Nonlinear sigma model of a spin ladder containing a static single hole

0.14 MB·English

by  A. R. Pereira

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n Nonlinear sigma model of a spin ladder
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4 containing a static single hole
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e A. R. Pereira1,2 ∗, E. Ercolessi2,3†, A.S.T. Pires4, ‡
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t 1 Departamento de F´ısica, Universidade Federal de Vic¸osa
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t 36570-000, Vic¸osa, Minas Gerais, Brazil
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2Physics Department, University of Bologna, Via Irnerio 46, I-40126, Bologna, Italy
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n 3INFN and CNISM, Bologna, Italy
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c 4 Departamento de F´ısica, ICEX, Universidade Federal de Minas Gerais
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Caixa Postal 702, 30123-970, Belo Horizonte, Minas Gerais, Brazil
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Abstract
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0 In this letter we extend the nonlinear σ model describing pure spin ladders with an
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a arbitrary number of legs to the case of ladders containing a single static hole. A simple
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immediate application of this approach to classical ladders is worked out.
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c PACS numbers: 75.50.Ee; 75.10.Jm; 75.10.Hk; 75.30.Ds
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Keywords: magnons; antiferromagnets; impurities.
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Corresponding author: A. R. Pereira; e-mail: [email protected]; Tel.: [phone],
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Fax: [phone].
∗E-Mail: [email protected]
†E-Mail:[email protected]
‡E-mail: antpires@fisica.ufmg.br
1
The interest in low dimensional quantum antiferromagnets has been great ever since
Haldane conjectured [1] that integer spin chains have a gap in their excitation spectrum
while half-integer spin chains do not. More recently, spin ladders (two or more coupled
spin chains) have also attracted much interest, mainly when the effects upon doping are
considered. Indeed, when one manages to remove spins from the system (leaving holes
behind)the existence of superconductivity is predicted [2] (and experimentally observed
[3]). In this letter we would like to study the presence of static holes in spin ladders
with an arbitrary number of legs by considering the nonlinear σ model continum limit
of the model.
The continuum limit of pure antiferromagnetic Heisenberg spin ladders with arbi-
trary number of legs has been derived by several authors [4, 5, 6, 7]. Here we apply the
approach of Ref.[4] for studying the case of ladders containing static holes. We start
with the Hamiltonian for a pure ladder system with n legs of length Na (a is the
0 0
lattice constant and N ≫ n) defined as
n N
H = J S~ (j)·S~ (j +1)+J˜ S~ (j)·S~ (j) , (1)
a a a a,a+1 a a+1
Xa=1Xj=1h i
where S~ (j) are the spin operators located in the ath leg at the position j = 1,...,N,
a
while J > 0 and J˜ > 0 are the antiferromagnetic exchange couplings along the
a a,a+1
leg and rung respectively. The partition function of the above Hamiltonian in the spin
coherent state path-integral representation is given by
β
Z(β) = [DΩ~]exp iS ω[Ω~ (j,τ)]− dτH(τ) , (2)
a
 
