Consequences of lattice imperfections and interchain couplings for the critical properties of spin-$1/2$ chain compounds
To allow for a comparison of theoretical predictions for spin chains with experimental data, it is often important
 to take impurity effects as well as interchain couplings into account. Here we present the field theory for finite
 spin chains at finite temperature and calculate expe...
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| Cite this: | Consequences of lattice imperfections and interchain couplings for the critical properties of spin-$1/2$ chain compounds / J. Sirker // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 449-462. — Бібліогр.: 40 назв. — англ. |
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| citation_txt | Consequences of lattice imperfections and interchain couplings for the critical properties of spin-$1/2$ chain compounds / J. Sirker // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 449-462. — Бібліогр.: 40 назв. — англ. |
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| description | To allow for a comparison of theoretical predictions for spin chains with experimental data, it is often important
to take impurity effects as well as interchain couplings into account. Here we present the field theory for finite
spin chains at finite temperature and calculate experimentally measurable quantities like susceptibilities and
nuclear magnetic resonance spectra. For the interchain couplings we concentrate on geometries relevant for
cuprate spin chains like Sr₂CuO₃ and SrCuO₂. The field theoretical results are compared to experimental as
well as numerical data obtained by the density matrix renormalization group.
Щоб мати можливiсть порiвняти теоретичнi передбачення для спiнових ланцюжкiв з експериментальними даними, часто важливо взяти до уваги ефекти домiшок, так само, як мiжланцюжковi взаємодiї. Тут ми представляємо теорiю поля для скiнчених спiнових ланцюжкiв при скiнченiй температурi i обчислюємо експериментально вимiрюванi величини як сприйнятливостi i спектри ядерного магнiтного резонансу. Для мiжланцюжкових взаємодiй ми зосереджуємося на геометрiях, iстотних для купратних спiнових ланцюжкiв як от Sr₂CuO₃ і SrCuO₂. Теоретико-польовi результати порiвнюються з експериментальними i чисельними даними, отриманими з використанням ренормалiзацiйної групи для матрицi густини.
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| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 3, pp. 449–462
Consequences of lattice imperfections and interchain
couplings for the critical properties of spin-1/2 chain
compounds
J. Sirker
1 Department of Physics and Research Center OPTIMAS, University of Kaiserslautern,
D–67663 Kaiserslautern, Germany
2 Max-Planck Institute for Solid State Research, Heisenbergstr. 1, D–70569 Stuttgart, Germany
Received May 7, 2009
To allow for a comparison of theoretical predictions for spin chains with experimental data, it is often important
to take impurity effects as well as interchain couplings into account. Here we present the field theory for finite
spin chains at finite temperature and calculate experimentally measurable quantities like susceptibilities and
nuclear magnetic resonance spectra. For the interchain couplings we concentrate on geometries relevant for
cuprate spin chains like Sr2CuO3 and SrCuO2. The field theoretical results are compared to experimental as
well as numerical data obtained by the density matrix renormalization group.
Key words: spin chains, impurities, thermodynamics, bosonization, density-matrix renormalization group
PACS: 75.10.Pq, 75.10.Jm, 11.10.Wx, 02.30.Ik
1. Introduction
A large number of materials are known which, over a certain temperature range, are well de-
scribed by simple spin chain or ladder models [1–11]. In all these Mott insulators the superexchange
constants are spatially very anisotropic so that a three-dimensional crystal effectively shows one-
dimensional magnetic properties. In one of the best known spin-1/2 chain compounds Sr2CuO3, for
example, the superexchange constant along the chain direction J ∼ 2200 K whereas the magnetic
couplings J⊥ along the other directions are at least three orders of magnitude smaller [1,8]. In such
a system one can, therefore, experimentally explore the physical properties of a spin chain over a
very wide temperature range J⊥ � T . J . From a theoretical perspective this is very exciting
because it allows us to study experimentally many aspects of one-dimensional field theories [12–14].
