Phase transitions in Bose-Fermi-Hubbard model in the heavy fermion limit: Hard-core boson approach
Phase transitions are investigated in the Bose-Fermi-Hubbard model in the mean field and hard-core boson approximations for the case of infinitely small fermion transfer and repulsive on-site boson-fermion interaction. The behavior of the Bose-Einstein condensate order parameter and grand canonical...
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| Zitieren: | Phase transitions in Bose-Fermi-Hubbard model in the heavy fermion limit: Hard-core boson approach / I.V. Stasyuk, V.O. Krasnov // Condensed Matter Physics. — 2015. — Т. 18, № 4. — С. 43702: 1–20. — Бібліогр.: 54 назв. — англ. |
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| citation_txt | Phase transitions in Bose-Fermi-Hubbard model in the heavy fermion limit: Hard-core boson approach / I.V. Stasyuk, V.O. Krasnov // Condensed Matter Physics. — 2015. — Т. 18, № 4. — С. 43702: 1–20. — Бібліогр.: 54 назв. — англ. |
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| description | Phase transitions are investigated in the Bose-Fermi-Hubbard model in the mean field and hard-core boson approximations for the case of infinitely small fermion transfer and repulsive on-site boson-fermion interaction. The behavior of the Bose-Einstein condensate order parameter and grand canonical potential is analyzed as functions of the chemical potential of bosons at zero temperature. The possibility of change of order of the phase transition to the superfluid phase in the regime of fixed values of the chemical potentials of Bose- and Fermi-particles is established. The relevant phase diagrams are built.
Дослiджено фазовi переходи в моделi Бозе-Фермi-Хаббарда в наближеннi середнього поля та жорстких
бозонiв при врахуваннi одновузлової бозон-фермiонної взаємодiї типу вiдштовхування та у випадку безмежно малого переносу фермiонiв. Проаналiзовано поведiнку параметра порядку бозе-конденсату та
термодинамiчного потенцiалу як функцiй хiмiчного потенцiалу бозонiв при нульовiй температурi. Встановлено можливiсть змiни роду фазового переходу до надплинної фази у режимi заданих значень хiмiчних потенцiалiв бозе- та фермi-частинок. Побудовано вiдповiднi фазовi дiаграми.
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Condensed Matter Physics, 2015, Vol. 18, No 4, 43702: 1–20
DOI: 10.5488/CMP.18.43702
http://www.icmp.lviv.ua/journal
Phase transitions in Bose-Fermi-Hubbard model in
the heavy fermion limit: Hard-core boson approach
I.V. Stasyuk, V.O. Krasnov
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
Received July 2, 2015
Phase transitions are investigated in the Bose-Fermi-Hubbard model in the mean field and hard-core boson
approximations for the case of infinitely small fermion transfer and repulsive on-site boson-fermion interaction.
The behavior of the Bose-Einstein condensate order parameter and grand canonical potential is analyzed as
functions of the chemical potential of bosons at zero temperature. The possibility of change of order of the
phase transition to the superfluid phase in the regime of fixed values of the chemical potentials of Bose- and
Fermi-particles is established. The relevant phase diagrams are built.
Key words: Bose-Fermi-Hubbard model, hard-core bosons, Bose-Einstein condensate, phase transitions,
phase diagrams
PACS: 71.10 Fd, 71.38
1. Introduction
Physics ofmany-particle systemswith strong short-range correlations is one of the important research
fields. A new outbreak of activity in this area is associatedwithmanifestations of peculiar properties of ul-
tracold atomic systems in traps at imposition of periodic field formed by the interference of coherent laser
beams. In the optical lattices formed this way, there occurs a transition to the state with Bose-Einstein (BE)
condensate at very low temperatures (T < 10−7 −10−8
K) in the case of Bose-atoms. Experimentally, this
effect was originally observed by Greiner and others in 2002–2003 [1, 2] in the system of
87
Rb atoms.
Bose-Einstein condensation occurs in this case by the phase transition of the 2nd order from the so-called
Mott insulator phase (MI-phase) to the superfluid phase (SF-phase). The theoretical description of the
phenomenon is based on the Bose-Hubbard model [3, 4], which takes into account two main factors that
determine the thermodynamics and energy spectrum of the system of Bose-particles — tunneling be-
tween the neighbouring minima of potential in the lattice and short-range on-site repulsive interaction
of Hubbard type. A lot of investigations [5–14] are dedicated to the construction of phase diagrams that
define the conditions of the existence of SF-phase at T = 0 and at non-zero temperatures as well as to the
study of different aspects of one-particle spectrum (see, also, reviews [15–17]).
Along with ultracold atomic Bose-systems, the boson-fermion mixtures in optical atomic lattices are
also actively researched. Their experimental implementation (e.g., spin-polarized mixtures of
87
Rb–
40
K
atoms [18–20]) made it possible to see the MI-SF transition in the presence of Fermi-atoms. An impor-
tant factor that has been observed is the fading of the coherence of bosons and the decay of condensate
fraction in SF-phase in a certain range of the values of thermodynamic parameter (chemical potential
of bosons or temperature) influenced by the interaction with fermions. This is reflected in the change
of conditions of existence of SF- and MI-phases and the shift of curve of SF-MI transition on the phase
diagram. Renormalization (due to self-trapping) of tunneling hopping of bosons [18, 20, 21], influence of
higher boson bands [22, 23], the intersite interactions (including the so-called correlated hopping) [24]
are considered to be possible explanations of this effect. An important feature of boson-fermion mix-
tures is the possible existence of new quantum phases such as CDW-type phase (with the particle density
© I.V. Stasyuk, V.O. Krasnov, 2015 43702-1
http://dx.doi.org/10.5488/CMP.18.43702
http://www.icmp.lviv.ua/journal
I.V. Stasyuk, V.O. Krasnov
modulation), supersolid (SS) phase (with the spatial modulation of density as well as the order parameter
of condensate), SFf phase (with the condensate of fermion pairs), and their various combinations (see, in
particular, [25]). Another interesting factor that should be taken into consideration is the formation of the
so-called fermion composites [26, 27], which are caused by the fermion pairing with one or more bosons
(or one or more boson holes) due to their effective attraction (or repulsion). Interaction of interspecies
in this case can be changed [28] using the Feshbach resonance [29]. An asymmetry between the attrac-
tion and repulsion cases in the behavior of boson-fermion mixtures and in the phase diagrams [18, 30] is
observed here.
The Bose-Fermi-Hubbard model is used to describe the boson-fermion mixtures in optical lattices.
This model and its microscopic justification were proposed in [30]. Following this, in [31, 32] the phase
diagrams at T = 0 (ground state diagrams) in the mean-field approximation were constructed. Moreover,
the atom transfer was taken into account using the perturbation theory; the effective static interaction
between bosons was included (in the cases JB = JF = 0; JB , 0, JF = 0; JB = JF = J , where JB, JF are
parameters of the atom transfer). The areas of the existence of phases with composite fermions that
contain a different number of bosons (or boson holes) were found. The analysis was performed in the
regime of fixed fermion density.
In [33] as well as in subsequent studies [34, 35], it was shown within the BFH model that the di-
rect boson-fermion interaction can lead to an effective dynamic interaction between bosons through
fermionic field. This gives the appearance of instability; when ~q = 0— in regard to phase separation, and
when ~q =~kDW— in regard to spatial modulation and SS phase formation (which in the case of half-filling
for fermions and the increase of energy of their repulsion off bosons becomes a CDW phase [36]). The
mechanisms of SS phase arising were studied in some other works (see, in particular [25]). Bose-Einstein
condensate enhances the s-wave pairing of fermions, while uncondensed bosons contribute to the ap-
pearance of CDW. At half-filling for fermions, SFf -phase competes with antiferromagnetic ordering [37].
On the other hand, at the spin degeneracy, the reverse effect is possible when the pairing of fermions
is induced by bosons. This situation is analogous to the formation of Cooper pairs in the BCS model, as
was shown in several papers (see, [38–40]). It results in the creation of a SFf phase, where the role of
the superfluid component belongs to fermion pairs; the corresponding phase diagrams are constructed
in [25]. Integration over fermionic variables also provides an additional static interactionUBB between
bosons, which promotes MI → SF transition or suppresses it. To a large extent it depends on the mass
ratio of Bose- and Fermi-atoms (i.e., on the ratio of the hopping parameters tF/tB). The phase diagram
obtained by functional integration and the Gutzwiller approach for the cases of “light” (a mixture of
87
Rb–
40
K atoms) and “heavy” (a mixture of
23
Na–
40
K atoms) fermions is presented in [34].