Z  Xj,a Z0 
where τ = it is the imaginary time variable, ω[Ω~ (j,τ)] is the Berry phase factor and
a
H(τ) is obtained by replacing the operator S~ (j) by the classical variable SΩ~ (j,τ) in
a a
theHamiltonian(1). Togetthecontinuumlimit,itisusualtoassumethatthedominant
2
contribution to the path integral comes from paths described by [1, 8]
1/2
|~l (j,τ) |2 ~l (j,τ)
Ω~ (j,τ) = (−1)a+jφ~(j,τ) 1− a + a . (3)
a S2 S
!
The field φ~(j,τ) is supposed to be slowly varying and the fluctuation field ~l (j,τ) is
a
supposedto besmall (~l (j,τ)/S << 1). The constraint Ω~2(j,τ) = 1 implies φ~2(j,τ) = 1
a a
and φ~(j,τ)·~l (j) = 0. Numerical works support the fact that the staggered spin-spin
a
correlation length is much greater than the total width of the ladder [9, 10]. Then,
assuming that φ~(j,τ) depends only on the site index j along the legs, Dell’Aringa et
al. [4] mapped the antiferromagnetic Heisenberg ladder system onto a (1+1) quantum
nonlinear σ model
1 β 1
Z = [Dφ~]exp(iΓ[φ~])exp − dτ dx (∂ φ~)2+v (∂ φ~)2 , (4)
σ τ s x
2g v
Z (cid:26) Z0 Z (cid:20) s (cid:21)(cid:27)
where Γ[φ~] = (θ/4π) βdτ dxφ~ ·(∂ φ~ ×∂ φ~) (with θ = 2πS) for n odd and Γ[φ~] = 0
0 τ x
for n even, reflectingRthe faRct that, for half-spin systems, the excitation spectrum has a
gap (is gapless) when n is even (odd). Besides, the nonlinear σ model parameters, the
coupling constant g and the spin wave velocity v , are defined by
s
1/2
g−1 = S J L−1 , (5)
 a b,c
a,b,c
X
 
1/2
J
v = S a a , (6)
s Pb,cL−b,c1!
where L−1 is the inverse of the matrixP
a,b
4J +J˜ +J˜ for a = b
a a,a+1 a,a−1
L = (7)
a,b
 L = J˜ for | a−b |= 1
 a,b a,a+1
with J˜ ≡ J˜ andJ˜ = J˜ = 0.
a,a+1 a−1,a 1,0 n,n+1
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Now we consider the system in the presence of static holes (spins removed from the
ladder). In two spatial dimensions, one of the simplest way of studying this problem in
the continuum limit is through a non-simply connected manifold [11, 12, 13]. In this
case a disk is removed from the magnetic plane, leaving a hole behind, and this hole is
interpreted as a nonmagnetic impurity (or a spin vacancy) since there is no magnetic
degrees of freedom insight it. This approach has good qualitative and quantitative
agreement with numerical calculations [11, 12, 13, 14]. However, in the case of ladders
as described by the nonlinear σ model given by Eq. (4), we cannot simply remove a
disc from the space because the problem becomes essentially one-dimensional. There is
no possibility of removing a part of the space without breaking the “effective” lattice.
Then, within the above approach a “hole” must affect the exchange interactions J and
a
J˜ . As a consequence, the matrix L becomes j-dependent. In such a way that
a,a±1 a,b
the lattice is not broken for a single defect. It means that the parameters g and v are
s
now functions of the position along the ladder. Depending on the number of legs, there
are more than one position to put a single vacancy which yields different results. Some
examples are shown in Fig.(1).
With the above considerations in mind, our approach for the nonlinear σ model
describing spin ladders with a static hole centered at j = x (along the legs), a = k
0
(along the rungs) are summarized as follows
β
Z = [Dφ~]exp(iΓ[φ~])exp − dτ dxL , (8)
σ,hole σ,hole
Z (cid:18) Z0 Z (cid:19)
with
L = dx 1 1 (∂ φ~)2+v (x−x )(∂ φ~)2 . (9)
σ,hole 2gk(x−x0) vs,k(x−x0) τ s,k 0 x
h i
R
Inordertoexplicitly definethenewparametersspacedependentsg (x−x )andv (x−
k 0 s,k
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x ), we first rewrite the n×n matrix L as follows
0 a,b
L L 0 0 0 ... 0 0 0 ... 0
1,1 1,2
 
L L L 0 0 ... 0 0 0 ... 0
2,1 2,2 2,3
 
 0 L3,2 L3,3 L3,4 0 ... 0 0 0 ... 0 
 
 ... ... ... ... ... ... ... ... ... ... ... 
 
