A unified approach to the thermodynamics and quantum scaling functions of one-dimensional strongly attractive $SU(w)$ Fermi Gases
Yi-Cong Yu Xiwen Guan









 





    G
        State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
        Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, ChinaUniversity of Chinese Academy of Sciences, Beijing 100049, China.[]xiwen.guan@anu.edu.au;xwe105@wipm.ac.cnState Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
        Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, ChinaCenter for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, ChinaDepartment of Theoretical Physics, Research School of Physics and Engineering,
        Australian National University, Canberra ACT 0200, Australia05.30.Fk, 02.30.Ik,03.75.Ss
        
        In this letter we present a unified derivation of the pressure  equation of states, thermodynamics and scaling functions for the one-dimensional (1D) strongly attractive  Fermi gases with SU(w) symmetry. These  physical quantities provide a rigorous understanding on a universality class of quantum criticality characterised by the critical exponents z=2 and correlation length exponent ν=1/2. Such a universality class of quantum criticality  can occur when  the Fermi sea of one branch of charge bound states starts to fill or become  gapped  at zero temperature.  The quantum critical cone   can be determined through  the double peaks     in  specific heat  which  serve to mark two crossover temperatures fanning out from the critical point.  Our method opens to further  study on  quantum phases and  phase transitions in strongly  interacting fermions with large SU(w)  and non-SU(w) symmetries in one dimension. 
        
        A unified approach to the thermodynamics and quantum scaling functions of  one-dimensional  strongly attractive SU(w) Fermi  Gases
            Xi-Wen Guan
            December 30, 2023
        ==================================================================================================================================
        
        
        
        
        
        
        The experimental realization of the 1D quantum gases, such as repulsive Bose gases <cit.>, Yang-Gaudin model <cit.>, multicomponent attractive Fermi gases <cit.>,  has provided a remarkable test ground for exactly solvable models. The mathematical theory of Bethe ansatz integrable models has become testable in ultracold atoms. The Bethe anssatz  has also found  success for other significant models like the Kondo physics <cit.>, BCS pairing model <cit.>, strongly correlated electronic systems <cit.>, spin ladders <cit.> and quantum degenerate  gases <cit.>. 
        
        Recent studies of the 1D Fermi gases with high spin symmetries <cit.> has given  many theoretical predictions on the existence of  bound states of multiparticles, quantum liquids and phase transitions. In this regard, exploring exactly solvable models of interacting fermions with high mathematical symmetries is highly desirable in order to understand new phases of matter.  However,  the thermodynamic properties of exactly solvable models with high symmetries at finite temperatures are notoriously difficult to extract and  present a formidable challenge. 
        Building on Yang-Yang thermodynamic Bethe ansatz equations,  here we present a unified approach to the thermodynamics and quantum critical scalings in  1D strongly attractive  Fermi gases with SU(w) symmetry.  Analytical results of the equation of states(EOS), dimensionless ratios and  scaling functions of thermal and magnetic properties  provide a rigorous understanding on a universality class of quantum criticality of free fermions. The quantum critical region   can be determined through  the double maxima    in  specific heat  which  characterize the  two crossover temperatures fanning out from the critical point.  These results suggest to experimentally test the universal nature of multicomponent quantum liquids  through the  1D strongly attractive  Fermi gases of ultracold atoms  <cit.>. 
        
        
         The model. The 1D  SU(w) Fermi gases  with δ-function interaction
        confined to length L  is described by the   following Hamiltonian <cit.>
            H   =   -ħ^2/2m∑_i=1^N∂^2/∂
            x_i^2+g_1D∑_1≤ i<j≤
            Nδ(x_i-x_j)
                  - E_z- μ N,
        
        and with the chemical potential μ and the  effective Zeeman energy E_z =
        ∑_r=1^w1/2r(w-r) n_rH_ r.
        Here  N is the total particle number.
        
