Quantum phases of a two-dimensional polarized degenerate Fermi gas in an optical cavity

1.4 MB·

by  Yanlin Feng

Quantum phases of a two-dimensional polarized degenerate Fermi gas in an optical
cavity
Yanlin Feng,1,2 Kuang Zhang,1,2 Jingtao Fan,1,2 Feng Mei,1,2,∗ Gang Chen,1,2,† and Suotang Jia1,2
1State Key Laboratory of Quantum Optics and Quantum Optics Devices,
Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China
2Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, China
Inthispaperweanalyticallyinvestigatetheground-statepropertiesofatwo-dimensionalpolarized
degenerate Fermi gas in a high-finesse optical cavity, which is governed by a generalized Fermi-
7
Dicke model with tunable parameters. By solving the photon-number dependent Bogoliubov–de–
1
Gennes equation, we find rich quantum phases and phase diagrams, which depend crucially on
0
the fermion-photon coupling strength, the fermion-fermion interaction strength, and the atomic
2
resonant frequency (effective Zeeman field). In particular, without the fermion-fermion interaction

n andwithaweakatomicresonantfrequency,wefindamixedphasethatthenormalphasewithtwo
a Fermi surfaces and the superradiant phase coexist, and reveal a first-order phase transition from
J
this normal phase to the superradiant phase. With the intermediate fermion-fermion interaction

3 and fermion-photon coupling strengths, we predict another mixed phase that the superfluid and
1 superradiant phases coexist. Finally, we address briefly how to detect these predicted quantum
phases and phase diagrams in experiments.
]
s
a PACSnumbers: 37.30.+i,42.50.Pq,67.85.Lm
g
-
t
n I. INTRODUCTION density, the Fermi surface displays a nesting structure
a and strongly enhances superradiance, which is, however,
u suppressedlargelyathighdensity,duetothePauliblock-
q TheexperimentalcombinationofaBose-Einsteincon-
ing effect. In addition, by introducing a cavity-assisted
. densate with a high-finesse optical cavity [1, 2] opens a
t spin-orbit coupling [24, 25], a topological SR phase has
a conceptually new regime of both cavity quantum elec-
been predicted [18]. Recently, the cavity-induced artifi-
m trodynamics and ultracold atoms. In this combination,
cialmagneticfield[20],chiralphases[21],andnon-trivial
- all ultracold bosons, occupying the same quantum state,
d interact identically with a single-mode quantized field, topological states [22] have been created. Moreover,
n when fermions are gauge coupled to a cavity mode, a
and thus, a strong collective matter-field interaction can
o SRphasewithaninfinitesimalpumpingthreshold,which
be achieved. Moreover, cavities can generate unconven-
c inducesadirectedparticleflow,hasbeenfoundforanin-
[ tional dynamical optical potentials, which induce rich
finite lattice [23].
nonequilibrium and strongly-corrected many-body phe-
3
nomena[3]. Forexample,whenpumpedtransversely,the In this paper, followed by the experimental scheme in
v
spinlessultracoldbosonsinthecavity-induceddynamical Ref. [26, 27], we consider a two-dimensional (2D) polar-
7
9 optical potentials undergo self-organization [4–6], which ized degenerate Fermi gas in a high-finesse optical cav-
8 has been observed experimentally [7, 8] and has been re- ity. When introducing two Raman transitions induced
2 gardedasan equivalencetothe well-knownsuperradiant bythequantizedcavityfieldandtwotransversepumping
0 (SR) phase transition in an effective Dicke model [9]. lasers,wefirstrealizeageneralizedFermi-Dickemodel,in
1. Motivated by near-term experimental prospects, which all parameters, including the fermion-photon cou-
0 another fundamental interaction between ultracold pling strength, the fermion-fermion interaction strength,
7 and the atomic resonant frequency (effective Zeeman
fermions and a high-finesse optical cavity has been in-
1 field), can be controlled independently. Then, based
vestigated theoretically. Since at lower temperature
:
v fermionsexhibitquitedifferentbehaviorthanbosons,ex- on a photon-number dependent Bogoliubov–de–Gennes
Xi oticphysicsisexpectedtoariseinthisnewplatform[10– (BdG) equation, we reveal rich quantum phases and
phasediagrams,whichdependcruciallyonthesetunable
23]. Inparticular,followedbytheexperimentalschemein
r parameters. In particular, without the fermion-fermion
a Ref. [7, 8], three groups have considered simultaneously
interaction and with a weak atomic resonant frequency,
spinless fermions in the cavity-induced dynamical opti-
we find a mixed phase that the normal phase with two
cal potential [15–17]. They have found that the Fermi
Fermi surfaces and the SR phase coexist, and reveal a
statistics plays a dominate role in the SR phase transi-
first-orderphasetransitionfromthisnormalphasetothe
tion at moderate and high densities. At the moderate
SR phase. With the intermediate fermion-fermion inter-
action and fermion-photon coupling strengths, we pre-
dict another mixed phase that the superfluid (SF) and
∗[email protected] SRphasescoexist. Finally, weaddressbrieflyhowtode-
†[email protected] tect the predicted quantum phases and phase diagrams
2
in experiments. PumpingLasers
This paper is organized as follows. In Sec. II, we
(cid:3) Photon
present an experimentally-feasible scheme to realize a
generalized Fermi-Dicke model with tunable parameters.
