Wilson ratio of Fermi gases in one dimension
X. W. Guan, X. G. Yin, A. Foerster, M. T. Batchelor, C. H. Lee, H. Q. Lin
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p Wilson ratio of Fermi gases in one dimension
X.-W. Guan,
1, 2, ∗ X.-G. Yin, A. Foerster,
2, 4
M. T. Batchelor,
5, 2, 6
C.-H. Lee, and H.-Q. Lin † State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China Department of Theoretical Physics, Research School of Physics and Engineering,Australian National University, Canberra ACT 0200, Australia Division of Materials Science, Nanyang Technological University, Singapore 639798 Instituto de Fisica da UFRGS, Av. Bento Goncalves 9500, Porto Alegre, RS, Brazil Centre for Modern Physics, Chongqing University, Chongqing 400044, China Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia State Key Laboratory of Optoelectronic Materials and Technologies,School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China Beijing Computational Science Research Center, Beijing 100084, China (Dated: September 5, 2018)We calculate the Wilson ratio of the one-dimensional Fermi gas with spin imbalance. The Wilsonratio of attractively interacting fermions is solely determined by the density stiffness and soundvelocity of pairs and of excess fermions for the two-component Tomonaga-Luttinger liquid (TLL)phase. The ratio exhibits anomalous enhancement at the two critical points due to the suddenchange in the density of states. Despite a breakdown of the quasiparticle description in one dimen-sion, two important features of the Fermi liquid are retained, namely the specific heat is linearlyproportional to temperature whereas the susceptibility is independent of temperature. In contrastto the phenomenological TLL parameter, the Wilson ratio provides a powerful parameter for testinguniversal quantum liquids of interacting fermions in one, two and three dimensions.
Fermi liquid theory describes the low-energy physics ofinteracting fermions, conduction electrons, heavy fermionmetals and liquid He [1]. It is remarkable that the Wil-son ratio, defined as the ratio of the magnetic suscepti-bility χ to specific heat c v divided by temperature T , R W = 43 (cid:18) πk B µ B g (cid:19) χc v /T (1)is a constant at the renormalization fixed point of thesesystems. Here k B is the Boltzmann constant, µ B is theBohr magneton and g is the Lande factor. For example, R W = 1 for noninteracting or weakly correlated elec-trons in metals [1], and R W = 2 in the Kondo regime forthe impurity problem [2]. The dimensionless Wilson ra-tio quantifies the interaction effect and spin fluctuationsand thus presents a characteristic of strongly correlatedFermi liquids [1]. R W > R W for 1D correlated electrons were consid-ered only in the scenario of spin-charge separation [4, 5].As far as the low energy physics is concerned, the fixedpoint critical Tomonaga-Luttinger liquid (TLL) behavesmuch like the Fermi liquid [6]. For instance, the Wil-son ratio of the quasi-1D spin-1/2 Heisenberg ladder nearthe critical point indicates a single component TLL with R W = 4 K , where K is the TLL parameter. Moreover, the Wilson ratio is always less than 2 as the band fill-ings tend towards the Mott insulator in the 1D repulsiveHubbard model [5]. For the 1D spin-1/2 Heisenberg chain R W = 2 as T → / T = 0 pairing phase [17–19] experimen-tally confirmed using finite temperature density profilesof trapped fermionic Li atoms [20, 21].In this context the Wilson ratio of the 1D attractiveFermi gas with polarization is particularly interesting dueto the coexistence of pairing and depairing under theexternal magnetic field. It is natural to ask if the Wilsonratio can capture a similar Fermi liquid nature of such aparticular pairing phase. Here we report our key resultfor the attractive Fermi gas, R W = 4 (cid:0) v b N + 4 v u N (cid:1) (cid:16) v b s + v u s (cid:17) (2)which holds throughout the two-component TLL phase.This result is in terms of the density stiffness v b , u N andsound velocity v b , u s for pairs b and excess single fermionsu. These parameters can be calculated from the ground b t R W CRCR FPP TLL H/ b t FIG. 1: (Color online) Contour plot of the Wilson ratio R W (1) of the attractive Fermi gas for dimensionless interaction | γ | = 10 as a function of the reduced temperature t = T /ε b and magnetic field. ε b is the binding energy. The result (2)provides a criterion for the