Gr¨uneisenParameters: origin, identity and quantum

Yi-Cong Yu,1 Shizhong Zhang,2, ∗ and Xi-Wen Guan1, 3, † 1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, IAPMST, Chinese Academy of Sciences, Wuhan 430071, China 2Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Hong Kong, China 3Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia In solid state physics, the Gr¨uneisenparameter (GP), originally introduced in the study of the effect of changing the volume of a crystal lattice on its vibrational frequency, has been widely used to investigate the characteristic energy scales of systems with respect to the changes of external potentials. On the other hand, the GP is little investigated in a strongly interacting quantum gas systems. Here we report on our general results on the origin of GP, new identity and caloric effects in quantum gases of ultracold atoms. We prove that the symmetry of the dilute quantum gas systems leads to a simple identity among three different types of GPs, quantifying caloric effect induced respectively by variations of volume, magnetic field and interaction. Using exact Bethe ansatz solutions, we present a rigorous study of these different GPs and the quantum refrigeration in one-dimensional Bose and Femi gases. Based on the exact equations of states of these systems, we obtain analytic results for the singular behaviour of the GPs and the caloric effects at quantum criticality. We also predict the existence of the lowest for cooling near a quantum phase transition. It turns out that the interaction ramp-up and -down in quantum gases provides a promising protocol of quantum refrigeration in addition to the usual adiabatic demagnetization cooling in solid state materials.

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I. INTRODUCTION ever, are much less explored. In addition to the ability to change the volume of the system by modifying the ex- ternal confining potential and the ability to change the The structure of energy spectrum of a quantum many- equivalent magnetic field by changing population imbal- body system and its evolution under external pertur- ance, it is also possible to change the interaction directly bation characterises essentially its possible phases. As by using Feshbach resonance [19, 20]. This possibility an example, the Gr¨uneisenparameter (GP) [1,2], which suggests a new avenue for studying a novel interacting was introduced by Eduard Gr¨uneisenin the beginning GP in addition to those defined by changes in volume of 20th Century in the study of the effect of volume and magnetic field [8, 11–16, 18, 21–26] . Furthermore, change of a crystal lattice on its vibrational frequencies, we establish an exact identity between these various GPs, has been extensively studied for the exploration of caloric making use of the scaling properties of the quantum gas effect of solids and phase transitions associated with vol- system. The interaction driven caloric effect in quantum ume change. Similarly, the magnetic GP quantifies the gases will also be discussed. magnetocaloric effect (MCE), establishing connection be- tween refrigeration and variation of magnetic field. So far, the GP has found diverse applications in geophysics [3,4], chemical physics[5,6], high pressure In this paper, we discuss the physical origin of GPs, physics and plasma physics [7–9]. Recently, experiments establish a new identity and investigate the efficiency also started to focus on GP in heavy-fermion systems of quantum refrigeration by modifying the interaction [10–12], in which the physical properties at low temper- strength in quantum gases. These GPs can be gener- atures are dominated by f-electrons and their antifer- ally described by the ratio between the energy-pressure arXiv:1909.05754v1 [cond-mat.quant-gas] 12 Sep 2019 romagnetic exchange J with conduction electrons [13]. (or energy-magnetization and energy-contact) covariance The heavy-fermion metals are extremely sensitive to a and energy fluctuation, similar to the the Wilson ratio small change of pressure and this pressure sensitivity is [27]. Using exact Bethe ansatz solutions, we show how reflected in highly enhanced values of the GP [14]. At interacting GP characterizes the quantum phase transi- low , divergence of the GP stronger than tions for 1D quantum gases and demonstrate that the logarithmic upon cooling in the quantum regime is used cooling effect is greatly enhanced near the quantum crit- for experimental identification of quantum critical points ical point. Our study shows promising route for studying [10, 15–18]. interaction driven quantum engine and refrigeration The corresponding GPs in dilute quantum gases, how- in ultracold atoms. 2

II. THEORY: THE ORIGIN, GENERALIZATION 3. The effective and magnetic Gr¨uneisenpa- AND NEW IDENTITY rameter . There is a widely used effective GP in exper- iment, defined as the ratio of thermal expansion param- 1 ∂V 1. The origin. There are many formulations of eter βT = V ∂T |p,N to the specific heat at a constant the GP to quantify the degree of anharmonicity on the volume [11, 12, 14, 21, 26, 32] structure of the energy spectrum in response to volume 2  −2 change. The original definition of the GP was introduced βT ∂ p ∂p κ Γeff = = Γ · 2 = Γ · 2 , (4) by E. Gr¨uneisenfor the Einstein model [1,2], cV /V ∂µ ∂µ n V ∂ω V ∂S Γ =: − 0 = , (1) here κ is the compressibility and n is the density. We de- ω0 ∂V CV ∂V note it by ”eff-Gr¨uneisenparameter ” since it is not equiv- where the excitations in a solid is described by N alent to the original definition (3). In the above equation, the thermal expansion parameter in grand canonical en- with the same frequency ω0. S is the and V de- notes the volume. In quantum statistical physics, the dif- semble can be given by ferential forms of the internal energy E and the pressure p  ∂2p ∂p ∂2p ∂p   ∂p −2 can be represented by the fluctuations and covariances of βT = 2 − . (5) thermodynamic quantities. If we regard the population ∂µ ∂T ∂µ∂T ∂µ ∂µ ai of the i-th energy level as a distribution function of Note that the usefulness of eff-GP is not well established a random variable and observable thermal quantities as in experiment; see the discussion on its divergent be- the expectation value with respect to this distribution, haviour at quantum critical points [12, 14, 25]. However, then one can obtain the following differential relations, it is clear that the eff-GP is not a dimensionless param- see supplementary material (SM) [28] eter and shows different scaling forms at the quantum dE = [−Cov(E,E)]dβ + [−p + βCov(p, E))]dV critical points. To clearly show the dimensionless nature of the Gr¨uneisen parameter , we define another dimen- dp = [Cov(p, E))]dβ + [E00 + βCov(p, p)]dV, sionless GP by [28]

2 00 P ∂ i where Cov denotes the covariance and E =: i ai ∂V 2 . ∂S V ∂V |N,T β = 1/(kBT ) and kB is the Boltzman constant and T is Γ = (6) T ∂S | the temperature. Then the GP is simply given by ∂T N,V

V Cov(p, E) V dp/dβ|V which is equivalent to the definition (1), (2) and (3) and Γ = − = . (2) is intimately related to the expansionary caloric effect Cov(E,E) dE/dβ|V Thus, in this case, Γ represents the relative importance ∂T T = Γ. (7) of energy-pressure covariance and the energy fluctua- ∂V S,N,H V tion in the system. In contrast to the susceptibility (or compressibility) Wilson ratio proposed in [27, 29, 30], In quantum statistics, the volume V of a system can be i.e., the ratio between the magnetization M (or particle regarded as an external filed that imposes a constrain number) fluctuation and the energy fluctuation, namely on the particles. Therefore, it is natural to investigate χ Cov(M,M) κ Cov(N,N) other potentials that impose different constraints on the RW ∝ Cov(E,E) (or RW ∝ Cov(E,E) ), the GP (2) pro- vides additional insights into the spectral information system. As a remarkable example, the well known mag- with respect to the change of the volume of the system. netic GP discussed in experiments [14–16, 18] can be in- 2. In grand canonical ensemble. In cold atom sys- troduced analogously by replacing the volume V by the tem, it is far more convenient to work in grand canonical magnetic field H in the definition (6) ensemble and it is useful to derive the form of Γ in the ∂S grand canonical ensemble. Let µ be the chemical poten- H ∂H |N,T,V Γmag = − ∂S (8) tial of the system and one finds T ∂T |N,B,V

dp V ∂2p ∂p ∂2p ∂p Here we added a minus sign following the former work dT 1 2 − Γ = V,N = ∂µ ∂T ∂µ∂T ∂µ , (3) [18, 33], and put the magnetic field H in the numerator in dE T ∂2p ∂2p ∂2p 2 dT V,N ∂µ2 ∂T 2 − ( ∂µ∂T ) order to keep the GP dimensionless. It is straightforward to obtain the explicit form of the magnetic GP in grand In deriving the above equations, we applied the Maxwell’s canonical ensemble relations and used the homogeneous assumption that the grand thermal potential is a linear function of the volume ∂2p ∂2p ∂2p ∂2p H ∂µ2 ∂H∂T − ∂µ∂H ∂µ∂T neglecting the surface effect in the thermodynamic limit Γmag = − 2 2 2 . (9) T ∂ p ∂ p − ( ∂ p )2 [31], i.e. Ω = −pV [63]. ∂µ2 ∂T 2 ∂µ∂T 3