 
 0 0 0 0 0 ... L 0 0 ... 0 
 k−2,k−1 
 
La,b =  0 0 0 0 0 ... Lk−1,k−1 Lk−1,k 0 ... 0 .(10)
 
 0 0 0 0 0 ... L L L ... 0 
 k,k−1 k,k k,k+1 
 
 0 0 0 0 0 ... 0 L L ... 0 
 k+1,k k+1,k+1 
 
 0 0 0 0 0 ... 0 0 L ... 0 
 k+2,k+1 
 .. .. .. .. .. .. .. .. .. .. .. 
 . . . . . . . . . . . 
 
 
 0 0 0 0 0 ... 0 0 0 ... Ln,n 
 
 
If we place the vacancy at (j,a) = (x ,k), the above matrix will be the same for all
0
j 6= x ,x −1 and equal to (7). For j = x −1 the matrix will be again of rank n with a
0 0 0
slightly different coefficient L . For j = x the matrix has zeroes along the k−th row
k,k 0
and the k−th column. Thus we define a new matrix K of order (n−1)×(n−1),
a,b
which has almost the same elements of the above matrix and without line k and column
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k. Explicitly:
L L 0 0 0 ... 0 0 ... 0
1,1 1,2
 
L L L 0 0 ... 0 0 ... 0
2,1 2,2 2,3
 
 0 L3,2 L3,3 L3,4 0 ... 0 0 ... 0 
 
 .. .. .. .. .. .. .. .. .. .. 
 . . . . . . . . . . 
 
 
 0 0 0 0 0 ... L 0 ... 0 
K =  k−2,k−1 , (11)
a,b  
 0 0 0 0 0 ... K 0 ... 0 
 k−1,k−1 
 
 0 0 0 0 0 ... 0 K ... 0 
 k+1,k+1 
 
 0 0 0 0 0 ... 0 L ... 0 
 k+2,k+1 
 ... ... ... ... ... ... ... ... ... ... 
 
 
 0 0 0 0 0 ... 0 0 ... Ln,n 
 
 
where K = L −J˜ and K = L −J˜ . This formula
k−1,k−1 k−1,k−1 k−1,k k+1,k+1 k+1,k+1 k+1,k
holds only in the region of the spin vacancy (| x−x |. a ). Therefore we may assume
0 0
that the coefficients g and v are given by:
g−1 for | x−x |& a ,
0 0
g−1(x−x ) = (12)
k 0  S J K−1 1/2 for | x−x |. a ,
 a6=k,b,c a b,c 0 0
(cid:16) (cid:17)
P

