Diagramma di fase

Recentemente sono stati pubblicati i risultati di alcuni studi in cui, attraver-so simulazioni numeriche [45] o attraverattraver-so considerazioni pi`u qualitative [46], viene ricostruito il diagramma di fase di un sistema costituito da due specie di atomi.

Il diagramma di fase ottenuto nel primo dei due articoli citati `e riportato in figura 4.9. In esso, viene rappresentata la differenza fra i potenziali chimici delle due specie δµ in funzione della costante di accoppiamento g fra le due specie interagenti, individuando le fasi del sistema per ogni coppia di valori.

Come gi`a annunciato, per piccole differenze fra i potenziali chimici, la fase condensata non viene rotta: essa quindi persiste per qualsiasi valore dell’ac-coppiamento, o come condensazione di coppie di Cooper (interazione debole) o di molecole (interazione forte).

Quando la differenza fra i potenziali chimici diventa abbastanza grande, la fase condensata non `e pi`u favorita energeticamente e viene rotta, o completa-mente o parzialcompleta-mente: nel primo caso, si passa direttacompleta-mente alla fase normale,

7 As a check of our results, we have determined the second order phase transition lines of the phase diagram by a Ginzburg-Landau (GL) expansion of the grand potential Ω, both in the homogeneous and in the LOFF phase. The use of this approximation is justified because also in the strong coupling regime one has ∆/δµ → 0 near the second order lines. Since we are interested to the second order transitions, it is enough to expand Ω up to the fourth order in ∆, so the grand potential can be written as

Ω = Ω0+^α 2^∆

2+ ^β 4^∆

4 , (14)

where Ω0is the free gas contribution and the coefficients are given by α = ² G^{+ T} ∞ ! n=−∞ " d3p (2π)3 2

(iωn− &1)(iωn+ &2) ^{, (15a)}

β = T ∞ ! n=−∞ " d3p (2π)3 2

(iωn− &1)2(iωn+ &2)2 . (15b) In Eqs. (15) the &σ are the dispersion laws of the quasi-particles,

&1= ^{(p + q)} 2

2m − µ1 , &2= (p − q)2

2m − µ2 (16)

(the homogeneous case is studied by putting q = 0 in the above expressions). The divergence in the integral defining the coefficient α is cured, by the introduction of the S-wave scattering length, as discussed in Section II. Using the GL expansion we reproduce within a few percent the second order transition lines obtained by the numerical evaluation of the free-energy minima in the full theory .

IV. PHASE DIAGRAM

We summarize our results in the phase diagram depicted in Fig. 5. In the following discussion of the phase diagram we will show that there is a correspondence between some regions and lines of the phase diagram and of the diagram depicted in Fig. 3. 0.12 0.15 0.18 0.95 1 1.05 -0.5 -0.25 0 0.25 0.5 g 0 0.5 1 1.5 2 ! µ/ " ⁰ BEC ^LOFF _BCS Normal P Gapless P

FIG. 5: Phase-diagram at T = 0. The full line (red online) indicates the first order phase transition between the homogeneous gapped superfluid phase and the LOFF phase for g <∼ 0.05 or the normal phase for 0.05<

∼ g<

∼ 0.13 or the gapless homogeneous superfluid phase for 0.13 <∼ g<

∼ 0.175. The dashed line (black online) indicates the second order phase transition between the LOFF phase and the normal phase. The dot-dashed line (blue online) indicates the second order phase transition between the homogeneous superconductive phase and the normal phase. The dotted (green online) line, which does not correspond to a phase transition, separates the homogeneous gapped phase from the homogeneous gapless phase. In the inset it is shown that the full line continues beyond the point where the dot-dashed line and the full line meet.