The magnetic GP (8) plays an important roles in the regard of the variations of volume, magnetic field and in- experimental study of solid state materials [13, 18, 24, teraction strength, respectively. Using the general ther- 32, 34]. mal potential [37], one can find a new identity for the One of the most important features of the magnetic three GPs for dilute system described by s-wave scatter- materials is the magnetocaloric effect, related to the mag- ing length as [19]. For these systems, one has the follow- netocaloric refrigeration. In low temperature physics, ing scaling transformations: L → eλL, c → eχλc, H → this is known as adiabatic demagnetization cooling; see e−2λH, where eλ is the scaling amplitute and the χ is recent new developments [16, 17]. By the definition of the exponent of the dependency of the coupling strength χ the Γmag in eq. (8), we further get and the scattering length by c ∝ as , then we find that the spectrum of such quantum many-body systems will ∂T T be changed by  → e−2λ , here  denotes the n-th = Γmag, (10) n n n ∂H S,N,V H energy level. Meanwhile, if the temperature transforms −2λ − n − i as T → e T , the population a = e T / P e T is in- which establishes an important relation between mag- n i variant under such scaling transformations, and so does netocaloric effect and the magnetic GP. Experimentally, the entropy S = − P a ln a , i.e. it is easier to measure the magnetocaloric effect and i i from Eq. (10), obtain Γ instead of using its origi- mag ∂S ∂S nal definition (8). One can obtain the magnetic entropy 0 = dS = dV + dT ∂V ∂T change ∂S/∂H | once we know the value of specific T,H,c V,H,c N,T,V heat. The magnetic GP contains information free of any ∂S ∂S + dH + dc. material-specific parameter [18]. ∂H V,T,c ∂c V,T,H 4. The interacting Gr¨uneisenparameter . In addition to the usual conjugate variables that one usu- Substituting the scaling transformations into the above ally encounters in , in ultracold atomic equation and noticing V = Ld with d being the dimension gases, it is also possible to define another set of conjugate of the system, we obtain an important identity variable related to the interaction between atoms. In the ∂S ∂S ∂S ∂S case of s-wave interacting system, the low-energy scat- dV = 2T + 2H − χc tering properties are determined entirely by the s-wave ∂V T,H,c ∂T V,H,c ∂H V,T,c ∂c V,T,H scattering length as. In one-dimensional system, the 1D coupling constant c is related to the scattering length that relates the entropy changes to the variations of the −1 interaction, magnetic field and the volume of the sys- (c ∝ as ). In reality, it is possible to change the scatter- tem. Using the definitions of GPs given in eqs. (1), (8) ing length as by using Feshbach resonance and one can define analogously another GP related to interaction and (11), we obtain a simple identity

∂2p ∂2p ∂2p ∂2p ∂S − dΓ + 2Γmag − χΓint = 2. (13) c ∂c |N,H,T,V ∂µ2 ∂c∂T ∂µ∂c ∂µ∂T c Γint = − = − . T ∂S | ∂2p ∂2p ∂2p 2 T ∂T N,H,c,V ∂µ2 ∂T 2 − ( ∂µ∂T ) In one dimension systems we have d = 1 and χ = −1 [38], (11) the identity above is reduced to Γ+2Γmag +Γint = 2. Al- though we obtained this identity through the scaling in- The physical significance of Γint is that it describes the variance of the entropy, it is universal and valid for quan- caloric effect due to modification of interaction strength. tum gases in 1D and higher dimensions. We can prove In particular, in an isentropic process, one can relate the this identity (13) in a more conventional way of quantum change of temperature to interaction strength given by statistical physics, see [28]. This identity eq. (13) has

∂T T many interesting applications. The term 2/d in eq. (13) = Γint. (12) ∂c c gives the nature of the scaling invariant spectrum of ideal S,N,V,H gases, also see the discussion on the Gr¨uneisen parame- 2 This is an interaction analog of the magnetocaloric ef- ter , where it is exactly 3 for 3D free gas [39], obviously fect. We observe that from eq. (12) that a corresponding to a special case of eq. (13) with d = 3 and and quantum refrigeration can be constructed by tuning Γmag = Γint = 0. A further study of the identity (13) will the interaction strength in quantum gases. Therefore the be published elsewhere [64]. interaction gradients are capable of cooling the system of interacting fermions like the magnetization gradient cool- ing [35, 36]. We shall further discuss interaction driven III. APPLICATIONS: QUANTUM CRITICALITY quantum refrigeration in next section. AND QUANTUM REFRIGERATION 5. Universal identity. So far we have presented three different GPs, i.e., Γ, Γmag and Γint, which quan- 1. Magnetic and interaction driven refrigera- tify the degrees of anharmonicity of spectral structures in tion. Through the study presented in last section, we 4 have shown that it is possible to reduce the tempera- L is given by [42] ture of a magnetic system by changing the external mag- netic field in an isentropic process. Based on this mag- 2 N 2 ˆ ~ X ∂ X netocaloric effect, it is possible to cool the systems into H = − 2 + 2c δ(xi − xj). (14) 2m ∂xi extremely low temperatures via either flipping or i=1 1≤i

(a) Schematics HH→ 2 S B 푇B > 푇퐶 > 푇퐴 > 푇퐷 Adiabatic HH→ 1 magnetization N Expel heat to the source A S The Magnetic Target Refrigeration Cycle Source N Absorb heat C from the target Adiabatic de-

magnetization HH→ 2 HH→ Working material D 1

(b) MCE cycle in the T-S plane (c) MCE cycle in the T-H plane

FIG. 1: Schematic representation for the magnetic refrigeration cycle: (a) The working circle between the target and the heat source absorbs heat from the target at a lower temperature Ttar and transfers heat to the source at a higher temperature Tsour. Different colors label different temperatures. At the point B and C, a strong magnetic field is applied and the material is polarized sufficiently. Whereas at the point A and D, the material is demagnetized. (b) The magnetic refrigeration cycle in the T − S plane and (c) in the T − H plane. B → C and D → A are isomagnetic processes with dH = 0, and A → B and C → D are isoentropic. The whole processes A → B → C → D → A were discussed in detail the main text. In D → A, the heat is absorbed by the working material from the target. In the process B → C, the heat is expelled from the working material to the heat source.

of such as cycle by dunning the interaction in the 1D heat ∆Q1 is removed from the working medium into the interacting Bose gas much likes the interaction driven hot ambient. C → D: The working medium is decoupled heat engine proposed in [46]. from the hot ambient. By performing a work, interaction ramp-down isentropic process takes place. The interac- The right panel of Fig.2 shows the four strokes in a tion strength decreases from cB to cD = cA, the working cooling cycle with the interacting bosons. A → B: The medium reaches the temperature TD. D → A: The work- working medium is initially in the thermal state A de- ing medium is coupled to the target cold reservoir keeping termined by the interaction strength cA = 1 and tem- the interaction strength constant until it reaches the ther- perature Ttar = 1. The isentropic ramp-up of interac- mal state (cA,Ttar). The heat ∆Q2 is extracted from the tion takes place and the interaction strength is finally target reservoir. The cooling efficiency is η = ∆Q2/∆Q1. enhanced to the value cB. After a quasi-adiabatic uni- tary evolution, the system reaches to a state with tem- We would like to stress that in a realistic cycle, A → B perature TB. B → C: Keeping cB constant, the working and C → D are likely not to be rigorously adiabatic. medium is coupled to the hot ambient at temperature However, the temperatures at the non-thermal state B Tsour and reaches the equilibrium state (cB,Tsour). The and D can be still well defined if the spectra of the 6

