Spin-1/2 isotropic XY chain with Dzyaloshinskii-Moriya interaction in random lorentzian transverse field

36 Pages·0.3 MB·English

by  O. V. Derzhko

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9 National Academy of Sciences of Ukraine
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Institute for Condensed Matter Physics

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ICMP - 96 - 25E
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m O.V. Derzhko, T.M. Verkholyak
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d Spin-1 isotropic XY chain with Dzyaloshinskii-Moriya
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n interaction in random lorentzian transverse field
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0 Abstract
0
2 The exact results for thermodynamical properties of one-di-
0 mensional spin-1 isotropic XY model with Dzyaloshinskii-Mo-
2
7 riyainteractioninrandomlorentziantransversefieldareobtained.
9 This permitsto discuss some approximatemethodsof disordered
/ spin systems theory. The approximate scheme of examining the
t
a thermodynamicsofone-dimensionalspin-1 XXZ Heisenbergmo-
m 2
del with Dzyaloshinskii-Moriya interaction in random lorentzian
- field is suggested.
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r L’viv - 1996
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Shortly after famous paper by E.Lieb, T.Schultz and D.Mattis [1], who
makinguseofJordan-Wignertransformationreformulatedthe Hamilto-
nian of spin s = 1 XY chain in terms of non-interacting fermions and
2
obtaineda seriesofrigorousresults,the randomversionsofsuchmodels
attractmuchattention. SomeexactresultswerederivedbyH.Nishimori
[2] for isotropic XY chain in random lorentzian transverse field. It be-
camepossiblebecauseofthefactthatafterfermionizationofsuchmodel
onecomesto asystemofelectronsonlatticethatmaytransferfromsite
to site with the random (lorentzian) energy at sites. The average one-
particle Green’s function for such model was obtained first by P.Lloyd
[3] (see also [4, 5, 6, 7]).
What follows is based on the notation that similarly to [2] one can
considerthecaseofmorecomplicateinterspininteractionincludingtoit
the so-called Dzyaloshinskii-Moriya interaction. It was introduced phe-
nomenologicallyby I.E.Dzyaloshinskii[8] and then derived by T.Moriya
[9] and it is widely used as one of microscopical mechanism (together
with ANNNI model) of appearance of incommensurate phase in crys-
tals [10]. V.M.Kontorovich and V.M.Tsukernik noted [11] that taking
into account of such interaction in s = 1 XY chain does not destroy
2
the consideration proposed in Ref. [1] since after Jordan-Wigner trans-
formation, as in the previous case, one comes to a quadratic in Fermi
operators form. In Ref. [11] the problem about the possibility of ap-
pearance of spiral spin structure in such a chain was examined; for this
purpose the pair equal-time spin correlation functions were evaluated.
Other papers dealing with statisticalmecanics ofs= 1 XY chains with
2
Dzyaloshinskii-Moriyainteraction [12, 13, 14, 15, 16, 17, 18, 19] like the
Ref. [11] are devoted to perfect (non-random) versions of the model.
Thepresentpapercontainssomeexactresultsofstatisticalmechanics
of1D s= 1 isotropicXY modelwithDzyaloshinskii-Moriyainteraction
2
in random lorentzian field. The paper is organized as follows. Fermion-
ization and Lloyd’s problem are considered in Section 1. Here the av-
erage one-fermion Green’s functions, the average spectral density and
theaveragefermioncorrelationfunctionsarederived. Section2contains
calculations of thermodynamical properties of the model in question.
A discussion of the estimation of static spin correlations is also given
in it. The comparison of exact results with the ones obtained within
different approximate approaches (Bose commutation rules approxima-
tion, Tyablikov-like approximation, coherent potential approximation)
are performed in Section 3. The developed in Sections 1 and 2 scheme
may be used for the approximate study of 1D s = 1 XXZ Heisen-
2
bergmodelwithDzyaloshinskii-Moriyainteractioninrandomlorentzian
1
field. This possibility is discussed in Section 4. Conclusions are given
in Section 5. Briefly the results of the present paper were reported in
[20, 21, 22].