Other aspects of the fermion influence on the MI→ SF phase transition in a boson subsystem were
investigated in [22–41]. It is shown, in particular, that virtual transitions of bosons under the effect of
interaction UBF through their excited states in the optical lattice potential minima lead to an extension
of the MI phase region at T = 0 (the shift of the curve of the phase transition in the (t , µ) plane towards
larger values of the t/U ratio). A similar role is played by retardation at the boson screening by fermions
and “polaron effect” [42]. This is manifested by the reduction of the parameter of the transfer of bosons
tB and by their slowing down. However, for heavy fermions, the SF phase region broadens for the case of
intermediate temperatures [35]. Interparticle interactions, in particular the so-called bond-charge inter-
action, can have a significant effect on the transfer of bosons as shown in [24]. This also can lead to the
shift of transition from MI to SF phase.
A separate direction of theoretical description of boson atoms and boson-fermion mixtures in opti-
cal lattices is associated with the use of the hard-core boson approach, where the occupation of on-site
states conforms to the Pauli principle. For Bose atoms on the lattice, this model is a limiting case of Bose-
Hubbard model for U → ∞ and is rather widely used [43–47]. It is suitable for the region 0 É nB É 1,
but also can describe the MI-SF phase transitions in the vicinity of the points µB = nU , n = 0,1,2, . . . at
finite values of U (in the case of strong coupling, tB¿U ) where n É nB É n +1 for T = 0 within the SF
phase region [13, 14]. In these cases, the model is also applicable to these phase transitions at non-zero
temperatures.
For boson-fermion mixtures, the BFH model in hard-core bosons limit remains less explored. In [48],
the phase diagrams and phase separation or charge modulation conditions at the ion intercalation in
semiconductor crystals were investigated based on the BFH-type model; in [49, 50], within pseudospin-
43702-2
Phase transitions in BFH model
fermion description (that corresponds to the abovementionedU →∞ limit) the conditions of the appear-
ance of SS and CDWphases under effective interparticle interactions were investigated. For the four-state
model, that arises in this case, the calculations for fermion band spectrum in Hubbard-I approximation
were performed, and its transformation during transition to the SF-phase and at the presence of a Bose-
Einstein condensate [51] was investigated.
The aim of this work is to more thoroughly study the thermodynamics of the above mentioned 4-state
model. We confine ourselves to the case of “heavy” fermions (i.e., extremely low values of the fermion
hopping parameter tF). Such a situation was partially considered in [23, 31]. It was argued, in particular,
that frozen fermions, as fixed subsystem when tF = 0, are capable of preventing the occurrence of long-
range correlations of superfluid-type and the appearance of BE condensate. There exists, however, the
critical fermion concentration below which this effect is absent (for d = 2, ncritp ∼ 0.59; for d = 3, ncritp ∼
0.31; see [23]).
We consider the equilibrium case, assuming that tf takes small values, but not so small that could lead
to the state of a glass type [15, 52, 53]. We shall use the mean-field approximation, basing, however, on an
accurate allowance for interspecies interaction in the spirit of the strong coupling approach. An analysis
of the MI-SF phase transition, based on the conditions of thermodynamic equilibrium, will be performed
(we do not restrict ourselves to the criterion of the normal (MI) phase stability in order to determine the
phase transition point). The study will be performed for the case of fixed chemical potentials of bosons
µB and fermions µF in the zero temperature limit (T = 0). Corresponding phase diagrams will be built,
taking into account the possibility of a change of the phase transition order from the second to the first
order. Our investigation will cover the case of a repulsive on-site interaction between hard-core bosons
and fermions (UBF > 0).
2. The four-state model
The Hamiltonian of the Fermi-Bose-Hubbard model is usually written in the form:
H =U
2
∑
i
nbi (nbi −1)+U ′∑
i
nbi nfi −µ
∑
i
nbi −µ′∑
i
nfi +
∑
〈i , j 〉
ti j b+
i b j +
∑
〈i , j 〉
t ′i j a+
i a j . (2.1)
Here,U andU ′
are constants of boson-boson and boson-fermion on-site interactions; µ and µ′
are chem-
ical potentials of bosons and fermions, respectively (we consider here the case of repulsive interactions
U > 0, U ′ > 0) and t , t ′ are tunneling amplitudes of bosons (fermions) describing the boson (fermion)
hopping between the nearest lattice sites.
Let us use, as in [54], the single-site basis of states
(nbi = n; nfi = 0) ≡ |n, i 〉, (nbi = n; nfi = 1) ≡ |ñ, i 〉, (2.2)
where nbi (nfi ) are occupation numbers of bosons (fermions) on the site i , and introduce the Hubbard op-
erators X n,m
i = |n, i 〉〈m, i |, X ñ,m̃
i = |ñ, i 〉〈m̃, i |, etc. Creation and destruction operators as well as operators
of occupation numbers will be expressed in terms of X -operators in the following way [14, 54]:
bi =
∑
n
p
n +1X n,n+1
i +∑
ñ
p
ñ +1X ñ,ñ+1
i ,
b+
i =∑
n
p
n +1X n+1,n
i +∑
ñ
p
ñ +1X ñ+1,ñ
i ,
ai =
∑
n
X n,ñ
i , a+
i =∑
n
X ñ,n
i ,
nbi =∑
n
nX n,n +∑
ñ
ñX ñ,ñ , nfi =
∑
ñ
X ñ,ñ , (2.3)
43702-3
I.V. Stasyuk, V.O. Krasnov
Then, the Hamiltonian in this new representation takes the form:
H = H0 +Hb
1 +H f
1, (2.4)
H0 =
∑
i ,n
λn X nn
i +∑
i ,ñ
λñ X ññ
i ,
λn = U
2
n(n −1)−nµ, λñ = U
2
ñ(ñ −1)−µñ −µ′+U ′ñ,
Hb
1 = ∑
〈i , j 〉
ti j b+
i b j , H f
1 =
∑
〈i , j 〉
t ′i j a+
i a j .
In the case of a hard-core boson approximation (U →∞), the single-site |nbi ,nfi 〉 basis consists of four
states:
|0〉 = |0,0〉, |0̃〉 = |0,1〉,
|1〉 = |1,0〉, |1̃〉 = |1,1〉. (2.5)
In this limit,
bi = X 01
i +X 0̃1̃, b+
i = X 10
i +X 1̃0̃,
ai = X 00̃
i +X 11̃, a+
i = X 0̃0
i +X 1̃1, (2.6)
nbi = X 11
i +X 1̃1̃
i , nfi = X 0̃0̃
i +X 1̃1̃
i ,
λ0 = 0, λ1 =−µ, λ0̃ =−µ′, λ1̃ =−µ−µ′+U ′
(2.7)
and in the expression (2.4) for Hamiltonian of the system, a restriction n = 0,1 and ñ = 0̃, 1̃ on occupation
numbers is imposed. As it was mentioned, we consider the case of the so-called heavy fermions, when the
inequalities t ′ ¿ t , t ′ ¿ U ′
are fulfilled. Our aim consists in the study of conditions at which the MI-SF
transition in such a model takes place, for the case when the fermion hopping between lattice sites can
be neglected. On this assumption, we shall start with Hamiltonian:
Ĥ =∑
i
(
λ0X 00
i +λ1X 11
i +λ0̃X 0̃0̃
i +λ1̃X 1̃1̃
i
)
+ ∑
〈i , j 〉
ti j b+
i b j . (2.8)
3. Mean field approximation
Let us introduce the order parameter of BE condensate ϕ = 〈bi 〉 = 〈b+
i 〉. In the case of mean field
approximation (MFA):
b+
i b j →ϕ(b+
i +bi )−ϕ2,∑
i j
ti j b+
i b j =ϕt0
∑
i
(b+
i +bi )−N t0ϕ
2, (3.1)
(here, t0 =∑
ti j =−|t0|, t0 < 0).
Then, for initial Hamiltonian after separating the mean field part we shall have:
H = HMF+
∑
i , j
ti j (b+
i −ϕ)(bi −ϕ). (3.2)
Here,
HMF =
∑
i
Hi −N t0ϕ
2, Hi =
∑
pr
Hpr X pr
i , (3.3)
43702-4
Phase transitions in BFH model
and
||Hpr || =
|0〉 |1〉 |0̃〉 |1̃〉
0 t0ϕ 0 0 |0〉
t0ϕ −µ 0 0 |1〉
0 0 −µ′ t0ϕ |0̃〉
0 0 t0ϕ −µ−µ′+U ′ |1̃〉
. (3.4)
Our next step is diagonalization:
Û T ∗ Ĥ ∗Û = ˆ̃H , (3.5)
where Û =
(
Û1 0̂
0̂ Û2
)
and Û1 =
(
cosψ −sinψ
sinψ cosψ
)
, Û2 =
(
cosψ̃ −sinψ̃
sinψ̃ cosψ̃
)
. Here,
sin2ψ= t0ϕ√
µ2/4+ t 2
0ϕ
2
, sin2ψ̃= t0ϕ√
(U ′−µ)2/4+ t 2
0ϕ
2
. (3.6)
Then we will get a diagonal single-site part (which is also a mean-field part) of the Hamiltonian:
Ĥi =
∑
p ′
εp ′ X p ′p ′
i −N t0ϕ
2, (3.7)
where p ′ = 0′,1′, 0̃′, 1̃′ are indices which denote the states of a new basis,
ε0′,1′ =−µ
2
±
√
µ2
4
+ t 2
0ϕ
2 ,
ε0̃′,1̃′ =−µ′− µ
2
+ U ′
2
±
√
(U ′−µ)2
4
+ t 2
0ϕ
2 . (3.8)
For Bose-operators, in a new basis we shall have:
bi =1
2
sin(2ψ)(X 0′0′
i −X 1′1′
i )+ 1
2
sin(2ψ̃)(X 0̃′0̃′
i −X 1̃′1̃′
i )+
+cos2ψX 0′1′
i − sin2ψX 1′0′
i +cos2 ψ̃X 0̃′1̃′
i − sin2 ψ̃X 1̃′0̃′
i .