v for |x−x |& a ,
s 0 0
vs,k(x−x0) =  S PPa6=KkJ−a1 1/2 for |x−x0 |. a0. (13)
(cid:18) b,c b,c(cid:19)
Thechangeintheparametersgk(x−x0)andvk,s(x−x0)inthezoneofinfluenceofthe
vacancy is associated with the discontinuous change in the number of legs in this region
(see Fig.(1)). Of course, the field φ~ must be continuous across the pure and impure
regions. Note that, in principle, only the coupling constant and spin wave velocity are
(locally) affected by the removed spin. As the Berry phases do not depend on these
values, they are not very sensitive to the presence of the defect (see Eq. (8)). It means
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that the ground state may not be very affected by the presence of the impurity. These
resultsareinagreementwithrecentnumericalcalculations[15],whichgiveevidencesthat
the ground state configuration of the entire ladder system is not changed significantly
by the impurity except for the local extraction of the missing bonds. Indeed, the energy
cost to remove a spin from a two-leg ladder with spin-1/2 and J = J˜ is E ≈ 1.215J
[15], which is almost completely accounted for by the missing energy bonds along the
legs (0.350J) and across the rungs (0.455J). Then, the method developed here is a
good approximation and can be generalized for ladders containing a low concentration
of impurities. Below we give a simple application of this approach. Our example is done
forclassical spinsystems becausethecalculations arealmostdirectinthiscase. Besides,
therearealsomanganesehalidecompounds[16]thatarequasi-one-dimensionalandtwo-
dimensional antiferromagnets. Furthermore, these Mn(II) compounds have spin 5/2 so
they are also nearly classical and therefore, potential systems to test our results.
At zero temperature, the Hamiltonian of spin ladders possesses, in the classical limit
(S → ∞), a minimum given by the antiferromagnetic vacuum solution φ~ (x) = φ zˆ= zˆ,
0 z
where zˆis an unit vector in the vertical direction. This solution breaks the O(3) invari-
ance of the model down to the subgroup O(2) of rotations around the z-axis. Conse-
quently there should appear two Goldstone modes, which are nothing but spin waves,
associated with φ and φ . Then a natural first step concerning the impurity systems
x y
is to study the interactions between spin waves and holes. In a quantum spin system,
the corresponding problem would be the interactions between triplons [17] (which are
a triplet of well defined spin-1 magnons) and holes. However, it will be considered in a
future work.
Expanding φ~(x,t) around the vacuum solution φ~(x,t) = φ~ (x) + ~η(x,t) and min-
0
imizing Hamiltonian (9), one obtains, in the linearized approximation, the following
scattering equation ∂2~η(x,t)−[1/v2 (x−x )]∂2~η(x,t) = U (x)~η(x,t), where the scat-
x s,k 0 t k
tering potential is U (x) = −{∂ ln[v (x−x )/g (x−x )]}∂ . However, the function
k x s,k 0 k 0 x
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v (x − x )/g (x − x ) is constant practically through all space (see Eqs.(12)) while
s,k 0 k 0
(13) varies only when entering the impurity regions. Therefore, U (x) is zero in al-
k
most all space and we have magnon solutions for the field equation in the three regions:
(x < x −a ), (x −a < x < x +a ) and (x > x +a ). The form of U (x) is not
0 0 0 0 0 0 0 0 k
explicitly known but its effects can be envisaged using the following simple analysis: if
a magnon (for instance, coming from the left) hits the zone of influence of the potential
U (x) (or the zone of the impurity) at x −a , then its velocity will be changed from v
k 0 0 s
to v = ( J / K−1)1/2 and after leaving behind this region at x +a , it will
k,s a6=k a b,c b,c 0 0
be changedPagain toPvs. Consequently, supposing a plane wave coming from −∞, and
assuming that the lowest order effect of U (x) is to cause elastic scattering centers for
k
magnons,thesolutionat+∞canbeapproximatedby~η exp[i(qx−ω t+δ (q)/2)]with
0 q n,k
frequency ω = qv , which of course, is precisely the dispersion relation for magnons in
q s
theabsenceofholes. Thefunctionδ (q)mayberegardedasaphase-shift(formagnons
n,k
in a ladder with n legs) which depends on the particular position (leg k) of the hole in
the spin ladder for a determined wave number q. Indeed, after passing the defect, the
wave is shifted from the original one due to the different velocity v acquired in the