To begin with we describe the full line (red online). At values of g <∼ 0.05 it indicates the first order phase transition between the homogeneous superconductive phase and the LOFF phase. For 0.05 <∼ g^<∼ 0.13 the full line indicates a

Figura 4.9: Diagramma di fase, ottenuto numericamente, di un gas costituito da due specie aventi diversi potenziali chimici, la cui differenza `e pari a δµ. In esso sono riportate le fasi in cui si trova il sistema per ogni valore di δµ e della costante di accoppiamento g.

nel secondo, si passa ad una fase intermedia, quella BP per accoppiamenti forti e quella LOFF per accoppiamenti deboli.

Per ogni valore di g, chiamiamo δµ_c la massima differenza che la fase su-perconduttiva omogenea pu`o sostenere, in modo che per δµ > δµ<sub>c</sub> il sistema non `e pi`u completamente condensato. δµ_c risulta una funzione crescente di g.

Dalla figura 4.9, si evince questo comportamento:

• −0.5 < g < 0.05: il sistema presenta una transizione dalla fase super-conduttiva a quella LOFF e successivamente una transizione dalla fase LOFF alla fase normale;

• 0.05 < g < 0.175: il sistema presenta una transizione dalla fase super-conduttiva a quella normale;

• 0.175 < g < 0.5: il sistema presenta una transizione dalla fase supercon-duttiva a quella di BP e successivamente a quella normale.

La figura 4.10 mostra quali tipi di transizioni avvengono per i diversi valori di g:

• g < 0: transizione di fase del primo ordine: ∆ passa da un valore finito (quello della fase BCS) a 0;

• 0 < g < 0.175: transizione di fase del primo ordine fra due valori di ∆ finiti, il primo relativo alla fase superconduttiva ed il secondo alla fase BP; successivamente, transizione del secondo ordine fino a ∆ = 0;

• g > 0.175: transizione di fase del secondo ordine fino a ∆ = 0. ₅

0.92 0.94 0.4 0.6 0.8 1 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 !µ/"_F 0 0.25 0.5 0.75 1 1.25 # /" ^F g=-0.1 g=0.135 g=0.2 0.92 0.94 0.96 0.2 0.4 0.6 0 0.5 1 1.5 2 !µ/"_F -0.2 0 0.2 0.4 0.6 0.8 1 µ/ " ^F g=-0.1 g=0.135 g=0.2 normal phase

FIG. 2: On the left: The gap ∆/!_F vs. δµ/!_F for three values of the dimensionless coupling g. From top to bottom the lines refer to g = 0.2 (purple online), g = 0.135 (red online) and g = −0.1 (black online). The g = 0.2 curve shows a second order phase transition to the normal phase at δµ/!_F "1.27. The g = −0.1 curve shows a first-order transition to the normal phase at δµ " 0.39!_F. The intermediate curve (g = 0.135), shown in more detail in the inset, shows the existence of two phase transitions. One phase transition is first-order. It leads to a superconductive phase with a different, smaller, value of the gap. The second transition leads smoothly to the normal phase. On the right: µ/!_F vs. δµ/!_F for the same three values of the dimensionless coupling g. The continuous upper curve (green online) refers to the normal phase (g → −∞). The other three curves from bottom to top refer to g = 0.2 (purple online), g = 0.135 (red online) and g = −0.1 (black online). The inset represents an enlargement of the curve at g = 0.135. In both panels the dotted parts of the g = 0.135 and of the g = −0.1 lines correspond to metastable states.

Ccorrespond to metastable points that are local minima of the free-energy. For g = 0.135 they were reported in the insets of Fig. 2 as dotted points in the upper curve and lower curve respectively. The points in region labeled as B correspond to unstable BP points that are maxima of the free-energy. The remaining parts of the diagram correspond to allowed regions. The white area corresponds to the stable gapped phase and the shadow area (yellow online), with the exclusion of the region C, to the stable gapless superconductive phase. In the shadow region δµ > !µ2+ ∆2 there are gapless excitations at one sphere in momentum space. All the regions meet at the point P, on the line corresponding to g = 0.175. The meaning of this point will be clarified below.

