Phase Transition of a Heavy-fermion Superconductor in a High Magnetic Field: Entanglement Analysis

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Journal of the Korean Physical Society, Vol. 63, No. 3, August 2013, pp. 284∼287

Phase Transition of a Heavy-fermion Superconductor in a High Magnetic Field: Entanglement Analysis

R. Afzali^∗

Physics Department, K. N. Toosi University of Technology, Tehran 15418, Iran N. Ebrahimian^†

Physics Department, Amirkabir University of Technology, Tehran 15914, Iran (Received 5 June 2012, in final form 23 October 2012)

When the magnetic field is only acting on the spin of electrons, a transition from a normal to a modulated superconducting state or Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state may occur at low temperatures. A FFLO superconducting state, which accompanies an order parameter that oscillates spatially, may be stabilized by a high applied magnetic field or a molecular field. Quantum multipartite entanglement is a new procedure for investigating quantum phase transitions. In this article, we deal with the phase transition of the FFLO state of CeCoIn5

to a normal state by obtaining quantum multipartite entanglement of the system. For this purpose, using normal and anomalous Green functions and the density matrix, we obtain concurrence, as a measure of bipartite entanglement. Then, the order parameter and the magnetic -field dependence of multipartite entanglement in momentum space is calculated. The phase transition is determined, and the behavior of the system based on order parameter is discussed. Furthermore, the phase transitions of both the Bardeen-Cooper-Schrieffer (BCS) and FFLO states to the normal state are compared.

PACS numbers: 03.67.-a, 03.65.Ud, 74.20.Fg

Keywords: Entanglement, FFLO superconductor, Phase transition DOI: 10.3938/jkps.63.284

I. INTRODUCTION

Superconductivity can be established in the presence of an exchange high field [1–9]. As an example, this occurs in Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductors. In FFLO superconductors, the Cooper pair of the FFLO superconductor is accompanied by a Cooper pair with opposite spins but a non-zero momentum. This so called FFLO state can only be observed for a very narrow set of parameters [4], which is extremely difficult to adjust in experiments. CeCoIn5 is an example of a heavy-fermion superconductor. Initially, experiments on CeCoIn5 indicated that in this substance, a FFLO state could be realized in a exchange field.

Quantum multipartite entanglement is a new procedure for investigating quantum phase transitions, which are one of the interesting topics in condensed matter [10, 11]. One important subject in quantum information, especially for indicating quantum phase transitions, is quantum entanglement. We use the average concurrence as the multipartite entanglement of system and then in-

∗E-mail: [email protected]

†E-mail: [email protected]

vestigate the quantum phase transition for FFLO superconductors. In this paper, a FFLO superconductor with order parameter ∆eⁱ^q·^x, whereq and ∆ are the transfer wave vector and the magnitude of gap of FFLO superconductor, respectively, is considered. qshould be small.

Finally, the magnetic dependence and the gap dependence of the concurrence are determined; thereby, the effect of the magnetic filed on the phase transition of the FFLO superconductor is determined. Also, the results obtained in this paper for the entanglement and the quantum phase transition of FFLO superconductors are compared to the results obtained in Ref. 12 for Bardeen- Cooper-Schrieffer (BCS) superconductors.

II. AVERAGE CONCURRENCE AND QUANTUM PHASE TRANSITION

The Hamiltonian of the system can be written as [13, 14] (throughout this paper we use units in whichkB = -284-

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Phase Transition of a Heavy-fermion Superconductor· · · – R.Afzaliand N.Ebrahimian -285-

=c= 1) Hˆ =

d³XΨˆ^†_α(X) 1

2m

−i∇+e A(X) 2

−µ

Ψˆα(X) +h^′

αΨˆ^†_α(X) ˆ Ψα(X)d ³X +

d³X

∆(X)^∗Ψˆ↑(X) ˆ Ψ↓(X) + ˆΨ^†_↓(X) ˆΨ^†_↑(X)∆( X) (1) The second term in the Hamiltonian is Zemann effect.

The Magnetic fieldh^′(X)(≡µ0H^′(X)) is assumed to be constant and to have no spatial dependence. ∆(X) is taken as ∆ exp(i Q·X)·Ψˆα(X), and ˆΨ^†_β(X) are annihilation and creation Fermionic field operators, respectively (subscriptsαandβindicate spin up or spin down). Also, the effective Hamiltonian of the FFLO superconductor in momentum space is described by [3]

H =

k,σ

ξσa^†_k,σak,σ

+

k

∆qa^†

k+(k/2),↑a^†

−k+(q/2),↓

+∆^∗_qa₋k+(q/2),↓ak+(q/2),↑

,

(2) where a^†

k,σ and ak,σ are creation and annihilation operators, respectively, Also, ξσ = ξk −σh, where h is the magnetic field, which is high for an FFLO state,ξk

is the kinetic energy with respect to the chemical po- tential (i.e., (k²/2m)−µ), and σ is ±1, (+) for spin up or (-) for spin down. Furthermore, ∆^∗_q is given by

−V ka^†

k+(q/2),↑a^†

−k+(q/2),↓, where V is the internal interaction between electrons.