4. Grand canonical potential and order parameter
The partition function in MFA is equal to:
ZMF = Spe−βHMF = eβN t0ϕ
2 ∏
i
Sp
{
exp
(
−β∑
pr
Hpr X pr
i
)}
= eβN t0ϕ
2 ∏
i
exp
(
−β∑
p ′
εp ′ X p ′p ′
i
)
= eβN t0ϕ
2
Z N
0 , (4.1)
where
Z0 = e−βε0′ +e−βε1′ +e−βε0̃′ +e−βε1̃′ . (4.2)
The grand canonical potential is:
ΩMF =−θ ln ZMF = N |t0|ϕ2 −NΘ ln Z0 (4.3)
43702-5
I.V. Stasyuk, V.O. Krasnov
or
ΩMF/N = |t0|ϕ2 −θ ln Z0 (4.4)
(here, we take into account that t0 =−|t0|). The equilibrium value of the order parameter ϕ can be found
from the global minimum condition of Ω.
We have an equation
∂(ΩMF/N )
∂ϕ
= 2|t0|ϕ− θ
Z0
∂Z0
∂ϕ
= 0 (4.5)
or
2|t0|ϕ+∑
p ′
〈X p ′p ′〉∂εp ′
∂ϕ
= 0. (4.6)
Here:
〈X p ′p ′〉 = 1
Z0
e−βεp′ (4.7)
Using that:
∂ε0′,1′
∂ϕ
=±t0 sin2ψ=∓|t0|sin2ψ,
∂ε0̃′,1̃′
∂ϕ
=±t0 sin2ψ̃=∓|t0|sin2ψ̃, (4.8)
from (4.6) we shall get:
ϕ= 1
2
sin2ψ
(
〈X 0′0′〉−〈X 1′1′〉
)
+ 1
2
sin2ψ̃
(
〈X 0̃′0̃′〉−〈X 1̃′1̃′〉
)
(4.9)
or in the explicit form,
ϕ= |t0|ϕ
2
〈X 1′1′〉−〈X 0′0′〉√
µ2
4 + t 2
0ϕ
2
+ 〈X 1̃′1̃′〉−〈X 0̃′0̃′〉√
(U ′−µ)2
4 + t 2
0ϕ
2
. (4.10)
This equation has trivial ϕ = 0 and non-trivial ϕ , 0 solutions, the second one can be obtained from
the equation:
1
|t0|
= 〈X 1′1′〉−〈X 0′0′〉
2
√
µ2
4 + t 2
0ϕ
2
+ 〈X 1̃′1̃′〉−〈X 0̃′0̃′〉
2
√
(U ′−µ)2
4 + t 2
0ϕ
2
. (4.11)
Whenwe have several solutions in this equation, we shall consider only those related to theminimum
of ΩMF.
Let us apply the unitary transformation Û T (. . .)Û to operators X 01
i and X 0̃1̃
i , which, in the matrix
form, are:
||X 01
i || =
0 1 0 0
0 0 0 0
0 0 0 0
0 0 0 0
, ||X 0̃1̃
i || =
0 0 0 0
0 0 0 0
0 0 0 1
0 0 0 0
. (4.12)
We shall get:
||Û T X 01
i Û || =
sinψcosψ cos2ψ 0 0
−sin2ψ −sinψcosψ 0 0
0 0 0 0
0 0 0 0
(4.13)
43702-6
Phase transitions in BFH model
and
||Û T X 0̃1̃
i Û || =
0 0 0 0
0 0 0 0
0 0 sinψ̃cosψ̃ cos2 ψ̃
0 0 −sin2 ψ̃ −sinψ̃cosψ̃
. (4.14)
In the transformed basis representation:
Û T X 01
i Û = cos2ψX 0′1′
i + sinψcosψ(X 0′0′
i −X 1′1′
i )− sin2ψX 1′0′
i ,
Û T X 0̃1̃
i Û = cos2 ψ̃X 0̃′1̃′ + sinψ̃cosψ̃(X 0̃′0̃′ −X 1̃′1̃′ )− sin2 ψ̃X 1̃′0̃′ . (4.15)
After averaging with the aid of HMF Hamiltonian:
〈X 01
i 〉 = 1
Z0
Sp
(
X 01
i e−βHMF
)
= 1
Z0
Sp
(
Û T X 01
i Û e−βHMF
)
. (4.16)
On the new basis, the Hamiltonian HMF is diagonal; that is why the averages of diagonal operators
X p ′p ′
i (p ′ = 0′,1′, 0̃′, 1̃′) will only be non-zero. Thus,
〈X 01
i 〉 = 1
2
sin2ψ
(
〈X 0′0′〉−〈X 1′1′〉
)
,
〈X 0̃1̃
i 〉 = 1
2
sin2ψ̃
(
〈X 0̃′0̃′〉−〈X 1̃′1̃′〉
)
. (4.17)
As a result, we have:
ϕ= 〈bi 〉 = 〈X 01
i 〉+〈X 0̃1̃
i 〉 = 1
2
sin2ψ
(
〈X 0′0′〉−〈X 1′1′〉
)
+ 1
2
sin2ψ̃
(
〈X 0̃′0̃′〉−〈X 1̃′1̃′〉
)
. (4.18)
Thus, we arrived at the same equation for ϕ as we have got from the extremum condition for grand
canonical potential.
5. Spinodals at T = 0
If in the equation (4.11) we substitute ϕ = 0, we shall have a condition for the second order phase
transition to SF phase (if this transition is possible). In general, it is the condition of instability of normal
(MI) phase with respect to the Bose-Einstein condensate appearance (in figures it corresponds to spinodal
lines).
The equation (4.11) can be rewritten as:
1
|t0|
= 〈X 1′1′〉−〈X 0′0′〉
ε0′ −ε1′
+ 〈X 1̃′1̃′〉−〈X 0̃′0̃′〉
ε0̃′ −ε1̃′
. (5.1)
If ϕ→ 0,
ε0′ =
{
λ0 , µ> 0,
λ1 , µ< 0;
ε0̃ =
{
λ1̃ , µ<U ′,
λ0̃ , µ>U ′;
(5.2)
ε1′ =
{
λ1 , µ> 0,
λ0 , µ< 0;
ε1̃ =
{
λ0̃ , µ<U ′,
λ1̃ , µ>U ′.
It is seen that we can divide the µ axis into three regions (1) µ < 0; (2) 0 < µ < U ′
; (3) µ > U ′
(when
U ′ > 0). For all these regions, the equation (5.1) takes the form:
1
|t0|
= 〈X 00〉−〈X 11〉
λ1 −λ0
+ 〈X 0̃0̃〉−〈X 1̃1̃〉
λ1̃ −λ0̃
. (5.3)
43702-7
I.V. Stasyuk, V.O. Krasnov
It can be rewritten as:
1
|t0|
= 〈X 11〉−〈X 00〉
µ
+ 〈X 0̃0̃〉−〈X 1̃1̃〉
U ′−µ . (5.4)
The equation (5.3) is the same as the one obtained in [50] from the condition of divergence of the
bosonic Green’s function (calculated in the random phase approximation) at ω = 0,q = 0. In this way, it
is the condition of instability of the phase with ϕ = 0. Therefore, the equation (5.3) is an equation for
spinodal line.
When T = 0, averages 〈X p ′p ′〉, 〈X p̃ ′ p̃ ′〉 are different from zero only for the lowest energy level. Here,
the three cases can be separated out.
1. µ′ < 0.
Here, at µ< 0, the ground state is |0〉, and at µ > 0 the ground state is |1〉. Respectively, in the first
one of these cases 〈X 00〉 = 1, while in the second one 〈X 11〉 = 1 (other averages are equal to zero).
The equation (5.4) can be written now as:
1
|t0|
=
{ − 1
µ , µ< 0,
1
µ , µ> 0.
(5.5)
It follows from here that:
µ=
{
t0, µ> 0,
−t0, µ< 0.
(5.6)
This is the spinodal equation for µ′ < 0
2. µ′ >U ′
.