In addition, the ideal Heisenberg spin-1/2 chain is integrable so that the compounds well described
by this model make it possible to experimentally address the question how physical properties,
in particular, transport, are affected by a nearby integrable point [2–4]. Here the infinite set of
constants of motion making the model integrable is expected to slow down the decay of current
correlations – or even prevent them from decaying completely – leading to anomalous transport
properties [15].
In real materials, however, we are always confronted with impurities and lattice imperfections
which weaken or even completely destroy a superexchange bond between two spins. Since a weak-
ening of a bond is a relevant perturbation in the renormalization group (RG) sense, we have to deal
– at least at low temperatures – with finite chains with open boundary conditions (OBCs). Mea-
surements on such systems correspond to taking averages over ensembles of finite chain segments
with lengths determined by a distribution function [16,17]. Furthermore, the description by a one
dimensional model will break down for temperatures T ∼ J⊥ where usually a three-dimensional
magnetic order sets in. However, even for temperatures above this ordering temperature interchain
couplings can have a significant effect which has to be taken into account.
In what follows, we will study the anisotropic spin-1/2 Heisenberg chain (XXZ model) with N
c© J. Sirker 449
J. Sirker
sites and OBCs
H = J
N−1∑
j=1
[
Sx
j S
x
j+1 + Sy
j S
y
j+1 + ∆Sz
j S
z
j+1
]
− h
N∑
j=1
Sz
j . (1.1)
Here J is the exchange constant and h is the applied magnetic field. Although exchange anisotropies
due to spin-orbit coupling are usually rather small so that experimentally only the isotropic case
∆ = 1 is relevant, the additional parameter ∆ is useful for the field theoretical calculations in
section 3. There are two important consequences of the OBCs. First, expectation values of local
operators become position dependent because translational invariance is broken. In the following
we will, in particular, study the local susceptibility defined as
χj =
∂
∂h
〈Sz
j 〉h=0 =
1
T
〈Sz
j S
z
tot〉h=0 , (1.2)
where Sz
tot =
∑
j S
z
j . Second, there are well defined boundary contributions to all thermodynamic
quantities. For N → ∞ the total free energy F is, for example, given by
F = Nfbulk + FB + O(1/N) , (1.3)
where FB is the boundary free energy. Similarly, one can define a boundary susceptibility χB [18–
22]. These boundary or surface terms will be studied here as well. The local susceptibility defined
in equation (1.2) can be related to the boundary susceptibility by
χB = lim
N→∞
N∑
j=1
χj −Nχbulk
, (1.4)
where χbulk is the bulk susceptibility defined analogously to the bulk free energy in equation (1.3).
For the interchain couplings we will consider two different cases relevant for many materials.
One is a simple ladder-like antiferromagnetic coupling between neighboring chains as, for example,
in Sr2CuO3 as well as in many other spin chain compounds. This case is shown in figure 1b.
The other is a zigzag ferromagnetic coupling between neighboring chains. This kind of interchain
coupling is sketched in figure 1c and is relevant, for example, for SrCuO2 [1].
Figure 1. A non-magnetic impurity in (a) a single spin chain, (b) in a spin chain coupled to
a neighboring chain by a ladder-like interchain coupling, and (c) in a spin chain coupled in a
zigzag fashion to a neighboring chain.
To provide an intuitive picture we start with some numerical results for semi-infinite spin
chains with geometries as shown in figure 1 obtained by the density matrix renormalization group
applied to transfer matrices (TMRG) in section 2. In section 3 we will present the field theory
for finite Heisenberg chains with OBCs at finite temperatures. In section 4 we use the results
obtained in the previous section to calculate the susceptibility as an ensemble average over finite
spin chain segments and show that the obtained results are in good agreement with experimental
measurements. In section 5 we derive, in a similar fashion, the nuclear magnetic resonance (NMR)
spectrum and show that it provides information about the interchain couplings. Finally, we give a
brief summary and present some conclusions.
