Hindawi Journal of ermodynamics Volume 2017, Article ID 3060348, 12 pages https://doi.org/10.1155/2017/3060348
Research Article Thermodynamics of Low-Dimensional Trapped Fermi Gases
Francisco J. Sevilla
Instituto de F´ısica, Universidad Nacional Autonoma´ de Mexico,´ Apdo. Postal 20-364, 01000 Ciudad de Mexico,´ Mexico
Correspondence should be addressed to Francisco J. Sevilla; [email protected]
Received 8 October 2016; Revised 2 December 2016; Accepted 5 December 2016; Published 26 January 2017
Academic Editor: Felix Sharipov
Copyright © 2017 Francisco J. Sevilla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The effects of low dimensionality on the thermodynamics of a Fermi gas trapped by isotropic power-law potentials are analyzed. Particular attention is given to different characteristic temperatures that emerge, at low dimensionality, in the thermodynamic functions of state and in the thermodynamic susceptibilities (isothermal compressibility and specific heat). An energy-entropy argument that physically favors the relevance of one of these characteristic temperatures, namely, the nonvanishing temperature at which the chemical potential reaches the Fermi energy value, is presented. Such an argument allows interpreting the nonmonotonic dependence of the chemical potential on temperature, as an indicator of the appearance of a thermodynamic regime, where the equilibrium states of a trapped Fermi gas are characterized by larger fluctuations in energy and particle density as is revealed in the corresponding thermodynamics susceptibilities.
1. Introduction temperaturerangesinvolved.Further,thecontrolachieved on the experimental settings has opened the possibility of The discovery of the quantum statistics that incorporate directly testing a variety of quantum effects such as Pauli Pauli’s exclusion principle [1], made independently by Fermi blocking [35] and designing experiments to probe condensed [2] and Dirac [3], allowed the qualitative understanding of matter models, though much lower temperatures are needed severalphysicalphenomena—inawiderangeofvaluesof to achieve the phenomena of interest. On this trend, exper- the particle density, from astrophysical scales to subnuclear imentally new techniques are being devised to cool further ones—in terms of the ideal Fermi gas (IFG). The success a cloud of atomic fermions [36–39]. Techniques based on of the explicative scope of the ideal Fermi gas model relies the giving-away of entropy by changing the shape of the on Landau’s Fermi liquid theory where fermions interacting trapping potential have become of great importance and, repulsively through a short range forces can be described in as in many instances, a complete understanding of trapped somedegreeasanIFG.Thesituationschangedramaticallyin noninteracting fermionic atoms would become of great value. low dimensions, since Fermi systems are inherently unstable In distinction with the ideal Bose gas (IBG), which towards any finite interaction [4–6]; thus the IFG in low suffers the so-called Bose-Einstein condensation (BEC) in dimensions becomes an interesting solvable model to study three dimensions, the IFG shows a smooth thermodynamic the thermodynamics of possible singular behavior. behavior as function of the particle density and temperature; On the other hand, the experimental realization of this, however, does not preclude interesting behavior as has quantum degeneracy in trapped atomic Fermi gases [7–11] been pointed out in [28, 40], where it is suggested that the triggered a renewed interest, over the last fifteen years, in IFG can suffer a condensation-like process at a characteristic 0 the study not only of interacting fermion systems [12–15] temperature 𝑇 .Argumentsbasedonathermodynamic but also of trapped ideal ones [16–34]. Indeed, the nearly approach in support of this phenomenon are presented in ideal situation has been experimentally realized by taking [40], where the author suggests that the change of sign advantage of the suppression of 𝑠-wave scattering in spin- of the chemical potential, which defines the characteristic 0 polarized fermion gases due to Pauli exclusion principle temperature 𝑇 , marks the appearance of the condensed and of the negligible effects of 𝑝-wave scattering for the phasewhenthegasiscooled. 2 Journal of Thermodynamics
105
2 104
103
1 F 2 /T F 10 ∗ E / T 𝜇 0 101 T /T T0/T 𝜇 F F 100 −1 −1 10 ∼ ∼ −2 0 1 2345 010.2 0.4 0.6 0.8
T/TF d/s
d/s = 1/4 d/s = 3/2 d/s = 1/2 d/s = 2 d/s = 3/4 d/s = 3 d/s = 1 (a) (b)
Figure 1: (a) Dimensionless chemical potential 𝜇/𝐸𝐹 as function of the dimensionless temperature 𝑇/𝑇𝐹 for different values of 𝑑/𝑠.The ∗ 0 ∗ crossings with the horizontal lines 𝜇/𝐸𝐹 =1and 𝜇/𝐸𝐹 =0mark the temperatures 𝑇 and 𝑇 , respectively. (b) The temperature 𝑇 as function 0 of 𝑠/𝑑 (solid line). Additionally the temperatures 𝑇 (dashed line) and 𝑇𝜇 (dash-dotted line) are included for comparison.
