- Email: [email protected]
Quantum phase transition in the two dimensional XY model with single ion and exchange anisotropies
Physica A xx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Quantum phase transition in the two dimensional XY model with single ion and exchange anisotropies Q1
A.S.T. Pires ∗ Departamento de Fisica, Universidade Federal de Minas Gerais, Belo Horizonte, MG, CP 702, 3012-970, Brazil
highlights • The SU(3) Schwinger boson formalism is used in this paper. • The phase diagram at zero temperature is obtained. • The energy gap as a function of the anisotropy parameter and temperature is calculated.
article
info
Article history: Received 7 January 2015 Received in revised form 15 April 2015 Available online xxxx Keywords: A. XY model A. Antiferromagnet D. Quantum phase transitions
abstract I study the spin-one XY antiferromagnet with easy plane single ion and exchange anisotropies on the square lattice using the SU(3) Schwinger boson formalism. The phase diagram at zero temperature is obtained. The gap, in the disordered phase, as a function of temperature is presented. © 2015 Published by Elsevier B.V.
1. Introduction
1
The study of low dimensional quantum magnetic models has played an important role in the understanding of phase transitions and critical behavior in condensed matter. As it is well known classical phase transitions are driven only by thermal fluctuations. In contrast, fluctuations driven by the uncertainty principle can drive phase transitions, in the ground state, even at zero temperature. These transitions are called quantum phase transitions (QPT). They take place by changing some parameters in the Hamiltonian, instead of the temperature. The subject of QPT is very interesting and has motivated a lot of papers and some books [1] (and references there in). In this context a model of great interest is the spin-one antiferromagnet with single-ion anisotropy. This model has been studied theoretically using analytical and numerical techniques, with the advantage that experimental works are also available [1–5]. These systems become quantum paramagnets for sufficiently high values of the easy-plane single-ion anisotropy. This is purely a quantum effect without a counterpart in the classical models [1]. The large D phase is disordered in the sense that ⟨S ⟩ = 0, but one has ⟨(S x )2 ⟩ = ⟨(S y )2 ⟩ ̸= ⟨(S z )2 ⟩ which implies a nematic order [1]. Since all kinds of anisotropies can be presented in a real compound, in this paper I will study the XY antiferromagnetic model with spin S = 1 on a square lattice described by the following Hamiltonian:
J x x (Sr Sr +δ + λSry Sry+δ ) + D H = (Srz )2 . 2 r ,δ
∗
Tel.: +55 31 34996224; fax: +55 31 34996600. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.physa.2015.05.114 0378-4371/© 2015 Published by Elsevier B.V.
r
(1)
Q2
2 3 4 5 6 7 8 9 10 11 12 13 14 15
16
2
A.S.T. Pires / Physica A xx (xxxx) xxx–xxx
10
As shown by Wang and Wang [6], when D > 0 the XY model has the same properties as the ones for the Heisenberg model and it has the advantage that we do not need to do a decoupling for four-operator terms. Here, I will consider the case 0 ≤ λ ≤ 1. When λ = 0 we have an Ising like system. z = 0 separated by a gap from the first In the limit of infinite D, the model (1) will be in a disordered ground state with Stotal z = ±1. Therefore there exists a critical DC denoting a quantum phase transition, excited states, which lie in the sectors Stotal described by the condensation of magnons, from the large-D phase to the ordered phase. Although the model with λ = 1 has been intensively studied [6–11], the case λ ̸= 1, as far as I know, has not being treated before. In this paper, I use the Su(3) Schwinger boson formalism to study the phase diagram and critical properties of the Hamiltonian (1). In Section 2, I present the formalism used and in Section 3, the results of my calculations.