        There are w possible hyperfine states |1⟩, |2⟩, …,
        |w⟩ that the fermions can occupy.
        Experimentally, g_1D=-2ħ^2/m a_1D, with a_1D the effective scattering length in 1D <cit.>,  can be tuned from a weak interaction to a strong coupling regime  via Feshbach resonances.
        For our convenience, from now on, we choose our units such that ħ^2=2m=1 unless we particularly use the units.  
        
        In this model,  the two-body charge bound states involve the Bethe ansatz roots
        {λ_j± i c/2}, j=1...M_2 and the three-body bound states
        {λ_j ± i c, λ_j},j=1...M_3, and so on, where M_2 and M_3
        are the numbers of charge bound states and three-body bound states,
        respectively <cit.>.  The thermodynamics of the model  are determined by the effective external fields H_r,
        chemical potential, interaction between different particles  and spin
        wave fluctuations. 
        
        
        
         The TBA  equations. The thermodynamics of  the Hamiltonian (<ref>)  is determined by  the following TBA equations <cit.>
            ϵ^(r)(k)   =   rk^2 - rμ- H_r - r(r^2-1)c^2/12
                  
            -∑_q=1^w â_rq∗ F[ϵ^(q)]
            +∑_q=1^∞ a_q ∗ F[η_r,q],
        
            η_r,l(k)    =    l ·(2 H_r - H_r - 1 - H_r + 1) 
            - a_l ∗ F[ϵ^(r)] 
                   - ∑_q=1^∞ U_lq∗ F[η_r,q] 
             + ∑_q=1^∞ S_lq∗ F[η_r-1,q]
                  + ∑_q=1^∞ S_lq∗ F[η_r+1,q],
        
        where we denote 
        
            a_n(x)   =   1/2πn|c|/(nc/2)^2+x^2,
            â_lj(x)   =   ∑_q=1,2q ≠ l+j^min(l,j) a_l+j-2q(x),
            
            F[ε]    ≜    -T ln[1+exp(-ε/T)].
        
        In the above equations, ∗ denotes the convolution 
        (f ∗ g)(λ) = ∫_-∞^∞ f(λ-λ') g(λ') dλ' and 
        the functions   U_lj(x) and S_lj(x) are given in <cit.>.
        From the dressed energies ϵ^(r)(k) for bound states of r-atoms with r=1,⋯, w, one can obtain the pressure 
        
            p=∑_r=1^w rT/2π∫_-∞^∞ 
              dk ln(1+e^-ϵ^(r)/T).
        
        The summation of the pressures of all charge bound states services as the EOS,
        from which we can obtain  full thermodynamics of the model at the  temperatures ranging from zero to high.  This form of the EOS gives rise to the additivity nature of quantum liquids in low temperatures <cit.>. 
        
        
        We are interested in the low temperature behaviour of interacting fermions with high symmetries  in 1D. 
        We can see from  the TBA equations (<ref>) that  the  fermomagnetic ordering (the second term in (<ref>)) drives the spin contributions η_r,l(k) to the dressed energies of charge bound states exponentially small in strongly attractive regimes.  Consequently, we can ignore all the string contributions in 
        the above TBA equations when temperature is much less than the binding energies of the charge bound states ε_r=1/48 r ( r^2 -1) g_1D^2. In the recent study <cit.>, it has been proved  that the dimensionless Wislon  ratios, i.e. either the ratio of the susceptibility χ   to the specific heat c_V divided by the temperature
        
            R_ W^ s=4/3(π k_B/μ_B g )^2χ/c_V/T,
        
        or the  ratio  of  the compressibility κ  to the specific heat c_V divided by the temperature
        
            R_ W^ c=k_B^2π^2 /3κ/c_V/T,
        
        essentially captures the  quasiparticle   nature of Fermi liquid  <cit.>,  as its value characterizes the interacting effect in the Fermi liquid.
         