B adiabatically B increase coupling contact source contact source T C adiabatically sour increase coupling Tsour C

adiabatically decrease coupling A T adiabatically Ttar tar contact A decrease coupling contact target target D D

FIG. 2: Schematic demonstration of the interaction driven refrigeration in the T − s and T − c planes. The cycle with four processes is an analog to the magnetic refrigeration which we discussed in the Fig.1. Here the processes: A → B: interaction ramp-up isentrope; B → C: an isochore by contacting a hot source (release heat to the hot source); C → D: interaction ramp-down isentrope; D → A: a isochore by contacting a cold source (absorb heat from the target source). The cycle is plotted by numerically solving the thermodynamic Bethe ansatz equations (TBAE) of the Lieb-Linger model, see [28]. The cycle begins at point A with n = 0.1, c = 1.0,T = 1.0, (here all the quantities are in the nature units ~ = 2m = kB = 1.) then the coupling strength is tuned to strong interacting region c = 10.0, after contacting with the heat source the coupling strength is tuned back to c = 1.0, finally the working material contacts sufficiently to the target, and the cycle is finished. working system are scaling invariant, see discussion in Bethe ansatz. Theoretical predictions for the existence of Ref.[46]. The exact solution of the working system allows a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing state us to determine working efficiency in this particular case. in the 1D interacting Fermi gas have emerged by using We would like to mention that the modulation of the cou- the exact solution [50–52]. The key features of this T = 0 pling strength in an interaction-driven cooling cycle can phase diagram were experimentally confirmed using fi- be associated with the coupling to external degrees of nite temperature density profiles of trapped fermionic 6Li freedom, also see a recent study of the quantized refrig- atoms [53]. The Hamiltonian of the Yang-Gaudin model erator [47]. In next section, we shall discuss the reachable X Z  2 d2  lowest temperature for an engineering refrigeration with Hˆ = φ† (x) − ~ + µ φ (x)dx σ 2m dx2 σ σ the 1D interacting fermions. σ=↓,↑ Z 2. The Gr¨uneisenparameter at quantum criti- † † +g1D φ (x)φ (x)φ (x)φ (x)dx cality. As discussed in the last section, the Gr¨uneisen ↓ ↑ ↑ ↓ parameters play the central role in this cooling process 1 Z   − h φ† (x)φ (x) − φ† (x)φ (x) dx (16) based on the equations (10) or (12). Since the Gr¨uneisen 2 ↑ ↑ ↓ ↓ parameters are second order derivatives with respect to describes a 1D δ-function interacting two-component free energy, it is expected that the Gr¨uneisenparam- Fermi gas of N fermions with mass m and an external eters will show divergent and scaling behaviors at the magnetic field H constrained by periodic boundary con- QCPs [13, 14, 17, 18, 24, 32, 34], leading to much en- ditions to a line of length L. Where g = −2 2/(ma ) hanced effects for quantum refrigeration. 1D ~ 1D is determined by an effective scattering length a1D In order to illustrate this idea and to analyse the via Feshbach resonances or confinement-induced reso- scaling behaviors of the GPs, we take the Yang-Gaudin nances [38, 43, 44]. g1D > 0 (< 0) represents repul- 2 model [48, 49] as an example to carry out rigorous cal- sive (attractive) interaction. Usually c = mg1D/~ = culations. This model is one of the most important −2/a1D denotes the effective interaction strength. exactly solvable quantum many-body systems. It was Here we show that the different GPs (3), (9) and (11) solved long ago by Yang [48] and Gaudin [49] using the not only signal quantum phase transitions but also quan- 7

be clearly understood by investigating the entropy of the system. At low temperatures, the state of the system away from the critical points usually behaves like Fermi liquid (or Tomonaga-Luttinger liquid region in 1D), see Fig.3. The entropy S ∝ T . In contrast, the entropy at the quantum criticality behaves as [54, 55]

S µ − µ ∝ T (d/z)+1−(1/νz)K( c ). (18) V T 1/νz For 1D system, z = 2 is the dynamic critical exponent and ν = 1/2 is the critical index for correlation length, whereas the µ presents an effective chemical potential, and the µc is the quantum critical point. K(x) is some analytica scaling function. Note that the entropy is ex- actly zero at zero temperature, so there is no background term in eq. (18)[20, 29, 30, 37,√ 56, 57]. For the Gaudin- Yang model, the entropy S ∝ T  T near the QCPs, which implies the local maximum of the entropy at QCPs. FIG. 3: Contour plot of the negative GP (3), i.e. −Γ, mapping out the full phase diagram of the Yang-Gaudin model with an If we plot the entropy in the T − H plane, see Fig.4, the attractive interaction in h−µ plane. It consists of three novel isentropic lines will be bent down significantly at QCPs. phases, fully paired state, Fully polarized state and a FFLO According to equation (10), the GPs are proportional to like state. Here the dimensionless temperature t = 0.001. The the slope of the isentropic line which leads to the diver- GP has a sudden enhancement near the phase boundaries, gence of the GP when T → 0. −1/2 giving a universal divergent scaling Γ ∼ t , see the main The local maximum of the entropy at low temperature text. reveals the essence of the QCPs when the low lying ex- citation becomes degenerate with the ground state [58]. tify various fluctuations in quantum systems. Using the In general, the divergence of the GPs is also present in exact TBA equations, a full critical phase digram of the generic models when quantum phase transition occurs. Yang-Gaudin model at t = 0.0001b is determined by This nature has been extensively studied both in theory the the GP expression (3), see supplementary material and experiments [10–16, 18, 21, 23, 32–34, 59, 60]. [28]. In this contour plot the rescaled units were used, 3. Quantum refrigeration near a phase transi- i.e. t˜ = t/(c2/2),µ ˜ = µ/(c2/2) and h˜ = h/(c2/2). We tion. For refrigeration, it is important to ask what the observe that the GP (3) characterizes the universal di- lowest temperature one can achieve. In Supplementary vergent scaling near the phases boundaries. It shows material [28], we answer the above question for the free that the energy-pressure covariance has a stronger fluc- Fermi gas. This question seems trivial in the common tuations than the energy fluctuation. This nature can refrigeration [23]. However, if the system approaches its be used to identify different quantum phases, i.e. novel quantum critical point, things can be significantly dif- Luttinger liquids of fully-paired state, FFLO-like pairing ferent. The divergent behaviour of the GPs near QCPs state and fully polarized state are determined at low tem- can lead to significant cooling of the system. In fact, the peratures, see Fig.3. We show that the phase boundaries feature of local maximum of the entropy leads to a local between the fully polarized phase and FFLO-like pairing temperature minimum in an isentropic process, see Fig.4. phase and between the fully paired phase and FFLO-like Consequently, one can take this advantage of the quan- pairing phase in Fig.3 can be cast into a universal scaling tum phase transition to enhance the MCE (or interaction (for a constant h) driven MCE). Using exact solution of the 1D attractive Fermi gas, we further demonstrate a magnetic (or inter- 2(µ − µ ) Γ = λt−1/2G c (17) action driven) refrigeration in the interacting Fermi gas t [48, 49]. √ √ Using the condition (see the equation (10)) with the factor λ = πn and λ = 2πn, respectively.