450
Spin-1/2 chain compounds
2. Numerical results
A method particularly suited to the calculation of the thermodynamic properties of one-
dimensional systems is the density matrix renormalization group applied to transfer matrices
(TMRG) [20–28]. To this end, the one-dimensional quantum system is mapped onto a two-dimen-
sional classical system by a Trotter-Suzuki decomposition [29–31]. Then, the additional dimension
corresponds to the inverse temperature β. For the classical model, a transfer matrix is defined
which evolves along the spatial direction. Importantly, one can show that even for a critical system
there is always a gap at finite temperatures between the leading eigenvalue Λ0 and the next-leading
eigenvalues Λα of the transfer matrix T with ξ−1
α = ln |Λ0/Λα| defining a correlation length. This
makes it possible to perform the thermodynamic limit, i. e., system size N → ∞, exactly. With
the TMRG one can treat impurity problems [21,22,32] as well as frustrated systems [33] making it
an ideal numerical tool to study the problem considered here.
We will start by investigating the local susceptibility as defined in equation (1.2) for a semi-
infinite chain. By this we mean a chain which is infinitely long but has one end with OBCs. To
obtain χj we calculate the local magnetization 〈Sz
j 〉 for small magnetic fields h/J ∼ 10−2 and take
a numerical derivative. Within the transfer matrix formalism the local magnetization is given by
lim
N→∞
〈Sz
j 〉 =
〈Ψ0
L|T (Sz)T j−1T̃ |Ψ0
R〉
Λj
0〈Ψ0
L|T̃ |Ψ0
R〉
. (2.1)
Here T̃ is a modified transfer matrix containing the broken bond, T (Sz) is the transfer matrix
with the operator Sz included, and |Ψ0
R〉 (〈Ψ0
L|) are the right (left) eigenstates belonging to the
largest eigenvalue Λ0, respectively. Far away from the boundary 〈Sz
j 〉 becomes a constant, the bulk
magnetization
m = lim
j→∞
lim
N→∞
〈Sz
j 〉 = lim
j→∞
∑
n〈Ψ0
L|T (Sz)T j−1|Ψn
R〉〈Ψn
L|T̃ |Ψ0
R〉
Λj
0〈Ψ0
L|T̃ |Ψ0
R〉
=
〈Ψ0
L|T (Sz)|Ψ0
R〉
Λ0
. (2.2)
By taking again a numerical derivative with respect to a small magnetic field, we can obtain the
bulk susceptibility χbulk from (2.2).
0 20 40 60 80 100 120
j
-1
-0.5
0
0.5
1
χ j -
χ bu
lk
Figure 2. χj − χbulk for a semi-infinite chain with ∆ = 1 at temperatures T/J =
0.2, 0.1, 0.05, 0.025. The numerical TMRG data (closed symbols) are compared to the field
theory formula (3.23) from section 5 (open symbols).
In figure 2 the susceptibility profile χj − χbulk for a single semi-infinite chain with ∆ = 1 is
shown. As might be expected, the boundary induces Friedel-like oscillations which become larger
with decreasing temperature. Interestingly, at low temperatures the oscillations first increase and
reach a maximum, before decaying at large distances. This phenomenon has been first studied by
451
J. Sirker
Eggert and Affleck in [34] and we will rederive their field theory result as a special case of our more
general considerations in section 3.
0 0.1 0.2 0.3 0.4
T/J
0
0.2
0.4
0.6
0.8
1
1.2
1.4
χ B
0 0.2 0.4
T/J
0.08
0.1
0.12
0.14
χ bu
lk
Figure 3. The boundary susceptibility χB for ∆ = 1 as obtained by TMRG (symbols) compared
to the field theory result (3.20) (line) derived in section 3 which is valid at low temperatures.
Inset: Numerical results for the bulk susceptibility χbulk for ∆ = 1 (symbols) compared to the
field theory formula (3.19) (line).
Next, we consider the boundary susceptibility which we can easily obtain from the susceptibility
profile using equation (1.4). The result is shown in figure 3. At low temperatures it was shown
analytically that χB ∼ [T ln(T0/T )]−1 with a known constant T0 [17–19]. The bulk susceptibility,
on the other hand, behaves like χbulk ∼ const + ln−1(T0/T ) [12], i. e., it goes to a finite value at
T = 0 with infinite slope. We will come back to this in section 3 but let us notice here that the
numerical data are well described by the field theory.
