Article

# Some Properties of the Model of a Superconductor with Pair Hopping and Magnetic Interactions at Half-Filling

Acta Physica Polonica Series a (Impact Factor: 0.53). 11/2011; 121(4):733. DOI: 10.12693/APhysPolA.121.733

Source: arXiv

### Full-text

Available from: Konrad Jerzy Kapcia, Jan 05, 2014Some properties of the model of a superconductor

with pair hopping and magnetic interactions at half-ﬁlling

Konrad Kapcia

∗

Electron States of Solids Division, Faculty of Physics,

Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland

(Dated: November 25, 2011)

We present our preliminary studies of an eﬀective model of a superconductor with short coher-

ence length involving magnetic interactions. The Hamiltonian considered consists of (i) the eﬀective

on-site interaction U, (ii) the intersite magnetic exchange interactions (J

z

, J

xy

) between nearest-

neighbors and (iii) the intersite charge exchange term I, determining the hopping of electron pairs

between nearest-neighbor sites. In the analysis of the phase diagrams and thermodynamic proper-

ties of this model for half-ﬁlling (n = 1) we have adopted the variational approach, which treats the

on-site interaction term exactly and the intersite interactions within the mean-ﬁeld approximation.

One ﬁnds that the system considered can exhibit very interesting multicritical behaviors (includ-

ing tricritical, critical-end and bicritical points) caused by the competition between magnetism and

superconductivity, even for n = 1. Our investigations show that, depending on the values of interac-

tion parameters, the system at half-ﬁlling can exhibit three homogeneous phases: superconducting

(SS), (anti-)ferromagnetic (F) and nonordered (NO). The transitions between ordered phases (SS,

F) and the NO phase can be ﬁrst order as well as second order ones, whereas SS–F transition is ﬁrst

order one. Temperature dependencies of the order parameters and thermodynamic properties of

the system at the sequence of transitions: SS→F→NO with increasing temperature for J/I = 0.3,

U/I

0

= 0.69 and n = 1 are also presented.

PACS numbers: 71.10.Fd, 71.10.-w, 74.20.-z, 74.81.-g, 75.30.Fv

I. INTRODUCTION

There has been much interest in superconductivity

with very short coherence length. This interest is due

to its possible relevance to high temperature supercon-

ductors (the cuprates, doped bismuthates, fullerenes and

iron-based) and also to the several other exotic super-

conducting materials (for a review, see Refs. 1 and 2 and

references therein). It can also give relevant insight into

a behavior of strongly bounded fermion pairs on the op-

tical lattices.

The interplay and competition between superconduc-

tivity and magnetic orderings is currently under intense

investigations (among others in iron chalcogenides and

cuprates, e. g. Refs. 3–5 and references therein). A con-

ceptually simple model for studying that competition will

be studied in this report.

The Hamiltonian considered has the following form:

ˆ

H = U

X

i

ˆn

i↑

ˆn

i↓

− I

X

hi,ji

ˆρ

+

i

ˆρ

−

j

+ ˆρ

+

j

ˆρ

−

i

+ (1)

− 2J

X

hi,ji

ˆs

z

i

ˆs

z

j

− µ

X

i

ˆn

i

,

where ˆn

i

=

P

σ

ˆn

iσ

, ˆn

iσ

= ˆc

+

iσ

ˆc

iσ

, ˆρ

+

i

= (ˆρ

−

i

)

†

= ˆc

+

i↑

ˆc

+

i↓

,

ˆs

z

i

= 1/2(ˆn

i↑

− ˆn

i↓

). ˆc

iσ

and ˆc

+

iσ

denote annihilation and

creation operators of an electron with spin σ =↑, ↓ at the

site i, which satisfy canonical anticommutation relations

{ˆc

iσ

, ˆc

+

jσ

0

} = δ

ij

δ

σσ

0

, {ˆc

iσ

, ˆc

jσ

0

} = {ˆc

+

iσ

, ˆc

+

jσ

0

} = 0, (2)

∗

e-mail: kakonrad@amu.edu.pl

where δ

ij

is the Kronecker delta.