In the above equation, n is the density, G(x) is the scal- Γmag = 0, (19) ing function and µ is an effective chemical potential [20]. More detailed study on the quantum scalings of the GPs we may answer the above question . For the Yang-Gaudin (17) will be published elsewhere. In addition, the mag- model [28], we expect an enhancement of the cooling ef- netic and interacting GPs (9) and (11) also give the full ficiency when the working system is approaching to a phase diagram at low temperatures. quantum critical point in the phase diagram Fig.3. Here The divergence of the GPs at T → 0 near QCPs can we focus on the low temperature region, i.e. T  Td, here 8

stituting this solution into TBA results given in [28], we get at the phase transitions point from a fully- paired phase to the FFLO-like state and from the FFLO QC 2 : 1/2 liked state to the fully paired Fermi gas, respectively S nkB TC2 = 21 1/2 LTd √ S m = λ · √ k3/2T 1/2, for H → H TLL 2： L 1 B c1 m1 QC 1 : S2 nk T ~ 2π TLL 1： S nk T 1/2 = B F2 √ S8 nk T =  B C1 LT3 = B F1 LT1 1/2 d S m 3/2 1/2 LT3 d d = λ1 · √ kB Tc2 , for H → Hm2, (21) L ~ π

TLL 2 x0 3 x0 TLL 1 where λ1 = x0Li1/2(−e )− 2 Li3/2(−e ) ≈ 1.3467. Hm1 and Hm2 are two critical fields corresponding to the two temperature minima in the isentropic contour lines. Us- ing TBA equation, we have the entropy in the liquid phases of pairs and fully-polarized fermions

FIG. 4: The contour plot of the entropy in t − h plane for S 4m 2 −1 = kBTL1n , for H < Hm1, the attractive Yang-Gaudin model at low temperatures. Here L 3~2 the magnetic field h = H/b and the temperature t = T/b S m = k2 T n−1, for H > H . (22) are rescaled by the binding energy with 2m = ~ = kB = 1. L 3 2 B L2 m2 We carried out our calculation through the TBA equations ~ [28] with a fixed density n = 0.1. hc1 and hc2 are the criti- Here TL1 and TL2 are the temperatures in the Luttinger cal points for the phase transitions from fully paired TLL to liquid regions, see the phases T LL1 and T LL2 in Fig.4. the FLLO like phase and from the FFLO like phase to the For the first equation in (22), we applied the strong cou- fully polarized phase at t = 0, respectively. The dash lines in pling condition γ = c/n  1. From these equations (21) different colors present the contour values of entropies at dif- ferent temperatures. The bending down of the contour lines and (22), we find two temperature minima of the refrig- indicates an entropy accumulation with a minimum temper- eration around the two phase transitions ature ( yellow dot). For h < h the system is in the TLL c1  2 of bound pairs obeying the state equation (22), whereas for Tc1 2 TL1 = 8λ2 · , h > hc2 the system is in a fully polarized TLL obeying the Td Td equation (22). These analytical results of the state equations T λ2 T 2 directly give the minima of the temperature during the adia- c2 = 2 · L2 (23) batic demagnetization processes, see (23). Td 2 Td

2π with λ2 = ≈ 1.5552. We further observe that the 2 2 3λ1 ~ n kBTd = ( 2m ) is the degenerate temperature. Fig.4 leading contribution to the entropy at the critical point shows that the condition Γmag = 0 gives solutions for Hm1 involves the excitations of the excess fermions [29]. each quantum phase transition. Like the free Fermi gas Whereas at the critical point Hm2, the leading contri- given in [28], the condition Γmag = 0 leads to two inde- bution to the entropy comes from the excitations of the pendent equations for the Yang-Gaudin model at the two bound pairs. In the isentropic process, the system thus quantum critical points, namely, can retain more entropy per unit temperature near the finite temperature critical point Hm2. This result re- ˜(r) 1 A˜(r)/t A A˜(r)/t veals an enhancement of the cooling efficiency. In cold − Li 1 (−e ) + Li− 1 (−e ) = 0, (20) 2 2 t 2 atom experiment, the temperature is usually much lower where r = 1 and r = 2 stand for the unpaired fermions than the degenerate temperature [61], i.e. TL1/Td  1 and bound pairs, respectively. This means that at the and TL2/Td  1. Trom equation (23), we thus have phase transition H the density of state of unpaired Tc1  TL1 and Tc2  TL2. Moreover, the ideal limit c1 of the temperature T for the refrigeration is twice in fermions is suddenly changed, whereas at the critical c1,2 point H the density of state of the paired fermions order of magnitude compared to the temperature of the c2 heat source T . is suddenly changed. While the effective chemical po- L1,2 tentials of unpaired fermion and pairs, here A˜(1) = ˜(2) 2 (µ + H/2)/b, A = (2µ + c /2)/b were rescaled by IV. SUMMARY 2 the bonding energy b = c /2. Here µ is the chemical potential, t = T/ is the rescaled temperature. b We have conducted a comprehensive investigation of Equation (20) is very similar to the equation Y(x) = x Li1/2(−e ) the GP for ultracold quantum gases, including its ori- x − x = 0 found in the free Fermi gas [28]. We 2Li−1/2(−e ) gin, new identity, caloric effects and quantum refriger- (r) thus have the same solution A˜ /t = x0 ≈ 1.3117. Sub- ation. We have proposed the interaction related GP 9 which reveals characteristic energy scales of quantum der extreme compression. Pramana-Journal of Physics, system induced by the variation of the interaction. To- 87(2), 2016. gether with the other two GPs related to the variations [9] S. A. Khrapak. Gr¨uneisen parameter for strongly coupled of volume and magnetic field, we have established a new Yukawa systems. Physics of Plasmas, 24(4), 2017. [10] R. Kuchler, N. Oeschler, P. Gegenwart, T. Cichorek, identity between them which characterises the univer- K. Neumaier, O. Tegus, C. Geibel, J. A. Mydosh, sal scalings of fluctuations and caloric effect in quantum F. Steglich, L. Zhu, and Q. Si. Divergence of the gases. Based on the entropy accumulation at the quan- Gr¨uneisen ratio at quantum critical points in heavy tum critical point, two promising protocols of quantum fermion metals. Physical Review Letters, 91(6), 2003. refrigeration driven either by interaction or by magnetic [11] R. K¨uchler, P. Gegenwart, J. Custers, O. Stock- field have been studied. Using Bethe Ansatz, we studied ert, N. Caroca-Canales, C. Geibel, J. G. Sereni, and the expansionary, the magnetic and the interacting GPs, F. Steglich. Quantum criticality in the cubic heavy- Fermion system CeIn3−xSnx. Physical Review Letters, quantum refrigeration, magnetocaloric effect and quan- 96(25):256403, 2006. tum critical phenomenon of the Lieb-Liniger model and [12] R. K¨uchler, P. Gegenwart, C. Geibel, and F. Steglich. Yang-Gaudin model. Our method opens to further study Systematic study of the Gr¨uneisenratio near quantum the GPs and quantum refrigeration for quantum gases of critical points. Science and Technology of Advanced Ma- ultracold atoms with different spin symmetries in 1D and terials, 8(5):428–433, 2007. higher dimensions. [13] P. Gegenwart. Gr¨uneisenparameter studies on heavy fermion quantum criticality. Reports on Progress in Physics, 79(11), 2016. Acknowledgement [14] A. Steppke, R. K¨uchler, S. Lausberg, E. Lengyel, L. Steinke, R. Borth, T. L¨uhmann,C. Krellner, M. Nick- las, C. Geibel, F. Steglich, and M. Brando. Ferromag- The author thanks Y.X. Liu, L. Peng, Y.Z. Jiang, Y.Y. netic quantum critical point in the heavy-Fermion metal Chen, F. He, H. Pu and R. Hulet for helpful discussions. ybni4(p1−xasx). Science, 339(6122):933–936, 2013. This work is supported by the National Key R&D Pro- [15] Y. Tokiwa, T. Radu, C. Geibel, F. Steglich, and gram of China No. 2017YFA0304500, the key NSFC P. Gegenwart. 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ety of Japan, 87(3):034712, 2018. [61] H. Pethick, C. J. Smith. Bose-Einstein Condensation in Dilute Gases. Cambridge University Press, Cambridge, 2 edition, 2008. [62] C. N. Yang and C. P. Yang. Thermodynamics of a one-dimensional system of Bosons with repulsive delta- function interaction. Journal of Mathematical Physics, 10(7):1115–1122, 1969. [63] Here we do not write out explicitly the verti- cal lines and corresponding variables in the par- tial derivatives, where the derivatives are calcu- lated in the grand canonical ensemble with vari- ables V (volume),T (temperature),H(magnetic filed),and c(coupling strength). [64] L. Peng, Y.-C. Yu and X.-W. Guan, in preparation, 2019. 2 00 Supporting Material for “Gr¨uneisenParameters: = NkB[(~ω0β) f (~ω0β)] (S3) origin, identity and quantum refrigeration”

Yi-Cong Yu, Shi-Zhong Zhang and Xi-Wen Gaun is again homogeneous, and the same for