two-component TLL phase in theregion below the dashed lines, where R W is temperature inde-pendent. The dashed lines indicate the crossover temperature T ∗ ∼ | H − H c | separating the relativistic liquid from the non-relativistic liquid. R W = 0 for both the TLL of pairs (PP)and the TLL of excess fermions (F). In the critical regimes(CR) R W gives a temperature-dependent scaling. However,near the two critical points, the ratio reveals anomalous en-hancement discussed further in the text. The inset shows theenhancement at the lower critical point. state energy. Fig. 1 shows that at finite temperaturesthe contour plot of R W can map out not only the two-component TLL phase but also the quantum criticalityof the attractive Fermi gas. The Wilson ratio thus gives asimple testable parameter to quantify interaction effectsand the competing order between pairing and depairing. The Model.-
The δ -interacting spin-1/2 Fermi gas with N = N ↑ + N ↓ fermions of mass m with external magneticfield H is described by the Hamiltonian [8, 9, 21] H = − ~ m N X i =1 ∂ ∂x i + g D N ↑ X i =1 N ↓ X j =1 δ ( x i − x j ) + E z (3)in which the terms are the kinetic energy, interactionenergy and Zeeman energy E z = − gµ B H ( N ↑ − N ↓ ).Here the inter-component interaction is determined by aneffective 1D scattering length g D = − ~ ma D which canbe tuned from the weakly interacting regime ( g D → ± )to the strong coupling regime ( g D → ±∞ ) via Feshbachresonances and optical confinement [22]. g D > <
0) isthe contact repulsive (attractive) interaction. The totaldensity n = n ↑ + n ↓ , the magnetization M = ( n ↑ − n ↓ ) / P = ( n ↑ − n ↓ ) /n , where n = N/L is the linear density and L is the length of the system.For convenience, we define the interaction strength as c = mg D / ~ and dimensionless parameter γ = c/n forphysical analysis. We set Boltzmann constant k B = 1and µ B g = 1.The thermodynamic properties of the model are de-termined by the thermodynamic Bethe ansatz (TBA)equations [23]. A high precision equation of state in thephysically interesting low temperature and strong cou-pling regime ( T ≪ ǫ b , H and γ ≫
1) has been derived[24, 25]. The hydrodynamic description of the attrac-tive gas (3) is restricted to the limit cases c → −∞ and c → − [26]. Susceptibility.-
In the Fermi liquid, the interaction en-ters the susceptibility and specific heat via the effectivemass and the Landau parameters [27]. Thus the specificheat increases linearly with the temperature T becauseonly the electrons within k B T near the Fermi surfacecontribute to the specific heat. The susceptibility is in-dependent of temperature since only the electrons within µ B gH near the Fermi surface contribute to the magne-tization. This is a consequence of the forward scatteringprocess between quasiparticles near the Fermi surface. Incontrast, in 1D many-body systems, all particles partic-ipate in the low energy physics and thus form collectivemotion of bosons, i.e., the TLL. However, the TLL is alsothe consequence of the forward scattering process involv-ing low-lying excitations close to Fermi points. Thereforeit is natural to expect that 1D many-body systems havea Fermi liquid nature in the low energy sector.Here we find such a Fermi liquid signature of the 1DFermi gas using the analytic results for the susceptibilityand specific heat obtained via the TBA equations [28].At zero temperature, the susceptibility can be calculatedfrom the dressed energy equations which are obtainedfrom the TBA equations in the limit T → H c and H c the zero temperaturesusceptibility of the gapless phase can be expressed inthe form 1 χ = 1 χ u + 1 χ b . (4)This result can be established on general grounds. Theeffective magnetic field H depends on the chemical po-tential bias H := ∆ µ = µ ↑ − µ ↓ . The magnetizationdepends on the difference ∆ n = n ↑ − n ↓ . We prove thatthe magnetic susceptibility χ = ∂ ∆ n/∂ ∆ µ can be writ-ten in terms of the charge susceptibilities of bound pairsand excess fermions χ b , u = ∂n b , u /∂µ b , u | µ u,b , where µ b = µ + ǫ b / µ u = µ + H/ n is fixed. Here n b and n u are the densities of pairs and ex-cess fermions. Physically, the system has two processesoccurring in parallel, namely the breaking of pairs andthe alignment of spins. The analog for the zero tem-perature susceptibility of the gapless phase is thus twoparallel resistors in a circuit.We also find that the effective