In Sec. III, we derive a photon-number dependent BdG
DrivingLaser
equation, and then obtain the ground-state energy and z(B)
the mean-field gap, particle number, and SR equations. y
x
In Secs. IV and V, we reveal rich quantum phases and
(a)
phase diagrams without or with the fermion-fermion
two-body interaction, respectively. The parameter
estimation and possible experimental observation are
2
addressed in Sec. VI, and the brief discussion and 1 D
D 2
conclusion are given in Sec. VII. 1
(cid:3) (cid:3)
g
g W W 2
1 2 1
II. MODEL AND HAMILTONIAN
w
¯ ¯
Figure 1 shows our proposed scheme that all ultra-
(b)
coldfermionsarecoupledwithahigh-finesseopticalcav-
ity supporting a single-mode photon. As illustrated in
Fig. 1(a), the fermions in the optical cavity are con-
fined in a far-of-resonance optical trap (yz plane) by a FIG. 1: (a) Our proposed schematic setup that all ultra-
tightly-radial confinement along the x direction. The cold fermions (black online), which are confined in a far-
of-resonance optical trap (yellow online) of the yz plane by
cavity mode is driven by a linearly-polarized laser and
a tightly-radial confinement along the x direction, are cou-
the fermionsare pumpedby twotransverse lasers, which
pled with a high-finesse optical cavity. The cavity mode is
are left- and right-handed circular polarized in the yz
driven by a linearly-polarized driving laser (with frequency
plane. In addition, each fermion has four levels, includ-
ω), which propagates along the x direction. Two transverse
l
ingtwogroundstates(|↑(cid:105)and|↓(cid:105))andtwoexcitedstates
pumpinglasers(withfrequenciesω andω ),whichareleft-
A B
(|1(cid:105)and|2(cid:105)),asshowninFig.1(b). Thequantizedcavity and right-handed circular polarized in the yz plane, propa-
field and the two transverse pumping lasers induce two gate along the x direction and induce two Raman processes
Raman processes; see more details in the caption. bycombiningthequantizedcavityfield. Inordertoobtaina
Formally, the total time-dependent 2D Hamiltonian is time-independent Hamiltonian, these frequencies should sat-
written as isfy the following condition: ωl = (ωA+ωB)/2; see the de-
tailed derivation in the main text. The magnetic field B is
Hˆ (t)=Hˆ +Hˆ +Hˆ (t)+Hˆ (t)+Hˆ +Hˆ . (1) appliedalongthepositivez directionandproducesaZeeman
T F P D AR AP INT shift between two hyperfine ground states. (b) The atomic
energy levels and their transitions. Each fermion has two
Here, the Hamiltonian of the free four-level fermions is
ground states (|↑(cid:105) and |↓(cid:105)) and two excited states (|1(cid:105) and
given by |2(cid:105)). The |↓(cid:105) ←→ |1(cid:105) and |↑(cid:105) ←→ |2(cid:105) transitions (red solid
lines) are caused by the quantized cavity field with fermion-
Hˆ = (cid:88) (cid:90) d2rψˆ†(r)(cid:18) pˆ2 −µ+ω (cid:19)ψˆ (r), (2) photoncouplingstrengthsg1 andg2. The|↑(cid:105)←→|1(cid:105)and|↓(cid:105)
F i 2M i i ←→ |2(cid:105) transitions (green dashed lines) are governed by the
i=1,2,↑,↓ transverse pumping lasers with Rabi frequencies Ω and Ω .
1 2
ω =ω −ω is the resonant frequency between the ground
↑↓ ↑ ↓
where ψˆ†(r) and ψˆ (r) (i = 1,2,↓,↑) are the creation states |↑(cid:105) and |↓(cid:105) with eigenfrequencies ω and ω . ∆ and
i i ↑ ↓ 1
and annihilation operators of the fermionic fields, M is ∆2 are the detunings from the excited states |1(cid:105) and |2(cid:105).
theatommass,µisthechemicalpotential,andω arethe
i
eigenfrequenciesofallquantumstates. TheHamiltonian
of the quantized cavity field, together with the driving fermionic fields and the two transverse pumping lasers,
laser, is written as reads
1(cid:90) (cid:104)
HˆP+HˆD(t)=ωcaˆ†aˆ+ε(cid:0)aˆeiωlt+aˆ†e−iωlt(cid:1), (3) HˆAR(t) = 2 d2r Ω1ψˆ1†(r)ψˆ↑(r)e−iωAt
(cid:105)
where aˆ† and aˆ are the creation and annihilation oper- +Ω2ψˆ2†(r)ψˆ↓(r)e−iωBt+H.c. , (4)
ators of the quantized cavity field with frequency ω ,
c
and ε (ω ) is the magnitude (frequency) of the driv- where Ω and Ω (ω and ω ) are the Rabi frequencies
l 1 2 A B
ing laser. Under the rotating-wave approximation, the (frequencies) of the transverse pumping lasers and H.c.
Hamiltonian,whichdescribestheinteractionbetweenthe denotes the Hermitian conjugate, whereas the Hamilto-
3
nian for governing the interaction between the fermionic whereλisthenegativeinteractionstrength, i.e., λ < 0.
and quantized cavity fields is given by
For the time-dependent Hamiltonian (1), we first per-
(cid:16) (cid:17)
(cid:90) (cid:110)(cid:104) (cid:105) (cid:111) formaunitarytransformationUˆ(t)=exp iHˆ(cid:48)t , where
Hˆ = d2r g ψˆ†(r)ψˆ (r)+g ψˆ†(r)ψˆ (r) aˆ+H.c. ,
AP 1 1 ↓ 2 2 ↑
(5)
ω (cid:104) (cid:105)
where g and g are both the fermion-photon coupling Hˆ(cid:48) = ω aˆ†aˆ+ B ψˆ†(r)ψˆ (r)−ψˆ†(r)ψˆ (r)
1 2 l 2 2 2 ↓ ↓
strengths. In addition, here we only consider the attrac-
ω (cid:104) (cid:105)
tive contact interaction between the ground states |↑(cid:105) + A ψˆ†(r)ψˆ (r)−ψˆ†(r)ψˆ (r) (7)
2 1 1 ↑ ↑
and |↓(cid:105) since the excited states are eliminated adiabati-
cally, as will be shown below. Therefore, the two-body
interacting Hamiltonian is given by with ωl = (ωB +ωA)/2, to obtain a time-independent
(cid:104) (cid:105)
Hamiltonian Hˆ =Uˆ(t)Hˆ (t)Uˆ†(t)+i ∂Uˆ(t)/∂t Uˆ†(t),
(cid:90) 1 T
Hˆ =λ d2rψˆ†(r)ψˆ†(r)ψˆ (r)ψˆ (r), (6) i.e.,
INT ↑ ↓ ↓ ↑
Hˆ = ω˜aˆ†aˆ+ε(cid:0)aˆ+aˆ†(cid:1)+ (cid:88) (cid:90) d2rψˆ†(r)(cid:18) pˆ2 −µ(cid:19)ψˆ (r)+(cid:90) d2r(cid:104)∆ ψˆ†(r)ψˆ (r)+∆ ψˆ†(r)ψˆ (r)(cid:105)
1 i 2M i 1 1 1 2 2 2
i=1,2,↑,↓
(cid:90) (cid:104) (cid:105) 1(cid:90) (cid:104) (cid:105)
+ d2r ω˜ ψˆ†(r)ψˆ (r)+ω˜ ψˆ†(r)ψˆ (r) + d2r Ω ψˆ†(r)ψˆ (r)+Ω ψˆ†(r)ψˆ (r)+H.c.
