Magnetic properties of the spin-1 heisenberg antiferromagnet on an interpolating square – triangular lattice in Popop-Fedotov representation

We study the S=1 anisotropic antiferomagnetic Heisenberg model on a square lattice for all temperatures using the Popov-Fedotov representation of spin operators. In this representation the constraint of single particle site occupation is rigorously fulfilled by introducing an imaginary valued chemical potential.

MAGNETIC PROPERTIES OF THE SPIN

MAGNETIC PROPERTIES OF THE SPIN 1 HEISENBERG 1 HEISENBERG

ANTIFERROMAGNET ON AN INTERPOLATING SQUARE

ANTIFERROMAGNET ON AN INTERPOLATING SQUARE –

TRIANGULAR LATTICE IN POPOP

TRIANGULAR LATTICE IN POPOP FEDOTOV REPRESENTATION FEDOTOV REPRESENTATION FEDOTOV REPRESENTATION

Pham Thi Thanh Nga 1 , Trinh Thi Thuy 2 , Nguyen Toan Thang 3

1

Thuyloi University

2

Hanoi National University of Education

3

Institute of Physics

Abstract:

Abstract: We study the S=1 anisotropic antiferomagnetic Heisenberg model on a square

lattice for all temperatures using the Popov-Fedotov representation of spin operators In this representation the constraint of single particle site occupation is rigorously fulfilled

by introducing an imaginary valued chemical potential We consider the model with antiferromagnetic interactions J 1 between nearest neighbors and J 2 bond between next nearest neighbors in only one of the diagonal (the same one) in each square plaquette

We investigate a Neel antiferromagnetic phase and an incomensurate spiral phase in one loop approximation The analytical results are derived and numerically investigated for some anisotropic parameter values

Key

Keywords: words: words: Spin-1, Heisenberg antiferromagnet, interpolating square-triangular lattice,

Popov-Fedotov representations

Email: nga_ptt@tlu.edu.vn

Received 6 September 2018

Accepted for publication 15 December 2018

1 INTRODUCTION

The properties of two-dimensional Heisenberg models are of considerable current interest.The interplay between fluctuations and frustration (either interaction or geometric)

of these models has been studied intensively [1] The properties of low dimensional Heisenberg systems are quite well understood in the absence of frustration [2] On the contrary, the case of frustrated models is less clear, for the interplay between frustration and quantum fluctuations in these systems can produce much richer and fascinating phenomena Both effects might lead to destroying the magnetic order of the spin lattice and

be responsible for new quantum phase One of the most studied of such models is the

spin-½ Heisenberg systems on the square lattice with competing nearest neighbor

antiferromagnetic interaction J1 and next nearest neighbor antiferromagnetic bonds J2 It is

generally believed that depending on value of α =J2/J1 different magnetic long-range order (LRO) may exist For α α< 1≈0.4the ground state is Neel LRO, whereas for

2 0.6

α α> ≈ there exists collinear stripe ordered state [3] Besides, an intermediate paramagnetic phase emerges in the interval α α< 1<α2 In principle, the various magnetic phases may appear depending on the type of lattice (square, triangular ), on the magnetic bonds and also on the magnitude of the spin quantum numbers of the atoms on the lattice sites [3] It is thus of great interest to study the spin system with spin S ≠1 / 2 The spin quantum number S can play an considerable role in the formation of the various magnetic phase A well-known example of this kind is the Haldane phase for S = 1 one dimensional Heisenberg model, which is absent in his S = ½ counter-parts [4] In this paper we study the S = 1 antiferromagnetic Heisenberg model on a square lattice with nearest neighbor

bond J1> 0 and next nearest neighbour J2 ≥0 along one of the diogonals of the squares as shown in Fig 1

Note that it is interpolating square-triangular lattice

model In the limit J2 =0 the model reduces to the square

lattice antiferromagnet while in the limit J2 =J1 the model

becomes topologicaly equipvalent to the isotropic

antiferromagnet Heisenberg model on triangular lattice

The spin-½ model has been studied recently by the linear

spin wave theory [3], the couple cluster method [5], the

series expansion [6,7], To our knowledge only a few

works has been done for the model with spin quantum

number S > ½ [7,8]

Fig 1:

Fig 1: Square lattice with bond

J 1 along the horizontal and vertical axis and J 2 along the

diagonal

One simple method used for investigating the magnetic order phases of the Heisenberg systems is linear spin wave theory based on the bosonic and fermionic representation for the spin operators The difficulty with the representation of spin operators is due to the fact that spin operators are neither Fermi or Bose operators The representation of spins as a bilinear form of auxiliary Fermi or Bose operators enlarges the Hilbert space, where these auxiliary operators act, so it introduces the unphysical states, which should be removed by some constraint condition The simplest way to resolve the constraint problem is the relaxiation of the local constraint on each lattice site by a so called global constraint, which

is fulfilled only on the average over all sites It is not clear whether such an approximation

is a good starting point for the description of Heisenberg model as temperature increases

Recently, Popov and Fedotovproposed an alternative approach for spin systems free of the

local constraintproblem, based on the fermionic representation for S = ½ and S = 1 with

imaginary chemical potential [9] Later, Veits et al have derived the generalization of the Popov-Fedotov method for arbitrary spin by introducing proper chemical potentials for spin fermions [10] However, the above papers are basically of a methodological nature Then, the Popov-Fedotov formalism has been applied to the spin-½ Heisenberg models on square lattice and on triangular lattice [11-14] In this paper, we study the spin-1 model by the Popov-Fedotov method following the results obtained in [15] We derive the analytical expressions for magnetization, free energy in one loop approximation and discuss the fluctuations around the mean-field results, taking into account the single occupacy condition We show that at finite temperature the exact local contraint changes results considerably

The paper is organized as follows In the Sec 2 we introduce the model and the

Popov-Fedotov formalism for S = 1 In the Sec.3 the results for sublattice magnetization,

free energy and specific heat are presented In the last section we shall conclude with our results

2 MODEL AND FORMALISM

Here, we study the Hamiltonian is given by:

ij

〈〈 〉〉

=∑ = ∑ + ∑ (1) where S i denotes the S = 1 spin vector operator We use the notation ij to denotenearest neighbor bonds with an antiferromagnetic exchange interaction J1and ij to denote

bond along the north-east diagonals with an antiferromagnetic exchange interaction J2 (See

Fig 1) The interactions J1and J2are competing and lead to magnetic frustration At first we consider the classical ground states, then we study the fluctuations around these states In two dimensional frustrated lattice described by Hamiltonian (1) the classical ground states have long range order, which are the collinear ordering and the spiral ordering Both

orderings are obtained in the following In the classical limit of S = ∞ we assume that the spins are planar in the plane Oxz and are described by some magnetic ordering vector Qas follows:

S i =usinQr i+vcosQr i (2)

where u v, are two orthogonal unit vectors in the plane Oxz The vector Q defines the relative orientation of the spins on the lattice, namely the angle between the vectors S i and

j

S is given by:

θij =θi−θj=Q r(i−r j) (3) Inserting S into Hamiltonian (1) we get the classical energy in terms of the ordering i

vector Q:

2

cl

E = NS J Q (4)

where N is site number of the lattice and J Q is the Fourier transform of the exchange ( )

integral:

1

i j

iQ r r

ij

here α =J2/J1 and we let the lattice constant a = 1 By minimizing (5) with respect to the

vector Qwe can obtain the magnetic ordering vector for each value of α By setting 0; ,

J

x z

Qλ λ

∂

∂ we find two kinds of ordered phases:i) Collinear Neel state is characterized by Q = ( π π , ) for the region 0 1

2

α

≤ ≤ For α = we have a square lattice 0 model.ii) Incommensurate spiral state with Q =( , )θ θ , where θ =arccos 1/ 2(− α) The case α = is topologicaly equipvalent to a triangular lattice which corresponds to the 1201 o spin structure For α = we have decoupled classical antiferromagnetic chains In order to 1 incorporate the fluctuations in a unique way for both ordered phases, it is convernient to

follow Miyake [16], introducing a local coordinate system on each site i, where the local z’ axis is oriented along its classical direction.Following Popov-Fedotov [12], the S = 1 spin

vector operators are written in terms of Fermi operators f iα+, f iα (α=1, 2, 3)

Si fiα αβ+τ fiβ ( α 1, 2, 3 )

= = (6) where the ταβare elements of S = 1 (3x3) matrices:

0 1 0 1

0 1 2

x

i

;

1

0 1 2

y

i

;

z S

(7)