The change of ground state takes place when µ=U ′
. For µ<U ′
, the state |0̃〉 is the ground one, and
for µ>U ′
it is the state |1̃〉. Respectively, at µ<U ′
, 〈X 0̃0̃〉 = 1 and at µ>U ′
, 〈X 1̃1̃〉 = 1. The equation
(5.4) takes now the form:
1
|t0|
=
{ 1
U ′−µ , µ<U ′,
1
µ−U ′ , µ>U ′.
(5.7)
In this case, the following lines will be the lines of spinodal:{
µ=U ′+ t0, µ>U ′,
µ=U ′− t0, µ<U ′.
(5.8)
3. 0 <µ′ <U ′
.
The change of ground state takes place now when µ = µ′
. For µ < µ′
, such a state is |0̃〉, and for
µ> µ′
it is the state |1〉. Respectively, at µ< µ′
, 〈X 0̃0̃〉 = 1 and for µ> µ′
, 〈X 11〉 = 1. For spinodal we
shall now have the equation:
1
|t0|
= 〈X 11〉
µ
+ 〈X 0̃0̃〉
U ′−µ . (5.9)
It follows from here that, µ=U ′−|t0| when µ< µ′
, and µ= |t0| when µ> µ′
. In the case |t0| <U ′
,
the solution µ= µ′
also appears. When µ′ >U ′/2, it exists forU ′−µ′ < |t0| < µ′
, and at µ′ <U ′/2 it
exists for µ′ < |t0| <U ′−µ′
. If |t0| ÊU ′
, the solution µ=µ′
disappears.
The lines of spinodales are broken andmuch differ for the cases |t0| ÊU ′
,U ′ > |t0| >U ′/2, |t0| <U ′/2.
As a result, the areas of absolute instability of the normal phase (calculated at T = 0) possess different
shapes. These areas are shown below [see (µ,µ′) diagrams in section 8].
43702-8
Phase transitions in BFH model
6. Phase transition of the first order to SF phase
Let us analyze now the dependences of the order parameter ϕ and grand canonical potentialΩ upon
the chemical potential of bosons at different values of temperature and µ′
using the equation (4.11). In the
limit T → 0, only the averages related to the ground state contribute to the right-hand side of the equation.
With the change of ϕ, the ground state can be reconstructed and this makes the problem of determining
the solutions for order parameter self-consistent. This problem can have a simple analytical solution,
when the states with fermion and without fermion are not mutually competitive. This is achieved when
µ′
takes the values which are off the [0,U ′] interval.
When we have the negative µ′
values, the state |1′〉 is the ground one and the equation (4.11) reduces
in this case:
1
|t0|
= 1
2
√
µ2
4 + t 2
0ϕ
2
. (6.1)
Then:
ϕ= 1
2
√
1−µ2/t 2
0 . (6.2)
In a positive region, when µ′ >U ′
, the state |1̃′〉 is the ground one; respectively, we have an equation:
1
|t0|
= 1
2
√
(U ′−µ)2
4 + t 2
0ϕ
2
. (6.3)
In this case:
ϕ= 1
2
√
1− (µ−U ′)2/t 2
0 . (6.4)
The dependences of functions (6.2) and (6.4) on µ are presented in figure 1.
In the region of intermediate values of µ′
(at 0 É µ′ ÉU ′
), the competition of “tilded” and “untilded”
states leads to deformation of the curve ϕ(µ). In figures 2 (a)–6 (a) one can see the plots of the order
parameter ϕ as function of chemical potential of bosons µ for different values of chemical potential of
fermions µ′
. These curves are obtained numerically from the equation (4.11) in the case T = 0.
One can see that in intervals 0 < µ′ < |t0| and U ′− |t0| < µ′ < U ′
(at |t0| < U ′/2) as well as in almost
the whole interval 0 <µ′ <U ′
(at |t0| >U ′/2) of µ′
values, the ϕ(µ) dependence has a reverse course and
S-like behaviour. This is an evidence of the possibility of the first order phase transition (rather than the
second order transition).
This conclusion can be confirmed by calculation of grand canonical potentialΩMF(µ) as function of µ.
0
0,0
0,5
|t0|-|t0|
(a)
0,0
0,5
U'-|t0| U'+|t0|U'
(b)
Figure 1. Order parameter ϕ as function of µ for µ′ < 0 (a) and for µ′ >U ′
(b).
43702-9
I.V. Stasyuk, V.O. Krasnov
'
-|t0|
'0-|t0| |t0|
N
(b)
Figure 2. Order parameter (a) and grand canonical potential (b) as functions of µ in the case |t0| <
U ′/2; 0 < µ′ < |t0|. Here, and in figures 3–6, the lines (α), (β) and (γ) are described by formulae (6.2),
(6.4) and (6.9), respectively.
-U'-|t0|
U' U'+|t0|U'-|t0|
-U'
- '
/N
(b)
Figure 3. Order parameter (a) and grand canonical potential (b) as functions of µ in the case |t0| <
U ′/2; U ′−|t0| <µ′ <U ′
.
In the case of negative values of chemical potential µ′
, when at T = 0, only the state |1′〉 remains:
ΩMF/N →|t0|ϕ2 −θ lne−βε1′ = |t0|ϕ2 − µ
2
−
√
µ2
4
+ t 2
0ϕ
2 . (6.5)
Using expression (6.2) we have:
ΩMF/N =− (µ+|t0|)2
4|t0|
. (6.6)
At the same time, at ϕ= 0, the ground state in the region µ′ < 0 is the state |0〉 for µ< 0 and the state
|1〉 for µ> 0. Then:
ΩMF/N |ϕ=0 =
{
0, µ< 0,
−µ, µ> 0.
(6.7)
One can see that ΩMF <ΩMF
∣∣
ϕ=0; the SF phase is more stable in the interval −|t0| < µ < |t0|. Deriva-
tives ∂ΩMF/∂µ and ∂ΩMF|ϕ=0/∂µ coincide in the limiting points µ=±|t0|. This verifies the second order
of phase transitions to the phase with BE condensate here.
The function ΩMF/N (µ) has a similar character in the case µ′ >U ′
. Here:
ΩMF/N =− (µ−U ′+|t0|)2
4|t0|
−µ′,
ΩMF/N
∣∣
ϕ=0 =
{ −µ′, µ<U ′
,
U ′−µ−µ′, µ>U ′
.
(6.8)
43702-10
Phase transitions in BFH model
0
'U'-|t0|
( )
( )
( )
U'/2
(a)
Figure 4. Order parameter (a) and grand canonical potential (b) as functions of µ in the case |t0| >
U ′/2; U ′−|t0| <µ′ <U ′/2.
|t0|U'-|t0| U'/2
( ) ( )
(a)
/N
0
- '
-U'
U'-|t0| U'/2 |t0|
(b)
Figure 5. Order parameter (a) and grand canonical potential (b) as functions of µ in the case when |t0| >
U ′/2; µ′ =U ′/2.
Here, the second order phase transitions take place in the points µ=U ′±|t0|.
The results of numerical calculations of function ΩMF/N (µ) in the case of intermediate values of µ′
[performed with the help of the earlier calculated ϕ(µ) dependences] are shown in figures 2 (b)–6 (b).
Here and hereafter, numerical values of parameters are given in theU ′
units.
The special character of the dependences of the order parameter ϕ on µ is quite understandable
taking into account the change of the ground state which occurs in the above mentioned intermediate
region of µ′
when ε1′ = ε1̃′ . The equation that follows from this condition has a solution:
ϕ=
√
µ′(U ′−µ′)(µ−µ′)(µ′−U ′+µ)
|t0||2µ′−U ′| , (6.9)
which in the plane (µ,ϕ) describes the line that separates the areas with different ground states: the state
|1̃′〉, (|1′〉) to the left (right) of the curve (at the given value of µ′
).
At µ′ =U ′/2, the line (6.9) is vertical and passes through the point µ=U ′/2. At µ′ <U ′/2, it is bent to
the left and lies on the x-axis when µ′ → 0; respectively, at µ′ >U ′/2 the bend is opposite, and curve lies
on the x-axis at µ′ →U ′
.
In the region where the ground state is |1′〉, the dependence ϕ(µ) is given by formula (6.2) (see fig-
ure 6), and when the ground state is |1̃′〉, it is described by the formula (6.4) (see figure 6). Depending on
the position of the line (6.9), certain fragments of graphs (6.2) and (6.4) remain on its both sides. This is
illustrated in figures 4 (a), 5 (a) and 6 (a).
In the cases when the function ϕ(µ) has a reverse course, one can see the so-called “fishtails” in the
behaviour of the grand canonical potential where the intersection points of the lowest curves correspond
to the first order phase transitions (phase transitions on the other side or on both sides of the interval
43702-11
I.V. Stasyuk, V.O. Krasnov
Figure 6. Order parameter ϕ and grand canonical potential Ω/N as function of µ in the case |t0| >
U ′/2; U ′/2 <µ′ < |t0|.
of non-zero ϕ values are of the second order). The values of µ at which the first order phase transitions
exist, are shifted relatively to the spinodal line points (at µ′ <U ′/2 to the left and at µ′ >U ′/2 to the right).