0 5 10 15 20 25 30
j
-0.4
-0.2
0
0.2
0.4
0.6
χ j -
χ bu
lk
Figure 4. χj −χbulk for ∆ = 1, J⊥ = 0.03 J and T = 0.09 J and a ladder-like interchain coupling
as shown in figure 1b. One of the chains has a non-magnetic impurity at site j = 0 (circles)
whereas the other one (diamonds) is infinitely long and does not have any impurities. The
numerical TMRG data (closed symbols) are compared to the field theory formula (5.4) from
section 5 (open symbols).
Finally, we have also calculated susceptibility profiles for the cases of coupled chains shown in
figure 1b and c. A ladder-like coupling of neighboring chains is, for example, realized in Sr2CuO3
[1]. In figure 4 it is shown that in this case an impurity in one chain has also a significant effect on
a neighboring chain without impurities. The Friedel-like oscillations are reflected in the impurity-
452
Spin-1/2 chain compounds
free chain due to the interchain couplings. Clearly, the size of the reflected Friedel-like oscillations
will depend on the ratio J⊥/T . However, this ratio is not the only relevant factor. The geometry
of the interchain couplings plays an important role as well. In SrCuO2, neighboring chains are
coupled by a ferromagnetic zigzag-type coupling. In this case an impurity in one chain leaves a
neighboring chain almost unaffected even if the temperature T ∼ J⊥ as is shown in figure 5. In
section 5 we will show that these differences can be easily understood if one starts from the field
theory results for a single chain derived in the next section and takes the interchain couplings into
account perturbatively.
0 5 10 15 20 25 30
j
-0.4
-0.2
0
0.2
0.4
0.6
χ j -
χ bu
lk
Figure 5. χj − χbulk for ∆ = 1, J⊥ = −0.1 J and T = 0.09 J with a zigzag interchain coupling
as shown in figure 1c. One of the chains has a non-magnetic impurity at site j = 0 (circles)
whereas the other one (diamonds) is infinitely long and does not have any impurities. For this
case the field theory predicts that there are no oscillations, χj −χbulk ≈ 0, in the infinitely long
chain.
3. Field theory for finite spin chains
In the limit of low temperatures and large chain length, the XXZ model (1.1) can be represented
by a field theory. The main step is a linearization of the dispersion around the two Fermi points. A
certain linear combination of particle-hole excitations around a Fermi point constitutes a collective
bosonic mode described by the Luttinger liquid Hamiltonian
H =
v
2
∫ L+a
0
dx
[
Π2 + (∂xΦ)2
]
− h
√
K
2π
∫ L+a
0
dx ∂xΦ , (3.1)
where v is the spin velocity, L = Na, and a is the lattice constant. The bosonic field Φ obeys the
standard commutation rule [Φ(x),Π(x′)] = iδ(x− x′) with Π = v−1∂tΦ. The Luttinger parameter
K is a function of the anisotropy ∆ and can be determined exactly by Bethe ansatz with K = 1
at the isotropic point. The spin operators can be directly expressed in terms of the boson Φ, in
particular, we have
Sz
j ≈
√
K
2π
∂xΦ + c(−1)j cos
√
2πKΦ (3.2)
at zero magnetic field. Here c is an amplitude which can also be obtained exactly [35–37]. The
separation of Sz
j into a uniform and a staggered part at low energies can be understood as follows.
In the equivalent spinless fermion representation of the XXZ model, obtained by a Jordan-Wigner
transformation, the Sz operator becomes the density operator. Due to the linearization of the
dispersion the electron and the hole can live either at the same Fermi point and have, therefore,
small momentum or they can be situated at different Fermi points in which case the associated
453
J. Sirker
momentum is 2kF. Zero magnetic field corresponds to half-filling for the spinless fermions. Thus,
2kF = π and we obtain the staggered contribution in (3.2). This also means that the local sus-
ceptibility defined in equation (1.2) can be separated at low temperatures into a uniform and a
staggered part
χj = χuni + (−1)jχst
j . (3.3)
In a bulk susceptibility measurement, the staggered part does not contribute. This part is, however,
measurable in probes of the local magnetism as, for example, in NMR. We will come back to this
point in section 5.