↑ ↑ ↑ ↓ ↓ ↓ 2 1 1 ↑ 2 2 ↓
(cid:90) (cid:90) (cid:110)(cid:104) (cid:105) (cid:111)
+λ d2rψˆ†(r)ψˆ†(r)ψˆ (r)ψˆ (r)+ d2r g ψˆ†(r)ψˆ (r)+g ψˆ†(r)ψˆ (r) aˆ+H.c. . (8)
↑ ↓ ↓ ↑ 1 1 ↓ 2 2 ↑
whereω˜ =ω −ω istheeffectivecavityfrequency, ∆ = large detunings (|∆ | (cid:29) {Ω ,g ,ω˜,ω }) are usually
c l 1 1,2 1,2 1,2 0
ω −ω /2 (∆ = ω −ω /2) is the detuning from the taken into account. In such case, the term ε(cid:0)a+a†(cid:1)
1 A 2 2 B
excited state |1(cid:105) (|2(cid:105)), and ω˜ = ω +ω /2 (ω˜ = ω + in the Hamiltonian (8) can be neglected and both the
↑ ↑ A ↓ ↓
ω /2) is the effective eigenfrequency of the ground state excitedstates|1(cid:105)and|2(cid:105)canbeeliminatedadiabatically
B
|↑(cid:105) (|↓(cid:105)). [28, 29]. Therefore, we obtain
In experiments [7, 8, 27], a weak driving (ε → 0) and
Hˆ = ω˜aˆ†aˆ+ (cid:88) (cid:90) d2rψˆ† (r)(cid:18) p2 −µ(cid:19)ψˆ (r)+(cid:90) d2r(cid:104)ω˜ ψˆ†(r)ψˆ (r)+ω˜ ψˆ†(r)ψˆ (r)(cid:105)
σ 2M σ ↑ ↑ ↑ ↓ ↓ ↓
σ=↑,↓
(cid:90) (cid:90) (cid:20)|g |2 |g |2 (cid:21)
+λ d2rψˆ†(r)ψˆ†(r)ψˆ (r)ψˆ (r)+ d2r 2 ψˆ†(r)ψˆ (r)+ 1 ψˆ†(r)ψˆ (r) aˆ†aˆ
↑ ↓ ↓ ↑ ∆ ↑ ↑ ∆ ↓ ↓
2 1
1(cid:90) (cid:26)(cid:20)g Ω∗ g Ω∗ (cid:21) (cid:27)
+ d2r 2 2ψˆ†(r)ψˆ (r)+ 1 1ψˆ†(r)ψˆ (r) aˆ+H.c. . (9)
2 ∆ ↓ ↑ ∆ ↑ ↓
2 1
When the parameters are chosen as the Hamiltonian (9) becomes
Hˆ = ωaˆ†aˆ+ (cid:88) (cid:90) d2rψˆ† (r)(cid:18) pˆ2 −µ(cid:19)ψˆ (r)
σ 2M σ
σ=↑,↓
(cid:90) (cid:104) (cid:105)
−ω d2r ψˆ†(r)ψˆ (r)−ψˆ†(r)ψˆ (r)
0 ↑ ↑ ↓ ↓
+√η (cid:90) d2r(cid:104)ψˆ†(r)ψˆ (r)+ψˆ†(r)ψˆ (r)(cid:105)(cid:0)aˆ+aˆ†(cid:1)
|g |2 |g |2 g Ω∗ g Ω∗ N ↓ ↑ ↑ ↓
1 = 2 , 1 1 = 2 2, (10) (cid:90)
∆1 ∆2 ∆1 ∆2 +λ d2rψˆ↑†(r)ψˆ↓†(r)ψˆ↓(r)ψˆ↑(r), (11)
4
√
where the factor 1/ N, with N being the total atom and η can be determined by the Rabi frequencies of
number, is introduced to ensure that the free energy per the transverse pumping lasers. Besides, λ can be tuned
fermion is finite in the thermodynamic limit [30, 31]. In by varying the s-wave scattering length a through the
s
the Hamiltonian (11), ω = (ω˜ −ω˜ )/2 is the effective Feshbach resonant technique [32]. See more detailed
0 ↓ ↑
resonantfrequencybetweenthegroundstates|↑(cid:105)and|↓(cid:105) discussions in Sec. VI.
andcanbeusuallyregardedasaneffectiveZeemanfield.
For simplicity, we take ω > 0 in the following discus-
√ 0 √
sions. η = Ng Ω∗/(2∆ ) = Ng Ω∗/(2∆ ) is the
1 1 1 2 2 2
effective fermion-photon coupling strength. ω =Nζ+ω˜ III. GROUND-STATE PROPERTIES
is the atom-number dependent cavity frequency, where
ζ = |g |2/∆ = |g |2/∆ . It should be emphasized that
1 1 2 2 In order to investigate the ground-state properties of
the choice of parameters in Eq. (10) has been used ex-
the Hamiltonian (11), we first expand operators of the
perimentally [7, 8, 27].