11
2. Schwinger boson formalism
1 2 3 4 5 6 7 8 9
12 13 14 15 16 17 18 19
20 21 22
23 24
25 26 27 28 29
30 31
32
33 34 35 36 37
38
39 40 41
42 43 44 45
46
Traditional spin wave Su(2) theory is a very good approach for treating quantum spin models that have a magnetically (dipolar) ordered ground state. The theory however is not adequate to treat nematic, quadrupolar, octupolar, or higher multipolar ordering, because, as pointed out by Muniz et al. [12], such ground state orderings do not have a classical counterpart at zero temperature. In these cases the local fluctuations of a spin are described by the SU(2S + 1) group of local unitary transformation, instead of the Su(2) group of rotations. (It should be noted that the N 2 − 1 components of the local SU(N ) order parameter can be decomposed in the different irreducible representations of Su(2).) To treat magnetic systems with single site anisotropy, Papanicolaou [13] derived a theory which is a generalization Su(3) of the Su(2) Schwinger boson approach. In this formalism we choose the following basis:
√ |y⟩ = (|1⟩ + | − 1⟩)/ 2,
√ |x⟩ = i(|1⟩ − | − 1⟩)/ 2,
|z ⟩ = −i|0⟩
(2)
where |n⟩ are eigenstates of S . The spin operators are then written via a set of three boson operators tα (α = x, y, z ) defined by z
tx+ |v⟩ = |x⟩ ,
ty+ |v⟩ = |y⟩ ,
tz+ |v⟩ = |z ⟩ ,
(3)
where |v⟩ is the vacuum state. To force single site occupancy on each site we impose the constraint tx+ tx + ty+ ty + tz+ tz = 1.
(4)
The constraint (4) projects the bosonic operators into the physical space of dimension 3. The Su(3) generators are bilinear forms: + Omn = trm trn ,
(5)
that satisfies the Su(3) commutation relations [12]:
[Omn , Oij ] = δni Omj − δ mj Oin .
(6)
In terms of the t operators we can write S x = −i(ty+ tz − tz+ ty ),
S y = −i(tz+ tx − tx+ tz ),
S z = −i(tx+ ty − ty+ tx ).
(7)
The states tx+ |v⟩ and ty+ |v⟩, both consist of S z = ±1 eigenstates and have the average ⟨S z ⟩ = 0. Papanicolaou has pointed out that the main motivation for this formalism was to eliminate certain difficulties of the standard semiclassical theory which become crucial for spin systems with single-site (or biquadratic) interactions. In contrast, the Su(2) Schwinger boson formalism does not lead to a quantum phase transition when used in Hamiltonian (1). To study the paramagnetic phase, it is convenient to introduce two bosonic operators u+ and d+ given by [6] 1 u+ = − √ (tx+ + ity+ ), 2
1 d+ = √ (tx+ − ity+ ), 2
(8)
and so
|1⟩ = u+ |v⟩ ,
|0⟩ = tz+ |v⟩ , +
+
|−1⟩ = d+ |v⟩ ,
(9)
+
with the constraint u u + d d + tz tz = 1. The spin operators are now written as S+ =
√
2(tz+ d + u+ tz ),
S− =
√
2(d+ tz + tz+ u),
S z = u+ u − d + d .
(10)
To treat ordered phases we can extend the Holstein–Primakoff representation from Su(2) to Su(3) by condensing one of the 3 Schwinger bosons (SB). The SB that is condensed is the one that creates the local state that minimizes the mean-field energy [12]: + trz , trz →
+ + 1 − trx trx − try try .
(11)
A.S.T. Pires / Physica A xx (xxxx) xxx–xxx
3
However for the disordered phase of a XY like system (as is the case studied here), Zhang et al. [1] and Wang [6] have shown a much better approximation is to condense the tz boson: i.e. ⟨tz ⟩ = ⟨tz+ ⟩ = t. (For a comparison of the two approximations see Ref. [1].) This approach is more appropriate for describing gapped phases that do not break any continuous symmetry. Substituting (10) into the Hamiltonian (1) and condensing the tz boson, as mentioned above, we obtain: 2
Jt
H =
2
+ + + [(1 + λ)(d+ r dr +δ + ur +δ ur + ur dr +δ + dr ur +δ + H .c .)