If one is only interested in correlation functions for infinite system size at finite temperatures
or correlation functions for finite systems at zero temperature, one can make use of the conformal
invariance of the field theory. It is then sufficient to calculate the correlation at zero temperature
and infinite system size and use a conformal mapping from the complex plane onto a cylinder
with the circumference corresponding either to inverse temperature or system size. Here, we are,
however, interested in the thermodynamics of finite chains and we will, therefore, have to use an
explicit mode expansion
Φ(x = ja, t) =
√
π
8K
+
√
2π
K
Sz
tot
j
N + 1
+
∞∑
n=1
sin (πnj/(N + 1))√
πn
(
e−i πnvt
L+a bn + ei πnvt
L+a b†n
)
(3.4)
which incorporates the OBCs. Here bn is a bosonic annihilation operator. Equation (3.4) is a dis-
crete version of the mode expansions used in [13,16] with x = ja becoming a continuous coordinate
for a→ 0,N → ∞ with L = Na fixed. Using this mode expansion, the local observables respect the
discrete lattice symmetry j → N +1− j corresponding to a reflection at the central bond (site) for
N even (odd), respectively. The sites 0 and N+1 are added to model (1.1) and we demand that the
spin density should vanish at these sites. Therefore, the upper boundary for the integrals in (3.1) is
L+a. The zero mode part (first two terms of equation (3.4)) fulfills
∑
j S
z
j ≈
√
K
2π
∫ L+a
0 ∂xΦ ≡ Sz
tot
and the oscillator part (last term of equation (3.4)) vanishes for j = 0, N + 1 as required.
For the free boson model, the uniform part of the susceptibility can easily be calculated and is
given by [13]
χuni
0 = − ∂2
∂h2
∣∣∣∣
h=0
f0 =
1
NT
∑
Sz
S2
ze−
πv
K(L+a)T
S2
z
∑
Sz
e−
πv
K(L+a)T
S2
z
= − 1
4NT
∂2
∂u2
∣∣∣∣
u=0
ln θ
(
e−
πv
K(L+a)T , u
)
. (3.5)
Here θ(q, u) is the elliptic theta function of the third kind θ = θ3(q, u) =
∑∞
n=−∞ qn2
ei2nu for
integer Sz (even N) and of the second kind θ = θ2(q, u) =
∑∞
n=−∞ q(n+1/2)2ei(2n+1)u for half-
integer Sz (odd N). Note, that χuni
0 has a simple scaling form as a function of NT . The lattice
parameter a appears here due to the OBCs. It leads to a boundary correction which we will consider
later. First, we set a ≡ 0 in (3.5) and concentrate on the following limiting cases
χuni
0 =
2
TN exp
[
− πv
KLT
]
NT/v → 0, N even,
1
4TN NT/v → 0, N odd,
K
2πv NT/v → ∞ .
(3.6)
The Curie-like divergence for N odd is caused by the degeneracy of the ground state Sz
tot = ±1/2.
At low temperatures the whole chain, therefore, behaves like a single spin. For N even, on the other
hand, the ground state is a singlet, Sz
tot = 0. At low temperatures, the chain becomes locked in this
state leading to an exponentially small susceptibility. In the thermodynamic limit, NT/v → ∞,
the susceptibility within the free boson approximation is just a constant.
The staggered part of the susceptibility, χst
j , for a finite chain with OBCs has been calculated
in [37]. It is given by
χst
j = − c
T
(
π
N + 1
)K/2 η3K/2
(
e−
πv
T L
)
θ
K/2
1
(
πj
N+1 , e
− πv
2T L
)
∑
mm sin[2πmj/(N + 1)]e−πvm2/(KLT )
∑
m e−πvm2/(KLT )
. (3.7)
454
Spin-1/2 chain compounds
Here η(x) is the Dedekind eta-function and θ1(u, q) the elliptic theta-function of the first kind.