The cost of the representation (6) is the extention of the Hilbert space into unphysical

sectors, which have to be eliminated by imposing a constraint for each site i:

i i

f fα α

α

=

∑ (8)

As it was shown by Popov-Fedotov [9], this can be done by means of introducing the projection operator ˆ i NˆF

eµ

Ρ = with ˆ

i

N f fα α

α

+

=∑ being number operator and µ=iπ / 3 , β β =k T B being the imaginary

chemical potential The partition of a spin system with Hamiltonian H s can be expressed in term of Fermi operators as follows

N

ˆ ˆ (H N

i

3

β µ

− −

  (9)

Here HˆF is Hamiltonian H s in the fermionic representation (6) The appearence of the imaginary chemical potential in (9) results in the modifying Fermi Matsubara frequences:

3

ω β

  (10)

The partition fuctionZ in (11) can be treated following the standard fuctional integral formalism similar to the case of S = ½ [14] The main differences are that instead of Pauli

(2x2) matrices and Matsubara frequencies 2 1

4

n

π µ β

  we have to deal with the spin S

= 1 (3x3) matrices (7) and Matsubara frequencies given by (10) The calculations are lenghthy but straightforward The explicit analytical results are obtained in [15], so we only list here the main steps Firstly, we express the partition fuction as part integral over Grassmann variables and transform the Heisenberg Hamiltonian into bilinear fermionic expression by introducing a Hubbard – Stratonovich boson vector field ϕi.Then we perform integration over the Grassmann variables to get the partition function in terms of auxiliry field.Following [13] we setϕi( )Ω =ϕio(Ω =0)+δϕ( )Ω , where ϕio(0) is the mean field part and δϕ Ω( ) is the fluctuation part of the auxilary field ϕ Ωi( ).Then one might derive explicit expresion for the effective action in the one-loop approximation The mean field ϕioα is chosen by minimizing the effective action in the application of the least action principle.Setting ( ) x( ) y( )

δϕ± Ω =δϕ Ω ±δϕ Ω , the partition function can be decomposed into a product of three terms Z=Z MF.Z Z zz +− where Z MF is the mean field contribution, Z zz

and Z +- are the fluctuation contributions from the longitudinal part δϕ ωz( ) and the transverse parts δϕ ω+( ), δϕ ω−( ) Consequently, the free energy may be written in the following way:

F =F MF +δF zz+δF+− (11)

3 MEAN-FIELD AND ONE-LOOP APPROXIMATIONS

Working in the local coordinate system, we are able to express in a unique form the results for both the Neel and the spiral state We introduce the common following notations

i) For Neel state : 0≤α≤ 1

2,

λ=2( 2−α)J 1 (12)

( ) ( )

( )

2

2

ω

α

α

α

−









x

x

x

p

p

p

(13)

ii) For the spiral state: 1

2

α >

1

J 1

2 (14)

( ) ( )

2 2

2 2 2

ω

α α

α α γ

α

+







−



x

x

x

p

p

p

(15)

Taking into acount the fact that in the local coordinate system ϕio=(0, 0, ϕo) for any

site i, we derive for the sublattice magnetization per site in the mean – field approximation:

2sinh

1 2cosh

o o

o

m m

m

βλ βλ

= + (16)

In the case of average constraint, instead of equation (16) one has:

Tanh

o o

m m

â

βλ

=  ɶ 

ɶ (17) Accordingly, the critical temperatures are different, namely, with the exact constraint

T C =2λ

3 , while T Co=λ

2 if the constraint is average It means

4 3

=

C Co

T

T , while for the case

of S = ½ one has C =2

Co

T

T [14 ] In both cases of S = 1 and S = ½ the critical temperature for the case of exact constraint is higher than for average constraint This is due to thermal fluctuations into unphysical spinless states, which reduce the magnetic moment In the case of spin S = ½, the number of physical states in the Fock space of the auxiliary fermion equals two (one particle states), while two states are unphysical (the vacuum and the two particle states) It results in the ratio C =2

Co

T

T For the case of spin S = 1, two states are unphysical (the vacuum and the three-particle states) Due to the particle-hole symmetry of the spin Hamiltonian those one particle states and three two-particle states are physical It results in the ratio 4