2 3 4
5
r ,δ
+ + (1 − λ)(dr dr +δ + dr u+ r +δ + ur ur +γ + ur dr +γ )] + + 2 +D (u+ µ r ( u+ r ur + d r d r ) − r ur + dr dr + t − 1). r
6
(12)
+ Ψq+ = (u+ q , dq , u−q , d−q ),
(13)
we find that the Hamiltonian (1), after a Fourier transformation, can be written as: 1 2
7
r
A temperature-dependent chemical potential µr has been introduced to impose the local constraint Sr2 = S (S + 1) = 2. In a mean-field approximation, we replace the local constraint by a global one and let [6] µr = µ. Defining the generalized Nambu spinor
H =
1
Ψq+ M Ψq + µN (1 − t 2 ) −
q
Λq ,
8 9 10
11 12
(14)
13
q
where the last term in Eq. (14) appears when we write the Hamiltonian in a symmetrical form, and
Γq + r Γ˜ q M = Γ˜ q Γq
Γ˜ q Γq + r Γq Γ˜ q
Γ˜ q Γq Γq + r Γ˜ q
Γq Γ˜ q , Γ˜ q Γq + r
14
(15)
with
15
16
t
Γq =
2
2 1
γq =
2
(1 + λ)γq ,
Γ˜ q =
t
2
2
(1 − λ)γq ,
r =
1 2
(D − µ),
(cos qx + cos qy ).
(16)
17
(17)
18
I have set J = 1. Green‘s function for Hamiltonian (1) is given by
19
G(k, iωn ) = (iωn σ z ⊗ I − Hαα )−1 ,
(18)
where ωn is the Matsubara frequency, σz is the Pauli matrix and I a 4 × 4 unit matrix. The theory of Green’s functions can be found in any book treating many bodies in condensed matter. From Eq. (15), writing z = iω, we obtain (1) 2
(2) 2
det G
(q, z ) = (z − (ωq ) )(z − (ωq ) ),
ωq(1) =
−1
2
2
(19)
with
20 21 22
23 24
ωq(2) =
Λ2q − ∆2q + δ ∆q (∆q + Λq ),
Λ2q − ∆2q − δ ∆q (∆q + Λq ),
(20)
where
∆q =
26
1 2
(1 + λ)zt 2 γq ,
Λq = −µ + D + ∆q ,
δ=
(1 − λ) 2(1 + λ)
.
(21)
The mean field ground state energy per site is e0 =
25
1 2
(ωq(1) + ωq(2) ) −
q
28
Λq + µ(1 − t 2 ).
(22)
29
q
The Gibbs free energy is then written as: G = Ne0 −
27
1
β
ln[1 + n(ωq(1) )] −
q
with n(ωq ) = 1/(eβωq − 1).
30
1
β
ln[1 + n(ωq(2) )],
(23)
31
q 32
4
A.S.T. Pires / Physica A xx (xxxx) xxx–xxx
Fig. 1. The critical anisotropy parameter DC as a function of λ. Below the line the system is in the Néel phase, while above it is in the disordered phase. 1
2
3 4 5
6
7 8 9 10
11 12 13 14
15
16
Defining the variable y by y = (1 + λ)zt 2 /(−µ + D) we can write
ωq(1) = (−µ + D) (1 + yγq )(1 + yδγq ),
ωq(2) = (−µ + D) (1 + yγq )(1 − yδγq ).
(24)
(1)
= ωq(2) for λ = 1 (as it should be). We see that (1) (2) taking ∆ = 1/λ we have δ = −(1 − ∆)/[2(1 + ∆)] and all the results are the same with the substitution ωq ↔ ωq . (2) For γq = 1, y = 1 (y is always less than 1 in the disordered phase) and taking z = 4 we can write ωq as ωq(2) = −µ + D − t 2 (1 − λ). (25) Both components of the dispersion relation vanish at q = (π , π ), and ωq
Since µ < 0, 0 ≤ λ ≤ 1, D ≥ DC and the maximum value of t 2 is 1 we see that the quasiparticle energies are always well (1) defined (the same discussion can be applied to ωq ). When y → 1, the energy gap at q = (π , π ) goes to zero, indicating a transition from the large-D phase to the Néel phase. We determine DC as the value of D where the gap vanishes. The energy gap is given by ) ) m = ωq(1=π = ωq(2=π ,
(26)
and is interpreted by some authors [14] as the mass of the quasiparticles. Extremizing the free energy (23) with respect to the parameters t 2 and µ we obtain the following self-consistent equations: 1
2 + yγq (1 + δ/2)
2 + yγ q (1 − δ/2)
+ (1 + n(q1) + n(q2) ), ( 1 + y γ )( 1 + y δγ ) ( 1 + y γ )( 1 − y δγ ) q q q q q 1 + δ(1 + 2yγq )/2 1 − δ(1 + 2yγq )/2 µ = 2(1 + λ) γq + (1 + n(q1) + n(q2) ). ( 1 + y γ )( 1 + y δγ ) ( 1 + y γ )( 1 − y δγ ) q q q q q 2
2−t =
4
(27)
(28)
18
We solve numerically these self consistent equations to find t 2 and µ for each value of λ and D (and T if we are doing calculations at T > 0).