The summation index m runs over all integers (half-integers) for N even (odd), respectively. In
the thermodynamic limit, N → ∞, we can simplify our result and obtain
χst
j =
cK
v
x
[
v
πT sinh
(
2πTx
v
)]K/2
(3.8)
with x = ja. This agrees for the isotropic Heisenberg case, K = 1, with the result in [34]. The
amplitude c, first introduced in equation (3.2), can be determined with the help of the Bethe ansatz
along the lines of [36]. This leads to c =
√
Az/2 with Az as given in equation (4.3) of [36]. The
formulas (3.7) and (3.8) are, therefore, parameter free.
To find the boundary contributions, such as the boundary free energy defined in equation (1.3),
one has to go beyond the free boson model (3.1). When deriving the low-energy effective theory
for the XXZ model, one finds in addition to the free boson model (3.1) infinitely many irrelevant
terms. These terms either stem from band curvature (corrections to the linear dispersions around
the Fermi points) or from the interaction term. For ∆ close to 1 the leading irrelevant term is given
by
δH = λ
∫ L+a
0
dx cos(
√
8πKφ). (3.9)
It is the bosonized version of Umklapp scattering where two left moving electrons get scattered
to right movers or vice versa interchanging a reciprocal lattice vector. This term becomes relevant
for ∆ > 1 and is responsible for the opening of an excitation gap in this regime. For the isotropic
case, ∆ = 1, Umklapp scattering is marginally irrelevant and leads to important corrections to the
results obtained for the free boson model. Due to the integrability of the XXZ model the amplitude
λ of the Umklapp term can be obtained exactly as well [38]. In [16,17] the free energy and the
susceptibility corrections to the free boson result up to the first order in the Umklapp scattering
have been calculated. For the susceptibility the following correction was obtained
δχuni
1 =
2λ
T 2
( π
N
)2K
η6K
(
e−
πv
T L
) ∫ 1/2
0
dy
g0
(
y, e−
πv
KLT
)
θ2K
1
(
πy, e−
πv
2T L
) (3.10)
with
g0 (y, q) = −
∑
Sz
S2
z cos(4πSzy)q
S2
z
∑
Sz
qS2
z
+
(∑
Sz
cos(4πSzy)q
S2
z
)(∑
Sz
S2
zq
S2
z
)
(∑
Sz
qS2
z
)2 . (3.11)
In addition, there is also a boundary correction related to the parameter a in (3.5). Expanding in
this parameter to the lowest order we find
δχuni
2 =
πva
KT 2L3
g2
(
e−
πv
KLT
)
, (3.12)
where
g2 (q) =
∑
Sz
S4
zq
S2
z
∑
Sz
qS2
z
−
(∑
Sz
S2
zq
S2
z
)2
(∑
Sz
qS2
z
)2 . (3.13)
a plays the role of a lattice constant and its value can be determined for ∆ < 1 (K > 1) by the
Bethe ansatz and is given by [20,21]
a = 2−1/2 sin [πK/(4K − 4)] / cos [π/(4K − 4)] . (3.14)
The uniform part of the susceptibility of a finite chain is, therefore, given by χuni(L, T ) = χuni
0 (a ≡
0) + δχuni
1 + δχuni
2 . The corrections to the uniform zeroth order susceptibility χuni
0 (a ≡ 0) are
455
J. Sirker
important here because in the thermodynamic limit they give the boundary susceptibility χB. In
this limit g2
(
e−πv/(KLT )
)
→ K2T 2L2/(2π2v2) and therefore
χB,1 = lim
L→∞
Lχuni
2 =
Ka
2πv
. (3.15)
This is just a constant contribution to the boundary susceptibility. Much more important is the
boundary contribution stemming from δχuni
1 . Here we find
χB,2 = lim
L→∞
Lχuni
1 = −λ
(
K
v
)2
B(K, 1 − 2K)[π2 − 2ψ′(K)]
(
2πT
v
)2K−3
, (3.16)
with B(x, y) = Γ(x)Γ(y)/Γ(x + y), ψ′(x) = dψ(x)/dx, and ψ(x) being the digamma function.