As a result, the region of SF phase existence at T = 0 is wider than the region limited by spinodals.
7. Phase diagrams at T = 0
A more detailed analysis shows that at |t0| <U ′/2 in the 0 < µ′ < |t0| region, the phase transition of
the first order to the SF phase is determined by the intersection of the branches of the grand canonical
potential
ΩMF
N
∣∣∣|0̃〉 =−µ′
(7.1)
and
ΩMF
N
∣∣∣|1′〉 =− (µ+|t0|)2
4|t0|
(7.2)
and is associated with the change (|0̃〉→ |1′〉) of the ground state of boson-fermion system. Consequently,
the relation
µ′ = (µ+|t0|)2
4|t0|
, (7.3)
arises which is an equation of the phase transition line on the plane (µ,µ′). In figure 7, where the regions
of the existence of different ground states are shown in (µ,µ′) diagram, this curve is depicted by heavy
solid line. SF phase exists here in the regions marked as |1′〉 and |1̃′〉; to distinguish between these two
cases, we shall use, respectively, the notations SF
|1′〉
and SF
|1̃′〉
.
In the case when U ′/2 < |t0| < U ′
, the curve (7.3) proceeds to describe the transition of the 1st or-
der until 0 < µ′ < (U ′)2/4|t0|. At the same time, it remains to the left of spinodal line (figure 8). When
(U ′)2/4|t0| < µ′ <U ′− (U ′)2/4|t0|, the line of the first order transitions is placed between spinodals. The
latter are described by equations µ = U ′− |t0| and µ = |t0|; at such values of µ, the second order phase
transitions to the SF phase occur. For U ′ − |t0| < µ < µ′
, this is a phase SF
|1̃′〉
and for µ′ < µ < |t0| — a
phase SF
|1′〉
(superscripts indicate the states of the system at T = 0). The transition between these two
phases, which is of the first order, is defined by the equality of grand canonical potentialsΩMF/N
∣∣|1′〉 and
ΩMF/N
∣∣|1̃′〉, here,
ΩMF
N
∣∣∣|1̃′〉 =− (µ−U ′+|t0|)2
4|t0|
−µ′. (7.4)
The line that describes this transition on the plane (µ,µ′) is:
µ′ = U ′
4|t0|
(2µ+2|t0|−U ′). (7.5)
43702-12
Phase transitions in BFH model
~
~
|0>
|1'> |1>
|1'>
|1>
U'-|t
0
|
|t
0
|
U'+|t
0
|U'-|t
0
|-|t
0
|
'
|t
0
| U'
U'
U'/2
|0>
~
1
2
Figure 7. (µ,µ′) diagram at |t0| <U ′/21.
~|0>
|0> |1'>
~
|1'>
~
|1>
|1>
-U'/2
U'
U'/2
U'+|t
0
|U'
'
U'/2
1
2
Figure 8. (µ,µ′) diagram at |t0| =U ′/2.
In the regionU ′− (U ′)2/4|t0| < µ′ <U ′
, the line of the first order phase transition continues; it separates
the phase SF
|1̃′〉
from the normal phase (for the latter, the ground state is |1〉). In this case, the equation of
the phase equilibrium curve follows from the condition:
ΩMF
N
∣∣∣
1̃′〉 =
ΩMF
N
∣∣∣|1〉 , (7.6)
where ΩMF/N
∣∣|1〉 =−µ and has the form
µ′ =µ− (µ−U ′+|t0|)2
4|t0|
. (7.7)
On the (µ,µ′) plane, this line is positioned to the right of the spinodal line.
Equations (7.3), (7.5) and (7.7) define the lines of phase transitions of the 1st order at |t0| > U ′
as
well. A full set of phase diagram (µ,µ′) types is supplemented by diagrams shown in figures 9 and 10. It
1
Here, as well as in figures 8–13, the following notations are used: heavy solid line for the 1st order PT; dashed line for the 2nd
order PT; dotted line for the spinodal; fine solid line for the border between different ground states which belongs to normal phase.
43702-13
I.V. Stasyuk, V.O. Krasnov
~
|0>
|0>
~
~
|1'>
|1>
|1'>
|1>
-|t
0
|
U'
U'-|t
0
|
|t
0
|
U'+|t
0
|U'U'-|t
0
| |t
0
|
'
1
2
3
Figure 9. (µ,µ′) diagram atU ′/2 < |t0| <U ′
.
~
~
|0>
|0>
|1'>
|1'>
|1>
|1>
-|t
0
| U'-|t
0
|
U'
U' |t
0
| U'+|t
0
|
'
~
Figure 10. (µ,µ′) diagram at |t0| >U ′
.
should be pointed out that the continuous line of PT of 1st order in the whole region 0 <µ′ <U ′
separates
the states (phases) with one fermion (“tilded” ones) and the states without fermions (“untilded” ones).
The distinction between the superfluid phases SF
|1̃′〉
and SF
|1′〉
is related to this fact. In the first case, BE
condensate exists at the full fermion filling
(
n̄ f = 1
)
, and in the second one, at the absence of fermions
(n̄ f = 0).
At nonzero temperatures, the dependence of n̄ f on µ
′
at crossing the line of PT of 1st order is smooth.
The concentration of fermions n̄ f gradually reduces in “tilded” areas and correspondingly increases in a
similar way in “untilded” areas. The principal difference between the phases of SF
|1̃′〉
and SF
|1′〉
vanishes.
The above described effect of the phase transition order change also takes place when chemical po-
tential µ′
is positioned in the middle of the [0,U ′] interval (see figure 9). The point µ′ =U ′/2 is a special
one. With a decrease of µ′
, the fragmentation of SF region into two parts takes place at this point.
Having the phase diagrams (µ,µ′) built for various values of |t0|, one can pass to diagrams on the
plane (µ, |t0|). Using formulae (7.3), (7.5) and (7.7) it is easy to get the relations between µ and |t0| (for
43702-14
Phase transitions in BFH model
Figure 11. (µ, |t0|) diagram at 0 <µ′ <U ′/2. Figure 12. (µ, |t0|) diagram at µ′ =U ′/2.
Figure 13. (µ, |t0|) diagram atU ′/2 <µ′ <U ′
.
fixed values of µ′
) on the lines of phase transitions of the first order:
µ=
√
4|t0|µ′−|t0|, (7.8)
— when µ′ <U ′/2;
µ=U ′+|t0|−
√
4|t0|(U ′−µ′), (7.9)
— when µ′ >U ′/2; and
µ= U ′
2
−|t0|+ 2|t0|µ′
U ′ (7.10)
— in both these cases when the phase transition occurs within SF-phase region.
Using formulae (7.8), (7.9) and (7.10), and the equation for spinodals at T = 0, we can get the diagrams
shown in figures 11, 12 and 13. In contrast to the (µ, |t0|) diagram for pure bosonic case, figure 12, known
for a hard-core boson system (see, for example, [44, 46]), these diagrams are asymmetric. At the smaller
(or larger) values of µ, the phase transition (under the effect of fermions) changes its order from the
second to the first one, depending on the value of the chemical potential µ′
.
There also appears the minimum value for bosons transfer parameter (|t0|min = µ′
at µ′ < U ′/2 or
|t0|min =U ′−µ′
at µ′ >U ′/2). SF phase can exist only if |t0| > |t0|min.
43702-15
I.V. Stasyuk, V.O. Krasnov
8. Phase separation at a fixed boson chemical potential
Besides a jump of a number of fermions, there is also a jump-like behaviour of boson concentration
on the line of phase transition of the 1st order, n̄B =−∂ (ΩMF/N )/∂µ. At zero temperature
n̄B
∣∣|0̃〉 = 0, n̄B
∣∣|1〉 = 1,
n̄B
∣∣|1̃〉 = 1, n̄B
∣∣|0〉 = 0,
n̄B
∣∣|1̃′〉 = µ−U ′+|t0|
2|t0|
, n̄B
∣∣|1′〉 = µ+|t0|
2|t0|
. (8.1)
Using these relations, the limiting values of n̄B (from the left or right sides of the transition lines), can be
written for a given µ′
. Using the formulae (7.3), (7.5) and (7.7), we find:
1) on the line (7.3)
n̄B
∣∣
l = 0, n̄B
∣∣
r =
√
µ′/|t0|; (8.2)
2) on the line (7.5)
n̄B
∣∣
l =
µ′
U ′ −
U ′
4|t0|
, n̄B
∣∣
r =
µ′
U ′ +
U ′
4|t0|
; (8.3)
3) on the line (7.7)
n̄B
∣∣
l = 1−
√
U ′−µ′
|t0|
, n̄B
∣∣
r = 1. (8.4)
The dependences n̄B
∣∣
l ,r on µ
′
are presented graphically in figures 14 and 15; instances |t0| < U ′/2 and
U ′/2 < |t0| < U ′
are shown separately. In thermodynamical regime of given values of chemical poten-
tials µ and µ′
, these plots illustrate the jump of the concentration of bosons in different points of the
phase equilibrium curve. On the other hand, in the regime of fixed n̄B and µ′
, they can be interpreted as
diagrams describing the phase separation on the regions with different concentrations n̄B.