𝑠 Truly, the significance of 𝜇 has motivated the discussion of energy spectrum of the form 𝜀∝𝑝, 𝑝 being the particle its meaning and/or importance at different levels and contexts momentum [59]. [41–52]. For the widely discussed—textbook—case, namely, In one dimension, the chemical potential of the IBG the three-dimensional IFG confined by an impenetrable box decreases monotonically with temperature, and as in the two potential, the chemical potential results to be a monotonic dimensional case, this behavior is related to the impossibility decreasing function of the temperature, diminishing from of BEC as shown by Hohenberg [53], at finite, nonzero, the Fermi energy, 𝐸𝐹,atzerotemperature,tothevaluesof temperature. In contrast, the chemical potential of the IFG the ideal classical gas for temperatures much larger than exhibits a nonmonotonic behavior: which starts rising quad- −1 2 2 𝑇 𝑘𝐵 (ℏ /𝑚𝜆𝑇),where𝑘𝐵 is Boltzmann’s constant, ℏ is Planck’s ratically with above the Fermi energy instead of decreas- constant divided by 2𝜋, 𝑚 is the mass of the particle, and ing from it and returns to its usual monotonic decreasing 𝜆 = √2𝜋ℏ2/𝑚𝑘 𝑇 behavior at temperatures that can be as large as twice the 𝑇 𝐵 is the thermal wavelength of de Broglie, 1 𝑇 Fermi temperature (see Figure 1; see also Figure in [60]). where denotes the system’s absolute temperature. A clear, This unexpected, and not well understood behavior, can be qualitative, physical argument of this behavior is presented exhibited mathematically by the Sommerfeld expansion [60, by Cook and Dickerson in [41]. In comparison, the chem- 61] or by other methods [62–64], though no intuitive physical ical potential of the IBG vanishes below a characteristic 𝑇 explanation of it, which predicts its appearance in the more temperature, called the critical temperature of BEC, 𝑐,and general case, seems to have been given before. (Indeed, the decreases monotonically for larger temperatures converging precise argument presented in [41] is only valid for the asymptotically to the values of the classical ideal gas. free IFG in three dimensions.) This forms the basis for the This picture changes dramatically as the dimensionality motivation of the present paper. of the system 𝑑 is lowered. In two dimensions the IBG shows After this excursus, one may conceive dimension two as no off-diagonal-long-range order at any finite temperature a crossover value for which the thermodynamic properties of [53]andthereforetheBECtransitiondoesnotoccur.Atthis ideal quantum gases are conspicuously distinct for 𝑑>2than quirky dimension, the chemical potential of both, the Fermi those for 𝑑<2. This can be seen in the specific heat, which and Bose ideal gases, decreases monotonically with tem- in the case of the IFG exhibits a no-bump → bump transition perature essentially in the same functional way [54], being as dimension is varied from 3 to 1 [60] analogous to the well- different only by an additive constant, expressly, the Fermi known cusp → no-cusp transition of the IBG specific heat. In energy. This results in the same temperature dependence of the latter case, the cusp marks the BEC phase transition while their respective specific heats at constant volume 𝐶V [54–56]. nophysicalmeaningisyetgivenforthebumpintheformer In general, this last outstanding feature occurs whenever the case. number of energy levels per energy interval is uniform as in In this paper we provide an analysis that attempts to the case of a one-dimensional gas in a harmonic trap [57, 58], explain the various features that are observed in the low- or the case 𝑠=𝑑where 𝑠 is the exponent of the single-particle dimensional, trapped IFG, focusing in the nonmonotonic Journal of Thermodynamics 3 dependence on 𝑇 ofthechemicalpotential.InSection2the thermodynamic limit, 𝑁→∞and V →∞with 𝑁/V = system under consideration is described, thermodynamics constant, [70] as quantities are calculated, and characteristics temperatures ∞ are introduced. In Section 3 a heuristic explanation of the Ω(𝑇,V,𝜇)=−𝑘𝐵𝑇 ∫ 𝑑𝜀 𝑔 (𝜀) nonmonotonic dependence of the IFG chemical potential on 0 (2) temperatureisgiven.InSections4and5thephysicalmeaning × ln [exp {𝛽 (𝜀 − 𝜇)}𝑓 (𝜀,) 𝑇 ], of two relevant characteristic temperatures is given. Finally, FD conclusions and final remarks conform Section 6. 