        Here k_B is Boltzmann's constant, μ_B is the Bohr magneton and g is the Lande factor.
        The two types of dimensionless ratios (<ref>) and  (<ref>)   characterise   a competition between the   fluctuations of two thermodynamic quantities.
        
        Thus  a constant  Wilson ratio    implies that  the two types of  fluctuations are on an equal footing with respect to the temperature, regardless of the microscopic details of many-body systems <cit.>. In Fig. <ref>, we demonstrate the compressibility Wilson ratio elegantly maps out the full  phase digram of the SU(2) Fermi gas <cit.> with a strong attraction at T=0.001ε_2/k_B, where ε_2 is the binding energy of a  bound pair. This  turns out that the low temperature SU(2) TBA equations (<ref>) provide rigorous results of quantum liquid behaviour and quantum criticality,  prompting  us  to explore universal thermodynamics and quantum scaling functions for the high symmetry Fermi gases with strong attractions through the SU(w) TBA equations. 
        
        
        
        
        
        
        
        Without loss of accuracy at  low temperature thermodynamics, we  simplify the TBA equations (<ref>)  into  the following form 
        
            ϵ^(r)(k)=V_r-∑_q=1^w â_rq∗ F[ϵ^(q)] , 
                 r=1,2,⋯,w
        
        with V_r=rk^2-rμ-H_r-1/12r(r^2-1) c^2,
        where  the last effective field H_w is set to zero due to the spin singlet charge bound states. 
        For example,
         the SU(3) case, i.e. w=3 in the Eq. (<ref>)
         determines  the low temperature properties of the 1D  three-component strong interacting fermions <cit.>. 
        
         Equation of states.
        In the strong coupling region, i.e. |c| ≫ E_F (E_F is the Fermi energy) and at  low temperatures, we have the following expansion form
        
            -a_n*F[ϵ^(m)](k) ≈n|c|/(nc/2)^2+k^2p^(m)/m
                      +1/2π32/3m√(m)1/n^3|c|^3Γ(5/2)T^5/2 Li_5/2(-e^A^(m)/T),
        
        which is obtained  by integration by parts. In (<ref>) the 
        Li(x) is the polylog function. Substituting  this relation into 
        the TBA equations <ref>, we have 
        
            ϵ^(r)(k)   =   V_r+∑_m=1^w ∑_^min(r,m)4/|c|p^(m)/m(r+m-2q)
                  -∑_m=1^w ∑_^min(r,m)16/|c|^3p^(m)/m(r+m-2q)^3k^2
                  +∑_m^w Z_rmT^5/2/|c|^3 Li_5/2 (-e^A^(m)/T).
        
        In the above equation, we defined matrix,
        
            Z_rm=∑_^min(r,m)4/√(π)1/(r+m-2q)^3 m^3/2
        
        and the A^(m) collects  the constant  terms (terms are independent of the k) in the dressed energy ϵ^(m)(k). We would like to mention that the polylog functions involve different modes of generating functions of Fermi integrals. In contrast to the Somerfield expansion with respect to the powers of the temperature t, here the polylog functions contain enough thermal and quantum fluctuations   that are acquired by the quantum criticality. Therefore, the polylog functions essentially characterise the singular behaviour of the 1D strongly interacting fermions even near quantum phase transitions. 
        If we  consider the first three orders in  the pressure, only the constant term and the 
        quadratic term (O(k^2) terms) contribute to the thermodynamic quantities, and we can safely 
        drop  high order terms  in k .
        Explicitly, we express the dressed energy as 
        
            ϵ^(r)=r D_r k^2- A^(r),
        
        Integrating  eq.(<ref>) by parts we have
        
            p^(m)=-√(m)/2√(π D_m) T^3/2 Li_3/2
                (-e^A^(m)/T).
        
        The parameter D_r is a  modification to  the quadratic term in  the dressed energy ϵ^(m)(k) and could be read off  from the Eq.(<ref>)
        
            D_r=1-∑_m=1^w ∑_^min(r,m)16/|c|^3p^(m)/rm(r+m-2q)^3 .
        