The values n̄B
∣∣
l and n̄B
∣∣
r correspond in this case to different phases (states), into which the system
segregates, depending on the value of chemical potential µ′
. This is shown in diagrams 14 and 15 (the
separation regions are enclosed by solid lines); the phases into which the separation takes place are also
shown.
In the caseU ′/2 < |t0| <U ′
(also at |t0| >U ′
), the separation occurs throughout the whole interval of
0 <µ′ <U ′
. When |t0| <U ′/2, the system decomposes into separate phases only at the µ′
values in regions
µ′ < |t0| andU ′−|t0| <µ′ <U ′
. There is no separation in the central region |t0| <µ′ <U ′−|t0| in this case;
Figure 14. (µ′, n̄B) diagram at |t0| <U ′/2.
43702-16
Phase transitions in BFH model
Figure 15. (µ′, n̄B) diagram atU ′/2 < |t0| <U ′
.
here, at a given fractional value of n̄B, the system exists in a mixed state, where the lattice sites are in the
state |1̃〉 (boson and fermion on site) or |0〉 (unfilled site) with probabilities determined by their weights
(n̄B or 1− n̄B, respectively).
9. Conclusions
To describe phase transitions in the boson-fermion mixtures in optical lattices we used the Bose-
Fermi-Hubbard model in the mean-field and hard-core boson approximations, for the case of infinitely
small fermion transfer and repulsive on-site boson-fermion interaction. Our aim was to study the con-
ditions, at which the MI-SF transition in such a model occurs, in the case when the fermion hopping be-
tween lattice sites can be neglected. The approach used in this work does not use the traditional scheme
of mean-field approximation based on the decoupling of the on-site interactionU ′nbi nfi . Instead, the for-
malism of Hubbard operator acting on the |nbi ,nfi 〉 basis of states is employed; this makes it possible to
exactly take into account the boson-fermion interaction U ′
(the case of repulsive interaction (U ′ > 0) is
considered in this work). The single-site problem is formulated with only one self-consistency parameter
ϕ (ϕ= 〈bi 〉 = 〈b+
i 〉), and the mean-field approach is used to describe the BE condensation.
On-site boson interactionU is treated as repulsive and infinitely large (U > 0,U →∞); this imposes a
restriction on the occupation numbers of bosons (nbi = 0 or 1). Nevertheless, this approximation makes it
possible, as is known for the pure Bose-Hubbard model, to describe the MI-SF transition in close vicinity
of the µ = nU points (where n are integer numbers) in the case of finite values of U . The investiga-
tion is performed in a thermodynamical regime of fixed values of chemical potentials of bosons (µ) and
fermions (µ′
).
The equilibrium values of the order parameter ϕ (related to the appearance of SF phase) were found
from the global minimum condition of grand canonical potential Ω and, in parallel, the averages of cre-
ation and destruction operators of bosons 〈b〉 and 〈b+〉 were calculated. From the obtained equation,
using substitution ϕ→ 0, we get the condition of 2nd order phase transition to SF phase (if this tran-
sition is possible). In general, it is the condition of instability of normal (MI) phase with respect to the
Bose-Einstein condensate appearance. This equation is the same as as the one obtained earlier from the
condition of divergence of the bosonic Green’s function (calculated in the random phase approximation)
at ω= 0, q = 0 (see [54]).
The spinodal lines are calculated at T = 0, and corresponding phase diagrams on the (µ,µ′
) planes
are built. Analyzing their shape, different cases are separated depending on the value of the chemical po-
tential of fermions µ′
, and thus the corresponding phase diagrams are built. Moreover, the dependences
43702-17
I.V. Stasyuk, V.O. Krasnov
of the order parameter ϕ and the grand canonical potential Ω on µ (at different chemical potential µ′
values) are derived.
Considering the order parameter dependence upon chemical potential of bosons (µ) we found that
in the region of intermediate values of µ′
(especially at 0 É µ′ É U ′
) the competition between “tilded”
and “untilded” states leads to a deformation of the curve ϕ(µ). These cases are distinguished when such
a dependence has a reverse course and S-like behaviour. This is an evidence of the possibility of the
first order phase transition between MI and SF phases (rather than the second order transition). This
conclusionwas confirmed by calculation of grand canonical potentialΩMF(µ) as function of µ. As a result,
the region of the existence of SF phase at T = 0 is wider than the limited region by spinodals. The above
described effect of the phase transition order change disappears when chemical potential µ′
takes the
values µ′ < 0 and µ′ > U ′
. In the first case, there are no fermions (nF = 0) while in the second case the
fermion states are fully occupied (nF = 1). At nF = 0, the model is reduced to the pure hard-core boson
model with the phase transitions of the 2nd order; at nF = 1, the picture of a MI-SF transition is the same
but the chemical potential of bosons is shifted (µ→µ+U ′).
The [0,U ′] interval of the µ′
values corresponds to the fractional (0 < nF < 1) fermion concentration.
BE condensation taking place in this case is influenced by the states which have different number of
fermions. The point µ′ =U ′/2 is a special one. With a decrease of µ′
, the fragmentation of SF region into
two parts takes place at this point.
The obtained results are illustrated with the help of (µ,µ′) and (µ, |t0|) phase diagrams. It should be
emphasized that in the case |t0| > U ′/2, when (U ′)2/(4|t0|) < µ′ < U ′− (U ′)2/(4|t0|), the 1st order phase
transition line is placed inside the SF region. It divides this region into parts with BE condensate of differ-
ent type (SF
|1̃′〉
region when all fermion states are filled, and SF
|1′〉
region when there are no fermions).
The asymmetry of (µ, |t0|) phase diagrams (contrary to the pure hard-core boson system) is another in-
teresting feature. There also exists a minimum value of the transfer parameter |t0| necessary for the SF
phase existence.
We have also briefly considered the regime of a fixed boson concentration. In this case, the phase
separation effect is possible. The phase diagrams (µ′,nB) are obtained where the separation regions are
shown. Depending on µ′
value, the system can segregate into the regions with different nB concentrations
and different phases (MI and SF or SF
|1̃′〉
and SF
|1′〉
).
It is not easy to directly compare the obtained phase diagrams with the available data concerning
the thermodynamics of the BFH model. In most cases, the investigations were performed for another
thermodynamical regime, namely the regime of fixed fermion concentration (besides the given chemical
potential of bosons). Starting with our scheme, the transition to the regime of the fixed nF values could
be made with the help of Legendre transformationΩ/N → Ω̃/N =Ω/N +µ′nF and subsequent transition
to the new thermodynamical variables. Nevertheless, one can see that the states with a certain fractional
value of nF are positioned on the (µ,µ′) diagram at T = 0 on the curve that separates the “tilded” and
“untilded” regions (the jump-like change from nF = 1 to nF = 0 value takes place at the crossing over
this line). Moving along the line (at the change of µ) we shall pass through the intervals of µ values
corresponding to the regions of the SF phase existence. When T = 0 and nF > 1/2 (nF < 1/2), the above
mentioned curve will be placed a little higher (lower) of the zero-temperature curve. The symmetry of the
diagram will be broken, and this leads to a situation when at a decrease of |t0|, the SF phase disappears
in the region near the µ= 0 point sooner (or later) than near the µ=U ′
point. Although this conclusion
is qualitative, it can be considered as a confirmation and explanation of the results obtained in [54] for
the full BFHM in the tF = 0 case at finite temperatures in the regime nF = const. We should stress that in
[54], as in a lot of works in this field (see for example [26, 34], the phase diagrams were built basing on
the conditions of the SF phase instability (the latter is determined by spinodals).
The fact of the minimum |t0| value existence necessary for the appearance of the SF phase at the
presence of fermions (which is seen on (µ, |t0|) diagrams, figures 11–13) is a consequence of an exact
treatment of the boson-fermion interaction U ′
in our approach. However, when such an interaction is
taken into account basing on a simple linearization scheme (in the spirit of Hartree-Fock mean-field de-
coupling), the minimum value |t0|min is equal at T = 0 to zero [26, 31, 34, 35]. This value is reached at
certain µ = µ∗
and the effect of fermions consists in the shift of the µ∗
point [34]. We have a similar
effect in our case: the position of non-zero |t0|min as function of µ depends on µ′
(see figures 11, 13). In
43702-18
Phase transitions in BFH model
this regard, the investigations of changes that could appear at finite temperatures and the comparison
of different approaches turn out to be an interesting task (at T , 0 |t0|min is non-zero even in a simple
mean-field approximation). For a heavy-fermion limit, as was shown in [35], the SF phase region becomes
broader near |t0|min point and the |t0|min value increases at the temperature growth.