1 Fermionization, Lloyd’s problem, average
one-fermion Green’s functions,average spec-
tral density, average fermion correlation
functions
A chain of N spins s = 1 with interaction between nearest neighbours,
2
thatareintransversefieldswithrandomcomponentdistributedaccord-
ing to lorentzian law is considered. The Hamiltonian of the model has
the form
N N 1
H = (Ω +Ω )sz +J − (sxsx +sysy )
0 j j j j+1 j j+1
j=1 j=1
X X
N 1
+D − (sxsy sysx )
j j+1− j j+1
j=1
X
N
1
= (Ω0+Ωj) s+j s−j − 2
j=1 (cid:18) (cid:19)
X
N 1
− J +iD J iD
+ 2 s+j s−j+1+ −2 s−j s+j+1 . (1.1)
j=1 (cid:18) (cid:19)
X
After Jordan-Wigner transformation
c1 =s−1,cj =(−2sz1)(−2sz2)...(−2szj−1)s−j , j =2,...,N,
c+ =s+,c+ =( 2sz)( 2sz)...( 2sz )s+, j =2,...,N,
1 1 j − 1 − 2 − j−1 j
c+,c =δ , c+,c+ =0, c ,c =0 (1.2)
{ i j} ij { i j } { i j}
one comes to the following quadratic in Fermi operators Hamiltonian
N
1
H = (Ω +Ω ) c+c
0 j j j − 2
j=1 (cid:18) (cid:19)
X
N 1
− J +iD J iD
+ c+c − c c+ , (1.3)
2 j j+1− 2 j j+1
j=1 (cid:18) (cid:19)
X
2
that can be treated like in Ref. [3].
Let’s introduce the following retarded and advanced temperature
two-times Green’s functions [23]
G (t) iθ( t)< c (t),c+(0) >. (1.4)
∓nm ≡∓ ± { n m }
ThegoaloffurtherconsiderationistofindtheaverageGreen’sfunctions
G∓nm(t), where the averagemeans
+ +
∞ ∞
(...) dΩ ... dΩ p(...,Ω ,...)(...),
1 N j
≡
Z−∞ Z−∞
N
1 Γ
p(...,Ω ,...)= (1.5)
j πΩ2+Γ2
j=1 j
Y
(i.e. Ω sareindependentlydistributedaccordingtothelorentzianprob-
j
ability distribution density centred at Ω =0 with the width Γ).
j
It is easy to get the equation of motion for (1.4), namely
d
i G (t)=δ(t)δ +(Ω +Ω )G (t)
dt ∓nm nm 0 n ∓nm
J +iD J iD
+ 2 G∓n+1,m(t)+ −2 G∓n−1,m(t). (1.6)
Using spectral representation of Green’s functions (1.4)
1 +∞
G (t)= dωexp( iωt)G (ω),
∓nm 2π − ∓nm
Z−∞
+
∞
G (ω)= dtexp(iωt)G (t) (1.7)
∓nm ∓nm
Z−∞
the equations (1.6) can be rewritten in the form
ωG (ω)=δ +(Ω +Ω )G (ω)
∓nm nm 0 n ∓nm
J +iD J iD
+ 2 G∓n+1,m(ω)+ −2 G∓n−1,m(ω), (1.8)
or in the form that is initial for locator expansion
G∓nm(ω)= ω (Ωδnm+Ω ) + J+2iDG∓n+1ω,m(ω()Ω++J−Ω2iD)G∓n−1,m(ω). (1.9)
0 n 0 n
− −
3
Formal solution as a series with respect to intersite interaction reads
δ
nm
G∓nm(ω)= ω (Ω +Ω )
0 n
−
J +iD 1 δ
n+1,m
+
2 ω (Ω +Ω )ω (Ω +Ω )
0 n 0 n+1
− −
J iD 1 δ
n 1,m
+ − − +.... (1.10)
2 ω (Ω +Ω )ω (Ω +Ω )
0 n 0 n 1
− − −
It is easy to average G (ω iǫ) presented as (1.10). Really, one
∓nm ±
should perform the averaging of [ 1 ]kj, i.e. to calculate the
integral ω±iǫ−(Ω0+Ωj)
1 +∞ Γ 1 kj
dΩ
π jΩ2+Γ2 ω iǫ (Ω +Ω )
Z−∞ j (cid:20) ± − 0 j (cid:21)
1 +∞ Γ
= dΩ .(1.11)
π Z−∞ j(Ωj +iΓ)(Ωj −iΓ)[ω±iǫ−(Ω0+Ωj)]kj