        Here we only consider the first three orders in the dressed energy equations, thus  D_r ≈ 1, and the EOS becomes
        
            p^(r)   =   -√(r)/2√(π) T^3/2 Li_3/2
                (-e^A^(r)/T), 
            
            A^(r)   =    A_0^(r)-∑_m=1^N D_rm p^(m)/|c| 
            - T^5/2Z_rm Li_5/2(-e^A^(m)/T)/|c|^3,
        
        where A^(r)_0 and D_rm are determined by the Eq.(<ref>)
        
            A^(r)_0   =   r μ+H_r+1/12r(r^2-1)c^2,
            
            D_rm   =   ∑_^min(r,m)4/m(r+m-2q).
        
        In order to simplify the EOS,  we define the dimensionless quantities and  parameters
        
            p̃^(r)   =   p^(r)/|c|^3,  Ã^(r)=A^(r)/|c|^2,
            μ̃   =   μ/|c|^2,  h_r=H_r/|c|^2,  
            t=T/|c|^2.
        
        Then the dimensionless EOS is given by 
        
            p̃^(r)=   -√(r)/2√(π) t^3/2 Li_3/2
                (-e^Ã^(r)/t), 
            Ã^(r)=   Ã_0^(r)-∑_m=1^N D_rmp̃^(m) 
            -Z_rm t^5/2 Li_5/2
            (-e^Ã^(m)/t).
        
        where  Ã_0^(r)=r μ̃+h_r+1/12r(r^2-1). 
        For simplifying our notations, we further define matrices 
        
            ( Li_s)_rm= Li_s
                (-e^Ã^(r)/t)δ_rm,  
                (D)_rm=D_rm,
                       (Ẑ)_rm=Z_rm,   (Ã)_r1=Ã^(r),  
             (p̃)_r1=p̃^(r),
                    (M_r)_rm=√(r)/2√(π)δ_rm,  (F_s)_rm≜ t^s  Li_s(-e^Ã^(r)/t)δ_rm,
                  (f_s)_rm≜ t^s  Li_s(-e^Ã_0^(r)/t)δ_rm.
        Li_5/2, Li_3/2,
             M_r,D and Z̃ are square matrices ( or the  column matrices  when they are at the most right of the related terms),  p̃ and 
             Ã are column matrices.
           For example, in the SU(2) case
        
            D =
            (
            [ 0 2; 4 1;   ]),
                Z=
            (
            [         0 √(2)/√(π);    4/√(π)  1/4√(2π);           ]),
        
        and for SU(3) case,
        
            D =
            (
            [    0    2  2/3;    4    1 16/9;    2  8/3    1;      ]),  Z=
            (
            [           0   √(2)/√(π)    1/6√(3π);      4/√(π)    1/4√(2π) 112/81√(3π);     1/2√(π)  56/27√(2π) √(3)/16√(π);             ]).
        
        With the help of these notations,
        we rewrite  a unified expression of EOS for the SU(w) strongly attractive Fermi gases
        
            p̃
            =   -M_rt^3/2 Li_3/2,  
            Ã
            =   Ã_̃0̃-Dp̃-Ẑ 
            t^5/2 Li_5/2.
        
        The last term in the function Ã is negligible in pressure. Nevertheless, it is 
        necessary in the  calculation of the scaling function or phase boundaries. 
        After a lengthy iteration, we get a close form  of the EOS 
         
            p̃
            =   -M_rF_3/2,
            
            Ã≈   Ã_̃0̃+DM_rf_3/2
            +DM_rf_1/2DM_rf_3/2.
        