It is also interesting to study, in the fixed chemical potentials (µ and µ′
) regime, the phase transition
picture at the finite fermion hopping. Such an investigation was done in many works for a given fermion
concentration; a rather full analysis was performed in [25] using a simple version of the mean-field ap-
proach forUbf interaction. The possibility of the band fermion pairing leads at certain conditions to the
appearance of phases with the pair fermion condensate. It is necessary, however, to exceed some critical
value of the hopping parameter tF for their thermodynamical preference [25, 37].
Summarizing, we can make the following main conclusion. The phase transition to the SF-phase in a
mixed system (consisting of bosons and heavy fermions), described by the hard-core model and consid-
ered in the µ= const and µ′ = const regime, becomes of the 1st order in such a region of the chemical po-
tential values where the BE condensation is influenced by competition between filled and empty fermion
states. As a consequence, the phase separation at fixed concentrations of particles can appear.
References
1. Greiner M., Mandel O., Esslinger T., Hänsch T.H., Bloch I., Nature, 2002, 415, 39; doi:10.1038/415039a.
2. Greiner M., Regal C.A., Jin D.S., Nature, 2003, 426, 537; doi:10.1038/nature02199.
3. Fisher M.P.A., Weichman P.B., Grinstein G., Fisher D.S.O., Phys. Rev. B, 1989, 40, 546;
doi:10.1103/PhysRevB.40.546.
4. Jaksch D., Bruder C., Cirac J.I., Gardiner C.W., Zoller P., Phys. Rev. Lett., 1998, 81, 3108;
doi:10.1103/PhysRevLett.81.3108.
5. Van Oosten D., van der Straten P., Stoof H.T.C., Phys. Rev. A, 2001, 63, 053601; doi:10.1103/PhysRevA.63.053601.
6. Elstner N., Monien H., Phys. Rev. B, 1999, 59, 012184; doi:10.1103/PhysRevB.59.12184.
7. Batrouni G.G., Scalettar R.T., Zimanyi G.T., Phys. Rev. Lett., 1990, 65, 1765; doi:10.1103/PhysRevLett.65.1765.
8. Clark S.R., Jaksch J., New J. Phys., 2006, 8, 160; doi:10.1088/1367-2630/8/8/160.
9. Dirkerscheid D.B.M., van Oosten D., Denteneer P.J.H., Stoof H.T.C., Phys. Rev. A, 2006, 68, 043623;
doi:10.1103/PhysRevA.68.043623.
10. Sengupta K., Dupuis N., Phys. Rev. A, 2005, 71, 033629; doi:10.1103/PhysRevA.71.033629.
11. Ohashi Y., Kitaura M., Matsumoto H., Phys. Rev. A, 2006, 73, 033617; doi:10.1103/PhysRevA.73.033617.
12. Huber S.D., Altman E., Büchler H.P., Blatter G., Phys. Rev. B, 2007, 75, 085106; doi:10.1103/PhysRevB.75.085106.
13. Konabe S., Nikuni T., Nakamura M., Phys. Rev. A, 2006, 73, 033621; doi:10.1103/PhysRevA.73.033621.
14. Stasyuk I.V., Mysakovych T.S., Condens. Matter Phys., 2009, 12, 539; doi:10.5488/CMP.12.4.539.
15. Bloch I., Nat. Phys., 2005, 1, 23; doi:10.1038/nphys138.
16. Bloch I., Dalibard J., Zwerger W., Rev. Mod. Phys., 2008, 80, 885; doi:10.1103/RevModPhys.80.885.
17. Lewensteine M., Sanpera A., Ahufinger V., Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body
Systems, Oxford University Press, Oxford, 2012.
18. Best T., Will S., Schneider U., Hackenmüller L., van Oosten D., Bloch I., Lühmann D.-S., Phys. Rev. Lett., 2009, 102,
030408; doi:10.1103/PhysRevLett.102.030408.
19. Gunter K., Stoferle T., Moritz H., Kohl M., Essling T., Phys. Rev. Lett., 2006, 96, 180402;
doi:10.1103/PhysRevLett.96.180402.
20. Ospelkans C., Ospelkans S., Humbert L., Ernst P., Sengst K., Bongs K., Phys. Rev. Lett., 2006, 97, 120402;
doi:10.1103/PhysRevLett.97.120402.
21. Lühmann D.-S., Bongs K., Sengstock K., Pfannkuche D., Phys. Rev. Lett., 2008, 101, 050402;
doi:10.1103/PhysRevLett.101.050402.
22. Tewari S., Lutchyn R.M., Das Sarma S., Phys. Rev. B, 2009, 80, 054511; doi:10.1103/PhysRevB.80.054511.
23. Mering A., Fleischhauer M., Phys. Rev. A, 2008, 77, 023601; doi:10.1103/PhysRevA.77.023601.
24. Jürgensen O., Sengstock K., Lühmann D.-S., Phys. Rev. A, 2012, 86, 043623; doi:10.1103/PhysRevA.86.043623.
25. Bukov M., Pollet L., Phys. Rev. B, 2014, 89, 094502; doi:10.1103/PhysRevB.89.094502.
26. Fehrmann H., Baranov M., Lewenstein M., Santos L., Opt. Express, 2004, 12, 55; doi:10.1364/OPEX.12.000055.
27. Cramer M., Eisert J., Illuminati F., Phys. Rev. Lett., 2004, 93, 190405; doi:10.1103/PhysRevLett.93.190405.
28. Feshbach H., Ann. Phys., 1958, 5, 337; doi:10.1016/0003-4916(58)90007-1.
29. Inouye S., Andrews M.R., Stenger J., Miesner H.-J., Stamper-Kurn D.M., Ketterle W., Nature, 1998, 392, 151;
doi:10.1038/32354.
43702-19
http://dx.doi.org/10.1038/415039a
http://dx.doi.org/10.1038/nature02199
http://dx.doi.org/10.1103/PhysRevB.40.546
http://dx.doi.org/10.1103/PhysRevLett.81.3108
http://dx.doi.org/10.1103/PhysRevA.63.053601
http://dx.doi.org/10.1103/PhysRevB.59.12184
http://dx.doi.org/10.1103/PhysRevLett.65.1765
http://dx.doi.org/10.1088/1367-2630/8/8/160
http://dx.doi.org/10.1103/PhysRevA.68.043623
http://dx.doi.org/10.1103/PhysRevA.71.033629
http://dx.doi.org/10.1103/PhysRevA.73.033617
http://dx.doi.org/10.1103/PhysRevB.75.085106
http://dx.doi.org/10.1103/PhysRevA.73.033621
http://dx.doi.org/10.5488/CMP.12.4.539
http://dx.doi.org/10.1038/nphys138
http://dx.doi.org/10.1103/RevModPhys.80.885
http://dx.doi.org/10.1103/PhysRevLett.102.030408
http://dx.doi.org/10.1103/PhysRevLett.96.180402
http://dx.doi.org/10.1103/PhysRevLett.97.120402
http://dx.doi.org/10.1103/PhysRevLett.101.050402
http://dx.doi.org/10.1103/PhysRevB.80.054511
http://dx.doi.org/10.1103/PhysRevA.77.023601
http://dx.doi.org/10.1103/PhysRevA.86.043623
http://dx.doi.org/10.1103/PhysRevB.89.094502
http://dx.doi.org/10.1364/OPEX.12.000055
http://dx.doi.org/10.1103/PhysRevLett.93.190405
http://dx.doi.org/10.1016/0003-4916(58)90007-1
http://dx.doi.org/10.1038/32354
I.V. Stasyuk, V.O. Krasnov
30. Albus A., Illuminati F., Eisert J., Phys. Rev. A, 2003, 68, 023606; doi:10.1103/PhysRevA.68.023606.
31. Fehrmann H., Baranov M.A., Damski B., Lewenstein M., Santov L., Opt. Commun., 2004, 243, 23;
doi:10.1016/j.optcom.2004.03.094.
32. Lewenstein M., Santos L., Baranov M.A., Fehrmann H., Phys. Rev. Lett., 2004, 92, 050401;
doi:10.1103/PhysRevLett.92.050401.
33. Büchler H.P., Blatter G., Phys. Rev. Lett., 2003, 91, 130404; doi:10.1103/PhysRevLett.91.130404.
34. Polak T.P., Kopeć T.K., Phys. Rev. A, 2010, 81, 043612; doi:10.1103/PhysRevA.81.043612.
35. Refael G., Demler E., Phys. Rev. B, 2008, 77, 144511; doi:10.1103/PhysRevB.77.144511.
36. Titvinidze I., Snoeck M., Hofstetter W., Phys. Rev. Lett., 2008, 100, 100401; doi:10.1103/PhysRevLett.100.100401.
37. Anders P., Werner P., Troyer M., Sigrist M., Pollet L., Phys. Rev. Lett., 2012, 109, 206401;
doi:10.1103/PhysRevLett.109.206401.