P

hi,ji

indicates the

sum over nearest-neighbor sites i and j independently.

U is the on-site density interaction, I is the intersite

charge exchange interaction between nearest neighbors

and J is the Ising-like magnetic interaction between near-

est neighbors. µ is the chemical potential, depending on

the concentration of electrons:

n =

1

N

X

i

hˆn

i

i, (3)

with 0 ≤ n ≤ 2 and N is the total number of lattice sites.

There are two competitive interaction parameters of

the model: (i) the pair hopping interaction I, deter-

mining the electron pair mobility and responsible for

the long-range superconducting order (nonlocal pairing

mechanism) and (ii) the Ising-like interaction J between

nearest neighbors responsible for magnetism in the sys-

tem. The on-site density-density interaction U con-

tributes (together with I) to the pair binding energy by

reducing (U > 0) or enhancing (U < 0) its value. More-

over, repulsive U > 0 favors magnetic ordering. To sim-

plify our analysis we do not include in Hamiltonian (1)

the single electron hopping term (

P

i,j

t

ij

ˆc

+

iσ

ˆc

jσ

) as well

as other inter-site interaction terms. This assumption

corresponds to the situation when single particle mobil-

ity is much smaller than the pair mobility and can be

neglected.

The interactions U, I and J will be treated as the

eﬀective ones and will be assumed to include all the pos-

sible contributions and renormalizations like those com-

ing from the strong electron-phonon coupling or from the

coupling between electrons and other electronic subsys-

arXiv:1111.7296v1 [cond-mat.str-el] 30 Nov 2011

Page 1

2

tems in solid or chemical complexes [1].

Ferromagnetic XY-order of pseudospins

ˆ

~ρ

i

(for I > 0)

corresponds to the SS phase (s-pairing superconduct-

ing), whereas the antiferromagnetic XY-order (for I < 0)

– to the Sη phase (η-pairing superconducting). For

t

ij

= 0 there is a well known isomorphism between the

SS and Sη cases (with an obvious redeﬁnition of the

order parameter: ∆ = ∆

SS

=

1

N

P

i

hˆρ

−

i

i, for I > 0 and

∆

ηS

=

1

N

P

i

exp (i

~

Q ·

~

R

i

)hˆρ

−

i

i, for I < 0,

~

Q is half of the

smallest reciprocal lattice vector) for lattices consisting of

two interpenetrating sublattices such as for example SC

or BCC lattices. One should also notice that, in the ab-

sence of the single electron hopping term, ferromagnetic

(J > 0) interactions are simply mapped onto the antifer-

romagnetic cases (J < 0) by redeﬁning the spin direction

on one sublattice in lattices decomposed into two inter-

penetrating sublattices. Thus, we restrict ourselves to

the case of I > 0 and J > 0.

We have performed extensive study of the phase dia-

grams of the model (1) for arbitrary n and µ [6]. In the

analysis we have adopted a variational approach (VA),

which treats the onsite interaction term (U ) exactly and

the intersite interactions (I, J) within the mean-ﬁeld ap-

proximation (MFA). In this paper we present our prelim-

inary results for the half-ﬁlling (n = 1).

The model (1) have been analyzed within VA only for

particular cases: (i) J = 0 [7–11] and (ii) I = 0 [12, 13]

till now. The rigorous results for I = 0 in ground state

have been also obtained [14]. Some preliminary study of

the I = 0 case in ﬁnite temperatures using Monte Carlo

simulations has also been done for a square lattice [13].

The ferromagnetic (F) phase is characterized by

nonzero value of the magnetic order parameter (magneti-

zation) deﬁned as m = (1/N)

P

i

hˆs

z

i

i (and ∆ = 0), in the

superconducting (SS) phase the order parameter ∆ 6= 0

(and m = 0) and in the nonordered (NO) phase m = 0

and ∆ = 0.