        
        Furthermore,  we could obtain all the 
        thermodynamic quantities of the system in equilibrium by standard thermodynamic relations via  the  pressure  Eq.(<ref>), which  serves as  the grand thermodynamic potential of
        the system. In this context,  the partial derivatives of the pressure by any chemical potential, external  fields and temperature are essential in our approach. Thus we take the 
        derivative of the pressure Eq.(<ref>)  with respect to the variable η:
        η=μ̃,h_1,h_2,....  It follows that 
        
            ∂𝐩̃/∂η
            =   -M_r𝐅_1/2∂𝐀̃/∂η,
            ∂𝐀̃/∂η
            =   ∂𝐀̃_0/∂η-𝐃∂𝐩̃/∂η
            -Ẑ𝐅_3/2∂𝐀̃/∂η.
        
        By solving the above two linear equations,  we obtain the first order derivative  thermodynamic properties
        
            ∂𝐩̃/∂η
            =   -M_r F_1/2(𝐈+DM_r F_1/2
            +(DM_r F_1/2)^2)
            ∂𝐀̃_0/∂η, 
            
            ∂Ã/∂η
            =   (𝐈+DM_r F_1/2
            +(DM_r F_1/2)^2)
            ∂𝐀̃_0/∂η.
        
        
        For the second order phase transitions, the second derivatives of the pressure, 
        the compressibility or the susceptibility for instance, give a deep insight into the quantum criticality of the systems. Similarly,  the second order thermodynamic quantities can  be obtained 
        
            ∂^2 p̃/∂η_1 ∂η_2   =   [-M_rF_1/2(I+DM_rF_1/2)
            DM_rF_-1/2.
                  .-M_rF_-1/2]
            (∂Ã/∂η_1∂Ã/∂η_2).
        
        
        On the other hand, the derivative of pressure with respect to t always imposes a tedious task. 
        After   carefully solving the above equations involving the derivatives of pressure,  we obtain the entropy s̃=∂p̃/∂ t
            ∂p̃/∂ t   =   -M_r3/2tF_3/2
            +M_rF_1/2Ã/t -M_rF_1/2(I+D M_r F_1/2)
            
                  × (-D M_r F_1/2Ã/t
            +D M_r3/2tF_3/2),
            
            ∂Ã/∂ t   =   [𝐈+DM_r F_1/2
            +(DM_r F_1/2)^2] (-D M_r F_1/2Ã/t.
            
                  . 
            +D M_r3/2tF_3/2
            +ẐF_3/2Ã/t
            -Ẑ5/2tF_5/2).
        
        This result contains not only the linear-temperature-dependent  behaviour of the entropy  in the Luttigner liquid (for T≪ E_F) but also the universal quantum scalings  of the entropy in the quantum critical region (for T≫ E_F) in the vicinity of the critical point. 
        Similarly, the second derivative of the pressure with respect to t is given by 
        
            ∂^2 p̃/∂ t^2   =   -M_r( 3/4t^2F_3/2 
            +1/tF_1/2B+F_-1/2B^2+.
                  . F_1/2D M_r(3/4t^2F_3/2 
            +1/tF_1/2B+F_-1/2B^2)
            )
        
        with B=∂Ã/∂ t-Ã/t. 
        In contrast to the previous studies on multicomponent  interacting fermions <cit.>, the above close forms of the thermodynamics are very useful for analyzing the behaviour of the  quantum liquids and critical scalings of the 1D interacting fermions with  SU(w) symmetry. 
        
         Quantum criticality.
        In the vicinities of  the phase boundaries in the phase diagrams of the the 1D interacting fermions with  SU(w) symmetries <cit.>,  a discontinuity emerges in the polylog functions Li_s(x) in the EOS, namely,
        
            lim_μ/t → 0^+ Li_s (-e^μ/t)=-(μ/t)^s/Γ(s+1),  lim_μ/t → 0^- Li_s (-e^μ/t)=0.
        