fermionic fields in terms of the plane waves, i.e.,
The Hamiltonian (11) is our required Hamiltonian
that governs two fundamental interactions, including
the fermion-photon and fermion-fermion two-body in- ψˆσ(r)= √1 (cid:88) Cˆk,σeik·r, (12)
S
teractions, and thus, is called a generalized Fermi-Dicke k,σ=↑,↓
model. This Hamiltonian has a distinct advantage that
all parameters can be controlled independently. For where Cˆ are the annihilation operators of fermions in
k,σ
example, ω and ω can be tuned by controlling the the momentum space and S is the gas area (hereafter
0
frequenciesofthedrivingandtransversepumpinglasers, S =1). After a straightforward calculation, we obtain
Hˆ = ωaˆ†aˆ+(cid:88)ξ Cˆ† Cˆ +ω (cid:88)(cid:16)Cˆ† Cˆ −Cˆ† Cˆ (cid:17)
k k,σ k,σ 0 k,↑ k,↑ k,↓ k,↓
k k
+λ(cid:88)Cˆ† Cˆ† Cˆ Cˆ + √η (cid:88)(cid:16)Cˆ† Cˆ +Cˆ† Cˆ (cid:17)(cid:0)aˆ+aˆ†(cid:1), (13)
k,↑ −k,↓ −k,↓ k,↑ n k,↑ k,↓ k,↓ k,↑
k k
where ξ = (cid:15) −µ, (cid:15) = k2/2M is the kinetic energy, erator aˆ [36, 37],
k k k
n = K2/(2π) = E M/π is the density of fermions in
F F
2D, and EF = KF2/(2M) is the Fermi energy. The i∂aˆ =(cid:104)aˆ,Hˆ(cid:105)−iκaˆ+γˆ (t), (16)
Hamiltonian (13) describes the interaction between two- ∂t in
component ultracold fermions and a high-finesse optical
where γˆ (t) is the quantum noise operator and satisfies
cavity in the momentum space. in
(cid:68) (cid:69)
For the attractive fermion-fermion two-body interac- the following conditions: γˆi†n(t)γˆin(t(cid:48)) = 2κδ(t−t(cid:48))
tion, the Cooper pairing with the opposite momentum and (cid:104)γˆ (t)γˆ (t(cid:48))(cid:105) = 0. In general, the fluctuation of
in in
and different spin is formed near to the Fermi surface quantum noise varies faster than 1/κ on the time scale
[33]. Inthemean-fieldapproximation,thecorresponding [38]. When the time scale of the atom dynamics in the
SF order parameter called the gap is assumed as [34, 35] motionaldegreeoffreedomislargerthan1/κ, thecavity
field can reach a steady state [6, 7], which is responsible
∆=λ(cid:88)(cid:68)Cˆ Cˆ (cid:69). (14) for obtaining the ground-state phase diagrams. In terms
−k,↓ k,↑
of Eqs. (13) and (16), the steady-state solution of aˆ is
k
given by
In such case, the two-body interacting Hamiltonian be-
(cid:68) (cid:69)
comes η(cid:80) Cˆ† Cˆ +Cˆ† Cˆ
k k,↑ k,↓ k,↓ k,↑
α=(cid:104)aˆ(cid:105)= √ . (17)
Hˆ =∆(cid:88)(cid:16)Cˆ Cˆ +Cˆ† Cˆ† (cid:17)− ∆2. (15) n(−ω+iκ)
INT −k,↓ k,↑ k,↑ −k,↓ λ
k Notice that when considering (cid:104)aˆ(cid:105), the noise term can be
neglected. In experiments [7, 8], the mean-photon num-
For simplicity, the mean-field gap is here assumed to be ber (cid:10)aˆ†aˆ(cid:11) = |α|2 governs the SR properties and is thus
real, i.e., ∆=∆∗.
called the SR order parameter.
In addition, our considered system usually exists the
Based on above discussions and in the basis of Nambu
cthaveiHtyesdieecnabyerwg-itLhanrgaeteviκn,eaqnudattihounsf,owrethsehcoauvldityinfiteroldduocpe- spinor Ψˆk = (cid:16)Cˆk,↑,Cˆk,↓,Cˆ−†k,↓,−Cˆ−†k,↑(cid:17)T, where T
5
standsforthetranspositionofamatrix,theHamiltonian the different regions [39]: E > 0 and E < 0. By
k,± k,±
(13) turns into means of Θ(x), the Hamiltonian (22) becomes
Hˆ = 1(cid:88)Ψˆ†M Ψˆ +(cid:88)ξ − ∆2 +ω|α|2, (18) Hˆ = (cid:88)Ek,±Θ(Ek,±)αˆk†,±αˆk,±
2 k k k k λ k,±
k k (cid:88) †
− E Θ(−E )αˆ αˆ +E , (23)
k,± k,± k,± k,± G
where the photon-number dependent BdG matrix is k,±
given by
with
ξk−ω0 η¯ ∆ 0  EG = (cid:88)[Ek,+Θ(−Ek,+)+Ek,−Θ(−Ek,−)]
η¯ ξ +ω 0 ∆
M = k 0 , k
k  ∆0 ∆0 −ξkη¯−ω0 −ξ η¯+ω  +(cid:88)(cid:18)ξ −(cid:113)ξ2 +∆2(cid:19)− ∆2 +ω|α|2.(24)
k 0 k k λ
√ (19) k
with η¯ = η(α+α∗)/ n. The BdG matrix (19) is also
In terms of Eq. (21), it is easy to see that E is always
written as k,+
positive, i.e., Θ(E ) = 1. Therefore, the ground-state
k,+
(cid:18) H ∆1 (cid:19) energy in Eq. (24) is simplified as a simple form
M = 0 , (20)
k ∆1 −σ H σ (cid:88)
y 0 y E = E Θ(−E )
G k,− k,−
k
where H = ξ +ω σ +η¯σ , σ and σ are the Pauli
matrices,0and1k isth0e2z×2unxitmxatrix. Tyhepropertyof +(cid:88)(cid:18)ξ −(cid:113)ξ2 +∆2(cid:19)− ∆2 +ω|α|2.(25)
the BdG matrix (20) implies that the Hamiltonian (18) k k λ
k
has the particle-hole symmetry.
According to the ground-state energy in Eq. (25),
By diagonalizing the BdG matrix M , we obtain the
k
three parameters ∆, µ, and |α| can be derived from
followingdispersionrelationsoftheBogoliubovquasipar-
the mean-field gap equation ∂E /∂∆ = 0, the particle
ticles: G
number equation ∂E /∂µ = −n, and the SR equation
G
(cid:18)(cid:113) (cid:113) (cid:19) ∂E /∂(|α|) = 0. If using the relation Θ(cid:48)(x) = δ(x),
Eυ =υ ξ2 +∆2± η¯2+ω2 , (21) G
k,± k 0 where δ(x) is the Dirac delta function, the above three
equations are given respectively by
where υ = ±1 correspond to the particle and (cid:34) (cid:35)
hole branches of the excitation spectra, and η¯2 = ∆ (cid:88) f(−Ek,−) −(cid:88) 1 − 2 = 0,(26)
4ω2η2|α|2/(cid:2)n(cid:0)ω2+κ2(cid:1)(cid:3). For each branch, there are (cid:112)ξ2 +∆2 (cid:112)ξ2 +∆2 λ
k k k k
two different excitations, due to the coexistence of the (cid:32) (cid:33)
fermion-photon and fermion-fermion two-body interac- (cid:88)ξkf(−Ek,−) +(cid:88) 1− ξk = n,(27)
(cid:112) (cid:112)
tions. In terms of Eq. (21), the Hamiltonian (18) is ξ2 +∆2 ξ2 +∆2
k k k k
rewritten as (cid:34) (cid:35)
Hˆ = (cid:88)(cid:16)E αˆ† αˆ +E αˆ† αˆ (cid:17) |α| (cid:88)−2ωη2f(−Ek,−)+nχ¯(cid:0)ω2+κ2(cid:1) = 0,(28)
k,+ k,+ k,+ k,− k,− k,− k
k
+(cid:88)(cid:18)ξ −(cid:113)ξ2 +∆2(cid:19)− ∆2 +ω|α|2,(22) w(cid:112)hη¯e2re+fω(−2.ENko,−ti)c=e tΘha(t−wEhke,−n)k−→Ek∞,−,δE(−q.E(k2,6−))dainvderχg¯e=s.