𝑓 (𝜀, 𝑇) ={ [𝛽(𝜀 − 𝜇)]−1 +1} where FD exp is the Fermi-Dirac distribution function that gives the average occupation of 2. General Relations, Calculation of the single-particle energy state 𝜀 at absolute temperature 𝑇. the Chemical Potential, and the As usual, 𝛽 denotes the inverse of the product of 𝑇 and 𝑘𝐵 Thermodynamical Susceptibilities Boltzmann’s constant. Theaveragenumberoffermions𝑁(𝑇, V,𝜇)in the system 𝑁 We consider an IFG of ,conserved,spinlessfermionsin is given by −(𝜕Ω/𝜕𝜇)𝑇,V [67] which gives arbitrary dimension 𝑑>0. We assume a single-particle 𝑁 𝑑 density of states (DOS) of the form [28, 59] =−𝐺 Γ( )(𝑘 𝑇)𝑑/𝑠 (−𝑒𝛽𝜇), V 𝑑,𝑠 𝑠 𝐵 Li𝑑/𝑠 (3) 𝑔 (𝜀) =𝐺 𝜀𝑑/𝑠−1, 𝑑,𝑠 (1) ∞ 𝑙 𝜎 where Li𝜎(𝑧) = ∑𝑙=1 𝑧 /𝑙 is the polylogarithm function of 𝜎 𝐺 =𝐺 /V 𝑁 where 𝜀 denotes the energy, 𝐺𝑑,𝑠 and 𝑠 are positive constants, order [71] and 𝑑,𝑠 𝑑,𝑠 . Expression (3) relates and the former depends on 𝑑 and on the specific energy and 𝜇,andforfixed𝑁,thechemicalpotentialisafunction spectrum of the system, while the latter is determined by the of the system temperature and volume. The internal energy particular system dynamics. 𝑈(𝑇, V,𝜇)per volume is given by Two instances lead to the power-law dependance in 𝑈 𝑑 expression (1): the first one is based on the generalized energy- =−𝐺 Γ( +1)(𝑘 𝑇)𝑑/𝑠+1 (−𝑒𝛽𝜇), 𝑠 V 𝑑,𝑠 𝑠 𝐵 Li𝑑/𝑠+1 (4) momentum relation [59, 65] 𝜀𝑘 = C𝑠𝑘 , 𝑘 being the magnitude k C >0 oftheparticlewave-vector and 𝑠 being a constant 𝑆(𝑇, V, 𝜇) = −(𝜕Ω/𝜕𝑇) whose particular form depends on 𝑠.Thephysicalcases𝑠= while the entropy V,𝜇 per volume by 2, 1 correspond, respectively, to the nonrelativistic IFG with 2 𝑆 𝑑 𝑑/𝑠 𝐶2 =ℏ/2𝑚 and to the ultrarelativistic IFG for which 𝐶1 = =−𝑘 𝐺 Γ( )(𝑘 𝑇) V 𝐵 𝑑,𝑠 𝑠 𝐵 𝑐ℏ, 𝑐 being the speed of light. In this case 𝐺𝑑,𝑠 takes the (5) V/[2𝑑−1𝜋𝑑/2Γ(𝑑/2)𝑠𝐶𝑑/𝑠] Γ(𝜎) 𝑑 𝜇 form 𝑠 ,with the gamma func- ⋅[( +1)× (−𝑒𝛽𝜇)− (−𝑒𝛽𝜇)] . V =𝐿𝑑 Li𝑑/𝑠+1 Li𝑑/𝑠 tion and the volume of the system. The second 𝑠 𝑘𝐵𝑇 instance is based on the 𝑑-dimensional IFG trapped by an 𝛼 isotropic potential of the form 𝑈(r)=𝑈0(𝑟/𝑟0) ,where In Figure 1(a) the temperature dependence of the ratio 𝜇/𝐸 𝑑/𝑠 𝑈0,𝑟0 are two constants that characterize the energy and 𝐹 is shown for different values of the ratio and for length scales of the trap. This trapping potential leads, in the 𝑁/V fixed, where 𝑇𝐹 denotes the Fermi temperature defined 𝐸 =𝑘 𝑇 𝐸 semiclassical approximation [66], to 𝐺𝑑,𝑠 = ((2/𝑠−1)Γ[𝑑(1/𝑠− through the relation 𝐹 𝐵 𝐹,where 𝐹 is explicitly given 𝑑 2 𝑑/2 𝑑(1/2−1/𝑠) −1 (𝑑/𝑠𝐺 )𝑠/𝑑(𝑁/V)𝑠/𝑑 𝑑 𝑑/𝑠 <1 1/2)]/Γ(𝑑/2)Γ[𝑑/𝑠]ℏ )(𝑚𝑟0 /2) 𝑈0 with 𝑠 =1/2+ by 𝑑,𝑠 in dimensions. For the −1 𝛼 .Noticethatinthelattercase,onecanimmediately nonmonotonic dependence on temperature is clearly shown 𝑑/𝑠 = 1/2 establish the thermodynamic equivalence between the IBG (the dashed line corresponds to the case ,while 𝑑/𝑠 = 1/4 and the IFG; namely, 𝛼 = 2𝑑/(2, −𝑑) implying that no such is presented with the only purpose of making the equivalence is possible in dimensions 𝑑>2for positive 𝛼.The effects of the system dimensionality more conspicuous). In 𝛼→∞ the limit of high temperatures, 𝑇≫𝑇𝐹,theclassicalresult equivalence does occur in two dimensions if ,which 𝑑/𝑠 corresponds to the infinite well potential and in one dimen- 𝜇→𝑘𝐵𝑇 ln[(𝑇/𝑇𝐹) Γ(𝑑/𝑠 + 1)] is recovered. sion if 𝛼=2, which corresponds to the harmonic potential. As occurs for the 2D ideal gas in a box potential (𝑠= The thermodynamical properties of the ideal quan- 𝑑=2), the DOS is a constant whenever 𝑠=𝑑,andthe tum gases are easily computed from the grand potential chemical potential has the well-known analytical dependence −𝑇𝐹/𝑇 Ω(𝑇, V,𝜇) ≡ 𝑈−𝑇𝑆−𝜇𝑁[67, 68], where 𝑈 and 𝑆 denote on the temperature 𝜇=𝐸𝐹 +𝑘𝐵𝑇 ln[1 − 𝑒 ].For𝑇≪ the internal energy and entropy, respectively. For the trapped 𝑇𝐹, the chemical potential lies below the Fermi energy by a gas, V denotes the appropriate thermodynamic variable that negligible, exponentially small correction. The low tempera- generalizes the volume of a fluid in a rigid-walls container (see ture behavior of 𝜇 for 𝑑 =𝑠̸ can be obtained approxi- [69] for the case of the three-dimensional harmonic trap), mately as a direct application of the Sommerfeld expansion 2 𝑑/2 𝑑(1/2−1/𝑠) 𝑇≪𝑇 which in this paper is taken as V =(𝑚𝑟0 /2) 𝑈0 for 𝐹 (see [72, pp. 45-46]); namely, −𝑑 which reduces to V =𝜔 for the isotropic harmonic trap 2 2 4 2 1/2 𝜋 𝑑 𝑇 𝑇 𝑈(r) = ℏ𝜔(𝑟/𝑟0) with 𝑟0 =(2ℏ/𝑚𝜔). For a gas of 𝜇≃𝐸 [1− ( −1)( ) ] + O ([ ] ) . 𝐹 6 𝑠 𝑇 𝑇 (6) noninteracting fermions, Ω(𝑇, V,𝜇) can be written in the 𝐹 𝐹 4 Journal of Thermodynamics