susceptibilities for theTLL of bound pairs and the TLL of excess fermionsare expressed as χ b = 1 / ( ~ πv bN ) and χ u = 1 / (4 ~ πv u N ).The density stiffness parameters are obtained from v rN = Lπ ~ ∂ E r ∂N r for a Galilean invariant system, with r = 1for excess fermions and r = 2 for bound pairs. Forthe strongly interacting regime ( γ > E r ≈ ~ m π N rL (cid:16) A r | c | + A r c (cid:17) with A = 4 n and A = 2 n + n . Here n and n are thedensity of excess fermions and pairs, respectively. Thus v b N = ~ πn m (cid:20) | c | ( n − n ) + 3 c (4 n − nn + 30 n ) (cid:21) v u N = ~ πn m (cid:20) | c | ( n − n ) + 4 c (3 n + 10 n − nn ) (cid:21) . The analytic expression (4) with these velocities is inexcellent agreement with the numerical results (see insetin Fig. 2).The onset susceptibility at the lower and upper criticalfields H c and H c is related to the collective nature ofthe pairs and excess fermions, with χ (cid:12)(cid:12) H → H c +0 = 1 ~ πv b N (cid:12)(cid:12)(cid:12)(cid:12) n = n = K b ~ πv b s (cid:12)(cid:12)(cid:12)(cid:12) n = n , (5) χ (cid:12)(cid:12) H → H c − = 14 ~ πv u N (cid:12)(cid:12)(cid:12)(cid:12) n = n = K u ~ πv u s (cid:12)(cid:12)(cid:12)(cid:12) n = n . (6)Here v rs and K r = v rs /v rN are the sound veloc-ities and effective TLL parameters of the boundpairs and excess single fermions. From the relation v rs = q Lmn ∂ E r ∂L , the velocities are given by v rs = ~ m πn r r (cid:0) A r / | c | + 3 A r /c (cid:1) .The separation of the susceptibility (4) naturally sug-gests that the low energy physics of the polarized pair-ing phase is described by a renormalization fixed pointof the two-component TLL class, where the interac-tion effect enters into the collective velocities, or equiv-alently the effective masses of the two TLLs are variedby the interaction. At finite low temperatures, the two-component TLL acquires a universal form F ( T, H ) ≈ E ( H ) − πk B T ~ (cid:0) /v b s + 1 /v u s (cid:1) of the free energy. Fortemperature T < H − H c and T < H c − H , the suscepti-bility is indeed independent of temperature provided that − ∂ (cid:0) /v b s + 1 /v u s (cid:1) /∂H ≈
0, see Fig. 2. We clearly seethat the T = 0 divergent susceptibility near the criticalpoint H c evolves into round peaks at low temperatures.The peak hight decreases as the temperature increases.Here the leading irrelevant operators gives a correctionof the order O ( T ) to the low energy in the vicinities ofthe two critical points.For the quantum critical regime ( T > H − H c and T >H c − H ) the susceptibility defines the universality class b | c | H/ b t=0.00001 t=0.0001 t=0.0004 t=0.0008 t=0.0012 t=0.002 | |=5| |=10 b | c | H/ b analytical energy relation numerics FIG. 2: (Color online) The dimensionless susceptibility vsmagnetic field for | γ | = 10 at different temperatures. Thesusceptibility is independent of temperature for T < H − H c and T < H c − H . Round peaks of the susceptibility in thevicinity of the two critical points are observed at low tem-peratures. The inset shows the susceptibility for | γ | = 5 and10 at T = 0. The pink crosses denote the analytic result (4)which is in excellent agreement with the numerical results ob-tained from the field-magnetization relation [19] (red circles)and from the dressed energy equations [28] (blue lines). for quantum criticality of nonrelativistic Fermi theory,with [28] χ ∼ | c | ǫ b (cid:20) λ + λ s t dz +1 − νz Li − (cid:18) − e α ( h − hc t νz (cid:19)(cid:21) . (7)Near the critical point h c = − µ + π √ (˜ µ + 1 / / wehave λ = 0 and λ ≈ √ π (cid:16) − π p ( h − h c ) / (cid:17) with α = 1 / t = T /ǫ b and h = H/ǫ b . Here the dynamicalcritical exponent z = 2 and correlation length exponent ν = 1 / h c the susceptibility defines a similarform as (7), but with the background susceptibility λ =0 [28]. Specific heat.-