19
3. Results and conclusions
17
20 21 22 23 24 25 26
Fig. 1 shows the critical parameter DC as a function of λ. As we can see we find it is a straight line that can be fitted to the equation DC = 2.75 + 2.968λ.
(29)
The parameter DC increases with the increasing of λ. This should be intuitively expected since the Ising model is almost a classical one and in the classical limit DC → ∞. Below the line the system is in the Néel phase. Above the line it is in Q3 the paramagnetic phase. In Fig. 2, I present the gap, given by Eq. (26), as function of λ, for D = 8 and D = 12. The gap decreases with the increasing of λ. I have found numerically that, for a given value of λ, the energy gap decreases linearly
A.S.T. Pires / Physica A xx (xxxx) xxx–xxx
5
Fig. 2. The gap m as a function of λ. Red line D = 12, black line D = 8. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. The gap as a function of temperature. Here D = 10. Red line λ = 0, black line λ = 0.5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
with decreasing D until DC , i.e. m ∝ (D − DC ). From Eq. (22) one finds that D − DC at λ = 0 is larger than at λ = 1. This explains the behavior found in Fig. 2. The energy gap, as given by Eq. (26), as a function of temperature is depicted in Fig. 3, for D = 10, and λ = 0, and λ = 0.5. The curve can be fitted to the following expression: m = m0 + c1 T 1/2 exp(−c2 /T ),
(30)
where c1 and c2 are constants which depend on D and λ and should be calculated numerically to fit the curve. The parameter m0 is the value of the gap at zero temperature. One important relation is that the correlation length ξ , for the spin–spin correlation function, is proportional to 1/m0 [14] and this quantity decreases with the temperature. A nonzero temperature induces an exponentially small density of thermally excited magnons. The expression (30) is in agreement with analytical calculations using scaling arguments [14]. (1)
(2)
The spin wave dispersions ωq and ωq , which can be experimentally measured using standard techniques, are shown in Figs. 4 and 5 for λ = 0 and D = 5. In conclusion I have studied the spin-one XY antiferromagnet with easy plane single ion and exchange anisotropies on the square lattice using the SU(3) Schwinger boson formalism. I have obtained the behavior of DC as a function of the exchange anisotropy parameter λ and calculated the energy gap as a function of λ and the temperature.
1 2 3 4
5
6 7 8 9 10 11 12 13 14 15
6
A.S.T. Pires / Physica A xx (xxxx) xxx–xxx
(1 )
Fig. 4. (Color on line) The dispersion relation ωq /(−µ + D) for λ = 0 and D = 5.
(2 )
Fig. 5. (Color on line) The dispersion relation ωq /(−µ + D) for λ = 0 and D = 5.
Acknowledgment
1
3
This work was partially supported by Conselho Nacional de Pesquisa e Fundacao de Amparo a pesquisa do estado de Minas Gerais.