1. THE DEFINITION OF THE GRUNEISEN¨ PARAMETER ∂S 2 00 ω0 = −NkB[(~ω0β) f (~ω0β)]. (S4) ∂ω0 In the microscopic point of view the Gr¨uneisenpa- rameter is related to the derivation of the model fre- quency and the bulk volume. The most simple con- In fact these two quantities have the simple relation ∂S ω0 = −CV . Because the frequency of the sideration about the thermodynamic quantities in solid ∂ω0 state physics is the Einstein model, where the excitation is decided by the size of the solid, we have the relation: in a solid is described by N phonon with the same fre- quency ω0, or equivalently saying, the solid is composed ∂S ω ∂V ∂S by N independent harmonic oscillator with the spectrum C = −ω = − 0 (V ), (S5) 1 V 0 ∂ω V ∂ω ∂V En = (n + 2 )~ω0. The partition function is 0 0

log Z = Nf(~ω0β), (S1) if we define Γ−1 = − ω0 ∂V , then according to equation where β is defined by β = 1/(k T ) with the Boltzmann V ∂ω0 B (S5) we have constant kB, and the analytic function f(x) is defined 1 x by f(x) = 2 x − log(e − 1). It is obvious that in Ein- stein model the temperature T and the frequency of the ∂S phonon ω are homogeneous in partition function Z and V 0 Γ = ∂V . (S6) in the entropy CV ∂ log Z S = kB(log Z − β ) ∂β This definition is from the original research of E. = NkB[f(~ω0β) − ~ω0βf(~ω0β)]. (S2) Gr¨uneisenwhen he was studying the Einstein model [1,2]. Notice that although the system is described by ∂ Although the energy E = − ∂β log Z loses the homogene- the frequency of the phonon which is a microscopic quan- ity for T and ω0, the specific heat tity, the definition (1) in the main text implies that the relationship between ω and V can be related to some ∂E ∂2 0 C = = k β2 log Z observable thermodynamic quantity. V ∂T B ∂β2

X V ∂ωi The analysis above implies that the Gr¨uneisenparam- = −( )N k [( ω β)2f 00( ω β)],(S9) ω ∂V i B ~ i ~ i eter is closely related to the homogeneity of the thermo- i i dynamic quantities, thus it can be easily extended to the

Deby model, in which a solid is described by indepen- where the ωi presents the i-th vibrational frequency in ∂S dent phonons with different frequencies. In this case the the solid, and for every ωi the specific heat and the ωi ∂ωi partition function: is related as equation (S5). If we define the Gr¨uneisen X parameter for i-th frequency log Z = Nif(~ωiβ), (S7) i d log ω Γ = − i , (S10) i d log V where the ωi presents the i-th vibrational frequency in the solid. Similarly, we have: we finally obtain the total Gr¨uneisenparameter by the Γi X 2 00 and C for each mode CV = NikB[(~ωiβ) f (~ωiβ)], (S8) V,i i P ∂S X ∂S ∂ωi i ΓiCV,i V = NiV Γ = P . (S11) ∂V ∂ω ∂V CV,i i i i

1 

change H

1 1 3 3 S = −ln − ln 1 1 2 2 3 3 6 6 0 S = −ln − ln − ln − ln 4 4 4 4 1 12 12 12 12 12 12 12 12 

change c

S=− aln a 5 5 7 7  ii S = −ln − ln 2 12 12 12 12 The magnetocaloric effect and the “coupling magnetocaloric effect”

FIG. S1: A diagramatic illustration of the magnetocaloric and interaction driven magnetocaloric effects for a system of 12 particles. A change of the magnetic field (upper panel) or a variation of interaction strength (lower panel) essentially influences the spectral structures of the quantum many-body system. The former leads to the changes of particle distribution among energy levels, whereas the later may cause energy level distribution changed, so that the entropies S1,2 for the two configures are respectively changed.

2. THE PROOF OF THE IDENTITY FOR obviously we have N = hni for the particle number, E = GRUNEISEN¨ PARAMETERS ∂n,m hn,mi for the interal energy, and p = −h ∂V i for the pressure. From the definition eq. (S12) and eq. (S13) we We give a brief proof of the identity (13) by general imediately obtain statistic approach. We begin with the partition function ∂a of the grand canonical ensemble n,m = a · (hni − n) ∂α n,m X ∂an,m Z = e−αn−βn,m (S12) = a · (h i −  ) ∂β n,m n,m n,m n,m ∂an,m ∂n,m ∂n,m = an,m · (h i − ) (S16) where the n,m presents the energy of the mth energy ∂y ∂y ∂y level of n particle Hamiltonian , β is the inversed tem- here y presents any external field, it can be volume, mag- perature β = 1/(k T ), and α = −βµ with chemical po- B netic field or the coupling, i.e. y = V, H, c., next we tential µ. The population function is introduce the covariance of two function fn,m, gn,m as 1 X a = e−αn−βn,m (S13) Cov(F,G) = a f g − hf ihg i (S17) n,m Z n,m n,m n,m n,m n,m n,m and the entropy is here the fn,m and gn,m is the value of thermodynamic X quantities F and G in the state an,m respectively. Apply S = − an,m ln an,m (S14) this notation and use eqs. (S12), (S13) and (S16) we have n,m dS = [βCov(E,N) + αCov(N,N)]dα Now for every function fn,m of the quantum index n, m, +[βCov(E,E) + αCov(E,N)]dβ we can define the expection hfn,mi by ∂E ∂E +β[βCov(E, ) + αCov(N, )]dV X ∂V ∂V hfn,mi = an,mfn,m (S15) ∂E ∂E n,m +β[βCov(E, ) + αCov(N, )]dH ∂H ∂H

2 ∂E ∂E +β[βCov(E, ) + αCov(N, )]dc 3. THE BETHE ANSATZ SOLUTIONS FOR ∂c ∂c LIEB-LINIGER MODEL dE = Cov(E,N)dα + Cov(E,E)dβ ∂E ∂E +[ + βCov(E, )]dV The first quantized form of the Lieb-Liniger Hamilto- ∂V ∂V nian is [42] ∂E ∂E +[ + βCov(E, )]dH ∂H ∂H N 2 X ∂ X ∂E ∂E Hˆ = − + 2c δ (x − x ) (S26) +[ + βCov(E, )]dc ∂x2 i j ∂c ∂c i=1 i i>j ∂E dN = Cov(N,N)dα + Cov(E,N)dβ + βCov(N, )dV ∂V The dimensionless interaction strength is defined by γ = ∂E ∂E c/n with c = −2/a1D, where n = N/L is the linear den- +βCov(N, )dH + βCov(N, )]dc (S18) ∂H ∂c sity. The interaction strength can be controlled by tuning either ω or a in experiments. Here we set Boltzmann Apply the (S18) we can get the thermodynamic quanti- ⊥ s constant k = 1, 2m = = 1. ties B ~ We consider c > 0 for repulsive interaction. The Bethe ∂S Cov(E,N)Cov(E,N) |N,V,H,c = β[Cov(E,E) − ] Ansatz equation for the ground state is given in term of ∂β Cov(N,N) the quasimomentum distribution ρ {k} ∂S ∂E Cov(E,N)Cov(N, ∂E ) | = β2[Cov(E, ) − ∂V ] ∂V N,β,H,c ∂V Cov(N,N) 1 1 Z kF 2c Cov(E,N)Cov(N, ∂E ) ρ (k) = + ρ (q) dq (S27) ∂S 2 ∂E ∂H 2 2 | = β [Cov(E, ) − ] 2π 2π −k c + (k − q) ∂H N,β,V,c ∂H Cov(N,N) F ∂E ∂S 2 ∂E Cov(E,N)Cov(N, ∂c ) Where k is the cut-off momentum, i.e. Fermi-like |N,β,V,H = β [Cov(E, ) − (S19)]. F ∂c ∂c Cov(N,N) momentum. The particle density is given by n = R kF Suppose the scaling behaviour of the spectrum: L → −k ρ (k) dk. The ground state energy can be expressed λ −λ −2λ −2λ F e L, c → e c, H → e H, n,m → e n,m, or in as other words the spectrums have the homogenous form  (L, H, c) = L−2f (L2H, Lc) where the f are Z kF n,m n,m n,m E = Nn2 ρ (k) k2dk (S28) analytical functions. We thus have the identity: −kF ∂ ∂ ∂ L n,m = −2 + 2H n,m + c n,m (S20) ∂L n,m ∂H ∂c In describing the thermodynamics of the model the key quantity is the dressed energy ∂S ∂S ∂n,m ∂n,m notice β ∂β = −T ∂T and L ∂L = d · V ∂V where the d is the dimension of the system, and notice that (k) = T ln(ρh(k)/ρ(k)) (S29) the Cov(F,G) is bilinear, then substitute eq. (S20) into (S19) we arrive which plays the role of excitation energy measured from ∂S ∂S ∂S ∂S the energy level (k ) = 0. The thermodynamics of the d · V = 2T + 2H + c (S21) F ∂V ∂T ∂H ∂c model in equilibrium follows from the Yang-Yang equa- substitute the definitions of GPs tion [62] ∂S ∞ V ∂V Z Γ = , (S22) (k) = 0(k) − µ − T dq a (k − q) ln(1 + e−(q)/T ) CV 2 −∞ ∂S H ∂H |N,T,V (S30) Γmag = − , (S23) 2 ∂S where 0(k) = ~ k2 is the bare dispersion, µ is the chem- T ∂T |N,B,V 2m ∂S ical potential and the integration kernel is denoted as c ∂c |N,H,T,V 1 2c Γint = − (S24) a2(x) = 2 2 is the integration kernel. T ∂S | 2π c +x ∂T N,H,c,V The pressure p(T )is given in terms of the dressed en- into (S21), we prove the identity ergy by