0 0.5 1 1.5 2

!µ/#

µ/

#

B

A _C

FIG. 3: Full (green online) lines are simultaneous solutions of the gap and number equations for different values of the coupling constant and of the mismatch between the Fermi spheres. The regions above the full (red online) line labeled with A, B and C correspond to phases where no physical solutions of the gap and number equations have been found. The phase in region Figura 4.10: Gap ∆/_F in funzione di δµ/_F per diversi valori di g: la figura

mostra che per g=0.2 si verifica una transizione del secondo ordine alla fase normale a δµ/_F ≈ 0.127; per g=0.135, si verifica prima una transizione del primo ordine fra due valori finiti del gap ∆, poi una transizione del secondo ordine alla fase normale; per g=-0.1, si verifica una transizione del primo ordine alla fase normale a δµ/_F ≈ 0.39 [45].

Quando g `e scelto nell’intervallo 0.13 ≤ g ≤ 0.175 e δµ `e entro un range di valori vicini al suo valore critico, la gap equation ha tre soluzioni, cio`e tre valori di ∆, di cui uno corrisponde ad un massimo e gli altri due a minimi dell’energia libera, come mostrato in figura 4.11.

Per δµ piccolo, il minimo dell’energia `e quello corrispondente al ∆ pi`u grande, al crescere di δµ i due valori diventano quasi degeneri finch´e il minimo si trova al ∆ minore. Per δµ ancora pi`u grandi, il ∆ minimo continuer`a a diminuire fino a 0 e si ha una transizione di fase del secondo ordine allo stato normale.

4 the other two to local minima. The minima are favored for different values of δµ. At small values of δµ the favored state is the one with ∆ = ∆0. For values of δµ larger than a critical value the favored state is the second one, with ∆ < ∆0. The transition between these two states is first order. We remark that such behavior of the free-energy takes place only in the range of the coupling 0.13 <∼ g^<∼ 0.175. For values smaller than ∼ 0.13 there is one phase transition from the homogeneous to the normal phase. For values of g large than ∼ 0.175 one of the minima of the free-energy disappears and, increasing δµ, one finds a second order phase transition from the normal phase to the unpaired phase.

In order to clarify the behavior in the above-mentioned range of g, we plot in Fig. 1 the free-energy difference F − F0 (F0 the value at ∆ = 0) as a function of ∆ for various values of δµ at g = 0.135, i.e. inside the interval [0.13, 0.175]. For each value of ∆, the value of µ is determined by the equation ∂F/∂µ = 0, corresponding to Eq. (9). We notice that, since the total number density is fixed, the average chemical potentials of the broken (∆ #= 0) and normal (∆ = 0) phases are in general different. For δµ = 0.936 #F the free-energy has a global minimum at ∆ = ∆0$ 0.95 #F and a local minimum at ∆ $ 0.75 #F; at δµ = 0.940 #F the two minima are almost degenerate, and the values of the gap at the local minima are ∆ = ∆0 and ∆ $ 0.625 #F; finally for δµ = 0.942 #F the former local minimum becomes the global one (and vice-versa), and the gap at the global minimum is ∆ $ 0.6 #F. For higher values of δµ the value of the gap decreases monotonically and for δµ = δµc∼ 0.955 #F the system has a second order phase transition to the normal phase.

0 0.5 1 !/"_F -0.002 0 0.002 (F - F 0 )/k F " ^F 3 #µ = 0.936 "_F #µ = 0.940 "_F #µ = 0.942 "_F

FIG. 1: Free energy difference F − F0 as a function of ∆ for various values of δµ at g = 0.135.