k,s
hole region. This phase-shift can be easily estimated considering the difference of paths
v ∆t and v ∆t, where the interval ∆t = 2a /v is the time necessary for the wave to
s k,s 0 k,s
leave behind the region of the hole (x −a < x < x +a ). In the lowest order, it is
0 0 0 0
given by
v J 1/2 K−1 1/2
δ (q) = −4qa s −1 = −4qa a a b,c b,c −1 . (14)
n,k 0(cid:18)vk,s (cid:19) 0 Pa6=kJa! Pb,cL−b,c1! 
 P P 
For the case of a spin ladder with two legs and N → ∞, the phase-shift does not
depend on the position of the vacancy, which can be put at the left (k = 1) or right
(k = 2) leg. For J = J and J˜ = J˜, it is given by
1 2 a,b
δ (q) = δ (q) = −4qa [(1+J˜/2J)1/2 −1]. (15)
2,1 2,2 0
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Spin ladders with three or more legs have two different situations. For example, for the
case with three legs, if the vacancy is placed at the first or last leg, we have:
1/2
(1+3R/4)
δ (q)= δ (q) = −4qa −1 , (16)
3,1 3,3 0
(1+R/2)(1+R/12)
( )
(cid:20) (cid:21)
where R = J˜/J. By the other hand, if the hole is placed at the central leg, the phase
shift is
1/2
(1+3R/4)
δ (q) = −4qa −1 . (17)
3,2 0
(1+R/4)(1+R/12)
( )
(cid:20) (cid:21)
Finally we discuss the magnon density of states in the impurity spin ladders. To do
this, we consider a large system of size Na and impose periodic boundary conditions
0
on the continuum (magnons) states ~η (x). This periodicity, together with ~η (x) ≈
q q x→±∞
~η exp[i(qx ± δ (q)/2)], gives the following condition for the allowed wave vectors:
0 n,k
Na q +δ (q ) = 2πm (m = 0,±1,±2,...). Clearly, the magnon density of states is
0 m n,k m
changed by the presence of a hole as follows
dm Na 1 dδ (q)
0 n,k
̺ (q) = = + , (18)
n,k
dq 2π 2π dq
and so, the change in the density of states ∆̺ (q) = ̺ (q)−Na /2π is given by
n,k n,k 0
1dδ (q)
n,k
∆̺ (q) = . (19)
n,k
π dq
Abovewehave multipliedbyafactor of2totake intoaccount thetwo Goldstonemodes.
For a classical two-leg spin ladder one has
−4a
∆̺ (q) = ∆̺ (q) = 0[(1+R/2)1/2 −1]. (20)
2,1 2,2
π
For three-leg ladder
1/2
−4a (1+3R/4)
0
∆̺ (q)= ∆̺ (q)= −1 (21)
3,1 3,3
π (1+R/2)(1+R/12)
( )
(cid:20) (cid:21)
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and
1/2
−4a (1+3R/4)
0
∆̺ (q) = −1 . (22)
3,2
π (1+R/4)(1+R/12)
( )
(cid:20) (cid:21)
In Fig.(2) we plot D = ∆̺ /a as a function of R = J˜/J for n = 2 and n = 3.
n,k 0
In general, the density of states does not depend on the wave-vector q (since the phase-
shift has a linear dependence on q). For all legs, the limit R → 0 implies D → 0. As
expected, it means that the presence of J˜is very important for the phase-shifts as well
as for the change in the density of states. In practice, it avoids a broken lattice and
leads to v 6= 0. For a three-leg ladder with the impurity placed at leg 1 or 3, ∆̺ is
k,s 3,1
very small for an appreciable range of R and becomes positive for R > 4. It increases
considerably for large values of R and in the limit R → ∞, ∆̺ → 1.27. Such a
3,1
general behavior is also expected for ladders with more than three legs with impurities
placed at the external legs (of course, it may have important qualitative changes as, for
example, the function is positive for R < R and then becomes positive for R > R ). If
c c
the vacancy is located at the central leg, the behavior of the three-leg ladder is similar
to the previous case but the change in the density of states is much larger. In this case,
∆̺ (q)/a becomes positive only for R > 20 (not shown in Fig.2). In addition , like
3,2 0
∆̺ (q)/a , ∆̺ (q)/a → 1.27 as R → ∞. On the other hand, for a two-leg ladder,
3,1 0 3,2 0
∆̺ (q)/a is always negative and its modulus increases monotonically as R increases.
2,1 0
The magnon density of states suffers a very expressive change for large values of J˜.
Therefore, in this circumstance, there is a clear distinction in the magnon spectrum
when a hole in the spin ladder is present or absent.
Ladders with a higher number of legs might be approached as well, using for ex-
ample the Mathematica program. In particular it is interesting to study how the ratio
w = v /v changesasweincreasethenumbernoflegs andapproachatwo-dimensional
s k,s
lattice. We have checked this for the isotropic case J = J˜ by putting the hole in the
center leg of the ladder, getting the following results for n = 11,31,51,71,91 respec-
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