k k λ 0
k Inordertoeliminatethisultravioletdivergence,λshould
be renormalized as [34, 35]
where αˆ are the operators of the Bogoliubov quasi-
k,±
particles and satisfy the anticommutation relations 1 (cid:88) 1
=− , (29)
(cid:110) (cid:111)
αˆ† ,αˆ =δ (l=l(cid:48) =±). λ 2(cid:15)k+Eb
k,± k,± ll(cid:48) k
If Ek,+ and Ek,− in Eq. (21) are both positive, the where Eb > 0 is the two-body binding energy in 2D. In
first two terms in the Hamiltonian (22) reflect the ex- the following discussions, we self-consistently solve the
citation energies and its rest term is called the ground- coupled equations (26)-(28) at a fixed atom density n
stateenergy. Infact,Ek,±arepositivedefiniteonlywhen to obtain three parameters ∆, µ, and |α|. Equations
(cid:112) (cid:112)
ξ2 +∆2 > η¯2+ω2. Inordertocorrectlywritedown (26) and (28) show that there always exists different
k 0
the Hamiltonian (22) as a sum of the excitation energies solutions about ∆ and |α|. In fact, we must consider the
and the ground-state energy, we should introduce the stability of the system to find the proper solutions, and
Heaviside step function, which is defined as Θ(x) = 1 then predict rich quantum phase and phase diagrams.
for x (cid:62) 0 and Θ(x) = 0 for x < 0. This Heaviside step For simplicity, we take E as the unit of energy.
F
function can help us separate the sum over momenta in
6
IV. PHASE DIAGRAMS FOR E =0 state, i.e., the system still remains the fundamental
b
properties of the N-I or N-II phases.
WhenE =0,thesystemhasnofermion-fermiontwo- If η becomes stronger, the fermion-photon interacting
body interabction. To better understand the relevant be- term,(η/√n)(cid:80) (cid:16)Cˆ† Cˆ +Cˆ† Cˆ (cid:17)(cid:0)aˆ+aˆ†(cid:1),domi-
k k,↑ k,↓ k,↓ k,↑
havior, we first consider the case of η = 0, i.e., the free
nates and the system acquires the macroscopic collective
Fermi gas. In this case, the scaled ground-state energy excitationwith|α|2 (cid:54)=0. Thisimpliesthatwhenincreas-
defined as E¯ =E /n (i.e., the ground-state energy per
G G ing η, the SR transition can be expected to occur. In
fermion) is given by
terms of Eq. (25), the corresponding scaled ground-state
energy is obtained by
1 (cid:104) (cid:105)
E¯ =− (µ+ω )2Θ(µ+ω )+(µ−ω )2Θ(µ−ω ) .
G 4EF 0 0 0 0 E¯ =− 1 (cid:2)µ2 +µ2Θ(µ )(cid:3)+ω|α¯|2, (36)
(30) G 4E + − −
F
We always assume Θ(µ+ω ) (cid:54)= 0, which implies µ +
0
ω > 0. If µ + ω < 0, E¯ = 0, in which the where µ =µ±χ¯ and |α¯|2 =|α|2/n is the scaled mean-
0 0 G ±
systemhasnodefinitephysicalmeaningsinceundersuch photon number. Since the Heaviside step function in
conditionnorealfermionscanbefound[40]. Whenω > Eq.(36)dependscruciallyonµ ,thefollowingdiscussion
0 −
µ, Θ(µ−ω )=0 and the scaled ground-state energy in oftheground-statepropertiesshouldbedividedintotwo
0
Eq. (30) becomes specific cases: µ <0 and µ (cid:62)0.
− −
(ω +µ)2
E¯ =− 0 . (31)
G 4E A. µ− <0
F
In order to describe the effects induced by the effective
When
Zeemanfield,weshouldintroducethescaledpolarization
[41] µ =µ−χ¯<0, (37)
−
m¯ = n↑−n↓ =−∂E¯G. (32) ΘEq(.µ(−3)6)=bec0omaensd the scaled ground-state energy in
n ∂ω
0
1
Based on Eqs. (27), (31), and (32), we obtain E¯ =− µ2 +ω|α¯|2. (38)
G 4E +
F
µ=2E −ω , m¯ =1. (33)
F 0 In addition, Eqs. (27), (28), and (32) turn into
Equation(33)showsthattheFermigasisfullypolarized (cid:113)
µ+ η¯2+ω2 = 2E , (39)
and the system only has a Fermi surface with µ↑ =2EF. 0 F
The corresponding normal phase is called the N-I phase.  (cid:16) (cid:112) (cid:17)
2ω2η2 µ+ η¯2+ω2
When ω < µ, Θ(µ−ω ) = 1 and the scaled ground- 0
state ene0rgy in Eq. (30) b0ecomes |α¯|2ω− E (ω2+κ2)(cid:112)η¯2+ω2 = 0, (40)
F 0
E¯G =−(cid:0)µ22E+ω02(cid:1). (34) ω0(cid:16)µ+(cid:112)η¯2+ω02(cid:17) = m¯. (41)
F (cid:112)
2E η¯2+ω2
F 0
We further obtain
By further solving Eqs. (39)-(41), we obtain
ω
µ=E , m¯ = 0 . (35)
F E µ=2E −ω , |α¯|=0, m¯ =1, (42)
F F 0
It is quite different from the N-I phase that in this case or
the Fermi gas is partially polarized and the system has  (cid:16) (cid:17)
tawnodFµe↓rm=i EsuFrfa−ceωs0d.efiTnheedcroersrpeescptoivnedliyngaspµh↑as=e EisFca+llωed0 µ|α¯=|=2(cid:113)EFη−2 ω−ω2+η2κω202(ω,2+κ2), (43)
the N-II phase. From above discussions, it can be seen ω2+κ2 4ω2η2
thatwhenvaryingω0,thesystemundergoesafirst-order  m¯ = ω0(2ωω2η+2κ2).