Within the VA the intersite interactions are decou-

pled within the MFA, what let us ﬁnd a grand canon-

ical potential per site ω(µ) (or free energy per site

f(n) = ω(µ) + µn) in the grand canonical ensemble. One

can also calculate the averages: n, ∆ and m, what gives

a set of three non-linear self-consistent equations (for ho-

mogeneous phases). This set for T > 0 is solved numer-

ically and one obtains ∆, m, and n (or µ) when µ (or

n) is ﬁxed. It is important to ﬁnd a solution correspond-

ing to the lowest ω(µ) (or f(n)). For n = 1 one obtains

µ = U/2 and two equations for ∆ and m need to be solved

numerically.

We also introduce the following denotation: I

0

= zI,

J

0

= zJ, where z is the number of nearest neighbors.

II. RESULTS AND DISCUSSION

There are two well deﬁned limits of the model (1):

(i) U → −∞ favoring superconductivity and

FIG. 1. (Color online) Phase diagrams k

B

T/I

0

vs. U/I

0

at half-ﬁlling (n = 1) for (a) J/I = 0.3, (b) J/I = 0.51 and

(c) J/I = 3. Dotted and solid lines indicate ﬁrst order and

second order boundaries, respectively. T , E and B denote

tricritical, critical-end and bicritical points, respectively.

(ii) U → +∞, where only magnetic orderings can

appear in the system.

For U → −∞ (states with single occupancy are ex-

cluded and only local pairs can exists in the system)

the model is equivalent with the hard-core charged bo-

son model on the lattice [9, 15, 16]. In this limit the

Page 2

3

SS–NO transition is second order one and is to the NO

phase being a state of dynamically disordered local pairs.

The SS–NO transition temperatures increase monotoni-

cally with decreasing |n − 1|. The maximum value of the

transition temperature is k

B

T/I

0

= 1 for n = 1 [9, 11].

In the opposite limit (i. e. U → +∞) the double oc-

cupied sites are excluded and only the magnetic states

can occur on the phase diagram [12, 13]. At suﬃciently

low temperatures the homogeneous phases are not states

with the lowest free energy and the PS state are stable (if

n 6= 1). On the phase diagram there is a second order line

at high temperatures, separating the F and NO phases,

whereas ﬁrst order transition takes place at lower tem-

peratures, leading to a phase separation of the F and NO

phases. The critical point for the phase separation (tri-

critical point) lies on the second order F–NO line and it

is located at k

B

T/J

0

= 1/3 and n = 1/3 [12]. The F–NO

(second order) transition temperature decreases with in-

creasing |n − 1| and its maximum value is k

B

T/J

0

= 1

for n = 1 [13].

A. The phase diagrams at half-ﬁlling

A few representative k

B

T/I

0

vs. U/I

0

phase diagrams

of the model (1) evaluated for various ratios of J/I at

half-ﬁlling (n = 1) are presented in Fig. 1.

The phase diagram k

B

T/I

0

vs. U/I

0

for J/I = 0.3 and

n = 1 is shown in Fig. 1a. Two ordered phases: the

SS phase and the F phase are separated by ﬁrst order

boundary on the diagram. Both order parameters change

discontinuous at the SS–F transition. With increasing

U/I

0

the SS–NO transition temperature decreases from

k

B

T/I

0

= 1 at U/I

0

→ −∞. At U/I

0

=

2

3

ln(2) ' 0.462

and k

B

T/I

0

= 1/3 the transition changes its type from

second order one to ﬁrst order one resulting in the tri-

critical point T on the phase diagram. The F–NO tran-

sition temperature is slightly dependent on U/I

0

and in-

creases to k

B

T/I

0

= 0.3 (k

B

T/J

0

= 1) at U/I

0

→ +∞.

The F–NO second order line ends at critical-end point E

on the ﬁrst order boundary of the SS phase occurrence.

