=mg_{1D}/\hbar^{2}$. Here $a_{\mathrm{1D}}$ is the effective scattering length in 1D. The Hamiltonian (26) has the symmetry of $U(1)\times SU(w)$ when the magnetic field is absent, where $U(1)$ and $SU(w)$ are the symmetries of the charge and spin degrees of freedom, respectively.

Using Bethe’s hypothesis, B Sutherland exactly solved the model by giving the energy $E=\sum_{j=1}^{N}k_{j}^{2}$, and the Bethe ansatz equations

Here $\{\lambda^{(r)}\}$, $r=1,2,\cdots,w-1$ denote the spin rapidities which are introduced to describe the motion of spin waves. The particle number $N_{r}$ in each spin state links to the quantum number $M_{\alpha}$ via the relation $N_{r}=M_{r}-M_{r-1}$ and $M_{0}=0$. For the repulsive interaction, i.e. $c>0$, there is no charge bound state, whereas each branch of spin rapidities $\{\lambda^{(r)}\}$ has complex roots

in the thermodynamic limit, which are now called spin wave bound states.

Following Yang-Yang’s grand canonical method Yang-Yang:1969 the thermodynamic Bethe ansatz equations for the Fermi gases (26) are given by Takahashi-b ; Guan:2013 ; Schlottmann:1997 ; Lee:2011 ; Jiang:2016

Where $\varepsilon^{\rm c}(k)$ and $\varepsilon^{r,n}(k)$ are the dressed energies for the charge sector and for the branch $r$ in the spin sector, respectively and $\varepsilon_{\pm}=T\ln(1+{\rm e}^{\pm\varepsilon/T})$. As a convention used in the TBA equations, $n$ labels the length of the strings, $\ast$ labels the convolution integral and the integral kernels are given by

The pressure is given by

which serves as the equation of states for the repulsive $SU(w)$ Fermi gases.

The Bethe ansatz equations (27)-(32) and the thermodynamic Bethe ansatz equations (34)-(37) impose a big challenge in obtaining physical properties of the model Batchelor:2007 ; Schlottmann:1997 ; Lee:2011 ; Guan:2012a ; Guan:2012b ; Jiang:2016 ; Yang-You:2011 . At low temperature, the behaviour of the Fermi gases with a repulsive interaction is described by spin-charge separated conformal field theories of an effective Tomonaga-Luttinger liquid and an antiferromagnetic $SU(w)$ Heisenberg spin chain, see experiment observation Pagano:2014 . It was found Jiang:2016 ; Liu:2014 ; Schlottmann:1993 that the sound velocity of the Fermi gases in the large $w$ limit coincides with that for the spinless Bose gas, whereas the spin velocity for balance gases vanishes quickly as $w$ becomes large. This novel feature of 1D multi-component Fermi gases shows a strong suppression of the Fermi exclusion statistics by the commutativity feature among the $w$-component fermions with different spin states in the Tomonaga-Luttinger liquid phase, which was for the first time proved by Yang and You Yang-You:2011 in 2011, and confirmed by Guan and coworkers Guan:2012b in 2012. This feature was recently demonstrated with ultracold atoms in 2D Song:2020 .

Moreover, the 1D multi-component Fermi gases exhibit significantly different features from usual interacting electrons, showing rich pairing phenomena in momentum space, see review Guan:2013 .