phase transition from the N-I phase to the N-II phase at
the critical point ωc =E [39]. In order to find the stable ground state, we should
For a weak 0η, thFe noninteracting terms in introduce the condition governed by ∂2E¯G/∂(|α¯|)2 > 0.
the Hamiltonian (13), ωaˆ†aˆ + (cid:80) ξ Cˆ† Cˆ + In terms of this stable condition, we find immediately
ω (cid:80) (cid:16)Cˆ† Cˆ −Cˆ† Cˆ (cid:17), play akdkomkin,σatek,σrole that |α¯|=0 for η <ηc(1), where
0 k k,↑ k,↑ k,↓ k,↓
(cid:114)
in the systematic dynamics. In this case, no fermion- ω (ω2+κ2)
photon interaction occurs and |α|2 = 0 for the ground ηc(1) = 0 2ω , (44)
7
(a)0 EF 0.5 ω0=0.8EF (a) 1 (b)
EF 0 ω0=1.2EF F 0.5
 0   0 ¯E/G −0.5 µ/E 0
c2 c1  −1 ηc(2) ηc(1) −0.5
0 5 0 5
η/E η/E
(b) E F F
0 F
(c) 1 (d)

 0  0 0.1 0.8
  2
1 2  ¯α|| 0.05 ¯m 0.6
c c 0.4
0
0.2
0 5 0 5
FIG.2: Thecomparisonofthecriticalpointsηc(1) andηc(2) for η/EF η/EF
(a) ω (cid:62)E and (b) ω <E . When ω (cid:62)E , η(1) >η(2),
0 F 0 F 0 F c c
and µ < 0 for both 0 < η < η(1) and η > η(1). When
− c c
ω < E , η(1) < η(2), and µ < 0 for η > η(2) and µ > 0 FIG. 3: (a) The scaled ground-state energy E¯ /E , (b) the
0 F c c − c − G F
for 0<η<ηc(2). chemical potential µ/EF, (c) the scaled mean-photon num-
ber|α¯|2,and(d)thescaledpolarizationm¯ asfunctionsofthe
effective atom-photon coupling strength η/E . The atom-
F
(cid:112) numberdependentcavityfrequencyandthecavitydecayrate
and |α¯| = η2/(ω2+κ2)−ω2(ω2+κ2)/(4ω2η2) [i.e.,
0 are given by ω = 10E and κ = 20E , respectively. When
Eq. (43)] for η >ηc(1). At the same time, we must notice the effective resonantFfrequency is choFsen as ω0 = 1.2EF,
the restrictive condition µ− = µ−χ¯ < 0 in Eq. (37), ηc(1) =5.48EF. When ω0 =0.8EF, ηc(2) = 5EF. In these fig-
which also induces a critical point ures,theblue-solidandred-dashedlinesrepresenttheanalyt-
ical results, while the open symbols correspond to numerical
(cid:114)
E (ω2+κ2) simulations.
η(2) = F . (45)
c 2ω
The analytical results in Eqs. (46)-(49) show two typi-
When η >ηc(2), µ− <0. cal properties. The first is that the first-order derivative
From the critical points in Eqs. (44) and (45), we find ofE¯ withrespecttoηiscontinuousbutitssecondorder
G
that when ω (cid:62) E , η(1) > η(2), and Eqs. (42)-(43) sat- is discontinuous, which means that a second-order phase
0 F c c
isfy the condition µ < 0 for both 0 < η < η(1) and transition from the N-I phase to the SR phase occurs at
η > η(1) [see Fig. −2(a)]. However, when ω c< E , the critical point ηc(1). This property is similar to the
c 0 F case of ultracold Bose atoms [30, 31]. The other is that
η(1) < η(2), and only Eq. (43) satisfy the condition
c c in the N-I phase no fermion-photon interaction occurs,
µ− < 0 for η > ηc(2) [see Fig. 2(b)]. In other words, whereas in the SR phase both the fermions and photons
when 0 < η < η(2), µ (cid:62) 0, in which we should intro- acquire the macroscopic collective excitations. Besides,
c −
duce a new scaled ground-state energy, as will be shown m¯ isinverselyproportionaltoη2 andµisdecreasedwith
in the next subsection. respecttoη2. InFigs.3(a)-3(d),weplotE¯ ,µ,|α¯|2,and
G
As a consequence, in the case of ω (cid:62) E , the scaled m¯ as functions of η. These figures show that our analyt-
0 F
ground-state energy is written as ical results agree well with direct numerical simulations.
Moreover, theabovetwotypicalpropertiesarerecovered
(cid:40)−E 0<η <η(1) naturally.
E¯G = −EF + ωη2 − ω02(ω2+κ2) η >η(1) c ,
F ω2+κ2 4ωη2 c
(46) B. µ (cid:62)0
−
and three parameters are given respectively by
When
(cid:40)2E −ω 0<η <η(1)
µ = 2(cid:16)FE −0 ωη2 (cid:17) η >η(1) c , (47) µ− =µ−χ¯(cid:62)0, (50)
F ω2+κ2 c
Θ(µ ) = 1 and the scaled ground-state energy in
(cid:40)0 0<η <η(1) Eq. (−36) becomes
c
|α¯| = (cid:113) ,(48)
ω2η+2κ2 − ω024(ωω22+η2κ2) η >ηc(1) E¯ =− 1 (cid:0)µ2 +µ2(cid:1)+ω|α¯|2. (51)
(cid:40)1 0<η <η(1) G 4EF + −
c
m¯ = ω0(ω2+κ2) η >η(1) . (49) According to the discussions in the subsection A of this
2ωη2 c section and considering the stable condition governed by
8
2 fully describe the fundamental properties of this mixed
phase, we should introduce the fractions of the N-II and
SR phases, x and 1−x . Moreover, we further obtain
0 0
1.5 N−I ηc(1)
x (cid:0)ω2+E2(cid:1)
E¯ = µ− 0 0 F
F G 2EF
E
/0 1 −(1−x0)(cid:18)E + ω02 (cid:19), (54)
ω 2 F E
SR F
µ = E , (55)
0.5 N−II ηc(2) (cid:115)F
E2 −ω2
|α¯| = F 0, (56)
2ωE
F
00 2 4 6 x ω ω (cid:0)ω2+κ2(cid:1)
m¯ = 0 0 +(1−x ) 0 . (57)
η/E E 0 2ωη2
F F
The detailed derivation of Eqs. (54)-(57) is given by the
FIG. 4: Phase diagram as a function of the effective reso-
Appendix A. Notice that since the atom densities are
nant frequency ω /E and the effective fermion-photon cou-
0 F
plingstrengthη/EF. Theatom-numberdependentcavityfre- equal in both the N-II and SR phases, x0 and 1 − x0
quency ω and the cavity decay rate κ are the same as those are the arbitrary values ranging from 0 to 1. In ad-
in Fig. 3. When the effective resonant frequency is chosen as dition, since m¯ (cid:54)= 0 in both the N-II and SR phases,
ω =E , η(1) =η(2). the nonzero polarization in the N-II-SR mixed phase is
0 F c c
caused by both the macroscopic collective excitation of
the fermions and photons and the effective Zeeman field.