The possible sequences of transitions with increasing

temperatures and the transition orders of them are listed

below (for J/I = 0.3):

(i) SS→NO: second order, for U/I

0

< 0.46 and ﬁrst

order, for 0.46 < U/I

0

< 0.63,

(ii) SS→F→NO: ﬁrst order and second order, respec-

tively, for 0.63 < U/I

0

< 0.7,

(iii) F→NO: second order, for U/I

0

> 0.7.

The phase diagram for J/I = 0.51 is qualitatively dif-

ferent than that for J/I = 0.3. For J/I = 0.51 the sys-

tem exhibits bicritical behavior (Fig. 1b) in contrary to

the tricritical behavior (and occurrence of E-point) for

J/I = 0.3. Similarly as for J/I = 0.3, the SS–F tran-

sition is ﬁrst order one while the SS–NO and F–NO

FIG. 2. (Color online) Temperature dependence of (a) su-

perconducting order parameter |∆| and (b) magnetic order

parameter m for J/I = 0.3, U/I

0

= 0.69 and n = 1.

transitions are second order ones. The two second or-

der boundaries and the ﬁrst order boundary merge at

bicritical point B.

The system exhibits the tricritical behavior for

|J/I| < 0.5, whereas the bicritical behavior occurs for

0.5 < |J/I| < 2. For |J/I| > 2 the system exhibits tri-

critical behavior again, however the tricritical point T is

located at the F–NO line at U/J

0

= −

2

3

ln(2) ' −0.462

and k

B

T/J

0

= 1/3 (cf. Fig. 1c). For |J/I| > 2 the F–NO

transition can be second order (for U/J

0

> −0.46) as well

as ﬁrst order (for U/J

0

< −0.46). Notice that the axis in

Fig. 1 are normalized by I

0

, not by J

0

.

One should notice that, for any J/I, with increas-

ing U/I

0

the SS–NO transition temperature decreases

monotonically from k

B

T/I

0

= 1 at U → −∞, whereas

the F–NO transition temperature is an increasing func-

tion of U/I

0

(to its maximum k

B

T/J

0

= 1 at U → +∞).

Let us concentrate now on temperature dependencies

of the order parameters and thermodynamic properties

of the system at the sequence of transitions: SS→F→NO

for J/I = 0.3, U/I

0

= 0.69 and n = 1.

Page 3

4

B. The order parameters

The temperature dependencies of the order parame-

ters: ∆ and m for J/I = 0.3, U/I

0

= 0.69 and n = 1 are

presented in Fig. 2. It is clearly seen that at the SS–F

transition (at k

B

T

c1

/I

0

= 0.16) the both order parame-

ters: superconducting order parameter ∆ and magneti-

zation m change discontinuously. In the SS phase ∆ 6= 0

and m = 0 whereas in the F phase m 6= 0 and ∆ = 0. The

F–NO transition (at k

B

T

c2

/I

0

= 0.24) is connected with

a continuous decay of m at the transition temperature.

C. The thermodynamic properties

Calculating the free energy per site f one can obtain

thermodynamic characteristics of the system for arbi-

trary temperature. The double occupancy per site D

is deﬁned as:

D =

1

N

X

i

hˆn

i↑

ˆn

i↓

i =

∂f

∂U

T

(4)

and it is related with the local magnetic moment γ by

the following formula:

γ =

1

N

X

i

hˆs

z

i

i =

1

2N

X

i

h|ˆn

i↑

− ˆn

i↓

|i (5)

=

1

2

n −

X

i

hˆn

i↑

ˆn

i↓

i =

1

2

n −

∂f

∂U

T

=

1

2

n − D,

because |ˆn

i↑

− ˆn

i↓

| = (ˆn

i↑

− ˆn

i↓

)

2

= ˆn

i↑

+ ˆn

i↓

− 2ˆn

i↑

ˆn

i↓

,

ˆn

2

iσ

= ˆn

iσ

= 0, 1 and |ˆn

i↑

− ˆn

i↓

| = 0, 1.