𝜀 𝑇0 𝑇∗ 𝑇 𝑇 𝑇 The power-law dependence on in expression (1) is mani- Table 1: The temperatures , , 𝜇, 𝐶V ,and 𝜅𝑇 for which fested itself in the last expression, where the ratio 𝑑/𝑠 appears 𝜇(𝑇0)=0 𝜇(𝑇∗)=𝐸 𝜇(𝑇 ) 𝐶 (𝑇 ) , 𝐹, 𝜇 is maximum, V 𝐶V is maximum, 𝑑/𝑠 <1 𝜅 (𝑇 ) 𝑑/𝑠 explicitly. Clearly, for , the chemical potential rises and 𝑇 𝜅𝑇 is maximum, for three characteristic values of , from the Fermi energy quadratically with 𝑇,andthenon- namely, 1/4, 1/2, and 3/4, at which a nonmonotonic behavior is monotonousness is a result of the fact that, for large enough observed. 𝜇(𝑇) temperatures, falls down with temperature to negative 𝑑/𝑠 𝑇0 𝑇∗ 𝑇 𝑇 𝑇 values close to those of the classical gas. As a consequence 𝜇 𝐶V 𝜅𝑇 of this “turning around,” 𝜇(𝑇) develops a maximum at 0.25 15.6729 13.2260 5.0286 0.6532 0.3751 temperature 𝑇𝜇 and equals 𝐸𝐹 at two distinct temperatures, 0.5 3.4797 1.8960 0.9365 0.8632 0.2906 ∗ at 𝑇 and 0, if 𝑑/𝑠,andonlyat <1 𝑇=0otherwise. Thus, the 0.75 1.9830 0.6666 0.4086 1.3893 0.2080 solution to the equation 𝜇(𝑇)𝐹 =𝐸 as function of the para- meter 𝑑/𝑠 bifurcates at the critical value 𝑑=𝑠as is shown ∗ in Figure 1(b). Note that, for 𝑠=2and 𝑑=1, 𝑇 is as 𝛽𝜇 1.896𝑇 𝑑/𝑠 →0 V Li𝑑/𝑠−1 (−𝑒 ) large as 𝐹 and diverges as .Thiscanbeshown 𝜅 = , ∗ 𝑇 𝛽𝜇 (8b) straightforwardly from (3) by putting 𝜇=𝐸𝐹,sincethen𝑇 𝑁𝑘𝐵𝑇 Li𝑑/𝑠 (−𝑒 ) −𝑇 /𝑇∗ −1 must satisfy the equation 1=[1+𝑒 𝐹 ] in that limit. 0 In addition, the temperatures 𝑇 and 𝑇𝜇 that mark the respectively. change of sign of 𝜇 and its maximum, respectively, are also 0 In Figure 2 the dimensionless 𝐶V(𝑇) 𝑠/𝑑𝑁𝑘𝐵 (a) and showninFigure1(b)(dashedlineanddashed-dottedline).𝑇 𝜅𝑇/𝜅0 (b) are shown as function of the dimensionless tem- 𝜇(𝑇0)=0 is determined from the equation , which explicitly perature 𝑇/𝑇𝐹 for different values of 𝑑/𝑠;clearly,for𝑑/𝑠 < gives 1, both quantities exhibit a nonmonotonous dependence 𝑑 𝑑 −𝑠/𝑑 on 𝑇. The specific heat clearly exhibits the universal linear 𝑇0 =[Γ( +1)𝜁( )(1−21−𝑑/𝑠)] 𝑇 ; (7) 𝑇 𝑠 𝑠 𝐹 dependence on in the low temperature regime and rises with temperature evidencing the effects of dimensionality. 3.48𝑇 1.44𝑇 this expression gives the approximated values 𝐹, 𝐹, In the high temperature regime all the curves converge to 0.989𝑇 𝑑/𝑠 = 1/2, 1, 3/2 and 𝐹 for , respectively. The tempera- 𝑑𝑁𝑘𝐵/𝑠 0 the classical result . Analogously, the isothermal com- ture 𝑇 diverges as exp{(𝑠/𝑑) ln 2} as 𝑑/𝑠 →0 and goes to pressibility exhibits the universal behavior in the low tem- 𝑠/𝑑 zero as [𝑒/(𝑑/𝑠)]/√2𝜋𝑑/𝑠 as 𝑑/𝑠,where ≫1 𝑒 is the Euler- perature regime, namely, a finite value due to the degeneracy Napier number. pressure. As temperature rises the effects of dimensionality It is clear from expression (6) that 𝑑<𝑠is required for are uncovered but are hidden again in the high temperature the anomalous behavior of 𝜇(𝑇) to take place; however, for regime, where the classical dependence on temperature physical systems with positive integer dimensions less than appears. threeimposesevererestrictionsonhowfastthetrapping The nonmonotonic dependence with temperature of both potential must grow with the system size, that is, on the values thermodynamic susceptibilities is manifested as a global 𝑇 𝑇 of the exponent 𝛼. For fermions in a box-like trap (𝑠=2) maximum at the temperatures 𝐶V and 𝜅𝑇 ,respectively(see the anomaly will be observed if 𝑑=1,acasewherethe solid lines in Figures 2(a) and 2(b)). One would be tempted to effects are conspicuously revealed even at large temperatures. proposethateitherofthesetemperatureswoulddistinguish Thiscaseindeedposesachallengetotrapdesigning,though, between two distinct behaviors of the IFG: one where the it could be realized experimentally by using the optical trap corresponding susceptibility behaves anomalously and the developed by Meyrath et al. [73]. In the typical experimental other where it behaves standardly. Notice, nevertheless, that situation of harmonically trapped Fermi gases (𝛼=2and such temperatures do not match between them or with 𝑇 𝑇∗ therefore 𝑠=1) studied intensively [16, 23, 26, 31] expression either the temperature 𝜇 or the temperature ,ascan (6) tells us that the anomaly is not observed for any integer be quantitatively appreciated in Table 1 and in Figure 2, 𝑑≥1. On the other hand, if one assumes 𝑑=1as the where solid triangles in both panels identify the values of 𝑇 minimum system dimensionality realizable experimentally the corresponding susceptibility evaluated at 𝜇, solid circles 𝐶 𝑇 (cigar shaped traps), then one should go beyond harmonic in (a) indicate the values of V at 𝜅𝑇 , and, analogously, 𝜅 𝑇 𝑑/𝑠 = trapping; that is, one has to choose 𝛼>2. solid squares in (b) mark the value of 𝑇 at 𝐶V for 1/4, 1/2 The nonmonotonicity of the chemical potential, just , and 3/4. Such discrepancy among all these tem- referred to during the previous paragraphs, is revealed in the peratures makes it difficult to consider them as points that thermodynamic susceptibilities. In this work we focus on the mark the separation of two distinct thermodynamic behav- specificheatatconstantvolume𝐶V = (𝜕𝑈/𝜕𝑇)V and the iors. 2 𝑑/𝑠 =1 isothermal compressibility 𝜅𝑇 =(1/𝑛)(𝜕𝑛/𝜕𝜇)𝑇,givenby For the signal value , expressions in terms of ele- 𝛽𝜇 mentary functions are possible (black-dashed lines in Fig- 𝐶 𝑑 𝑑 Li𝑑/𝑠+1 (−𝑒 ) V = ( +1) ure 2); namely, 𝛽𝜇 𝑁𝑘𝐵 𝑠 𝑠 Li𝑑/𝑠 (−𝑒 ) (8a) 2 𝛽𝜇 𝛽𝜇 𝑑 Li𝑑/𝑠 (−𝑒 ) 𝐶 Li2 (𝑒 ) −( ) , V =2 − (1+𝑒−𝛽𝜇) (1+𝑒𝛽𝜇) 𝛽𝜇 𝛽𝜇 ln (9a) 𝑠 Li𝑑/𝑠−1 (−𝑒 ) 𝑁𝑘𝐵 ln (1 + 𝑒 ) Journal of Thermodynamics 5
1.5 1.25
1.25 1
1 ) s
/ 0.75 B 0
0.75 /𝜅 T dNk ( 𝜅 / 0.5 𝒱
C 0.5
0.25 0.25
0 0 10−2 10−1 100 101 102 10−1 100 101 102
T/TF T/TF d/s = 1/4 d/s = 3/2 d/s = 1/4 d/s = 3/2 d/s = 1/2 d/s = 2 d/s = 1/2 d/s = 2 d/s = 3/4 d/s = 3 d/s = 3/4 d/s = 3 d/s = 1 d/s = 1
(a) (b)
Figure 2: Normalized thermodynamic susceptibilities as function of the dimensionless temperature 𝑇/𝑇𝐹 for different values of 𝑑/𝑠,say, 1/4, 1/2, 3/4, 1, 3/2, 2, and 3. (a) Specific heat per particle at constant generalized volume; the circles mark the corresponding values of 𝐶V 𝑇 𝜅 𝑇 𝜇 at 𝜅𝑇 , that is, at the temperature at which 𝑇 has a maximum value; analogously, the triangles do the same at 𝜇 where has a maximum. 𝑑/𝑠 𝜅 =(𝑑/𝑠)2(𝜋𝑑Γ(𝑑/2)/2𝜋𝑑/2)𝑠𝐶 𝐸−(𝑑/𝑠+1) 𝜅 𝑇 (b) Isothermal compressibility scaled with 0 𝑠 𝐹 ; rhombus marks the corresponding values of 𝑇 at 𝐶V that corresponds to the temperature at which 𝜅𝑇 has a maximum value and the triangles do the same as in (a). Notice the nonmonotonic dependence on 𝑇 for 𝑑/𝑠. <1
V 𝑒𝛽𝜇 𝜅 = , argument just qualitative. In fact, the difficulty in quantifying 𝑇 𝛽𝜇 𝛽𝜇 (9b) 𝑁𝑘𝐵𝑇 (1 + 𝑒 ) ln (1 + 𝑒 ) those quantities arises from the use of the thermodynamic relation wherewehaveusedthefactthatthepolylogarithmfunctions 𝜕𝑈 0, 1, 2 𝜇 (𝑆, V,𝑁) =( ) (10) of order correspond to the elementary functions 𝜕𝑁 𝑆,V Li0(𝑧) = ln(1 + 𝑥),Li1(𝑧) = 𝑧/1,andLi +𝑧 2(𝑧) = 𝑧 which requires the knowledge of 𝑈(𝑆, V,𝑁),rarelycon- ∫ 𝑥d𝑥/1 + 𝑥,respectively.For𝑑/𝑠,thevariationwith >1 0 sidered for analysis in the variables 𝑆, V,𝑁.From(2)the temperature of the thermodynamic susceptibilities is stand- functions 𝑁=𝑁(𝜇,𝑇,V) (3) and 𝑆=𝑆(𝜇,𝑇,V) (5) are ard. obtainedandsolvedinordertoobtain𝑈(𝑆, V,𝑁).InFigure3 the internal energy at constant entropy is plotted as function 3. Heuristic Explanation of the Nonmonotonic oftheparticledensity𝑁/V for 𝑆/𝑘𝐵V = 0.1 and for different Dependence of 𝜇 on 𝑇 for 𝑑/𝑠 <1 values of the ratio 𝑑/𝑠. The slope of the curves gives the value of the chemical potential as given by expression (10). The monotonic decreasing behavior of the chemical potential Also in the same Figure 3, but in (b), the temperature of the 𝑑/𝑠 ≥1 with temperature for is understood from the argu- system, scaled with the Fermi temperature, as function of 𝑈 mentbasedonthefactthattheinternalenergy diminishes theparticledensityisshownfor𝑆/𝑘𝐵V = 0.1.Clearly,the 𝐸 from its zero temperature value 𝐹 after adiabatically adding systems cool regardless of the ratio 𝑑/𝑠, when adding particles 𝑇≪𝑇 a fermion at the small temperatures 𝐹.QuotingCook to the system in an isentropic way. andDickerson[41],thesystemcoolsbyredistributingthe It is possible to obtain an expression for 𝜇(𝑆, V,𝑁) particlesintotheavailableenergystatesinsuchawaythat from (10) by the use of the asymptotic behavior of the the particle added goes into “... alowlying,vacantsingle 𝜎 2 Polylogarithm functions −Li𝜎(−𝑧) ≃ ln(𝑧) /Γ(𝜎 + 1) + (𝜋 / particle state, which will be a little below 𝐸𝐹,”Thisisa 𝜎−2 6) ln(𝑧) /Γ(𝜎−1)+⋅⋅⋅;aftersomealgebrawehaveexpressed consequence, as we will show below that in the three-dimen- that in the degenerate regime sional case the change of the Helmholtz free energy is dominated by the change of entropy in the low temperature 𝜇 (𝑆, V,𝑁) limit; however, the argument provided in [41] does not give 3 𝑆 2 V 2 𝑠 𝑠 (11) the amount of the energy change involved in the process ≃𝐸 [1 + ( ) ( ) ( −1)], 𝐹 2 orthechangeintemperature,makingthenatureofthe 2𝜋 𝑘𝐵V 𝑁 𝑑 𝑑 6 Journal of Thermodynamics
108
S/kB𝒱 = 0.1 4 10 S/kB𝒱 = 0.1 106
4 2 10
F 10 F 𝒱E / T/T
U 102 100 100
10−2 10−2 10−2 10−1 100 10−2 10−1 100 N/𝒱 N/𝒱
d/s = 1/4 d/s = 3/2 d/s = 1/4 d/s = 3/2 d/s = 1/2 d/s = 2 d/s = 1/2 d/s = 2 d/s = 3/4 d/s = 3 d/s = 3/4 d/s = 3 d/s = 1 d/s = 1
(a) (b)