∂2E¯ /∂(|α¯|)2 >0, we obtain We also find two first-order phase transitions from the
G
N-II-SR phase to the N-II phase or the SR phase. These
ω
µ=E , |α¯|=0, m¯ = 0 , (52) resultsarequitedifferentfromthoseintheN-Iphaseand
F E
F the ultracold Bose atoms [30, 31].
for 0 < η < η(2). Since Eq. (52) should satisfy the con-
c
dition µ− (cid:62) 0 when 0 < η <ηc(2), we find that ω0 <EF C. Phase diagram
in this case. When η > η(2), µ < 0 [see Fig. 2(b)], in
c −
which we should combine with the previous discussions In Fig. 4, we plot the whole phase diagram, including
in the subsection A of this section. the N-I phase, the N-II phase, the N-II-SR mixed phase,
As a consequence, in the case of ω0 < EF, the scaled and the SR phase, for ω0 < EF and ω0 (cid:62) EF. As
ground-state energy is written as predicted previously, the phase transition from the N-I
 phase to the SR phase is of the second order, whereas
E¯ =−E2F − 2ωE02F 0<η <ηc(2) . the phase transition from the N-II phase to the SR
G −EF + ωω2+η2κ2 − ω02(4ωω2η+2κ2) η >ηc(2) pNh-IaIseanisdoSfRthpehfiarssets oartdtehre, dcruietictaoltlhinee.coIenxiasdtednitcieono,ftthhies
(53) phase diagram has a tricritical point (the red dot), at
In addition, when 0 < η < η(2), µ, |α¯|, and m¯ are gov- which the phase transition changes from the first order
c
erned by Eq. (52), which means that the system is lo- to the second order.
cated at the N-II phase. When η > η(2), µ, |α¯|, and m¯
c
are governed by Eq. (43), which implies that the system
is located at the SR phase. The scaled ground-state en-
ergyinEq.(53)showsthatthephasetransitionfromthe V. PHASE DIAGRAMS FOR Eb (cid:54)=0 AND η(cid:54)=0
N-II phase to the SR phase is of the first order. More-
over, |α¯|2 exhibits a sudden change at the critical point WhenbothηandE exist,thepropertiesoftheBogoli-
b
η(2), as shown by the red-dashed line in Fig. 3(c). ubov quasiparticle states are determined by both χ¯ and
c
Interestingly, when η = η(2), the scaled ground-state ∆,asshownintheHamiltonian(22). Ifχ¯>∆,thequasi-
c (cid:112)
energies of the N-II and SR phases are equal, which particlestatesareoccupiedformax{0,µ− χ¯2−∆2}<
(cid:112) (cid:112)
meansthatthesetwophasescoexistandthecorrespond- (cid:15) <µ+ χ¯2−∆2 (µ+ χ¯2−∆2 >0), and thus, the
k
ing phase is called the N-II-SR mixed phase. In order to scaled ground-state energy is obtained by
9
(cid:34) (cid:32) (cid:112) (cid:33)(cid:35)
E¯ = E¯SF+ω|α¯|2− 1 2χ¯(cid:112)χ¯2−∆2−∆2ln χ¯+ χ¯2−∆2 Θ(µ¯ )Θ(cid:0)χ¯2−∆2(cid:1)
G G 4EF χ¯−(cid:112)χ¯2−∆2 −
(cid:34) (cid:32) (cid:112) (cid:33)(cid:35)
− 1 χ¯(cid:112)χ¯2−∆2+µ(cid:16)2χ¯−(cid:112)µ2+∆2(cid:17)−∆2ln −χ¯+ χ¯2−∆2 Θ(−µ¯ )Θ(cid:0)χ¯2−∆2(cid:1), (58)
4EF µ−(cid:112)µ2+∆2 −
(cid:112)
where µ¯ =µ− χ¯2−∆2 and A. µ¯ <0 and χ¯2−∆2 (cid:62)0
− −
(cid:32)(cid:112) (cid:33)
∆2 µ2+∆2−µ
E¯SF = ln When
G 4E E
F b
µ (cid:16)(cid:112) (cid:17) ∆2 µ¯ =µ−(cid:112)χ¯2−∆2 <0 (60)
− µ2+∆2+µ − (59) −
4E 8E
F F
is the fully-paired (m¯ = 0) SF energy. If χ¯ < ∆, E¯ = and
G
E¯SF. ItcanbeseenclearlyfromEq.(58)thattheground-
G
state properties, including ∆, µ, and |α¯|2, are governed χ¯2−∆2 (cid:62)0, (61)
by both η and E . When E = 0, µ¯ = µ−χ¯ and the
b b −
corresponding scaled ground-state energy is the same as Θ(µ¯ ) = 0 and Θ(cid:0)χ¯2−∆2(cid:1) = 1. Thus, the scaled
−
Eq. (36). When η = 0, χ¯ = ω and Eq. (58) reduces
0 ground-state energy in Eq. (58) becomes
to Eq. (14) in Ref. [42] and Eq. (8) in Ref. [43]. When
E (cid:54)=0 and η (cid:54)=0, a strong competition between the SF
b
and SR properties occurs. Consequently, rich quantum E¯ = E¯SF+ω|α¯|2− 1 (cid:104)χ¯(cid:112)χ¯2−∆2
phases can be predicted. Similarly, the Heaviside step G G 4EF
functioninEq.(58)dependscruciallyonµ¯−andχ¯2−∆2, (cid:32) χ¯+(cid:112)χ¯2−∆2(cid:33)
and thus, the following discussion of the ground-state −∆2ln − (cid:112)
µ− µ2+∆2
properties should also be divided into four specific cases:
µ¯− < 0 and χ¯2 −∆2 (cid:62) 0, µ¯− < 0 and χ¯2 −∆2 < 0, +µ(cid:16)2χ¯−(cid:112)µ2+∆2(cid:17)(cid:105). (62)
µ¯ (cid:62)0 and χ¯2−∆2 (cid:62)0, and µ¯ (cid:62)0 and χ¯2−∆2 <0.
