научная статья по теме SPIN NEMATIC STATES IN SPIN-1 ANTIFERROMAGNETS WITH EASY-AXIS ANISOTROPY Физика

Текст научной статьи на тему «SPIN NEMATIC STATES IN SPIN-1 ANTIFERROMAGNETS WITH EASY-AXIS ANISOTROPY»

Pis'ma v ZhETF, vol. 97, iss. 2, pp. 114-118 © 2013 January 25

Spin nematic states in spin-1 antiferromagnets with easy-axis anisotropy

A. V. Sizanov*1, A. V. Syromyatnikov*+ 1 * Konstantinov Petersburg Nuclear Physics Institute, National Research Centre "Kurchatov Institute", 188300 Gatchina, Russia

+ Department of Physics, St.-Petersburg State University, 198504 St.-Petersburg, Russia

Submitted 17 December 2012

It is well known that spin nematic phases can appear in either frustrated magnets or in those described by Hamiltonians with large exotic non-Heisenberg terms like biquadratic exchange. We show in the present study that non-frustrated spin-1 1D, 2D, and 3D antiferromagnets with single-ion easy-axis anisotropy can show nematic phases in strong magnetic field. For 1D case we support our analytical results by numerical ones.

DOI: 10.7868/S0370274X13020094

1. Introduction. Frustrated spin systems have offered in recent years a wealth of opportunities for the study of a broad range of novel types of states and phase transitions. Spin nematic phases form a class of objects in this area which has received much attention. Spin nematic states are spin-liquid-like states which show a multiple-spin ordering without the conventional long-range magnetic order. The two-spin ordering can be generally described by the tensor [1]

QJ = (SfS'i;) - Sap{SjS,)/3. The symmetric part of Q^i describes a quadrupolar order which has been extensively studied both theoretically and experimentally in frustrated systems with ferromagnetic (FM) and anti-ferromagnetic (AF) nearest-neighbor and next-nearest-neighbor couplings, respectively, in strong magnetic field h (see, e.g., Ref. [2] and references therein) and in magnets with large non-Heisenberg spin couplings such as biquadratic exchange (SiS2)2 [3]. It has been also shown recently that quantum fluctuations accompanied by a sizable single-ion easy-axis anisotropy can also stabilize a nematic phase in the kagome spin-1 antiferromagnet at h = 0 [4].

It is well established that the attraction between magnons caused by frustration is the origin of quadrupo-lar and multipolar phases in quantum magnets [5]. In particular, the bottom of the one-magnon band lies above the lowest multi-magnon bound state at h = hs, where hs is the saturation field, as a result of this attraction in magnets with FM and AF couplings between nearest- and next-nearest neighbors, respectively. Then, transitions to nematic phases at h < hs in such systems are characterized by a softening of the multi-magnon bound-state spectrum rather than the one-magnon spectrum.

We show in the present paper that the magnon attraction arises also in spin-1 1D, 2D, and 3D non-frustrated AFs with easy-axis single-ion anisotropy that leads to stabilization of nematic phases in strong magnetic field. We support our analytical results by numerical ones in the particular case of AF chain.

2. Model and technique. We discuss axially symmetric spin-1 systems described by the Hamiltonian

H = £ JijSiSj + D ^(SZ)2 - h £ SZ, (1) (i,j) i where {i,j) denote spin pairs coupled with exchange constants Jij0, and D < 0 is the value of the singleion easy-axis anisotropy.

We examine in the present paper the possibility of the nematic phase formation below the saturation field hs by considering the transition from the fully polarized state that is discussed using the Holstein-Primakoff transformation

Sf — S — a] a,

(2)

S-

a]\ 2S — a]a,.

Expanding the square root in Eq. (2), putting all operators a] to the left of all a, using commutation relations, discarding terms containing more than three operators a] and a, (this is reasonable because one can neglect interaction of more than two particles in a dilute gas of magnons which arises at h « hs) [6], and substituting the resulting expressions for S- and S+ into Hamilto-nian (1) one obtains

H — Ho + %2 + H4 + ...,

(3)

H2 —Y<[SJp - SJo - D(2S - 1) + h] ap ap, (4)

1)e-mail: alexey.sizanov@gmail.com; syromyat@thd.pnpi.spb.ru

1,2,3,4

D+^J^-TS^+h)

a!a2a_3a_4, (5)

where N is the number of spins, I = 1 - y/1 - 1/25', the momentum conservation law i Pi = 0 is implied in Eq. (5), we omit some indexes p, and we set H0 =0 in the subsequent discussion.

At h > hs, the one-magnon spectrum and the magnon Green's function are given solely by H2, they are exact and have the form

ep = SJp - SJ0 - D(2S - 1) + h,

G(w, p) =

1

w — ep + iS

(6) (7)

Two-magnon bound states are examined via analysis of the pole structure of the two-particle vertex function r(w, q, p, k) (see Fig. 1), which can be found analyti-

p + q/2 k + q/2

z + q/2

r("""k,= X =X+>0<

-p + q/2 -k + q/2

-z + q/2

Fig. 1. Bethe-Salpeter equation for the four-particle vertex. Black points stand for bare vertices given by Eq. (5)

cally at h > hs. It can be shown that r(w, q, p, k) is given by a series of ladder diagrams in the fully polarized phase which lead to the Bethe-Salpeter equation for the vertex shown in Fig. 1. One can represent Jp in the following way on any Bravais lattice:

Jp = 2 Ja cos pa

(8)

where a enumerates exchange constants and summation over repeated Greek indexes is implied here and below. Using Eq. (8) we can represent the equation shown in Fig. 1 as follows:

r(w, q, p, k)=TV q, p, k) -

/D + Ja COS Pa COS Za-2 J7 SJa COS V(c os'Pa + COS Za)