38. Bijlsma M.J., Herings B.A., Stoof H.T.C., Phys. Rev. A, 2000, 61, 053601; doi:10.1103/PhysRevA.61.053601.
39. Heiselberg H., Pethick C.J., Smith H., Viverit L., Phys. Rev. Lett., 2000, 85, 2418; doi:10.1103/PhysRevLett.85.2418.
40. Viverit L., Giorgini B., Phys. Rev. A, 2002, 66, 063604; doi:10.1103/PhysRevA.66.063604.
41. Lutchyn R.M., Tewari S., Das Sarma S., Phys. Rev. A, 2009, 79, 011606; doi:10.1103/PhysRevA.79.011606.
42. Mathey L., Wang D.W., Hofstetter W., Lukin M.D, Delmer E., Phys. Rev. Lett., 2004, 93, 120404;
doi:10.1103/PhysRevLett.93.120404.
43. Mahan G.D., Phys. Rev. B, 1976, 14, 780; doi:10.1103/PhysRevB.14.780.
44. Micnas R., Ranninger J., Robaszkiewicz S., Rev. Mod. Phys., 1990, 62, 113; doi:10.1103/RevModPhys.62.113.
45. Sengupta K., Dupuis N., Majumdur P., Phys. Rev. A, 2007, 75, 063625; doi:10.1103/PhysRevA.75.063625.
46. Stasyuk I.V., Velychko O.V., Condens. Matter Phys., 2011, 14, 13004); doi:10.5488/CMP.14.13004.
47. Stasyuk I.V., Vorobyov O., Condens. Matter Phys., 2013, 16, 23005; doi:10.5488/CMP.16.23005.
48. Mysakovych T., Krasnov V.O., Stasyuk I.V., Condens. Matter Phys., 2008, 11, 663; doi:10.5488/CMP.11.4.663.
49. Mysakovych T.S., J. Phys.: Condens. Matter, 2010, 22, 355601; doi:10.1088/0953-8984/22/35/355601.
50. Mysakovych T.S., Physica B, 2011, 406, 1858; doi:10.1016/j.physb.2011.02.042.
51. Krasnov V.O., Ukr. J. Phys., 2015, 5, 60.
52. Sanpera A., Kantian K., Sanchez-Palencia L., Zakrzewski J., Lewenstein M., Phys. Rev. Lett., 2004, 93, 040401;
doi:10.1103/PhysRevLett.93.040401.
53. Schulte T., Drenkelforth S., Kruse J., Ertmer W., Arlt J.J., Kantian A., Sanchez-Palencia L., Santos L., Sanpera A.,
Sacha K., Zoller P., Lewenstein M., Zakrzewski J., Acta Phys. Pol. A, 2006, 109, 89.
54. Stasyuk I.V., Mysakovych T.S., Krasnov V.O., Condens. Matter Phys., 2010, 13, 13003; doi:10.5488/CMP.13.13003.
Фазовi переходи у моделi Бозе-Фермi-Хаббарда в
наближеннi важких фермiонiв: пiдхiд жорстких бозонiв
I.В. Стасюк, В.О. Краснов
Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
Дослiджено фазовi переходи в моделi Бозе-Фермi-Хаббарда в наближеннi середнього поля та жорстких
бозонiв при врахуваннi одновузлової бозон-фермiонної взаємодiї типу вiдштовхування та у випадку без-
межно малого переносу фермiонiв. Проаналiзовано поведiнку параметра порядку бозе-конденсату та
термодинамiчного потенцiалу як функцiй хiмiчного потенцiалу бозонiв при нульовiй температурi. Вста-
новлено можливiсть змiни роду фазового переходу до надплинної фази у режимi заданих значень хiмi-
чних потенцiалiв бозе- та фермi-частинок. Побудовано вiдповiднi фазовi дiаграми.
Ключовi слова: модель Бозе-Фермi-Хаббарда, жорсткi бозони, бозе-конденсат, фазовi переходи, фазовi
дiаграми
43702-20
http://dx.doi.org/10.1103/PhysRevA.68.023606
http://dx.doi.org/10.1016/j.optcom.2004.03.094
http://dx.doi.org/10.1103/PhysRevLett.92.050401
http://dx.doi.org/10.1103/PhysRevLett.91.130404
http://dx.doi.org/10.1103/PhysRevA.81.043612
http://dx.doi.org/10.1103/PhysRevB.77.144511
http://dx.doi.org/10.1103/PhysRevLett.100.100401
http://dx.doi.org/10.1103/PhysRevLett.109.206401
http://dx.doi.org/10.1103/PhysRevA.61.053601
http://dx.doi.org/10.1103/PhysRevLett.85.2418
http://dx.doi.org/10.1103/PhysRevA.66.063604
http://dx.doi.org/10.1103/PhysRevA.79.011606
http://dx.doi.org/10.1103/PhysRevLett.93.120404
http://dx.doi.org/10.1103/PhysRevB.14.780
http://dx.doi.org/10.1103/RevModPhys.62.113
http://dx.doi.org/10.1103/PhysRevA.75.063625
http://dx.doi.org/10.5488/CMP.14.13004
http://dx.doi.org/10.5488/CMP.16.23005
http://dx.doi.org/10.5488/CMP.11.4.663
http://dx.doi.org/10.1088/0953-8984/22/35/355601
http://dx.doi.org/10.1016/j.physb.2011.02.042
http://dx.doi.org/10.1103/PhysRevLett.93.040401
http://dx.doi.org/10.5488/CMP.13.13003
Introduction
The four-state model
Mean field approximation
Grand canonical potential and order parameter
Spinodals at T=0
Phase transition of the first order to SF phase
Phase diagrams at T=0
Phase separation at a fixed boson chemical potential
Conclusions
|
| id | nasplib_isofts_kiev_ua-123456789-155277 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T13:36:44Z |
| publishDate | 2015 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Stasyuk, I.V. Krasnov, V.O. 2019-06-16T15:23:24Z 2019-06-16T15:23:24Z 2015 Phase transitions in Bose-Fermi-Hubbard model in the heavy fermion limit: Hard-core boson approach / I.V. Stasyuk, V.O. Krasnov // Condensed Matter Physics. — 2015. — Т. 18, № 4. — С. 43702: 1–20. — Бібліогр.: 54 назв. — англ. 1607-324X PACS: 71.10 Fd, 71.38 DOI:10.5488/CMP.18.43702 arXiv:1512.07798 https://nasplib.isofts.kiev.ua/handle/123456789/155277 Phase transitions are investigated in the Bose-Fermi-Hubbard model in the mean field and hard-core boson approximations for the case of infinitely small fermion transfer and repulsive on-site boson-fermion interaction. The behavior of the Bose-Einstein condensate order parameter and grand canonical potential is analyzed as functions of the chemical potential of bosons at zero temperature. The possibility of change of order of the phase transition to the superfluid phase in the regime of fixed values of the chemical potentials of Bose- and Fermi-particles is established. The relevant phase diagrams are built. Дослiджено фазовi переходи в моделi Бозе-Фермi-Хаббарда в наближеннi середнього поля та жорстких бозонiв при врахуваннi одновузлової бозон-фермiонної взаємодiї типу вiдштовхування та у випадку безмежно малого переносу фермiонiв. Проаналiзовано поведiнку параметра порядку бозе-конденсату та термодинамiчного потенцiалу як функцiй хiмiчного потенцiалу бозонiв при нульовiй температурi. Встановлено можливiсть змiни роду фазового переходу до надплинної фази у режимi заданих значень хiмiчних потенцiалiв бозе- та фермi-частинок. Побудовано вiдповiднi фазовi дiаграми. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Phase transitions in Bose-Fermi-Hubbard model in the heavy fermion limit: Hard-core boson approach Article published earlier |
| spellingShingle | Phase transitions in Bose-Fermi-Hubbard model in the heavy fermion limit: Hard-core boson approach Stasyuk, I.V. Krasnov, V.O. |
| title | Phase transitions in Bose-Fermi-Hubbard model in the heavy fermion limit: Hard-core boson approach |
| title_full | Phase transitions in Bose-Fermi-Hubbard model in the heavy fermion limit: Hard-core boson approach |
| title_fullStr | Phase transitions in Bose-Fermi-Hubbard model in the heavy fermion limit: Hard-core boson approach |
| title_full_unstemmed | Phase transitions in Bose-Fermi-Hubbard model in the heavy fermion limit: Hard-core boson approach |
| title_short | Phase transitions in Bose-Fermi-Hubbard model in the heavy fermion limit: Hard-core boson approach |
| title_sort | phase transitions in bose-fermi-hubbard model in the heavy fermion limit: hard-core boson approach |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155277 |
| work_keys_str_mv | AT stasyukiv phasetransitionsinbosefermihubbardmodelintheheavyfermionlimithardcorebosonapproach AT krasnovvo phasetransitionsinbosefermihubbardmodelintheheavyfermionlimithardcorebosonapproach |