− −
Moreover, wewilldrawtwoconclusionsforω <E and
0 F
ω (cid:62)E . From Eqs. (26)-(28) and (32), we obtain
0 F
(cid:32)(cid:112) (cid:33) (cid:16)χ¯+(cid:112)χ¯2−∆2(cid:17)(cid:16)(cid:112)∆2+µ2+µ(cid:17)
∆2+µ2−µ
∆ln +∆ln  = 0, (63)
E ∆2
b
µ+χ¯ = 2E , (64)
F
(cid:34) (cid:112) (cid:35)
ω2η2 µ+ χ¯2−∆2
− +ω |α¯| = 0, (65)
E (ω2+κ2) χ¯
F
(cid:16)(cid:112) (cid:17)
χ¯2−∆2+µ
2E m¯
= F . (66)
χ¯ ω
0
By further solving Eqs. (63)-(66), we obtain or

∆=0,
µ=2(cid:16)EF − ωω2+η2κ2(cid:17),
(cid:113) (68)
∆=0, µ=2EF −ω0, |α¯|=0, m¯ =1, (67) |m¯α¯|==ω0(ωω22η++2κκ22)−, ω024(ωω22+η2κ2),
2ωη2
10

or
 (cid:112) 1.9
∆= E (2ω −E ),
µ=2E b−ω0, b 0.8
F 0 (69) 1.8
|α¯|=0,
m¯ = Eb−22ωE0F+2EF, F 1.7 0.6
or E
/ 1.6
µ∆==2(cid:114)E√EFb−(cid:104)22ωω2ωωηηη2η2(cid:114)+2+2((((22ωωEE22FF++++κκE2E2))bbEE))FF,−Eb(cid:105), (70) Eb 11..45 00..24
|m¯α¯|==ω0((ωω222++κκ22)). (cid:104)2ωηη2(+2E(ωF2++Eκ2b))EF(cid:105)2− ωω20η22, 1.3 0
2ωη2 5.5 6 6.5 7
Sinceherethesystemhastwodependentorderparam- η/E
F
eters ∆ and |α¯|, the ground-state stability should be de-
terminedbya2×2Hessianmatrix[44], whichisdefined
FIG.5: ThefractionoftheSFphase,x ,asafunctionofthe
as 2
two-body binding energy E /E and the effective fermion-
b F
(cid:34) ∂2E¯G ∂2E¯G (cid:35) photon coupling strength η/EF, when the effective resonant
M = ∂∆2 ∂∆∂|α¯| . (71) frequencyischosenasω =1.2E . Theatom-numberdepen-
∂∂|α2¯E|¯∂G∆ ∂∂(2|αE¯¯|G)2 dent cavity frequency ω0and theFcavity decay rate κ are the
same as those in Fig. 3.
If M is positive definite (i.e., two eigenvalues of M are
positive),E¯ haslocalminimaandthesystemislocated
G
at the stable phase. If M is indefinite (i.e., one eigen-
and
values is positive, while the other is negative), E¯ has
G
saddle points and the system is dynamically unstable.
χ¯2−∆2 <0, (73)
If M is negative definite (i.e., two eigenvalues of M are
negative), E¯G has a local maximum and the system is Θ(µ¯ ) = 0 and Θ(cid:0)χ¯2−∆2(cid:1) = 0. Thus, the scaled
extremely unstable. −
ground-state energy in Eq. (58) becomes
IntermsofthestabilityconditiongivenbytheHessian
matrix(71),thegroundstatescorrespondingtothesolu-
E¯ =E¯SF+ω|α¯|2. (74)
tions (69) or (70) are unstable, whereas for the solutions G G
(67) or (68) they become stable. Since ∆ ≡ 0 in both
From Eqs. (26)-(28) and (32), we obtain
Eqs. (67) and (68), we can use the similar discussions
in the subsection A of Sec. IV. For example, using the
(cid:32)(cid:112) (cid:33)
stable condition governed by ∂2E¯ /∂(|α¯|)2 > 0, we ob- ∆2+µ2−µ
G ∆ln = 0, (75)
tain the superradiant critical point ηc(1), which separates Eb
the solutions (67) and (68). In addition, the restrictive (cid:112)
∆2+µ2+µ = 2E , (76)
conditions in Eqs. (60) and (61) lead to another critical F
point η(2). Comparing η(1) with η(2), we find that when 2ω|α¯| = 0, (77)
c c c
ω (cid:62) E , i.e., η(1) > η(2), µ¯ < 0 and χ¯2 −∆2 (cid:62) 0, m¯ = 0. (78)
0 F c c −
and thus, ∆, µ, |α¯|, and m¯ are governed by Eq. (67) for
By further solving Eqs. (75)-(78), we obtain
0 < η < η(1), and for η > η(1), they are governed by
c c
Eq. (68). When ω < E , i.e., η(1) < η(2), µ¯ (cid:62) 0 and
0 F c c − ∆=0, µ=E , |α¯|=0, m¯ =0, (79)
F
χ¯2 − ∆2 (cid:62) 0, and thus, for 0 < η < η(2), the scaled
c
ground-stateenergychangesandwewilldiscusstherele- or
vantresultsinthesubsectionDofthissection. However,
|fαo¯r|,ηan>dηm¯c(2)a,rµe¯−sti<ll g0oavnerdneχ¯d2−by∆E2q.(cid:62)(608,)a.nd thus, ∆, µ, ∆=(cid:112)2EFEb, µ=EF − E2b, |α¯|=0, m¯ =0. (80)
Since |α¯|≡0 in Eqs. (79) and (80), we should introduce
B. µ¯− <0 and χ¯2−∆2 <0 the stable condition governed by ∂2E¯G/∂∆2 >0 to find
the stable ground state. According to this stable condi-
tion and the restrictive condition in Eqs. (72) and (73),
When
we find that the ground state, with the solution (80), is
(cid:112)
µ¯ =µ− χ¯2−∆2 <0 (72) stable for all E and η.
− b