For retarded (advanced) Green’s function the integrand has one pole
in the lower (upper) half of the complex plane Ω and two poles in the
j
upper(lower)halfofthecomplexplaneΩ . Thusexpandingthecontour
j
of integration in the lower (upper) half of the complex plane and using
the residuum theory one ends up with
1 kj 1 kj
= (1.12)
ω iǫ (Ω +Ω ) ω iǫ (Ω iΓ)
(cid:20) ± − 0 j (cid:21) (cid:20) ± − 0∓ (cid:21)
that is the famous property of lorentzian distribution. In result the
average series (1.10) can be summed up with the result
δ
nm
G∓nm(ω iǫ)=
± ω iǫ (Ω iΓ)
0
± − ∓
+J+2iDG∓n+1,m(ω±iǫ)+ J−2iDG∓n−1,m(ω±iǫ). (1.13)
ω iǫ (Ω iΓ)
0
± − ∓
Since the average Green’s functions are translationally invariant the
equations (1.13) can be solved with the help of transformation
1
G∓nm(ω iǫ)= exp[i(n m)κ]G∓κ(ω iǫ), (1.14)
± N − ±
κ
X
4
κ= 2πn,n= N, N +1,...,N 1 for even N or n= N 1, N 1 +
N −2 −2 2 − − 2− − 2−
1,...,N2−1 for odd N; the solution of algebraic equation for G∓κ(ω±iǫ)
reads
1
G∓κ(ω iǫ)= ,
± ω [Ω +√J2+D2cos(κ+ϕ)] i(ǫ+Γ)
0
− ±
J D
cosϕ , sinϕ . (1.15)
≡ √J2+D2 ≡ √J2+D2
The averageGreen’s functions in site representationcan be found in re-
sultofsummationoverκin(1.14)thatinthermodynamicallimitreduces
to the following integration
1 π
G∓nm(ω iǫ)= dκexp[i(n m)κ]
± 2π − ×
Z−π
1
.(1.16)
ω (Ω + √J2+D2 exp[i(κ+ϕ)]+exp[ i(κ+ϕ)] i(ǫ+Γ))
− 0 2 { − }±
Setting z = exp[i(κ+ϕ)] one comes to the contour integral on unit
circle in complex plane z
exp[iϕ(n m)]
G∓nm(ω iǫ)= −
± − 2πi
zn m
−
dz . (1.17)
× √J2+D2z2 [ω Ω i(ǫ+Γ)]z+ √J2+D2
I 2 − − 0± 2
The denominator in the integrand in (1.17) should be presented as a
product a(z z )(z z ) where
1 2
− −
√J2+D2
a= ,
2
ω Ω i(ǫ+Γ) ω Ω i(ǫ+Γ) 2
0 0
z = − ± + − ± 1,
1
√J2+D2 s √J2+D2 −
(cid:20) (cid:21)
2
ω Ω i(ǫ+Γ) ω Ω i(ǫ+Γ)
0 0
z = − ± − ± 1, (1.18)
2
√J2+D2 −s √J2+D2 −
(cid:20) (cid:21)
and since z z = 1 z (z ) is over (in) the unit circle. For n m the
1 2 1 2
≥
integration yields
exp[iϕ(n m)]{ω−√ΩJ02±+i(Dǫ+2Γ) − [ω−√ΩJ02±+i(Dǫ+2Γ)]2−1}n−m
− (1,.19)
√J2+D2 [ω−√qΩJ02±+i(Dǫ+2Γ)]2−1
q
5
whereas for n m
≤
exp[iϕ(n m)]{ω−√ΩJ02±+i(Dǫ+2Γ) − [ω−√ΩJ02±+i(Dǫ+2Γ)]2−1}m−n
− (1..20)
√J2+D2 [ω−√qΩJ02±+i(Dǫ+2Γ)]2−1
q
Combing (1.19), (1.20) one finally gets for (1.17)
G∓nm(ω iǫ)=
±
exp[iϕ(n m)]{ω−√ΩJ02±+i(Dǫ+2Γ) − [ω−√ΩJ02±+i(Dǫ+2Γ)]2−1}|n−m|
− (1..21)
√J2+D2 [ω−q√ΩJ02±+i(Dǫ+2Γ)]2−1
q
The obtained result (1.21) gives the average elementary excitation
spectral density
1 1 1
ρ(E) δ(E Λj)= ImG−κ(E+iǫ)
≡ N − −πN
j κ
X X
1 1 1
= ImG−nn(E)= Im (.1.22)
−π −π (E Ω +iΓ)2 (J2+D2)
0
− −
It will be used in the next Section fopr examination of thermodynamical
properties of the model in question.