The dependence on g of the order of the phase transitions is shown on the left panel in Fig. 2 by three representative values of the dimensionless coupling constant, one inside the interval [0.13, 0.175], another one on the left, and a third one on the right of the interval. The lowest curve refers to g = −0.1. We have not considered here the possibility of inhomogeneous superconductivity and therefore we have a first order phase transition from the superconductive to the normal state. It occurs at δµ $ 0.79∆0. For 0.79 <∼ δµ/∆0≤ 1 the superconductive phase becomes metastable and is shown as a dotted line. The highest curve is computed at g = +0.2: for this value the transition from the superconductive to the normal phase is second order. The intermediate curve is obtained at g = +0.135 and shows, in agreement with the results of Fig. 1, two phase transitions: a first order phase transition from the value 0.95 to the value 0.65 of the gap parameter, and a second order phase transition to the normal phase. The values corresponding to the metastable phases are depicted as dotted curves. An enlarged picture of this case is in the inset. On the right panel in Fig. 2 we show the behavior of the average chemical potential µ as a function of δµ, for the same values of the dimensionless coupling constant g. In the figure, the upper curve (green online) represents the average chemical potential in the normal phase The inset refers again to g = 0.135.

It is also worth mentioning that the first order phase transition between the two minima of the free-energy cor-responds to a phase transition between a gapped and gapless phase. The gapless phase is characterized by having one zero in the quasiparticle dispersion law at one sphere in momentum space. Had the dispersion laws two zeros then the system could live in the Breached Pairing phase [9, 10], but this possibility is not realized in this model at least within the present approximations. To illustrate this point we have reported in Figure 3 the results for µ/∆ vs. δµ/∆ as lines (green online) labeled with various values of g. Since for some values of g there are first order phase transitions, some regions of this diagram are never reached by stable physical states, which is why in some cases the lines are interrupted. Such regions are above the thick full (red online) line, which has been determined comparing the energies of the various phases, and have been labeled with the letters A, B and C. The regions labeled as A and

Figura 4.11: Energia libera in funzione del gap ∆/_F per diversi valori di δµ e a g=0.135 [45].

Nel secondo articolo citato, invece, viene ipotizzato il diagramma di fase riportato in figura 4.12. Sull’asse y `e riportato η = <sup>H</sup> ∆ <sup>con H =</sup> 1 2<sup>(µ1</sup>− µ<sub>2</sub>). ∆ `e il gap del gas non polarizzato. Sull’asse x `e riportato

k = − ¹

na3.

La zona I `e quella della fase condensata, con i limiti BCS e BEC ed un crossover fra i due; la zona II `e quella della fase normale, la zona III della fase BP e infine la zona IV `e della fase LOFF.

La differenza fondamentale fra i due diagrammi di fase `e data dalla zona di confine fra le due fasi intermedie: in quest’ultimo grafico, infatti, le due regioni sono separate da una linea che congiunge i punti S e L: il punto S `e un punto di splitting, poich´e da esso parte una linea che divide la zona (III) in due sottoregioni, mentre il punto P `e un punto di Lifshitz, punto che separa, per definizione, una fase ordinata spazialmente uniforme (I), una fase disordinata (II) e una una fase ordinata spazialmente non uniforme (IV).

arXiv:cond-mat/0507586 v1 25 Jul 2005

INT-PUB 05-18

Phase Diagram of Cold Polarized Fermi Gas D. T. Son1

and M. A. Stephanov2

1

Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195-1550, USA

2

Department of Physics, University of Illinois, Chicago, Illinois 60607-7059,USA (Dated: July 2005)

We propose the phase diagram of cold polarized atomic Fermi gas with zero-range interaction. We identify four main phases in the plane of density and polarization: the superfluid phase, the normal phase, the gapless superfluid phase, and the modulated phase. We argue that there exist a Lifshitz point at the junction of the normal, the gapless superfluid and the modulated phases, and a splitting point where the superfluid, the gapless superfluid and the modulated phases meet. We show that the physics near the splitting point is universal and derive an effective field theory describing it. We also show that subregions with one and two Fermi surfaces exist within the normal and the gapless superfluid phases.