        The sign change of the Ã^(r) leads to a sudden change of the Polylog functions at the 
        critical point of a phase transition, for example, see Fig. <ref>, where a sudden enhancement  in the Wilson ratio is observed when the driving parameters is tuned across any  phase boundary. 
        The condition A^(r)>0 implies the  exist of the bound states of r-fermions in this certain  quantum phase. 
        As a consequence, in the vicinities of  phase boundaries, any  thermodynamic quantity can be separated into 
         two parts: 
        
          *  the background part and 
        
          *  the discontinuous part. 
        
        The background part involves the states which do not have a sudden change and
        the calculation of this part is cumbersome <cit.>.
        
        The discontinuous part can be obtained by analyzing  the first order 
        of divergence in the EOS, i.e. the part involves a sudden change in the density of  a certain branch of bound states. For example, at the phase transition from the fully paired states into the FFLO phase, the regular part mainly relates to  the thermal fluctuation presented by functions with A^(2), whereas the singular part results in a sign change of  A^(1) ,which indicates the crossover of the density of unpaired fermions from zero to non-zero 
        while the Fermi sea of the unpaired fermions starts to fill with particles. 
        
        We further calculate the phase transition driven by chemical potential or external fields. Supposing  that  the quantum phase transition is driven by
        the sign change of  Ã^(r) ,  
        we observe that the thermodynamic quantities are naturally  split into background and discontinuous parts as:
        
            ∂𝐩̃/∂η
            =   -M_r F_1/2(𝐈+DM_r F_1/2
            +(DM_r F_1/2)^2)
            ∂𝐀̃_0/∂η
            
            =   ∂𝐩̃/∂η|_0
            -M_r F_1/2∂𝐀̃_0/∂η,
        
        where ∂𝐩̃/∂η|_0 denotes  the backgroud part, i.e. 
        
            ∂𝐩̃/∂η|_0
            =-M_r F_1/2(𝐈+DM_r F_1/2
            +(DM_r F_1/2)^2)
            ∂𝐀̃_0/∂η|_F_s^(r)=0.
        
        Then we obtain the explicit scaling form of the first derivative thermodynamic quantities of the SU(w)  Fermi gases 
        
            ∂p̃/∂η 
            =   ∂p̃/∂η|_0^(r)
            -√(r)/2√(π)∂Ã^(r)_0/∂η 
            t^1/2 Li_1/2(-e^r(μ̃-μ_cr)/t),
        
        where the μ_cr is the critical chemical potential for the phase transition induced by the change of  the 
        r-atoms bond states.  This method can  be further applied to  the second order thermodynamic quantities, 
        explicitly 
        
            ∂^2 p̃/∂η_1 ∂η_2   ≈   ∂^2 p̃/∂η_1 ∂η_2|_0^(r)
                  
            -√(r)/2√(π)∂Ã^(r)_0/∂η_1∂Ã^(r)_0/∂η_2
            t^-1/2 Li_-1/2(-e^r(μ̃-μ_cr)/t).
        
        We thus read off the critical exponents from these scaling functions, i.e. the dynamic critical exponent z=2 and the correlation critical exponent ν=1/2. 
        In particular,  the specific heat is given by 
        
            c̃_V/t   ≈   ∂^2 p̃/∂ t^2≈∂^2 p̃/∂ t^2|_0
                  
            -M_r( 3/4t^2F_3/2 
            -1/tF_1/2Ã_0/t
            +F_-1/2(̃Ã_0)^2/t^2)
        
        that gives 
        
            ∂^2 p̃/∂ t^2≈   ∂^2 p̃/∂ t^2|_0^(r) 
            -√(r)/2√(π t)ℋ(r(μ̃-μ_cr)/t),
        
        where the function 
        
            ℋ (x) = 3/4 Li_3/2 ( - e^x) - x
             Li_1/2 ( - e^x) + x^2  Li_- 1/2 ( - e^x).
        
        Solving  the equation d/dxℋ (x)=0, we get two solutions:
        y_1≈ 0.639844, y_2 ≈ 0.276201 that  determine  the two peaks of the 
        specific heat at quantum criticality in the SU(w) Fermi gases, i.e. 
        
            t^*_1=-y_1 r(μ̃-μ_cr),   t^*_1=y_2  r(μ̃-μ_cr).
        