d · Γ + 2Γmag + Γcoupling = 2. (S25) T Z ∞ p(T ) = dk ln(1 + e−(k)/T ). (S31) The spectral structures of quantum gases are sensitive 2π −∞ to system volume, external fields and interactions, see Fig. S1. This identity quantifies universal scalings of the and the grand potential is Ω(µ, T, c) = −pL, then all the GPs in quantum systems of ultracold atoms in all dimen- equilibrium thermodynamic quantities can be obtained sions. by differential of Ω.

3 4. THE BETHE ANSATZ SOLUTIONS FOR THE and in the strong coupling limit c  1, applying the YANG-GAUDIN MODEL method similar to the Bose gas case, we have

(u) 2 1 2 (b) We consider the Yang-Gaudin model [48, 49], which  (k) = D1 · k − µ − H + p 1 2 |c| describs a 1D of the N δ -interacting spin- 2 fermions. It is constrained by periodic boundary conditions to a line 2 5/2 A(b)/T +√ T Li 5 (−e 0 ), of length L and subject to an external magnetic field H. 2π|c|3 2 The Hamiltonian reads 1 4 1 (b)(k) = D · 2k2 − 2µ − c2 + p(u) + p(b) 2 2 |c| |c| N ∂2 N √ ˆ X X 5/2 H = − 2 + 2c δ(xi − xj). (S32) 4 2T A(u)/T ∂xi + √ Li 5 (−e 0 ) i=1 i

1 b only consider the first three terms of the series in the b(k) = 2(k2 − µ − c2) + T a ∗ ln(1 + e (k)/T ) 4 2 thermodynamic quantities.The analytical equations for u(k)/T the Gaudin-Yang model is +T a1 ∗ ln(1 + e ), √ 1 b u(k) = k2 − µ − H + T a ∗ ln(1 + e (k)/T ) (r) r 3 A(r)/T 1 p = − √ T 2 Li 3 (−e ), 2 2 π 2 ∞ X −1 N (m) −T al ∗ ln(1 + η (k)), (r) X p A(r) =A − D (S39) l=1 0 rm |c| m=1 lH −(u) ln ηl(λ) = + al ∗ ln(1 + e /T ) T where r = 1, 2 is the index for unpaired and paired ∞ (1) (2) 2 X −1 fermions, A0 = µ + H/2 and A0 = 2µ + c /2,and the + Tlm ∗ ln(1 + ηm (λ)) (S34) Drm is the matrix D11 = 0,D12 = 2,D21 = 4,D22 = 1. m=1 Next we normaliz equation (S39) by the binding energy The pressure in terms of these finite temperature dressed  = c2/2,define energies is given by (r) (r) (r) p (r) A µ Z ∞ p˜ = , A˜ = , µ˜ = , T −b(k)/T 1 3 1 2 1 2 p = dk ln(1 + e ) 2 |c| 2 |c| 2 |c| π −∞ H T Z ∞ h = , t = . (S40) T −u(k)/T 1 2 1 2 + dk ln(1 + e ). (S35) 2 |c| 2 |c| 2π −∞ then the equation (S39) becomes Note that in the TBAE for Gaudin-Yang model (S34) √ there are ”stings” ηl(λ) which present the excitation (r) r 3 A˜(r)/t p˜ = − √ t 2 Li 3 (−e ), of spin flipping. In the low temperature limit where 2 2π 2 H/T  1, the contributions of strings to the thermo- N dynamic potentials are restrained, then we can ignore ˜(r) ˜(r) X (m) A =A0 − Drmp˜ , (S41) the strings and simplify the TBAE m=1

b 2 1 2 b(k)/T ˜(1) ˜(2)  (k) = 2(k − µ − c ) + T a2 ∗ ln(1 + e ) where A0 =µ ˜ + h/2, A0 = 2˜µ + 1. Then up to the 4 first three terms the result of the pressures are u(k)/T +T a1 ∗ ln(1 + e ), 1 b (1) 1 3 A˜(1)/t u(k) = k2 − µ − H + T a ∗ ln(1 + e (k)/T ).(S36) p˜ = − √ t 2 Li 3 (−e ), (S42) 2 1 2 2π 2

4 √ (1) (2) (2) (2) (2) 1 3 A˜(2)/t p˜ = − √ t 2 Li 3 (−e ), (S43) 2f 1 f 3 f 1 f 3 2 π 2 + 2 2 + 2 2 . (S45) (2) √ (1) (2) (2) (2) π 4π f 3 2f 3 f 1 f 1 f 3 ˜(1) ˜(1) 2 2 2 2 2 A = A0 + √ + + ,(S44) (r) π π 2π (r) s A˜ /t Here the definition fs , t Lis(−e 0 ),r = 1, 2. And √ (1) (2) (1) (2) 2f 3 f 3 f 3 f 1 for further discussions, we list the result of all the first A˜(2) = A˜(2) + √ 2 + √2 + 2√ 2 and higher order partial derivatives of the pressure 0 π 2 π 2π