4
References
2
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Q4
[1] Z. Zhang, K. Wierschem, I. Yap, Y. Kato, C.D. Batista, P. Sengupta, Phys. Rev. B 87 (2013) 174405. [2] A.S.T. Pires, Physica A 391 (2012) 5433; A.S.T. Pires, Physica A 373 (2007) 387; A.S.T. Pires, Physica A 390 (2011) 2787; A.S.T. Pires, B.V. Costa, Physica A 388 (2009) 3779. [3] J. Oitmaa, C.J. Hamer, Phys. Rev. B 77 (2008) 224435. [4] T. Roscilde, S. Haas, Phys. Rev. Lett. 99 (2007) 047205. [5] L. Yin, J.S. Xia, V.S. Zapf, N.S. Sullivan, A. Paduan-Filho, Phys. Rev. Lett. 101 (2008) 187205. [6] H.T. Wang, Y. Wang, Phys. Rev. B 71 (2005) 104429. [7] H.T. Wang, Y. Wang, J. Phys.: Condens. Matter 19 (2007) 386227. [8] D. Reys, A. Paduan-Filho, M.A. Continentino, Phys. Rev. B 77 (2008) 052405; D. Reys, M. Continentino, Phys. Rev. B 76 (2007) 075114. [9] H.T. Wang, J.L. Shen, Z.B. Su, Phys. Rev. B 56 (1997) 14435. [10] B.V. Costa, A.S.T. Pires, Phys. Rev. 64 (2001) 092407; A.S.T. Pires, J. Magn. Magn. Mater. 323 (2011) 1977; A.S.T. Pires, J. Magn. Magn. Mater. 370 (2014) 106. [11] A.S.T. Pires, Solid State Commun. 152 (2012) 1838; A.S.T. Pires, Solid State Commun. 196 (2014) 24; A.S.T. Pires, Solid State Commun. 186 (2014) 1; A.R. Moura, A.S.T. Pires, A.R. Pereira, J. Magn. Magn. Mater. 357 (2014) 45. [12] R.A. Muniz, Y. Kato, C.D. Batista, Prog. Theor. Exp. Phys. (2014) 083101. [13] N. Papanicolaou, Nuclear Phys. B 305 (1988) 367. [14] S. Sachdev, Quantum Phase Transitions, Cambridge, University Press, Cambridge, UK, 1999.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Quantum phase transition in the two dimensional XY model with single ion and exchange anisotropies Q1
A.S.T. Pires ∗ Departamento de Fisica, Universidade Federal de Minas Gerais, Belo Horizonte, MG, CP 702, 3012-970, Brazil
highlights • The SU(3) Schwinger boson formalism is used in this paper. • The phase diagram at zero temperature is obtained. • The energy gap as a function of the anisotropy parameter and temperature is calculated.
article
info
Article history: Received 7 January 2015 Received in revised form 15 April 2015 Available online xxxx Keywords: A. XY model A. Antiferromagnet D. Quantum phase transitions
abstract I study the spin-one XY antiferromagnet with easy plane single ion and exchange anisotropies on the square lattice using the SU(3) Schwinger boson formalism. The phase diagram at zero temperature is obtained. The gap, in the disordered phase, as a function of temperature is presented. © 2015 Published by Elsevier B.V.
1. Introduction
1
The study of low dimensional quantum magnetic models has played an important role in the understanding of phase transitions and critical behavior in condensed matter. As it is well known classical phase transitions are driven only by thermal fluctuations. In contrast, fluctuations driven by the uncertainty principle can drive phase transitions, in the ground state, even at zero temperature. These transitions are called quantum phase transitions (QPT). They take place by changing some parameters in the Hamiltonian, instead of the temperature. The subject of QPT is very interesting and has motivated a lot of papers and some books [1] (and references there in). In this context a model of great interest is the spin-one antiferromagnet with single-ion anisotropy. This model has been studied theoretically using analytical and numerical techniques, with the advantage that experimental works are also available [1–5]. These systems become quantum paramagnets for sufficiently high values of the easy-plane single-ion anisotropy. This is purely a quantum effect without a counterpart in the classical models [1]. The large D phase is disordered in the sense that ⟨S ⟩ = 0, but one has ⟨(S x )2 ⟩ = ⟨(S y )2 ⟩ ̸= ⟨(S z )2 ⟩ which implies a nematic order [1]. Since all kinds of anisotropies can be presented in a real compound, in this paper I will study the XY antiferromagnetic model with spin S = 1 on a square lattice described by the following Hamiltonian:
J x x (Sr Sr +δ + λSry Sry+δ ) + D H = (Srz )2 . 2 r ,δ
∗
Tel.: +55 31 34996224; fax: +55 31 34996600. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.physa.2015.05.114 0378-4371/© 2015 Published by Elsevier B.V.
r
(1)
Q2
2 3 4 5 6 7 8 9 10 11 12 13 14 15
16
2
A.S.T. Pires / Physica A xx (xxxx) xxx–xxx
10
As shown by Wang and Wang [6], when D > 0 the XY model has the same properties as the ones for the Heisenberg model and it has the advantage that we do not need to do a decoupling for four-operator terms. Here, I will consider the case 0 ≤ λ ≤ 1. When λ = 0 we have an Ising like system. z = 0 separated by a gap from the first In the limit of infinite D, the model (1) will be in a disordered ground state with Stotal z = ±1. Therefore there exists a critical DC denoting a quantum phase transition, excited states, which lie in the sectors Stotal described by the condensation of magnons, from the large-D phase to the ordered phase. Although the model with λ = 1 has been intensively studied [6–11], the case λ ̸= 1, as far as I know, has not being treated before. In this paper, I use the Su(3) Schwinger boson formalism to study the phase diagram and critical properties of the Hamiltonian (1). In Section 2, I present the formalism used and in Section 3, the results of my calculations.