We turn now to the specific heat of theattractive Fermi gas. The low temperature expansion ofthe TBA equations with respect to T ≪ H, ǫ b gives c v = πk B T ~ (cid:18) v b s + 1 v u s (cid:19) . (8)The linear T -dependence of the specific heat is a conse-quence of linear dispersions in branches of pairs and sin-gle fermions. The breakdown of this linear temperature-dependent relation defines a crossover temperature T ∗ which charaterizes a universal crossover from a relativis-tic dispersion into a nonrelativistic dispersion [24, 29].We see clearly in Fig. 3 that at low temperatures apeak evolves in the specific heat near each of the twocritical points, i.e., near P = 0 and P = 1 due to asudden change in the density of states. We also notethat the peak positions mark the TLL specific heat curve(8). The two peaks merge at the top of the TLL phasein Fig. 1. Thus the peak position in turn gives the TLLphase boundary in the c v − P or c v − H plane. The specificheat obtained from the equation of state [25] also definesa scaling behaviour c v ∼ r mε b ~ t (cid:20) ν + ν s t dz +1 − νz Li − (cid:18) − e α ( h − hc ) t νz (cid:19)(cid:21) (9)where ν , ν s and α are constants which can be determinedfrom the closed form of the specific heat if necessary [28].The two-component TLL specific heat (8) is clearly man-ifest in Fig. 3 from the numerical result obtained usingthe equation of state. c V b / ( T | c | ) P TLL t=0.00001 t=0.0001 t=0.0004 t=0.0008 t=0.0012 t=0.002 c V b / ( T | c | ) P t=0.00001 FIG. 3: (Color online) Dimensionless specific heat vs polar-ization for | γ | = 10 at different temperatures. The deviationfrom linear temperature dependence (8) (red crosses) indi-cates the breakdown of the two-component TLL. The insetshows a round peak evolved near H c at T = 0 . ǫ b . Wilson ratio.-
The linear temperature-dependent na-ture of the specific heat and the separable feature of thesusceptibility give the Wilson ratio (2) for the effectivelow energy physics of the two-component TLL. This Wil-son ratio for the 1D attractive Fermi gas is significantlydifferent from the ratio obtained for the field-inducedgapless phase in the quasi-1D gapped spin ladder [3],where the gapless phase is a single-component TLL [4, 6]and the ratio gives a renormalization fixed point of a lin-ear spin-1 / W R (cid:12)(cid:12) H → H c = 4 K b (cid:12)(cid:12) n → n , W R (cid:12)(cid:12) H → H c = K u (cid:12)(cid:12) n → n . P TLL t=0.00001 t=0.0001 t=0.0004 t=0.0008 t=0.0012 t=0.002 R W R W Pt=0.00001
FIG. 4: (Color online) Wilson Ration vs polarization for | γ | = 10 at different temperatures. The numerical result ob-tained from the equation of state fully agrees with the Wilsonratio (2) (red crosses) for the two-component TLL phase. Thedeviations from the result (2) characterise the crossover tem-perature T ∗ . Anomalous behaviour is found near P = 0 and P = 1 (see inset for near the critical point H c ). Here we find K b ≈ | c | n + 3 c n (3 n + 4 n ) K u ≈ | c | n + 4 c n ( n + 2 n ) . (10)Note that the values in the limit of infinitely strong cou-pling are W R = 4 at H c and W R = 1 at H c .The anomalous enhancement of the Wilson ratio nearthe onset values is shown in Fig. 4. Anomalous en-hancement of the Wilson ratio has been observed nearthe metal-insulator transition in simulations of a three-dimensional quantum spin liquid [30]. Here for the 1D at-tractive Fermi gases this anomalous divergence is mainlydue to sudden changes in the density of states either inthe bound state or excess fermion branch. Again, devi-ation from the Wilson ratio (2) gives the crossover tem-perature T ∗ ∼ | H − H c | separating the TLL from the freefermion liquid near the critical points. In addition to theanomalous divergence of the onset Wilson ratio, a roundpeak is observed near P ≈ . R W < < P < R W =2 / (1 + v σ /v c ) which simply gives a fixed point of the TLLin the context of spin-charge separation. Here the chargeand spin velocities v c,σ can be calculated following [31].The Wilson ratio of 1D Fermi gases can in principle bemeasured in experiments. The finite temperature den-sity profiles of a 1D trapped Fermi gas of Li atomshave been measured [20]. Most recently, the suscepti-bility has been directly obtained from the density profileof the trapped atomic cloud in higher dimensions [32].High precision measurements of thermodynamic quanti-ties have also been reported [33]. For the 1D case, thepredicted susceptibility could be tested from the densityprofiles n ↑ , ↓ and the chemical potential bias.The Wilson ratio of the 1D attractive Fermi gaseswhich we have obtained provides a measurable param-eter to quantify different phases of quantum liquids in1D interacting fermions with polarization. At low tem-peratures, the Fermi liquid nature is retained in 1Dmany-body systems of interacting fermions. Our analysiscan be adapted to different systems, such as interactingfermions, bosons and mixtures composed of cold atomswith higher spin symmetry. Acknowledgments.