Now, we proceed to calculate the density matrix that is needed to obtain the concurrence. For investigating the entanglement of the system, we write the two-electron space-spin density matrix as follows [15]:

ρ⁽²⁾(x1, x2;x^′₁, x^′₂) = 1

2Ψˆ^†(x^′₂) ˆΨ^†(x^′₁) ˆΨ(x1) ˆΨ(x2), (3) where · · · is the quantum mechanical average at zero temperature and x = (r, s). The two-particle Green’s function is defined by [4]

G(x1t1, x2t2;x^′₁t^′₁, x^′₂t^′₂)

=−T

ΨˆH(x1t1) ˆΨH(x2t2) ˆΨ^†_H(x^′₂t^′) ˆΨ^†_H((x^′₁t^′₁)) ,(4) where the subscriptH denotes the Heisenberg represen- tation, and T is the time-ordering operator. Then, the relation between the two-electron space-spin density matrix and the two-particle Green’s function is

ρ⁽²⁾(x1, x2;x^′₁, x^′₂) =−1

2G(x1t1, x2t2;x^′₁t⁺₁, x^′₂t⁺₂), (5)

wheret⁺denotes a time infinitesimally later thant. The two-particle Green’s function for the FFLO superconductor at zero temperature in momentum space can be written as

ρ⁽²⁾ = a^†_k+q

2,↑a^†_−k+q

2,↓a_−k+^q

2,↓a_k+^q

2,↑

= a^†_k+^q

2,↑a^†_−k+^q

2,↓a−k+^q₂,↓ak+^q₂,↑

−a^†_k+q

2,↑a−k+^q

2,↓a^†_−k+q 2,↓ak+^q

2,↑

+a^†_k+q 2,↑a_k+^q

2,↑a^†_−k+q

2,↓a_−k+^q

2,↓, (6)

where in the right-hand side of Eq. (7),can be related to the single-particle Green’s functions via, for example,

G↑↑≡G↑=a^†_k+^q

2,↑ak+^q₂,↑ (7)

or in space-time one can write

Gαβ(x1t1, x^′₁t^′₁)≡ −iT[ΨαH(x1t1)Ψ^†_β

H(x^′₁t^′₁)]. (8) Also, we can introduce the other average appearing in Eq. (7) in terms of the anomalous Green’s function.

Now, we introduce the following Bogoliubov transformations [15]:

γ+,k = uka↑,k+^q₂ −νka^†_↓,−k+^q

2, (9)

γ_−,k^† = νka↑,k+^q

2 +uka^†_↓,−k+^q

2 . (10)

Furthermore, the inverse transformations can be written as

a↑,k+^q₂ = ukγ+,k+νkγ_−,k^† , (11) a^†_↓,−k+^q

2 = ukγ_−,k^† −νkγ+,k. (12)

With Eq. (7) and Eqs. (12)-(13) , the two-electron density matrixρ⁽²⁾ or 12|ρ⁽²⁾|1^′2^′becomes

12|ρ⁽²⁾|1^′2^′

=

δσ^′₁σ^′₂,↑↓−δσ^′₁σ^′₂,↓↑

(δσ₁σ₂,↑↓−δσ₁σ₂,↓↑)

×

−ukν_k^∗γ_+,k^† γ+,k+ukν_k^∗γ−,kγ^†_−,k

×

−νku^∗_kγ_+,k^† γ+,k+νku^∗_kγ−,kγ_−,k^† +

δσ₁σ₁^′,↑↑+δσ₁σ^′₁,↓↓ δσ₂σ^′₂,↓↓ δσ₂σ^′₂,↑↑+δσ₂σ^′₂,↓↓

×

|uk|²γ_+,k^† γ+,k+|νk|²γ−,kγ_−,k^†

×

|uk|²γ_−,k^† γ−,k+|νk|²γ+,kγ^†_+,k

, (13)

where

|uk|²= 1− |νk|²= 1 2

1 +ξ^(s)_k

Ek

, (14)

Ek =

∆²_kq+ξ^(s)_k ² , (15)

ξ^(s)_k = ξk+^q₂ +ξk−^q₂

2 . (16)

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-286- Journal of the Korean Physical Society, Vol. 63, No. 3, August 2013

Also, we can define the following expressions:

nk↑=γ_+,k^† γ+,k, nk↓=γ_−,k^† γ−,k, (17) wherenk,σis the Fermi distribution function and is given by

nkσ= 1

1 +e^E^σ⁽^k)/T , (18)

in whichEσ(k) is

Eσ(k)≡Ekσ=Ek+σ

ξ_k^(α)+h^′

, (19)

withξ_k^(α) being defined by ξ^(α)_k = ξk+^q

2 −ξk−^q

2

2 . (20)