In order to estimate spin correlations one should calculate average
fermioncorrelationfunction <c+(0)c (t)> that canbe found fromthe
m n
relation [23]
<c+(0)c (t)>= 1Im ∞ dωexp( iωt)G−nm(ω+iǫ)(.1.23)
m n
−π − exp(βω)+1
Z−∞
Inparticular,thestaticaveragefermioncorrelationfunctionatlowtem-
pereture limit comes out from the following calculation
<c+(0)c (0)> <c+c >=<c+ (0)c (0)>
m m+p ≡ m m+p n−p n
1 0
=−πIm dωG−n,n−p(ω+iǫ)
Z−∞
1 −Ω [y+iγ (y+iγ)2 1]|p|
= Imexp(iϕp) dy − −
−π (y+iγ)2 1
Z−∞ p −
1 p
= π p Im exp(iϕp) y+iγ−p(y+iγ)2−1 | | |yy==−−Ω∞
| | (cid:26) h p i (cid:27)
6
1
= Im exp(iϕp)
π p {
| |
(ω2 γ2 1)2+4γ2ω2+ω2 γ2 1
ω + 0 − − 0 0 − −
0
×− s 2
p
 p
| |
(ω2 γ2 1)2+4γ2ω2 ω2+γ2+1
+iγ i 0 − − 0 − 0 ,(1.24)

− s 2 
p 

where y ω ω , ω Ω0 , γ ǫ+Γ . Thederived
≡ √J2+D2 − 0 0 ≡ √J2+D2 ≡ √J2+D2
results(1.15),(1.21),(1.22),(1.24)remindthecorrespondingexpressions
obtained in slightly different cases in Refs. [24, 25].
2 Thermodynamical properties. The influ-
ence of Dzyaloshinskii-Moriya interaction
The obtained in the previous Section results are of great use in un-
derstanding the thermodynamical properties of the model in question.
Really, consider a model with a certain realization of transverse fields
at sites. After exploiting Jordan-Wigner transformation (1.2) for the
Hamiltonian(1.1)onecomestoaquadraticinFermioperatorsformthat
N
can be diagonalized by a canonical transformation η = (g c +
k j=1 kj j
h c+) [1] (see also [26, 27]) with the result H = N Λ (η+η 1).
kj j k=1 Pk k k − 2
Elementary excitation spectrum Λ and the coefficients g and h are
k kj kj
P
determined from Λ g = N g A , Λ h = N h A , where
A = (Ω +Ω )δ k+knJ+iDδi=1 +ki JiniD−δ k k.nThe cia=l1culkaitio∗inn of free
ij 0 i ij 2 Pj,i+1 −2 j,i−1 P
energy per site for this realization is straightforward
1 1 βΛ βΛ
k k
f = lim ln exp +exp
N→∞N (−β k (cid:20) (cid:18)− 2 (cid:19) (cid:18) 2 (cid:19)(cid:21))
Y
1 βE
= dEρ(E)ln 2cosh ,
−β 2
Z (cid:18) (cid:19)
1
ρ(E) lim δ(E Λ ) (2.1)
k
≡N N −
→∞ k
X
and the result of its averaging over configuration is given by
1 βE
f = dEρ(E)ln 2cosh
−β 2
Z (cid:18) (cid:19)
7
1 Ω
0
= dEρ(E)ln[1+exp( βE)] , (2.2)
−β − − 2
Z
where ρ(E) is the average spectral density that has been found in the
Section 1 (formula (1.22)).