PACS numbers: 03.75.Ss

Introduction.—Fermi gas in the regime of large scat-tering length a [1] has attracted much interest due to its universal behavior. The regime can be achieved in atom traps by using the technique of Feshbach resonance [2]. Most attention is focused on systems consisting of two species of fermions (e.g., two spin components of a spin-1 2

fermion) with equal number density. When the effective range r0is small compared to n−1/3, where n is the total number density, many properties of the system depend on n and a only through the dimensionless diluteness pa-rameter

κ = − ¹

na3. (1)

When one varies κ the system interpolates between the Bose-Einstein condensation (BEC) regime and the Bardeen-Cooper-Schrieffer (BCS) regime. For all values of κ the ground state is believed to be a superfluid.

In contrast, the case of unequal number density (or un-equal chemical potentials) of the two species is much less understood. In the case of spin-1

2 fermions one refers to a polarized gas. We follow this terminology, understanding “polarized” in the sense of asymmetry between the two species.

In this Letter we propose a phase diagram for a polar-ized Fermi gas in the whole range from the BEC to the BCS regime. Our proposal is summarized in Fig. 1. The variable η on the vertical axis roughly corresponds to the degree of polarization. The phase diagram must have four main regions, corresponding to the gapped super-fluid (BEC/BCS) phase (I), the normal phase (II), the gapless superfluid phase (III), and a phase with spatially varying condensate (IV). There are two special points on the phase diagram. Point S (the splitting point) is a point where phases (I), (III) and (IV) meet. Point L is a Lifshitz point where (II), (III) and (IV) meet. The physics in the vicinity of point S is long-distance, i.e., universal, and can be studied within an effective field theory. Furthermore, phases (II) and (III) are each di-vided (by the dashed line) into two subregions which

dif-I II II III IV L S ! " BEC BCS 1 "₀

FIG. 1: The proposed phase diagram.

fer from each other by the number of Fermi surfaces. On the left of the dashed line there is one Fermi surface, on the right there are two. Region (IV) is most likely di-vided into phases with different patterns of breaking of the rotational symmetry (not shown in Fig. 1).

The proposal is an educated guess anchored on a few reliable facts: the phases in the BEC and BCS limits, the existence of the points S and L, and the structure of the phase diagram around S.

Axes on the phase diagram.—A particular system is characterized by three parameters: the scattering length a, the chemical potentials of the two species µ_↑ and µ_↓. Because of universality (corresponding to rescaling in-variance a → e−sa, µi→ e2sµi) the whole phase diagram can be captured in a two-dimensional plot. We introduce the notation

µ = ¹

2^{(µ↑+ µ↓), H =} 1

2^(µ↑− µ↓) . (2) Then parameter κ on the horizontal axis is defined by (1) where n = n(µ, a) is the density of an unpolarized gas at chemical potential µ and scattering length a. Thus κ is the inverse diluteness parameter of an unpolarized system with chemical potential equal to the average of µ↑ and µ↓ and with the same scattering length a.

Figura 4.12: Secondo diagramma di fase di un sistema disomogeneo, in cui si rappresenta la fase del sistema per ogni valore della differenza fra i poten-ziali chimici delle specie (rappresentata da H) e la costante di accoppiamento (rappresentata da k) [46].

Come gi`a spiegato in precedenza, la transizione dalla fase BEC alla fase BP avviene attraverso la curva η = 1, in quanto corrisponde alla condizione ∆ = δµ.

Lo studio delle ragioni per cui il risultato delle simulazioni numeriche (fi-gura 4.9) ha, nella zona intorno a g=0, un comportamento diverso da quello ipotizzato da Son e Stephanov (figura 4.12) `e appena iniziato. `E possibile che le differenze derivino dall’uso dell’approssimazione di campo medio nell’analisi i cui risultati conducono alla figura 4.9, ma non `e escluso che si tratti di effetti puramente numerici, senza significati fisici pi`u profondi [45].