For low-momentum transferq, one can ignoreξ^(α)_k , and ξ_k^(s) can be written as

ξ^(s)_k = 2k² 2m+ q²

4m ≈2k²

2m . (21)

Then, using Eq. (18), we can rewrite Eq. (14) as follows:

12|ρ⁽²⁾|1^′2^′

=

δσ^′₁σ^′₂,↑↓−δσ^′₁σ^′₂,↓↑

(σσ₁σ₂,↑↓−δσ₁σ₂,↓↑)

×(−ukν_k^∗nk↑+ukν_k^∗(1−nk↓))

×(−νku^∗_knk↑+νku^∗_k(1−nk↓)) +

δσ₁σ^′₁,↑↑+δσ₁σ^′₁,↓↓ δσ₂σ^′₂,↑↑+δσ₂σ^′₂,↓↓

×|

|uk|²nk↑+|νk|²(1−nk↓)

×

|uk|²nk↓+|νk|²1−nk↑

. (22)

It should be noted that in FFLO superconductors,Ekσ

can be positive or negative. For the negative energy case, we must use the following transformation [15]:

ak+^q

2

a^†_−k+q 2

=

u^∗_k νk

−ν_k^∗ uk

γ_+k^† γ_−k^†

Ek+<0,(23) a_k+^q

a−k+2^q 2

=

u^∗_k νk

−ν_k^∗ uk

γ+k

γ−k

Ek− <0 .(24) In this paper, the calculation is done for positive energy case. ˜ρis a Werner state;i.e., it satisfies the following expression [10,16]:

ρ⁽²⁾=1−p

4 I+p|Ψ⁻Ψ⁻|, (25) where I is the 4 × 4 identity matrix and |Ψ⁻ = (| ↑↓ − | ↓↑)/√

2. Using the properties of a Werner state [16], one can calculate the entanglement through, for example, the Peres-Horodecki separability criterion [16]. Specifically, a 2 × 2 Werner state is entangled if

Fig. 1. (Color online) Concurrence in terms ofξk/εF atT

= 0.

and only ifρ⁽²⁾ can be written as p > 1/3. We obtain

˜ ρ⁽²⁾ as

˜

ρ⁽²⁾= ρ⁽²⁾

T rρ⁽²⁾ = 1 T rρ⁽²⁾

⎛

⎜

⎝ ρ11

ρ22 ρ23

ρ32 ρ33

ρ44

⎞

⎟

 , (26) with

ρ11=ρ44=

|uk|²nk↑+|νk|²(1−nk↓)

×

|uk|²nk↓+|νk|²(1−nk↑)

ρ23=ρ32=−(−ukν_k^∗nk↑+ukν_k^∗(1−nk↓)) ρ22=ρ11−ρ23

From Eq. (26), we can obtainp. Concurrence can be calculated via [16]

c=max{0,(3p−1)/2} , (27) Figure 1 shows the curve of concurrence versus ξk/εF

(εF is the Fermi energy) atT = 0. The average ofC is defined as

∆C= ^kC

k1 . (28)

Figure 2 shows the curve of average ofC, as a multipartite of entanglement, versusT /Tc at different fixed magnetic fields. In Figs. 1 and 2, we have used εF = 15K,

∆0 = 5K (∆0 is the energy gap when the magnetic field is zero) and Tc = 2.3K [17]. Meanwhile, Fig. 2 was plotted for two different fixed magnetic fields. The value of one of magnetic fields is 0.750∆0 (along with

∆ ∼= 0.8∆0), and the value of the other one is 0.710∆0

(along with ∆∼= 0.9∆0) [3].

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Phase Transition of a Heavy-fermion Superconductor· · · – R.Afzaliand N.Ebrahimian -287-

Fig. 2. (Color online) Concurrence in terms of T /Tc at different fixed magnetic fields.

III. CONCLUSIONS

In this article, Using normal and anomalous Green functions and the density matrix, we obtained and plotted the concurrence, as a measure of bipartite entanglement, in terms ofξk/εF. Then, the average concurrence, as a measure of multipartite entanglement, was plotted at different fixed magnetic fields. The behavior of the concurrence for the FFLO superconductor atT = 0 and nearTc (the temperature at which the concurrence vanishes) is the same as the behavior of the concurrence for the BCS superconductor. The average concurrence decreases as temperature increases, and at the critical temperature (at which the gap vanishes), the average concurrence becomes zero. Also Fig. 2 shows the effect of magnetic field strength on quantum entanglement. At a fixed temperature, the larger the value of magnetic field, the larger is the average concurrence. At larger fixed magnetic field strength, the slope of the average concurrence and the phase transition change.

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