11
2. Schwinger boson formalism
1 2 3 4 5 6 7 8 9
12 13 14 15 16 17 18 19
20 21 22
23 24
25 26 27 28 29
30 31
32
33 34 35 36 37
38
39 40 41
42 43 44 45
46
Traditional spin wave Su(2) theory is a very good approach for treating quantum spin models that have a magnetically (dipolar) ordered ground state. The theory however is not adequate to treat nematic, quadrupolar, octupolar, or higher multipolar ordering, because, as pointed out by Muniz et al. [12], such ground state orderings do not have a classical counterpart at zero temperature. In these cases the local fluctuations of a spin are described by the SU(2S + 1) group of local unitary transformation, instead of the Su(2) group of rotations. (It should be noted that the N 2 − 1 components of the local SU(N ) order parameter can be decomposed in the different irreducible representations of Su(2).) To treat magnetic systems with single site anisotropy, Papanicolaou [13] derived a theory which is a generalization Su(3) of the Su(2) Schwinger boson approach. In this formalism we choose the following basis:
√ |y⟩ = (|1⟩ + | − 1⟩)/ 2,
√ |x⟩ = i(|1⟩ − | − 1⟩)/ 2,
|z ⟩ = −i|0⟩
(2)
where |n⟩ are eigenstates of S . The spin operators are then written via a set of three boson operators tα (α = x, y, z ) defined by z
tx+ |v⟩ = |x⟩ ,
ty+ |v⟩ = |y⟩ ,
tz+ |v⟩ = |z ⟩ ,
(3)
where |v⟩ is the vacuum state. To force single site occupancy on each site we impose the constraint tx+ tx + ty+ ty + tz+ tz = 1.
(4)
The constraint (4) projects the bosonic operators into the physical space of dimension 3. The Su(3) generators are bilinear forms: + Omn = trm trn ,
(5)
that satisfies the Su(3) commutation relations [12]:
[Omn , Oij ] = δni Omj − δ mj Oin .
(6)
In terms of the t operators we can write S x = −i(ty+ tz − tz+ ty ),
S y = −i(tz+ tx − tx+ tz ),
S z = −i(tx+ ty − ty+ tx ).
(7)
The states tx+ |v⟩ and ty+ |v⟩, both consist of S z = ±1 eigenstates and have the average ⟨S z ⟩ = 0. Papanicolaou has pointed out that the main motivation for this formalism was to eliminate certain difficulties of the standard semiclassical theory which become crucial for spin systems with single-site (or biquadratic) interactions. In contrast, the Su(2) Schwinger boson formalism does not lead to a quantum phase transition when used in Hamiltonian (1). To study the paramagnetic phase, it is convenient to introduce two bosonic operators u+ and d+ given by [6] 1 u+ = − √ (tx+ + ity+ ), 2
1 d+ = √ (tx+ − ity+ ), 2
(8)
and so
|1⟩ = u+ |v⟩ ,
|0⟩ = tz+ |v⟩ , +
+
|−1⟩ = d+ |v⟩ ,
(9)
+
with the constraint u u + d d + tz tz = 1. The spin operators are now written as S+ =
√
2(tz+ d + u+ tz ),
S− =
√
2(d+ tz + tz+ u),
S z = u+ u − d + d .
(10)
To treat ordered phases we can extend the Holstein–Primakoff representation from Su(2) to Su(3) by condensing one of the 3 Schwinger bosons (SB). The SB that is condensed is the one that creates the local state that minimizes the mean-field energy [12]: + trz , trz →
+ + 1 − trx trx − try try .