We thank M. Cazalila and W. Vin-cent Liu for helpful discussions. This work has been sup-ported by the NNSFC under the grant No. 91230203and the National Basic Research Program of China underGrants No. 2012CB922101 No. 2012CB821300, and No.2011CB922200. The work of XWG and MTB has beenpartially supported by the Australian Research Coun-cil. XWG thanks Chinese University of Hong Kong forkind hospitality. M. T. B. is supported by the 1000 Tal-ents Program of China. AF acknowledges financial sup-port from Coordena¸c˜ao de Aperfei¸coamento de Pessoalde Nivel Superior (Proc. 10126-12-0). ∗ e-mail:[email protected] † e-mail:[email protected][1] A. C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, 1997).[2] K. G. Wilson, Rev. Mod. Phys. , 773 (1975).[3] K. Ninios et al. , Phys. Rev. Lett. , 097201 (2012).[4] H. J. Schulz, Int. J. Mod. Phys. B , 57 (1991).[5] T. Usuki, N. Kawakami and A. Okiji, Phys. Lett. A ,476 (1989). [6] Y.-P. Wang, Int. J. Mod. Phys. B , 3465 (1998).[7] D. C. Johnston et al. , Phys. Rev. B , 9558 (2000).[8] M. Gaudin, Phys. Lett. A , 55 (1967).[9] C. N. Yang, Phys. Rev. Lett. , 1312 (1967).[10] A. E. Feiguin and F. Heidrich-Meisner, Phys. Rev. B et al. , Phys. Rev. B , 245105 (2008).[12] E. Zhao and W. V. Liu, Phys. Rev. A , 063605 (2008).[13] J.-Y. Lee and X.-W. Guan, Nucl. Phys. B , 125(2011).[14] P. Schlottmann and A. A. Zvyagin, Phys. Rev. B ,205129 (2012).[15] C. J. Bolech et al. , Phys. Rev. Lett. , 110602 (2012).[16] H. Lu, L. C. Baksmaty, C. J. Bolech and H. Pu, Phys.Rev. Lett. , 225302 (2012).[17] G. Orso, Phys. Rev. Lett. , 070402 (2007).[18] H. Hu, X.-J. Liu and P. D. Drummond, Phys. Rev. Lett. , 070403 (2007).[19] X.-W. Guan, M. T. Batchelor, C. Lee and M. Bortz,Phys. Rev. B , 085120 (2007).[20] Y. Liao et al. , Nature , 567 (2010).[21] For a review, see X.-W. Guan, M. T. Batchelor and C.Lee, arXiv:1301.6446, to appear in Rev. Mod. Phys.[22] M. Olshanii, Phys. Rev. Lett. , 938 (1998).[23] M. Takahashi, Thermodynamics of One-DimensionalSolvable Models , Cambridge University Press, Cam-bridge, (1999)[24] E. Zhao, X.-W. Guan, W. V. Liu, M. T. Batchelor andM. Oshikawa, Phys. Rev. Lett. , 140404 (2009).[25] X.-W. Guan and T.-L. Ho, Phys. Rev. A , 023616(2011).[26] T. Vekua, S. T. Matveenko and G. V. Shlyapnikov, JETPLett. , 289 (2009).[27] A. J. Schofield, Contemporary Physics , 95 (1999).[28] See Supplemental Material for detailed calculations of thesusceptibility and specific heat.[29] Y. Maeda, C. Hotta and M. Oshikawa, Phys. Rev. Lett. , 057205 (2007).[30] G. Chen and Y. B. Kim, Phys. Rev. B , 165120 (2013).[31] J.-Y. Lee, X.-W. Guan, K. Sakai and M. T. Batchelor,Phys. Rev. B , 085414 (2012).[32] Y.-R. Lee et al. , Phys. Rev. A , 043629 (2013).[33] M. J. H. Ku, A. T. Sommer, L. W. Cheuk and M. W.Zwierlein, Science , 563 (2012). Supplementary material