Figure 3: The dependence of the scaled internal energy 𝑈/V𝐸𝐹 (a) and scaled temperature 𝑇/𝑇𝐹 (b) as function of the dimensionless particle density 𝑁/V,whereV denotes the systems volume scaled with an arbitrary volume V0. where the nonmonotonic dependence on 𝑇 is evident when as a consequence, the number of available excited states is 𝑑/𝑠. <1 On the other hand, for the sake of completeness we reduced considerably in comparison with excitable number compute the system temperature as function of the particle of particles. We may conclude that when adding a particle in density, in the degenerate limit, which is given by an adiabatically way, the probability of occupying an energy state below 𝐸𝐹 is very small and therefore, it will occupy an 3 𝑠 𝑆 V 𝑇 (𝑆, V,𝑁) ≃ 𝑇 energy state above 𝐸𝐹. 2 𝐹 (12) 𝜋 𝑑 𝑘𝐵V 𝑁 In order to quantitatively characterize the incomplete accommodation described above, we consider the ratio 𝑅(𝑇) −1 and, as is shown in Figure 3(b), decreases as (𝑁/V) . ofthenumberofparticlesintheenergyinterval[𝐸𝐹,𝐸𝐹 +Δ] How can we understand the rising of the chemical tothenumberofavailablestatesinthesameenergyinterval, potential when 𝑑/𝑠? <1 Consider the number of particles 𝐸𝐹+Δ that can be excited by the energy 𝑘𝐵𝑇≪𝐸𝐹 from the ∫ 𝑑𝜀 𝑔 (𝜀) 𝑓 (𝜀,) 𝑇 𝐸 FD 𝑑-dimensional Fermi sphere. This number is approximately 𝑅 (𝑇) = 𝐹 . (13) 𝐸𝐹+Δ given by 𝑁𝑘𝐵𝑇/𝐸𝐹 while the number of available states above ∫ 𝑑𝜀 𝑔 (𝜀) 𝐸𝐹 the Fermi energy can be approximated by 𝑔(𝐸𝐹)𝑘𝐵𝑇.The quotient between both quantities is exactly 𝑠/𝑑.Thissimple ThisquantityisshowninFigure4asfunctionoftemperature and heuristic argument shows that there are more single- with ratio Δ/𝐸𝐹 = 0.001, for different values of 𝑑/𝑠.The 𝑑/𝑠 >1 particle excited states than excitable particles for , choice Δ∼𝑘𝐵𝑇≪𝐸𝐹 guarantees that a negligible number of which is evident because of the monotonic increasing behav- particles occupy states out of the interval [𝐸𝐹,𝐸𝐹 +Δ]. Under 𝑅(𝑇) 𝑓 (𝐸 ,𝑇) ior of the DOS. In principle all the excited particles can this condition we can approximate by FD 𝐹 and be accommodated into the available states without violating for temperatures 0<𝑇≪𝑇𝐹 we have that 𝑅(𝑇) ≃ (1/2)[1 − 2 Pauli’s principle. The accommodation, however, is not arbi- (𝜋 /12)(𝑑/𝑠 − 1)(𝑇/𝑇𝐹)],and,therefore,theoccupation trary.Theprobabilityofoccupationoftheavailablestatesin probability of the energy states in [𝐸𝐹,𝐸𝐹 +Δ]is smaller than thermal equilibrium must follow the Fermi-Dirac distribu- 1/2 for 𝑑/𝑠,greaterthan >1 1/2 for 𝑑/𝑠,andequalto <1 1/2 tion and therefore just a fraction of the excitable fermions for 𝑑=𝑠. areexcitedintotheinterval[𝐸𝐹,𝐸𝐹 +𝑘𝐵𝑇] (in fact, the occu- 0 pation probability for the states with energy larger than 𝜇 is 4. The Physical Meaning of 𝑇 smaller than 1/2). For this case we can certainly apply the argument given by Cook and Dickerson in [41] to infer that A condensation-like phenomenon has been suggested to when adding adiabatically an extra particle to the system, the occur in the IFG in [28, 40]; this can be understood as the internal energy will decrease from 𝐸𝐹. formation of a “core” in momentum-space, reminiscent of 0 In contrast, Pauli exclusion principle prohibits complete the Fermi sea, that starts forming at 𝑇 and that grows up to accommodation when 𝑑/𝑠, <1 since in this case the DOS form the Fermi sea as temperature is diminished to absolute 𝑛 has a monotonic decreasing dependence on energy and, zero.Thenumberofparticlesinthecore, core,iscomputed Journal of Thermodynamics 7
0.8 1.5
0.6 1 ) /n
T 0.4 ( core R n 0.5 0.2
0 0 10−2 10−1 100 101 010.2 0.4 0.6 0.8 0 T/TF T/T
d/s = 1/4 d/s = 3/2 d/s = 1/4 d/s = 3/2 d/s = 1/2 d/s = 2 d/s = 1/2 d/s = 2 d/s = 3/4 d/s = 3 d/s = 3/4 d/s = 3 d/s = 1 d/s = 1
Figure 4: The ratio of the number of particles in the energy interval Figure 6: Fraction of particles in the Fermi-sphere-like condensate 0 [𝐸𝐹,𝐸𝐹 +Δ]to the number of available states in the same energy as function of temperature scaled with 𝑇 for different values of 𝑑/𝑠. interval, 𝑅(𝑇) (see (13)), as function of temperature for different values of 𝑑/𝑠. exists for 𝜇(𝑇 ,𝑛 )>0and 𝑑/𝑠, ≥1 since the isotherm 𝜇0(𝑛) is a concave function of the particle density; in other words, d/s = 1/2 isotherms 𝜇(𝑇, 𝑛) of higher temperature are situated below 2 𝜇0(𝑛) (see [40] where the case 𝑑/𝑠 = 3/2 is discussed). In con- trast, for 𝑑/𝑠, <1 the zero temperature isotherm is a convex Isotherms T >T∗ function of 𝑛 asshowninFigure5,andtwopossibilitiesmay ∗ 𝑇 <𝑇 𝑛 <𝑛 1 ncore