The entropy s and the speciﬁc heat c can be derived

as:

s = −

∂f

∂T

, c = −T

∂

2

f

∂T

2

. (6)

The temperature dependencies of the thermodynamic

parameters for J/I = 0.3, U/I

0

= 0.69 and n = 1 are

shown in Fig. 3.

The concentration of paired electrons n

p

= 2D (nor-

malized to the total electron concentration n) as a func-

tion of temperature is presented in Fig. 3a. At the SS–F

transition large amount of electron pairs is destroyed.

Thus n

p

has a sharp break at the SS–F transition tem-

perature T

c1

and a substantial fraction of single particles

exists above T

c1

. As temperature is lowered, the conden-

sate growths both from a condensation of pre-existing

pairs and from binding and condensation of single parti-

cles. At the F–NO transition (at T

c2

) n

p

is continuous.

In the NO phase it increases to n

p

→ 0.5 at T → +∞

(two electrons at the site is one of four equal probable

conﬁgurations at the site and n = hˆn

i

i = 1).

The temperature dependencies of the entropy s and the

speciﬁc heat c are shown in Figs. 3b and 3c, respectively.

s increases monotonically with increasing T . At T

c1

the

entropy s is discontinuous whereas it is continuous at T

c2

.

FIG. 3. (Color online) Thermodynamic parameters (a) the

concentration of paired electrons n

p

/n = 2D/n, (b) the en-

tropy s and (c) the speciﬁc heat c as a function of k

B

T/I

0

for

J/I = 0.3, U/I

0

= 0.69 and n = 1.

One can notice that in the high-temperature limit the en-

tropy s/(k

B

I

0

) → ln(4) ≈ 1.386 (there are four possible

conﬁgurations at each site). The peak in c(T ) is asso-

ciated with the ﬁrst order transition (at T

c1

), while the

λ-point behavior is typical for the second order transition

(at T

c2

).

Page 4

5

III. FINAL REMARKS

We have studied a simple model of a magnetic super-

conductor with very short coherence length (i. e. with the

pair size being of the order of the radius of an eﬀective

lattice site) and considered the situation where the single

particle mobility is much smaller than the pair mobility

and can be neglected.

One has found that the system considered for n = 1 ex-

hibits various multicritical behaviors (determined by the

ratio J/I) including tricritical, critical-end and bicritical

points. It has been shown that, depending on the values

of interaction parameters, three homogeneous phases: su-

perconducting, (anti-)ferromagnetic and nonordered oc-

cur on the phase diagrams of the model (1) at half-ﬁlling.

The transitions between ordered phases (SS, F) and the

NO phase can be ﬁrst order as well as second order ones,

whereas the SS–F transition is ﬁrst order one. For n 6= 1

several types of phase separated states could be also sta-

ble in deﬁnite ranges of model parameters [6].

The other result of the interplay between magnetism

and superconductivity could be appearance of triplet

pairing [17]. Such a solution could appear together with

ferromagnetic spin ordering. However, in the model (1)

which assumes t

ij

= 0 such a state cannot be found. To

investigate the possibility of occurrence of a supercon-

ducting state with triplet pairing, the model should be

extended to the case of ﬁnite bandwidth (t

ij

6= 0) and

be analyzed taking into account intersite pairing (in par-

ticular triplet pairing), e. g. using Hartree-Fock broken

symmetry framework [17–19].

The mean-ﬁeld approximation used to the intersite

term is best justiﬁed if the I

ij

interactions are long-

ranged or if the number of nearest neighbors is relatively

large. The derived VA results are exact in the limit of

inﬁnite dimensions d → +∞, where the MFA treatment

of the intersite interactions I and J terms becomes the

rigorous one.