The Gaudin-Yang model [8, 9] is exactly solved by means of the nested Bethe ansatz. The thermodynamics ofthe model is given explicitly in Takahashi’s book [23]. At finite temperatures, the density distribution functions ofpairs, unpaired fermions and spin strings involve the densities of ‘particles’ ρ r ( k ) and ‘holes’ ρ hr ( k ) ( r = 1 , Z = tr(e −H / T ) = e − G/T in terms of the Gibbs free energy G = E − HM z − µn − T S with respect to themagnetic field H , chemical potential µ and entropy S . In terms of the dressed energies ǫ b ( k ) := T ln( ρ h ( k ) /ρ ( k ))and ǫ u ( k ) := T ln( ρ h ( k ) /ρ ( k )) for paired and unpaired fermions, the equilibrium states are determined by theminimization condition of the Gibbs free energy, which gives rise to the set of coupled nonlinear integral equations interms of the dressed energies ǫ b and ǫ u ǫ b ( k ) = g b ( k ) + K ∗ f ǫ b ( k ) + K ∗ f ǫ u ( k ) (11) ǫ u ( k ) = g u ( k ) + K ∗ f ǫ b ( k ) − ∞ X ℓ =1 K ℓ ∗ f T ln η ℓ ( k ) (12) T ln η ℓ ( λ ) = ℓH + K ℓ ∗ f ǫ u ( λ ) + ∞ X n =1 T ℓm ∗ f T ln η m ( λ ) (13)with ℓ = 1 , . . . , ∞ . The driving terms g b ( k ) = 2( k − µ − c /
4) and g u ( k ) = k − µ − H/
2. Here ∗ denotes the convolutionintegral K m ∗ f x ( λ ) = R ∞−∞ K m ( λ − λ ′ ) f x ( λ ′ ) dλ ′ with K m ( λ ) = π m | c | ( mc/ + λ and f x ( k ) = T ln (cid:0) − x ( k ) /T (cid:1) . Thefunction η ℓ ( λ ) = ξ hℓ ( λ ) /ξ ℓ ( λ ) is the ratio of the string densities. The function T ℓm ( k ) is given explicitly by [14, 19, 23] T mn ( x ) = (cid:26) a | m − n | ( x ) + 2 a | m − n | +2 ( x ) + . . . + 2 a m + n − ( x ) + a m + n ( x ) , for n = m a ( x ) + 2 a ( x ) . . . + 2 a n − ( x ) + a n ( x ) , for n = m .The Gibbs free energy per unit length is given by G = p b + p u where the effective pressures of the unpaired fermionsand bound pairs are given by p r = rT π Z ∞−∞ dk ln(1 + e − ǫ r ( k ) /T )with r = 1 for unpaired fermions and r = 2 for paired fermions.The thermodynamics of the model can be calculated from the standard thermodynamic relations. The density,magnetization, entropy, susceptibility and specific heat are given by n = (cid:18) ∂p ( µ, H, T ) ∂µ (cid:19) T,H , M z = (cid:18) ∂p ( µ, H, T ) ∂H (cid:19) T,µ ,s = (cid:18) ∂p ( µ, H, T ) ∂T (cid:19) µ,H , χ = (cid:18) ∂M z ∂H (cid:19) n,T , c v = T (cid:18) ∂s∂T (cid:19) n,p . (14)The TBA equations provide the full thermodynamics of the model, including the Tomonaga-Luttinger liquid physicsand quantum criticality. At zero temperature, the quantum phase diagram in the grand canonical ensemble can beanalytically determined from the dressed energy equations [19, 23] ǫ b ( k ) = g b ( k ) − Z Q − Q K ( k − Λ) ǫ b (Λ) d Λ ′ − Z Q − Q K ( k − k ′ ) ǫ u ( k ′ ) dk ′ ǫ u ( k ) = g u ( k ) − Z Q − Q K ( k − Λ) ǫ b (Λ) d Λwhich are obtained from the TBA equations (11)-(13) in the limit T →