V(q,z)

where |0) denotes the fully polarized state. Eq. (11) coincides with the conventional definition of the two-particle Green's function [7]

{T[aq/2+p(t1 )aq/2-p (t2) aq/2+k(t3)«q/2-k(t4)]) at

t1 = t2 = t and t3 = t4 = 0, where ap(t) denote operators in the Heisenberg representation. If there is a two-magnon bound state |2; q) with momentum q and energy e2(q), the function G11 has a pole at w = e2(q) near which it has the form

Gn[w ~ e2(q), q, p, k]

w - e2(q) + iS'

(12)

where

(P)

(2; q| al /2lpa\

q/2+^ q/2-p I"/

is a bound state wave function in momentum notation. Two-particle Green's function (11) is equal to the vertex r(w, q, p, k) multiplied by

JdwpdwkG(p+q/2)G(-p+q/2)G(k+q/2)G(-k + q/2)

= l/(w - £p+q/2 - £-p+q/2)(w - ek+q/2 - e-k+q/2)-

Consequently, r(w, q, p, k) has the form

r[w - e2(q), q, p, k]

/9(p)/9*(k) w - 62 (q) + iS

(13)

and the wave function (not normalized) is related to fq(p) as follows:

(P)

fq (P)

62(q) - 6p+q/2 - 6-p+q/2

(14)

Certainly, it can be obtained also from Eq. (9) that the vertex r[w ~ e2(q), q, p, k] near the pole can be represented in the form (13). It is also evident from Eq. (9) that fq(p) has the form

fq(p) = Aq + Ba cosp,

(15)

x r(w, q, z, k)

(2n)d

(9)

D(q, z) =

ill

= 2SJfi cos ^ cos z,3 - SJ0 - D(2S - 1) + h - (10)

where d is a lattice dimensionality. Let us consider the following two-particle Green's function:

Gn(w, q, p, k) = dt x Jo

X (0|aq/2+paq/2_pe-i(H-w)iaq /2lka.

(11)

q/2+kal/2-kl

because a counts independent Fourier modes. Substituting Eqs. (13) and (15) into Eq. (9) and neglecting r0(w, q, p, k) we obtain the following set of equations

for Aq and Bq :

M (q, D, h, w)

Aq

Aq

Br1

Ba=2

q

\

(16)

/

where M(q, D, h, w) is a square matrix with the following elements:

z

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8*

Moo = 1 + Dg- 2TSJ° cos \va,

a?

Moa = Dva - 'ITSJ1 COS \lfia.,

Mao = Java — 2gFSJa cos

aa

(17)

aa

Map = 5aj3 + JaIafi - 2TSJa cos —vp,

where

I a?

(2n)d D(a, z)' ddz cos za

WW

z cos z cos z

(2n)d D(a, z)

(18)

and D(a, z) is given by Eq. (10). Then, if determinant of M is zero at some w, Eq. (16) has a non-trivial solution and the vertex has a pole at the corresponding w and q. Otherwise, the residue is zero and there is no bound state at the corresponding w and q.

One notes that parameters (18) depend on D, h, and w only in the combination

h = h — w/2 — D(2S — 1)

and M(a, D, h, w) is actually a function of (D, h). Considering D and h as independent variables, we note that det M(D, h) is a linear function of D. It signifies that there exists a unique D value at which det M(D, h) = 0 for any h larger than some value ho such that denominator (10) is positive at h > h0. Then, the two-magnon bound state exists at h > h0 and the critical field at which the two-magnon bound state spectrum becomes gapless is higher than the critical field of the one-magnon mode (if these critical fields exist 2)).

The above equations are applicable for consideration of two-magnon bound states in strong magnetic field on any Bravais lattice with any exchange interactions and any S > 1. They can be easily extended also to the case of anisotropic exchange. We apply the above equations now to the particular spin-1 model on a tetragonal lattice with different unfrustrated exchange constants along z-axis (Jz) and in xy plane (Jxy) which are parametrized as follows:

Jz = cos 6, Jxy = sin 6, — n < 6 < n.

(19)

The cases of 6 = 0 and n correspond to 1D AF and FM chains, respectively, 6 = ±n/2 correspond to 2D AFs and FMs on the square lattice. Other 6 describe axially symmetric 3D magnets.

3. General consideration. One has Jp = = 2JZ cospz + 2Jxy(cospx + cospy) and we find using the above formulas that the bound state at strong field exists at any 6 if D < Dc(6). The bound state spectrum is always quadratic near it's minimum p = 0: 62(p) « 2(h — hs) + Czpl + Cxy(p2x + p2y). Function Dc(6) is shown in Fig. 2 by red solid line. Behavior of

2) In a range of parameters bound states are gapped at any h.

Fig. 2. Solid red line (black dots) represents Dc(6) curve (see text). Colored area is the nematic phase stability region

Dc(6) near its singularities which correspond to 1D and 2D systems can be found using (18) with the following result:

Dc{6 - 0) « -4/3 + const x ^

Dc(0 ~ tt/2) « -8/3 + ,C°nSt; ,, V ' ' ' ln|0 - 7T/2|

const

DJ6--tt/2 « —-—,

V ' ' In \9 — 7t/2|

Dc(6 - ±tt) « const X y/\e=fir\.

The bound state can formally become gapless upon the field decreasing if 6 lies in the range —0.197n < 6 < < 0.905n. In this interval, Bose condensate of bound state quasiparticles

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