𝑍𝑁,V (𝛽) =𝑘𝐵𝑇 ln [ ]. (16b) ∗ 𝑍𝑁+1,V (𝛽) coincides with 𝑇 , which is different from zero when 𝑑/𝑠 < ∗ 1. This suggests the possibility of interpreting 𝑇 as a cri- Δ𝑈 − The rhs of expression (16a) can be explicitly written as tical temperature at which a phase transition occurs. 𝑇Δ𝑆 Δ𝑈 = 𝑈(𝑇, V,𝑁+1)−𝑈(𝑇,V,𝑁) Δ𝑆 = ,where and In order to show the validity of these ideas we first 𝑆(𝑇, V,𝑁 + 1)− 𝑆(𝑇,V,𝑁) are the internal energy change use expression (16b) to compute 𝜇 in two distinct one- 𝑇Δ𝑆 of the system and the heat produced when adding, dimensional systems each consisting of 𝑁 spinless fermions. isothermally, exactly one more fermion (expression (16a) One corresponds to the IFG trapped by a box-like potential arises from the forward difference discretization of the con- (𝑑/𝑠 =) 1/2 and the other to the experimentally feasible sys- ventional definition (15). The physical interpretation is simple tem of and IFG trapped by a harmonic trap (𝑑/𝑠). =1 We and the same for the backward difference: they simply give show that, for the former case, 𝜇 rises above 𝐸𝐹,𝑁+1 and even- the change in the free energy when adding an extra particle tually returns to its decreasing behavior as the system tem- to the system. Experimental realization of such situation is, perature is increased from zero. For the latter, we show that for instance, Bose-Einstein condensation at constant temper- 𝜇<𝐸𝐹,𝑁+1 for all 𝑇>0. ature by increasing the particle number of the system [76] For exactly 𝑁 noninteracting fermions, the partition and injection of electrons into low-dimensional systems. The function satisfies the recursive relation [77, 78]. negative of such difference gives the change in the free energy when removing a particle from the system. Another defini- 1 𝑁 𝑍 (𝛽) = ∑ (−1)𝑛+1 𝑍 (𝑛𝛽) 𝑍 (𝛽) , tion of 𝜇 is given by the average of the forward and backward 𝑁 𝑁 1 𝑁−𝑛 (19) differences which is suitable when both processes, adding 𝑛=1 and removing a particle, are present at finite temperatures as where 𝑍1(𝛽) = ∑𝜀 exp{−𝛽𝜀} is the single-particle partition occurs in the grand canonical ensemble). function with 𝜀 being the single-particle energy spectrum and At zero temperature, the chemical potential is given by the 𝑍0(𝛽) ≡. 1 change in internal energy only, whose value coincides with Expression (19) can be reduced to the calculation of the Fermi energy of 𝑁+1fermions; that is, 𝑍1(𝑚𝛽),with𝑚 being a positive integer, by noting that 𝑍𝑁(𝛽) can be written as a sum of the product over the distinct 𝜇 (V,𝑁) =𝐸𝐹,𝑁+1 , (17) parts of all the partitions {(𝜆1,𝜆2,...,𝜆𝑟)} of 𝑁 (a partition where 𝐸𝐹,𝑁 denotes the explicit dependence of the Fermi is defined as a nonincreasing sequenceofpositiveintegers 𝜆 ,𝜆 ,...,𝜆 ∑𝑟 𝜂 𝜆 =𝑁 𝜂 energyontheparticlenumber.Ifthisvalueissubtractedfrom 1 2 𝑟 such that 𝑖=1 𝑖 𝑖 ,where 𝑖 denotes the 𝜆 (16a) we have that multiplicity of the part 𝑖 in a given partition, that is, the partition 2+2+2+1of 7; the part 2 has multiplicity 3; see Δ𝜇 = Δ𝑈 −𝑇Δ𝑆, (18) [79, pp.1]); thus 𝑍 (𝛽) where Δ𝜇 = 𝜇(𝑇, V,𝑁)−𝐸𝐹,𝑁+1 and Δ𝑈 =Δ𝑈−𝐸𝐹,𝑁+1 . 𝑁 Δ𝜇 ≤ 0 In this way, if for a given temperature we have that 𝑟 𝜂 (−1) 𝑚 𝜂 (20) (i.e., the chemical potential lies below the Fermi energy), = (−1)𝑁 ∑ ∏ [𝑍 (𝜆 𝛽)] 𝑚 . 𝜂𝑚 1 𝑚 𝜆𝑚 𝜂 ! then the relative change in the internal energy is smaller than {(𝜆1,𝜆2,...,𝜆𝑟)} 𝑚=1 𝑚 the respective heat exchange by adding the particle. In other words, the effects of the addition of a particle to the system, in The first four terms can be checked straightforwardly and are an isothermal way, are such that the entropic effects dominate shown in Table 2. over the energetic ones at that 𝑇. This argument accounts Computationally, evaluation of expression (20) is faster for the monotonic decreasing behavior of 𝜇 with 𝑇 and is than evaluating expression (19) since recursion is avoided and equivalent with argument given in [41]. Further, if Δ𝜇 > 0 only the algorithm for computing the unrestricted partitions for a given 𝑇, then the energetic changes are the ones that of the integer 𝑁 is needed. Such algorithm forms part of the dominate over the entropic ones, which give origin to the rise MATHEMATICA software package distribution. The com- of 𝜇 above the Fermi energy as has been shown in the pre- putation time and memory requirements grow with number vious section. The temperature that separates both regimes of partitions 𝑝(𝑁) of 𝑁, which grows asymptotically as Journal of Thermodynamics 9 exp{√𝑛} thus limiting computation to 𝑁∼10.Surprisingly, the calculation exhibits a fast convergence to the well-known result obtained from (3) for 64 particles (see Figure 7 for the box-like trap). 