0. The dressed energy ǫ b (Λ) ≤ ǫ u ( k ) ≤ | Λ | ≤ Q ( | k | ≤ Q ) correspond to the occupied states. The positive part of ǫ b ( ǫ u ) corresponds to the unoccupiedstates. The integration boundaries Q and Q characterize the Fermi surfaces for bound pairs and unpaired fermions,respectively. The pressures of pairs and excess fermions are given by p b = − π Z Q − Q d Λ ǫ b (Λ) , p u = − π Z Q − Q dk ǫ u ( k ) . The zero temperature susceptibility is obtained from these pressures using the standard statistical physics relations.In terms of the dimensionless quantities ˜ µ := µ/ε b , h := H/ε b , t := T /ε b and ˜ n := n/ | c | = γ − , where ε b = ~ m c isthe binding energy, the equation of states for the strongly attractive gas is [25]˜ p ( t, ˜ µ, h ) := p/ ( | c | ε b ) = ˜ p b + ˜ p u , (15)where the pressures of the bound pairs and unpaired fermions are given by˜ p b = − t √ π F b / (cid:20) p b p u (cid:21) + O ( c )˜ p u = − t √ π F u / (cid:2) p b (cid:3) + O ( c )in terms of the functions F bn , F un , f bn , and f un defined by F b,un := Li n (cid:0) − e X b,u /t (cid:1) and f b,un := Li n (cid:0) − e ν b,u /t (cid:1) , with thenotation ν b = 2˜ µ + 1, ν u = ˜ µ + h/
2. Here Li s ( z ) = P ∞ k =1 z k /k s is the polylog function, with I ( x ) = P ∞ k =0 1( k !) ( x ) k and X b t = ν b t − ˜ p b t − p u t − t √ π (cid:18) f b / + √ f u / (cid:19) X u t = ν u t − p b t − t √ π f b / + e − h/t e − K I ( K ) . From the equation of states (15), the susceptibility ˜ χ = χε b / | c | at finite temperatures is given by˜ χ = − √ π ∆ ( √ t F u − " √ t √ π F A b / + 2 √ tπ F b F u + 2 √ √ tπ F b − (cid:16) F u (cid:17) ) where ∆ = 1 − √ t √ π F b − t √ π F b F u + t √ π F b / . In the quantum critical regime, i.e., in the vicinity of the critical point and for temperature
T > | H − H c | , the universalscaling form can be evaluated analytically, with χ ∼ | c | ǫ b (cid:20) λ + λ s t dz +1 − νz Li − (cid:18) − e α ( h − hc ) t νz (cid:19)(cid:21) as given in the text. Near the critical point h c ≈ π ) / (˜ µ + 1 / / − µ + 1 / λ ≈ / (8 √ π p λ u ), λ s ≈ λ u / ( π √ π ), α ≈ √ π (cid:0) √ π (2˜ µ + 1) (cid:1) / and λ u ≈ (cid:0) √ π (2˜ µ + 1) / (cid:1) / −
16 (˜ µ + 1 / / / (3 √ π ).Morover, by iteration, the specific heat can be obtained from the equation of states (15) as c v = c bv + c uv where c bv | c | = 1 √ π ( − √ tF b + √ tF b ˜ ν b t + 58 t (4˜ p u + ˜ p b ) + √ ν u √ πt F u + ˜ ν b √ πt F b ! − √ t F b − ˜ ν b t (4˜ p u + ˜ p b ) + ˜ ν b t + 2 √ ν b ˜ ν u √ πt F u + 3˜ ν b √ πt F b + ˜ ν b √ πt F u !) c uv | c | = 1 √ π (cid:26) − √ √ tF u + √ t √ F u (cid:18) ˜ ν u t + 52 t ˜ p b + 2˜ ν b √ πt F b (cid:19) − √ √ t F u − (cid:18) ˜ ν u t + 2˜ ν u t ˜ p b + 2˜ ν b ˜ ν u √ πt F b + 2˜ ν u √ πt F b (cid:19)(cid:27) . The scaling form of the specific heat in the quantum critical regime, i.e.,
T > | H − H c | , can be worked out from theseclosed form expressions in a straightforward way, with the result given in the text.The anomalous enhancement of the Wilson ratio exhibited near the two critical points is further demonstrated inFig. 5. R W P FIG. 5: (Color online) Wilson ratio vs polarization for | γ | = 10 at temperature T = 0 . ǫ b . The numerical result isobtained from the equation of states (15). The ratio exhibits anomalous enhancement near the two critical points due to thesudden change of the density of states, where the values R W = 5 .
53 and R W = 1 ..