(11)
A.S.T. Pires / Physica A xx (xxxx) xxx–xxx
3
However for the disordered phase of a XY like system (as is the case studied here), Zhang et al. [1] and Wang [6] have shown a much better approximation is to condense the tz boson: i.e. ⟨tz ⟩ = ⟨tz+ ⟩ = t. (For a comparison of the two approximations see Ref. [1].) This approach is more appropriate for describing gapped phases that do not break any continuous symmetry. Substituting (10) into the Hamiltonian (1) and condensing the tz boson, as mentioned above, we obtain: 2
Jt
H =
2
+ + + [(1 + λ)(d+ r dr +δ + ur +δ ur + ur dr +δ + dr ur +δ + H .c .)
2 3 4
5
r ,δ
+ + (1 − λ)(dr dr +δ + dr u+ r +δ + ur ur +γ + ur dr +γ )] + + 2 +D (u+ µ r ( u+ r ur + d r d r ) − r ur + dr dr + t − 1). r
6
(12)
+ Ψq+ = (u+ q , dq , u−q , d−q ),
(13)
we find that the Hamiltonian (1), after a Fourier transformation, can be written as: 1 2
7
r
A temperature-dependent chemical potential µr has been introduced to impose the local constraint Sr2 = S (S + 1) = 2. In a mean-field approximation, we replace the local constraint by a global one and let [6] µr = µ. Defining the generalized Nambu spinor
H =
1
Ψq+ M Ψq + µN (1 − t 2 ) −
q
Λq ,
8 9 10
11 12
(14)
13
q
where the last term in Eq. (14) appears when we write the Hamiltonian in a symmetrical form, and
Γq + r Γ˜ q M = Γ˜ q Γq
Γ˜ q Γq + r Γq Γ˜ q
Γ˜ q Γq Γq + r Γ˜ q
Γq Γ˜ q , Γ˜ q Γq + r
14
(15)
with
15
16
t
Γq =
2
2 1
γq =
2
(1 + λ)γq ,
Γ˜ q =
t
2
2
(1 − λ)γq ,
r =
1 2
(D − µ),
(cos qx + cos qy ).
(16)
17
(17)
18
I have set J = 1. Green‘s function for Hamiltonian (1) is given by
19
G(k, iωn ) = (iωn σ z ⊗ I − Hαα )−1 ,
(18)
where ωn is the Matsubara frequency, σz is the Pauli matrix and I a 4 × 4 unit matrix. The theory of Green’s functions can be found in any book treating many bodies in condensed matter. From Eq. (15), writing z = iω, we obtain (1) 2
(2) 2
det G
(q, z ) = (z − (ωq ) )(z − (ωq ) ),
ωq(1) =
−1
2
2
(19)
with
20 21 22
23 24
ωq(2) =
Λ2q − ∆2q + δ ∆q (∆q + Λq ),
Λ2q − ∆2q − δ ∆q (∆q + Λq ),
(20)
where
∆q =
26
1 2
(1 + λ)zt 2 γq ,
Λq = −µ + D + ∆q ,
δ=
(1 − λ) 2(1 + λ)
.
(21)
The mean field ground state energy per site is e0 =
25
1 2
(ωq(1) + ωq(2) ) −
q
28
Λq + µ(1 − t 2 ).
(22)
29
q
The Gibbs free energy is then written as: G = Ne0 −
27
1
β
ln[1 + n(ωq(1) )] −
q
with n(ωq ) = 1/(eβωq − 1).
30
1
β
ln[1 + n(ωq(2) )],
(23)
31
q 32
4
A.S.T. Pires / Physica A xx (xxxx) xxx–xxx
Fig. 1. The critical anisotropy parameter DC as a function of λ. Below the line the system is in the Néel phase, while above it is in the disordered phase. 1
2
3 4 5
6
7 8 9 10
11 12 13 14
15
16
Defining the variable y by y = (1 + λ)zt 2 /(−µ + D) we can write
ωq(1) = (−µ + D) (1 + yγq )(1 + yδγq ),
ωq(2) = (−µ + D) (1 + yγq )(1 − yδγq ).