1 For the box potential in one dimension, the single- 1.03 particle partition function is given in terms of the Jacobi theta 1.02 ∞ 𝑛2 F,N+1 E function 𝜗3(𝑢, 𝑞) = 1 +𝑛=1 2∑ 𝑞 cos(2𝑛 ∗ 𝑢) as 𝑍1(𝛽) = / 1.01 −𝛽𝜀 𝜇 (1/2)[𝜗 (0, 𝑒 0 )−1] 𝜀 0.5 3 ,wheretheenergyscale 0 in the argu- 1 2 2 2 ment of the exponential is ℏ 𝜋 /2𝑚𝐿 . For few particles, the 2 0 0.1 0.2 chemical potential does not rise as (𝑇/𝑇𝐹) as is expected Sommerfeld approx. from the grand canonical ensemble result, but it grows 0 much more slower as is shown in the inset of Figure 7. 0123 This is consequence of the low temperature behavior of T/TF the partition function, which satisfies that 𝑍𝑁/𝑍𝑁+1 ∼ 𝛽𝐸𝐹,𝑁+1 −𝛽(2𝑁+1)𝜀0 𝑒 [1 + 𝑒 ] for 𝑇/𝑇𝐹 ≪1,leadingthustoΔ𝜇 ∼ N=2 N=16 −𝛽(2𝑁+1)𝜀 𝑘 𝑇𝑒 0 N=4 N=32 𝐵 . N=8 From equation (3) For the harmonic potential in dimension one, an exact 𝑍 (𝛽) analytical expression for 𝑁 is known [24, 80]; namely, Figure 7: The chemical potential (16b), scaled with 𝐸𝐹,𝑁+1 as function of the dimensionless temperature 𝑇/𝑇𝐹,with𝑇𝐹 = ℏ𝜔 𝑁 𝐸 /𝑘 𝑁=24 8 16 32 𝑍 (𝛽) = [−𝑁2𝛽 ] ∏ [1 − (−𝛽ℏ𝜔𝑗)]−1 . 𝐹,𝑁+1 𝐵,forthenumberofparticles , , , ,and . 𝑁 exp exp (21) Note that the results with 32 particles (double dashed-dotted line) 2 𝑗=1 are close to the grand canonical ensemble result (black line in long dashes) computed from (3). A comparison with the Sommerfeld A direct application of (16b) leads to approximation for low temperatures is shown in the inset.
𝜇=𝐸𝐹,𝑁+1
𝛽ℏ𝜔 (22) ∗ +𝑘 𝑇 [1 − (−𝛽𝐸 ) (− )] . system 𝑛.Thissuggeststhat𝑇 can be considered as the 𝐵 ln exp 𝐹,𝑁+1 exp 2 relevant temperature of the isotropically trapped IFG, as is Clearly (22) is a monotonically decreasing function of 𝑇 supported by the energy-entropy-like argument presented in Section 5. The region in the 𝜇-𝑇 plane, for which 𝜇>𝐸𝐹 agreeing with the grand canonical ensemble result; expres- 𝑇≤𝑇∗ −𝑇𝐹/𝑇 for , represents the set of thermodynamic states for sion 𝜇=𝐸𝐹 +𝑘𝐵𝑇 ln[1 − 𝑒 ] is recovered in the limit which the change in the Helmholtz free energy, when increas- 𝑁→∞. ing the particle density of the system, is dominated by the changes of the internal energy and would correspond to 6. Conclusions and Final Remarks an ordered phase. In the complementary region for which ∗ 𝑇≥𝑇, the thermodynamic states are characterized by Inthispaperwehavepresentedadiscussiononthemeaning changes in 𝐹(𝑇, V,𝑁)dominatedbyheatexchangebychang- of the nonmonotonic dependence on temperature of the ing entropy and can be considered as a “disordered phase.” thermodynamics properties of low-dimensional, trapped, Though, heuristic energy-entropy arguments have been used IFGs, with focus on the chemical potential (a similar behavior to uncover the possibility of a phase transition [74], we want has been predicted for weakly repulsively interacting Bose to emphasize that we are not claiming the existence of a phase gases [81] in that a hard core Bose gas behaves, at least transition in the IFG, on the basis that thermodynamic quan- qualitatively, as an ideal Fermi gas). The parameter used to tities do not show a singular behavior of the thermodynamic characterize the trapping and dimensionality 𝑑 of the system 𝑇∗ 𝑑/𝑠 susceptibilities at . is merely that explicitly appears in the single-particle Thoughthechemicalpotentialisnotdirectlymeasured density of states (1). Thus low-dimensional trapped systems 𝑑/𝑠 <1 in current experiments, development on imaging techniques are characterized by values of .Inthisrangeofvalues, of ultracold gases [82–85] has opened the possibility of the chemical potential, the specific heat at constant volume, experimentalists to measure the local particle density in situ and the isothermal compressibility exhibit a nonmonotonic and from the data to extract 𝜇 and 𝑇. dependence on temperature which have been characterized 𝑇 𝑇 𝑇 by the temperatures 𝜇, 𝐶V ,and 𝜅𝑇 [60], respectively. We 0 Competing Interests also have computed 𝑇 as function of 𝑑/𝑠 [28, 40] and ∗ 0 introduced a new characteristic temperature 𝑇 ≤𝑇,which The author declares that they have no competing interests. corresponds to the nonzero value of the temperature at which ∗ 𝜇(𝑇 , V,𝑁)=𝐸𝐹. ∗ Acknowledgments We found that 𝑇 marks the temperature at which the 𝑛 particle density of a Fermi-like core core that starts forming The author wants to express their gratitude to Mauricio Fortes 0 𝑇 saturatesatthevalueofthetotalparticledensityofthe and Miguel Angel Sol´ıs for valuable discussions and to Omar 10 Journal of Thermodynamics
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