Further one finds the internal energy
∂f E Ω
0
e=f +β = dEρ(E) , (2.3)
∂β 1+exp(βE) − 2
Z
the entropy
∂f βE
s=β2 = dEρ(E) ln[1+exp( βE)]+ (2,.4)
∂β − 1+exp(βE)
Z (cid:26) (cid:27)
and the specific heat
∂s E2
c= β =β2 dEρ(E) . (2.5)
− ∂β (2coshβE)2
Z 2
Using magical property of ρ(E) (1.22)
∂ ∂ 1 1
ρ(E)= Im
∂Ω0 ∂Ω0 "−π (E Ω0+iΓ)2 (J2+D2)#
− −
∂
p
= ρ(E) (2.6)
−∂E
one finds the transverse magnetization
N
1 ∂f 1 1
< sz >= = dEρ(E) (2.7)
N j ∂Ω 1+exp(βE) − 2
j=1 0 Z
X
and static transverse susceptibility
∂< 1 N sz > 1
χ = N j=1 j = β dEρ(E) . (2.8)
zz P∂Ω0 − Z (2coshβ2E)2
Note, that since at T 0 1 0 if E > 0 and 1 1 if
→ 1+exp(βE)→ 1+exp(βE)→
E <0, the transverse magnetization and static transversesusceptibility
at Ω =0 in low tempereture limit are given by
0
N
1
< sz >
N j
j=1
X
−Ω0 1 1
= dE′ Im , (2.9)
Z0 "−π (E′+iΓ)2−(J2+D2)#
p
8
1 1
χ = . (2.10)
zz
−π√Γ2+J2+D2
The results of numerical investigation of thermodynamical proper-
ties of the model (1.1) are depicted in Figs.1-5. The dependence on
Dzyaloshinskii-Moriya interaction (D = 0, D = 0.5J, D = J) of the
average spectral density ρ(E)(Ω = 0) for different values of the width
0
of lorentzian distribution Γ (Γ = 0, Γ = 0.25J, Γ = 0.5J, Γ = J) is
showninFig.1. InFigs.2,3it isshownthe temperaturebehaviourofen-
tropy and specific heat for Ω =0 for different values of Dzyaloshinskii-
0
Moriya interaction (D = 0, D = 0.5J, D = J) and different values
of Γ (Γ = 0, Γ = J). The changes in the dependence of transverse
magnetization on the value of transverse field Ω that are caused by
0
Dzyaloshinskii-Moriya interaction (D = 0, D = 0.5J, D = J) for sev-
eral values of Γ (Γ = 0, Γ = J) at T = 0 can be seen in Fig.4. In
Fig.5 the temperature behaviour of static transverse susceptibility in
zero transverse field for few values of Dzyaloshinskii-Moriya interaction
(D =0, D =0.5J, D =J) and Γ(Γ=0, Γ=J) are presented.
As it can be easily seen from the formula for ρ(E) (1.22) and in
Fig.1 the presence of Dzyaloshinskii-Moriya interaction formally causes
the change of J2 to J2 + D2. This leads to effective increase of in-
terspin interaction and results in broadening of zones both with sharp
edges(whenΓ=0)andwithsmoothonesbecauseofrandomness(when
Γ = 0). Rather small quantitative changes in temperature behaviour
6
of entropy and specific heat (Figs.2,3) reveal the tendency caused by
Dzyaloshinskii-Moriya interaction. Dzyaloshinskii-Moriya interaction
decreases the transverse magnetization for a given value of transverse
field in both cases when Γ = 0 and when Γ = 0 (Fig.4) and decreases
6
thevalueofstatictransversesusceptibilityinzerotransversefieldinlow
temperature region. The obtained result are in agreement with the de-
rived ones for non-random version of 1D s = 1 isotropic XY model in
2
transverse field [18, 19].
Unfortunately, the obtained results do not permit to get any exact
estimations for static spin correlation functions. Considering the sim-
plest one <szsz >, using the relation sz = c+c 1 and exploiting
j j+n j j j − 2
Wick-Bloch-de Dominicis theorem one finds
<szsz >
j j+n
1 1 1
=<c+c c+ c > <c+c > <c+ c >+
j j j+n j+n − 2 j j − 2 j+n j+n 4
=<c+c ><c+ c > <c+c+ ><c c >
j j j+n j+n − j j+n j j+n
9