√ (1) (2) (1) (2) (2) 2 (1) 2 (2) (1) (2) 2 (2) 3 n ∂p˜ F 1 F 1 2F 1 F 1 (F 1 ) (F 1 ) F 1 3F 1 (F 1 ) (F 1 ) n˜ = = = − √2 − √2 − 2 2 − 2 − 2 2 − √2 2 − 2 , (S46) |c| ∂µ˜ 2 2π π π 2π 2π3/2 2π3/2 4π3/2 (1) (1) (2) (1) 2 (2) (1) (2) 2 m ∂p˜ F 1 F 1 F 1 (F 1 ) F 1 F 1 (F 1 ) m˜ = = = − √2 − 2√ 2 − 2 2 − 2√ 2 , (S47) |c| ∂h 4 2π 2 2π 4π3/2 4 2π3/2 (1) (2) √ (1) (2) (1) (2) (2) (2) b 2 F 2F 3 2F F 3F F 3F F  ∂ p˜ − 1 − 1 1 − 1 − 1 1 − 1 1 κ˜ = κ = = − √ 2 − √ 2 − 2 2 − √2 2 − 2 2 |c| ∂µ˜2 2 2π π π 2π π (1) (1) (2) (1) (1) (2) (1) (2) (2) (1) (2) (2) (2) (2) (2) 3F 1 F 1 F− 1 3F− 1 F 1 F 1 21F 1 F− 1 F 1 15F− 1 F 1 F 1 3F− 1 F 1 F 1 − 2 2 2 − 2 2 2 − √2 2 2 − √2 2 2 − 2 2 2 , (S48) π3/2 2π3/2 2π3/2 2 2π3/2 π3/2 (1) (1) (2) (1) (1) (2) (1) (1) (2) (1) (2) (2) b 2 F F F F F F 3F F F F F F  ∂ p˜ − 1 − 1 1 1 1 − 1 − 1 1 1 − 1 1 1 χ˜ = χ = = − √ 2 − √2 2 − 2 2 2 − 2 2 2 − 2√ 2 2 , (S49) |c| ∂h2 8 2π 4 2π 4π3/2 8π3/2 8 2π3/2 (1) (2) ˜(1) (1) ˜(2) (2) S˜ ∂p˜ S 3 F 3 3 F 3 1 A F 1 1 A F 1 s˜ = = = = − √ 2 − √ 2 + √ 2 + √ 2 L ∂t L|c| 4 2π t 4 π t 2 2π t 2 π t (1) (2) (1) (2) (2) (2) ˜(1) (1) (2) ˜(2) (1) (2) ˜(2) (2) (2) 3F 3 F 1 3F 1 F 3 3F 1 F 3 A F 1 F 1 A F 1 F 1 A F 1 F 1 − √2 2 − √2 2 − 2 2 + √ 2 2 + √2 2 + 2 2 2 2πt 4 2πt 8πt 2πt 2 2πt 4πt (1) (1) (2) (1) (2) (2) (1) (2) (2) (2) (2) (2) ˜(1) (1) (1) (2) 3F 1 F 3 F 1 3F 3 F 1 F 1 15F 1 F 1 F 3 3F 1 F 1 F 3 A F 1 F 1 F 1 − 2 2 2 − 2√ 2 2 − √2 2 2 − 2 2 2 + 2 2 2 4π3/2t 4 2π3/2t 8 2π3/2t 16π3/2t 2π3/2t ˜(1) (1) (2) (2) ˜(2) (1) (2) (2) ˜(2) (2) (2) (2) A F 1 F 1 F 1 5A F 1 F 1 F 1 A F 1 F 1 F 1 + √2 2 2 + √2 2 2 + 2 2 2 , (S50) 2 2π3/2t 4 2π3/2t 8π3/2t c˜ c b ∂2p˜ V = V = t T |c| ∂t2 ˜(1) 2 (1) ˜(1) (1) (1) ˜(2) 2 (2) ˜(2) (2) (2) (A ) F− 1 A F 1 3F 3 (A ) F− 1 A F 1 3F 3 = − √ 2 + √ 2 − √ 2 − √ 2 + √ 2 − √ 2 2 2πt2 2 2πt2 8 2πt2 2 πt2 2 πt2 8 πt2 √ ˜(1) ˜(2) (1) (2) ˜(2) 2 (1) (2) ˜(2) (1) (2) ˜(1) 2 (1) (2) 2A A F 1 F− 1 (A ) F 1 F− 1 3A F 3 F− 1 (A ) F− 1 F 1 − 2 2 − √ 2 2 + √ 2 2 − √ 2 2 πt2 2 2πt2 2πt2 2πt2 √ ˜(1) ˜(2) (1) (2) ˜(1) (1) (2) ˜(2) (1) (2) (1) (2) A A F− 1 F 1 2A F 1 F 1 A F 1 F 1 9F 3 F 1 − √ 2 2 + 2 2 + √ 2 2 − √2 2 2πt2 πt2 2πt2 4 2πt2 ˜(2) 2 (2) (2) ˜(2) (2) (2) ˜(1) (1) (2) (1) (2) 3(A ) F− 1 F 1 A F 1 F 1 3A F− 1 F 3 9F 1 F 3 − 2 2 + 2 2 + √ 2 2 − √2 2 4πt2 2πt2 2 2πt2 8 2πt2 ˜(2) (2) (2) (2) (2) ˜(1) 2 (1) (1) (2) ˜(1) ˜(2) (1) (1) (2) 3A F− 1 F 3 9F 1 F 3 (A ) F 1 F 1 F− 1 A A F 1 F 1 F− 1 + 2 2 − 2 2 − 2 2 2 − 2 2 2 4πt2 16πt2 π3/2t2 π3/2t2 ˜(1) (1) (1) (2) ˜(2) (1) (1) (2) (1) (1) (2) ˜(1) 2 (1) (1) (2) 3A F 1 F 3 F− 1 3A F 1 F 3 F− 1 9F 3 F 3 F− 1 (A ) F− 1 F 1 F 1 + 2 2 2 + 2 2 2 − 2 2 2 − 2 2 2 π3/2t2 2π3/2t2 4π3/2t2 π3/2t2