Let us point out that in the MFA, which does not

take into account collective excitations, one obtains the

same results for the U-I-J

z

model, i. e. model (1),

and the U-I-J

xy

model, where the term 2J

P

ˆs

z

i

ˆs

z

j

is re-

placed with J

P

(ˆs

+

i

ˆs

−

j

+ ˆs

+

j

ˆs

−

i

), describing interactions

between xy-components of spins at neighboring sites,

ˆs

+

i

= ˆc

+

i↑

ˆc

i↓

= (ˆs

−

i

)

†

. In both cases the self-consistent

equations have the same form, only a magnetization

along the z-axis becomes a magnetization in the xy-

plane [12].

ACKNOWLEDGMENTS

The author is indebted to Professor Stanisław

Robaszkiewicz for very fruitful discussions during this

work and careful reading of the manuscript. The work

has been ﬁnanced by National Science Center (NCN)

as a research project in years 2011-2013, under grant

No. DEC-2011/01/N/ST3/00413. We would also like to

thank the European Commission and Ministry of Science

and Higher Education (Poland) for the partial ﬁnancial

support from European Social Fund – Operational Pro-

gramme “Human Capital” – POKL.04.01.01-00-133/09-

00 – “Proinnowacyjne kształcenie, kompetentna kadra,

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**ABSTRACT:**We have studied the extended Hubbard model with pair hopping in the atomic limit for arbitrary electron density and chemical potential. The Hamiltonian considered consists of (i) the effective on-site interaction U and (ii) the intersite charge exchange interactions I, determining the hopping of electron pairs between nearest-neighbour sites. The model can be treated as a simple effective model of a superconductor with very short coherence length in which electrons are localized and only electron pairs have a possibility of transferring. The phase diagrams and thermodynamic properties of this model have been determined within the variational approach, which treats the on-site interaction term exactly and the intersite interactions within the mean-field approximation. We have also obtained rigorous results for a linear chain (d = 1) in the ground state. Moreover, at T = 0 some results derived within the random phase approximation (and the spin-wave approximation) for d = 2 and 3 lattices and within the low-density expansions for d = 3 lattices are presented. Our investigation of the general case (as a function of the electron concentration n and as a function of the chemical potential μ) shows that, depending on the values of interaction parameters, the system can exhibit not only the homogeneous phases, superconducting (SS) and nonordered (NO), but also the phase separated states (PS: SS-NO). The system considered exhibits interesting multicritical behaviour including tricritical points. - [Show abstract] [Hide abstract]
**ABSTRACT:**We have studied the extended Hubbard model with pair hopping in the atomic limit for arbitrary electron density and chemical potential and focus on paramagnetic effects of the external magnetic field. The Hamiltonian considered consists of (i) the effective on-site interaction U and (ii) the intersite charge exchange interactions I, determining the hopping of electron pairs between nearest-neighbour sites. The phase diagrams and thermodynamic properties of this model have been determined within the variational approach (VA), which treats the on-site interaction term exactly and the intersite interactions within the mean-field approximation. Our investigation of the general case shows that the system can exhibit not only the homogeneous phases-superconducting (SS) and non-ordered (NO)-but also the phase separated states (PS: SS-NO). Depending on the values of interaction parameters, the PS state can occur in higher fields than the SS phase (field induced PS). Some ground state results beyond the VA are also presented. - [Show abstract] [Hide abstract]
**ABSTRACT:**We present studies of an effective model which is a simple generalization of the standard model of a local pair superconductor with on-site pairing (i.e., the model of hard core bosons on a lattice) to the case of finite pair binding energy. The tight binding Hamiltonian consists of (i) the effective on-site interaction U, (ii) the intersite density-density interactions W between nearest-neighbours, and (iii) the intersite charge exchange term I, determining the hopping of electron pairs between nearest-neighbour sites. In the analysis of the phase diagrams and thermodynamic properties of this model we treat the intersite interactions within the mean-field approximation. Our investigations of the U<0 and W>0 case show that, depending on the values of interaction parameters, the system can exhibit three homogeneous phases: superconducting (SS), charge-ordered (CO) and nonordered (NO) as well as the phase separated SS-CO state.