        The two crossover temperatures fanning out from the critical point indicate  the quantum critical region beyond  the quantum liquid phases, see Fig. <ref>.
        Recent studies on the quantum criticality in 1D  Heisenberg spin chain <cit.> and 1D Bose gas <cit.> confirm such a novel  existence of the critical cone while a quantum phase transition occurs. 
        
        
        
        
        
        For clarity and possible experimental use, we present the explicit  scaling forms of the thermodynamic properties for the 1D strongly attractive SU(w) Fermi gases. The polarization of the systems is m̃=∑_k^w 1/2 k(w-k) ñ_k, 
        where ñ_r=∂p̃/∂ h_r presents the density of the 
        r-atoms . 
        For the phase transition related to the sign change of  Ã^(r), i.e. the phase transition occurs when  the Fermi sea of the charge bound states of r-atoms starts to fill or to be gapped  at zero temperature. Based on the above unified scaling forms (<ref>)–(<ref>), 
        we can directly present  the result of the scaling functions of  physical quantities
        
            ñ≈   ñ_0 -r√(r)/2√(π) t^1/2ℱ (r(μ̃-μ_cr)/t)  
            m̃≈   m̃_0 -1/2 (w-r)r
            √(r)/2√(π) t^1/2ℱ (r(μ̃-μ_cr)/t) 
            κ̃≈   κ̃_0 
            -r^2√(r)/2√(π) t^-1/2𝒢 (r(μ̃-μ_cr)/t) 
            χ̃≈   χ̃_0
             -1/2 r(w-r) √(r)/2√(π) t^-1/2𝒢 (r(μ̃-μ_cr)/t) 
            c̃_V/t≈   c̃_V0/t
             -√(r)/2√(π) t^-1/2ℋ(r(μ̃-μ_cr)/t)
        
        with 
        
            ℱ(x) =    Li_1/2 ( - e^x) 
            𝒢(x) =    Li_-1/2 ( - e^x) 
            ℋ (x) =    3/4 Li_3/2 ( - e^x) - x
             Li_1/2 ( - e^x) + x^2  Li_- 1/2 ( - e^x).
        
        Here every first term in the above quantities denote the background parts. 
        Notice that for the  SU(2) case, we conventionally  take the magnetic field H=2H_1, and the dimensionless form now p̃^(r)=p^(r)/1/2|c|^3, 
        Ã^(r)=A^(r)/1/2|c|^2,
        μ̃=μ/1/2|c|^2,
        h=H/1/2|c|^2,
        t=T/1/2|c|^2.
        These scaling functions provide exact result of  quantum critical phenomena of  the 1D SU(w) Fermi gases, also see the theory of quantum criticality   <cit.>.
        
        
        In summary, we have presented a unified approach to the thermodynamics and quantum scaling functions for the 1D strongly attractive Fermi gases with SU(w) symmetry. In particular, we have obtained the two crossover temperature lines fanning out from the critical point that confirm the existence of the critical cone at quantum criticality. While the quantum liquids can be measured through the dimensionless ratios, revealing the  important free fermion nature of 1D interacting fermions. Our results pave a way to experimentally study quantum criticality of 
        the fermionic alkaline-earth atoms that display an exact SU(w) spin symmetries with w=2I+1<cit.>. Here I is the nuclear spin. The study of critical phenomena and quantum correlations in ultracold atoms with  high symmetries has become a new frontier in atomic physics. 
        
         Acknowledgments.  The authors  thank Yu-Peng Wang,  Yu-Zhu Jiang, Feng He, Song Cheng   and Peng He   for helpful discussions. This work is supported by the NSFC under grant numbers 11374331 and the key NSFC grant No. 11534014. XWG has been partially supported by the Australian Research Council.
        
        
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