5 ˜(1) (1) (1) (2) ˜(1) (1) (1) (2) (1) (1) (2) ˜(1) ˜(2) (1) (2) (2) A F 1 F 1 F 1 3A F− 1 F 3 F 1 3F 3 F 1 F 1 3A A F 1 F− 1 F 1 + 2 2 2 + 2 2 2 − 2 2 2 − √ 2 2 2 π3/2t2 2π3/2t2 2π3/2t2 2π3/2t2 ˜(2) 2 (1) (2) (2) ˜(2) (1) (2) (2) ˜(1) ˜(2) (1) (2) (2) 5(A ) F 1 F− 1 F 1 9A F 3 F− 1 F 1 5A A F− 1 F 1 F 1 − √ 2 2 2 + √2 2 2 − √ 2 2 2 2 2π3/2t2 2 2π3/2t2 2 2π3/2t2 √ ˜(2) 2 (1) (2) (2) ˜(1) (1) (2) (2) ˜(2) (1) (2) (2) ˜(2) (1) (2) (2) (A ) F− 1 F 1 F 1 A F 1 F 1 F 1 A F 1 F 1 F 1 2A F 1 F 1 F 1 − √ 2 2 2 + √2 2 2 + √2 2 2 + 2 2 2 2 2π3/2t2 2π3/2t2 2 2π3/2t2 π3/2t2 (1) (2) (2) ˜(2) 2 (2) (2) (2) ˜(2) (2) (2) (2) ˜(1) (1) (2) (2) 3F 3 F 1 F 1 5(A ) F− 1 F 1 F 1 A F 1 F 1 F 1 3A F 1 F− 1 F 3 − 2√ 2 2 − 2 2 2 + 2 2 2 + √2 2 2 2 2π3/2t2 8π3/2t2 4π3/2t2 2 2π3/2t2 ˜(2) (1) (2) (2) (1) (2) (2) ˜(1) (1) (2) (2) ˜(2) (1) (2) (2) 15A F 1 F− 1 F 3 9F 3 F− 1 F 3 15A F− 1 F 1 F 3 3A F− 1 F 1 F 3 + √ 2 2 2 − 2√ 2 2 + √ 2 2 2 + √ 2 2 2 4 2π3/2t2 4 2π3/2t2 4 2π3/2t2 2 2π3/2t2 (1) (2) (2) ˜(2) (2) (2) (2) (2) (2) (2) (1) (2) (2) (2) (2) (2) 15F 1 F 1 F 3 9A F− 1 F 1 F 3 3F 1 F 1 F 3 9F− 1 F 3 F 3 9F− 1 F 3 F 3 − √2 2 2 + 2 2 2 − 2 2 2 − √2 2 2 − 2 2 2 , (S51) 4 2π3/2t2 8π3/2t2 8π3/2t2 8 2π3/2t2 32π3/2t2 (1) (1) (1) (2) (2) (2) (1) (1) (1) (1) (1) 2 A˜ F F A˜ F F A˜ F F F F ∂ p˜ 1 − 1 1 1 1 − 1 1 1 − 1 1 1 1 = √ 2 − √ 2 + √ 2 − √ 2 + 2 2 − 2 2 ∂µ∂t 2 2π t 4 2π t π t 2 π t 2πt 4πt √ ˜(1) (1) (2) ˜(2) (1) (2) (1) (2) ˜(1) (1) (2) ˜(2) (1) (2) (1) (2) 2A F 1 F− 1 A F 1 F− 1 3F 3 F− 1 A F− 1 F 1 A F− 1 F 1 F 1 F 1 + 2 2 + √ 2 2 − √2 2 + √ 2 2 + √ 2 2 − √2 2 πt 2πt 2πt 2πt 2 2πt 2πt ˜(2) (2) (2) (2) (2) (1) (2) (2) (2) ˜(1) (1) (1) (2) (1) (1) (2) 5A F− 1 F 1 F 1 F 1 3F− 1 F 3 3F− 1 F 3 A F 1 F 1 F− 1 3F 1 F 3 F− 1 + 2 2 − 2 2 − √2 2 − 2 2 + 2 2 2 − 2 2 2 2πt πt 4 2πt 4πt π3/2t 2π3/2t ˜(1) (1) (1) (2) ˜(2) (1) (1) (2) (1) (1) (2) (1) (1) (2) 11A F− 1 F 1 F 1 A F− 1 F 1 F 1 9F 1 F 1 F 1 3F− 1 F 3 F 1 + 2 2 2 + 2 2 2 − 2 2 2 − 2 2 2 4π3/2t 2π3/2t 8π3/2t 4π3/2t √ √ ˜(1) (1) (2) (2) ˜(1) (1) (2) (2) ˜(2) (1) (2) (2) ˜(2) (1) (2) (2) A F 1 F− 1 F 1 2 2A F 1 F− 1 F 1 13A F 1 F− 1 F 1 2A F 1 F− 1 F 1 + √2 2 2 + 2 2 2 + √ 2 2 2 + 2 2 2 2π3/2t π3/2t 2 2π3/2t π3/2t (1) (2) (2) ˜(1) (1) (2) (2) ˜(2) (1) (2) (2) (1) (2) (2) ˜(2) (2) (2) (2) 15F 3 F− 1 F 1 A F− 1 F 1 F 1 5A F− 1 F 1 F 1 13F 1 F 1 F 1 3A F− 1 F 1 F 1 − 2√ 2 2 + √ 2 2 2 + √ 2 2 2 − √2 2 2 + 2 2 2 2 2π3/2t 2 2π3/2t 4 2π3/2t 4 2π3/2t π3/2t (2) (2) (2) (1) (1) (2) (1) (2) (2) (1) (2) (2) (2) (2) (2) 7F 1 F 1 F 1 3F− 1 F 1 F 3 15F 1 F− 1 F 3 15F− 1 F 1 F 3 15F− 1 F 1 F 3 − 2 2 2 − 2 2 2 − 2√ 2 2 − √2 2 2 − 2 2 2 , (S52) 8π3/2t 4π3/2t 4 2π3/2t 8 2π3/2t 8π3/2t (1) (1) (1) (1) (1) (1) (1) (1) (2) (1) (2) (2) (1) (2) (1) (1) (1) (2) 2 A˜ F F A˜ F F F F A˜ F F A˜ F F A˜ F F F ∂ p˜ − 1 1 − 1 1 1 1 1 − 1 − 1 1 1 1 − 1 = √ 2 − √2 + 2 2 − 2 2 + √2 2 + √ 2 2 + 2 2 2 ∂h∂t 4 2πt 8 2πt 4πt 8πt 2 2πt 4 2πt 2π3/2t (1) (1) (2) ˜(1) (1) (1) (2) ˜(2) (1) (1) (2) (1) (1) (2) (1) (1) (2) 3F 1 F 3 F− 1 7A F− 1 F 1 F 1 A F− 1 F 1 F 1 5F 1 F 1 F 1 3F− 1 F 3 F 1 − 2 2 2 + 2 2 2 + 2 2 2 − 2 2 2 − 2 2 2 4π3/2t 8π3/2t 4π3/2t 16π3/2t 8π3/2t ˜(2) (1) (2) (2) ˜(2) (1) (2) (2) (1) (2) (2) (1) (1) (2) (1) (2) (2) 5A F 1 F− 1 F 1 A F− 1 F 1 F 1 F 1 F 1 F 1 3F− 1 F 1 F 3 3F 1 F− 1 F 3 + √2 2 2 + √ 2 2 2 − 2 √ 2 2 − 2 2 2 − 2√ 2 2 4 2π3/2t 8 2π3/2t 2 2π3/2t 8π3/2t 8 2π3/2t (1) (2) (2) 3F− 1 F 1 F 3 − √2 2 2 , (S53) 16 2π3/2t (1) (1) (2) (1) (2) (1) (1) (2) (1) (1) (2) (1) (2) (2) 2 F F F F F F F F 3F F F 3F F F ∂ p˜ − 1 1 − 1 − 1 1 1 1 − 1 − 1 1 1 1 − 1 1 = − √ 2 − 2√ 2 − √2 2 − 2 2 2 − 2 2 2 − 2 √ 2 2 ∂µ∂h 4 2π 2π 2π π3/2 4π3/2 2 2π3/2 (1) (2) (2) 3F− 1 F 1 F 1 − √2 2 2 , (S54) 2 2π3/2

(r) s A˜(r)/t where Fs , t Lis(−e ),r = 1, 2

6 x0 3 x0 5. QUANTUM REFRIGERATION FOR THE where λ1 = x0Li1/2(−e )− 2 Li3/2(−e ) ≈ 1.3467. This FREE FERMI GAS means that we obtain the explicit form of the density and entropy at the minimum point of the temperature for an For a low temperature quantum refrigeration, we may isentropic process. We also observe that at this critical raise up a question what is the low temperature limit for point, the entropy is of order O(T 1/2) and independent a given temperature of the hot source in the magnetic of the density. In the expression eq. (S58), Tc denotes refrigeration? In order to answer this question, we con- the temperature at the minimum of the entropy. We sider the simplest case, i.e. the 1D two component free can take eq. (S58) as the state function of the D in the Fermi gas. The grand canonical potential for this system refrigeration cycle depicted in Fig.1. Whereas the point is simply given C, which determines the temperature of the heat source, √ locates deeply in the Tomonaga-Luttinger liquid(TLL) 2mL h 3/2 (µ+h/2)/(kB T ) region. From the results of Luttinger liquid [20, 29], we Ω = √ (kBT ) Li 3 (−e ) 2~ 2π 2 have the following relation among the density, entropy i (µ−h/2)/(kB T ) and temperature +Li 3 (−e ) , (S55) 2 nS m 2 where the h is the magnetic field, µ is the chemical po- = kBTF , (S59) L 3~2 tential, L the scale of the system, T is the temperature, here TF denotes the initial temperature of the free ~ is the Planck constant and the kB the Boltzman con- fermions. The expressions (S58) and (S59) directly build stant. For finding the local minimum of the entropy, we up a useful relation between the temperatures at the min- only need to solve the equation imum entropy and at any state in the TLL region through an isentropic process, namely, Γmag = 0. (S56)  2 Tc 2 TF Using eq. (8), it is not difficult to get the condition ≈ λ2 · . (S60) Td Td µ + h/2 µ − h/2 Y( ) = Y( ), (S57) 2π 2n2 Here the Td = ~ is the degenerate temperature and kBT kBT mkB λ = 2π/3λ ≈ 1.5552 is a constant. The relation (S60) x 2 1 Li1/2(−e ) here the analytic equation Y(x) = x − x . Next gives the lowest limit of the temperature for a magnetic 2Li−1/2(−e ) we apply the low temperature limit kBT  µ, which refrigeration with a fixed temperature of the heat source. implies that µ+h/2  1. Because the analytic property We see clearly that in the quantum gases the thermal kB T wave length is usually much smaller than its de Broglie of the function Y(x) shows Y(x) → π x−1 when x → ∞, 6 wave length. We always have a relation T  T [61]. then the solution of equation (S57) at low temperature F d µ−h/2 Therefore, according to (S60), we have Tc  TF  Td. is simply = x0, where the x0 ≈ 1.3117 is the only kB T Here T is the Fermi temperature of the heat source, zero point of function Y(x). Having this solution in mind, F i.e. T → T . Whereas the T is the temperature of we then have the density and the entropy from canonical F sour c the target system, i.e. T → T , see Fig. 4 in the main thermodynamic potential (S55) c tar text. We ought to point out that the equation (S60) holds √ √ only in the case that the temperature TF is lower than 2m 2 p S 2m 3/2 1/2 n = 2µ, = λ1 · √ kB Tc , (S58) the temperature of quantum degenerate temperature Td. 2~ π L 2~ π

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