(24)
(1)
= ωq(2) for λ = 1 (as it should be). We see that (1) (2) taking ∆ = 1/λ we have δ = −(1 − ∆)/[2(1 + ∆)] and all the results are the same with the substitution ωq ↔ ωq . (2) For γq = 1, y = 1 (y is always less than 1 in the disordered phase) and taking z = 4 we can write ωq as ωq(2) = −µ + D − t 2 (1 − λ). (25) Both components of the dispersion relation vanish at q = (π , π ), and ωq
Since µ < 0, 0 ≤ λ ≤ 1, D ≥ DC and the maximum value of t 2 is 1 we see that the quasiparticle energies are always well (1) defined (the same discussion can be applied to ωq ). When y → 1, the energy gap at q = (π , π ) goes to zero, indicating a transition from the large-D phase to the Néel phase. We determine DC as the value of D where the gap vanishes. The energy gap is given by ) ) m = ωq(1=π = ωq(2=π ,
(26)
and is interpreted by some authors [14] as the mass of the quasiparticles. Extremizing the free energy (23) with respect to the parameters t 2 and µ we obtain the following self-consistent equations: 1
2 + yγq (1 + δ/2)
2 + yγ q (1 − δ/2)
+ (1 + n(q1) + n(q2) ), ( 1 + y γ )( 1 + y δγ ) ( 1 + y γ )( 1 − y δγ ) q q q q q 1 + δ(1 + 2yγq )/2 1 − δ(1 + 2yγq )/2 µ = 2(1 + λ) γq + (1 + n(q1) + n(q2) ). ( 1 + y γ )( 1 + y δγ ) ( 1 + y γ )( 1 − y δγ ) q q q q q 2
2−t =
4
(27)
(28)
18
We solve numerically these self consistent equations to find t 2 and µ for each value of λ and D (and T if we are doing calculations at T > 0).
19
3. Results and conclusions
17
20 21 22 23 24 25 26
Fig. 1 shows the critical parameter DC as a function of λ. As we can see we find it is a straight line that can be fitted to the equation DC = 2.75 + 2.968λ.
(29)
The parameter DC increases with the increasing of λ. This should be intuitively expected since the Ising model is almost a classical one and in the classical limit DC → ∞. Below the line the system is in the Néel phase. Above the line it is in Q3 the paramagnetic phase. In Fig. 2, I present the gap, given by Eq. (26), as function of λ, for D = 8 and D = 12. The gap decreases with the increasing of λ. I have found numerically that, for a given value of λ, the energy gap decreases linearly
A.S.T. Pires / Physica A xx (xxxx) xxx–xxx
5
Fig. 2. The gap m as a function of λ. Red line D = 12, black line D = 8. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. The gap as a function of temperature. Here D = 10. Red line λ = 0, black line λ = 0.5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
with decreasing D until DC , i.e. m ∝ (D − DC ). From Eq. (22) one finds that D − DC at λ = 0 is larger than at λ = 1. This explains the behavior found in Fig. 2. The energy gap, as given by Eq. (26), as a function of temperature is depicted in Fig. 3, for D = 10, and λ = 0, and λ = 0.5. The curve can be fitted to the following expression: m = m0 + c1 T 1/2 exp(−c2 /T ),
(30)
where c1 and c2 are constants which depend on D and λ and should be calculated numerically to fit the curve. The parameter m0 is the value of the gap at zero temperature. One important relation is that the correlation length ξ , for the spin–spin correlation function, is proportional to 1/m0 [14] and this quantity decreases with the temperature. A nonzero temperature induces an exponentially small density of thermally excited magnons. The expression (30) is in agreement with analytical calculations using scaling arguments [14]. (1)
(2)
The spin wave dispersions ωq and ωq , which can be experimentally measured using standard techniques, are shown in Figs. 4 and 5 for λ = 0 and D = 5. In conclusion I have studied the spin-one XY antiferromagnet with easy plane single ion and exchange anisotropies on the square lattice using the SU(3) Schwinger boson formalism. I have obtained the behavior of DC as a function of the exchange anisotropy parameter λ and calculated the energy gap as a function of λ and the temperature.
1 2 3 4
5
6 7 8 9 10 11 12 13 14 15
6
A.S.T. Pires / Physica A xx (xxxx) xxx–xxx
(1 )
Fig. 4. (Color on line) The dispersion relation ωq /(−µ + D) for λ = 0 and D = 5.
(2 )
Fig. 5. (Color on line) The dispersion relation ωq /(−µ + D) for λ = 0 and D = 5.
Acknowledgment
1
3
This work was partially supported by Conselho Nacional de Pesquisa e Fundacao de Amparo a pesquisa do estado de Minas Gerais.
4
References
2
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Q4
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