3

C Co

T

T = Howerver, at zero temperature both constraint methods give the same result:limT→0m o =limT→0mɶo=1 In Fig 2 and Fig 3 we plot the temperature dependence of the mean field sublattice magnetization for Neel and spiral states, respectively Fig 2 and Fig 3 show the difference of the results for the case of exact constraint and average one

The mean field free energy per site can be expressed in term of the magnetization m oas follows:

2

MF

F

m

β

= − + (18)

In the case of average projection instead of (18) one has:

ln 2cosh

o MF

o

F

m N

βλ λ

β

= ɶ −  ɶ  (19)

Fig.2 2 2:::: Temperature dependence of mean field

magnetization m o (for exact constraint: full line,

from right to left) and (for: average constraint:

dashed line, from right to left) for Neel state with

0

α = , α = 3 8/ , α = 0.5

Fig.

Fig.3 3 3:::: Temperature dependence of mean field

magnetization m o (for exact constraint: full line, from left to right) and mɶo (for: average

constraint: dashed line, from left to right) for incommensurate spiral phase with α =0.75,

1

α = , α =1.25

Again, at T = 0K, (18) and (18) lead to the same result

In one loop approximation we obtain the the contributions of the longitudinal and transverse fluctuations to the energy as follows:

0

1 ln 2

zz

p

δ

β

(20)

and

0

( )

s h

ln

s h 2

p

E p F

m

β δ

β λ β

(21)

while the longitudinal δm zz and transerve δm+− parts for the magnetization are given by:

1 2

p o

B m

1 1

1 coth 1

o p

o

o

m

m

β

ω

βλ

λ

+−

+

−

∑

(23)

We use the following notations:

2( )

( ) 1

w

x

p

p

γ

γ

−

βλ γ

o

C

2

( )

3

1 2

4 3

o

m

(27)

o o

o

C m

C

β βλ

The magnon energy reads:

E ( p ) = λ m ω ( p ); ω ( p ) = 1 − γ p 1 − γ p

(29) When the single occupancy condition is disregarded on the equations (20) – (24) the following replacement should be done:

1

2

;

1

o

o

C

C

β βλ









ɶ

ɶ

ɶ (30)

From the explicit expressions (18)-(21) of the free energy on may derive thermodynamic quantities by conventional means.It is easy to check that at zero temperature limit it does not matter whether the constraint is treated exactly or on the average: the magnetization and the free energy are the same for both cases of the constraint conditions Taking the limit of zero temperature T →0K for the equations (24), (25), we get the same sublattice magnetization obtained in linear spin wave approximation by mean

of Holstein-Primakov representation Letting T →0K in the equations (22), (23) one can see that the ground state energy in the same as the result obtained in linear spin wave approach [3].On the contrary, at finite temperature the exact constraint leads to the different values of the thermodynamic quantities, e.g., the magnetization, the internal energy, the specific heat In Fig.4 and Fig.5 we depict the temperature dependense for the

specific The difference in the finite temperatureresults of the two methods of treating the constraint conditions for both ordered states is obvious

Fig.

Fig.4 4 4:::: Temperature dependence of specific heat

(for exact constraint: full lineand for average

constraint: dashed line) for Neel state with

3 / 2

α = , α =1,α =0.75

Fig.

Fig.5 5 5:::: Temperature dependence of specific

heat (for exact constraint: full line and for average constraint: dashed line) for incommensurate spiral phase with α = 0,

3 / 8

α = , α =1/ 2

4 CONCLUSIONS

The Eqs (18), (22) - (31) are the main analytical results of the paper, where the free energy, the magnon energy and the sublattice magnetization are derived for both Neel and

spiral phases for S=1 Letting α=0 and α=1 at T=0K limit we recover the results for square and triangular lattice in Holstein-Primakov representation [17] At T=0K we get the ground

state properties of the antiferromagnet on the interpolating lattices consistent with the results obtained by mean of other linear spin wave theories [3] At the finite temperatures Popov-Fedotov approach gives significant differences in comparison with the other auxiliary particle methods treating the constraint on average At the parameter range of interest the present results agree with those of numerical study [7,8] It would be interesting to investigate the spin liquid phase of the spin -1 systems taking into account the exact constraint condition

Acknowledgments: This work was financially supported by the Vietnam Academy for

Science and Technology in the framework of Support project for high rank researchers under Grant NVCC05.02/18-18