Note that for 1 < K < 3/2 (1/2 < ∆ < 1) the boundary spin susceptibility χB shows a divergent
behavior ∼ 1/T 3−2K as temperature decreases.
3.1. The isotropic point
At the isotropic point, Umklapp scattering becomes marginal and simple perturbation theory is
no longer sufficient. In this case we have to replace the Umklapp scattering amplitude by a running
coupling constant g(L, T ) which obeys a known set of renormalization group equations [38]
1/g + ln(g)/2 = ln
(√
2/πe1/4+γmin[L, v/T ]
)
. (3.17)
Here γ ≈ 0.577 is Euler’s constant and for the isotropic case considered here, the spin velocity is
v = Jπ/2. Then, the uniform susceptibility is given by
χuni(N,T ) = χuni
0 + δχuni
1 , (3.18)
where K → 1+ g(L, T )/2 in the exponentials of (3.5) and λ → g(L, T )/4 in (3.10). In this case the
parameter a in (3.5) is not determined by (3.14) and has to be used as a fitting parameter.
In the thermodynamic limit we can again split the susceptibility (3.18) into a bulk and a
boundary part. For the bulk susceptibility this yields the result first derived by Lukyanov [38]
χbulk =
1
π2
(
1 +
g(T )
2
+
3g3(T )
32
+
√
3
π
T 2
)
. (3.19)
Here we have also added the g3 correction from Umklapp scattering as well as a T 2-term which
stems from irrelevant operators with scaling dimension 4 describing the band curvature. The run-
ning coupling constant g(T ) is given by (3.17) with L = ∞. In the inset of figure 3 this formula is
displayed in comparison with the numerical results. For the boundary susceptibility, on the other
hand, we find
χB =
a
π2
+
g
12T
+
g2
8T
(
0.66− Ψ′′(1)
π2
)
+ O(g3) , (3.20)
where we included the second order corrections in g as derived in [17]. The comparison of this
formula with a = 1.5 and numerical results is shown in the main panel of figure 3.
Finally, the result for the staggered part of the susceptibility given in equation (3.7) has to be
modified at the isotropic point. Here we find [37]
χst
j = − 1
(2π3g̃)1/4T
(
π
N + 1
)1/2 η3/2
(
e−
πv
T L
)
θ
1/2
1
(
πj
N+1 , e
− πv
2T L
)
∑
mm sin[2πmj/(N + 1)]e−πvm2(1−g̃)/(LT )
∑
m e−πvm2(1−g̃)/(LT )
.
(3.21)
Now the running coupling constant g̃ also depends on the distance of site j from the boundary and
is given by
1/g̃ + ln(g̃)/2 = ln
(
min[C0x,C0(L− x),
√
π/2e1/4+γ/T ]
)
, (3.22)
456
Spin-1/2 chain compounds
where the constant C0 is not known and has to be used as a fitting parameter. Note, however, that
for low temperatures and x, L − x � 1 the value of C0 becomes irrelevant and our result for χj ,
therefore, is again parameter-free. If we consider the thermodynamic limit of equation (3.21) we
find
χst
j =
(
23
π7g̃
)1/4
1
1 − g̃
x√
1
2T sinh (4Tx)
. (3.23)
A comparison of this formula with numerical data is shown in figure 2. The agreement is good
but not perfect. The main problem here is that the temperature and the length scale set by the
distance from the boundary are competing. The renormalization group equations, however, cannot
be solved with both scales present. The formula (3.22) is derived in the limit when only one of
those scales matter and the corrections can be significant if this is not the case.
4. The averaged susceptibility and a comparison with experimental data
In an actual crystal we have imperfections and impurities which limit the length of a spin chain
segment. It is important to emphasize again that any weakening of a link is a relevant perturbation
in the renormalization group sense. It is, therefore, expected to be a good approximation to assume
that at low temperatures we have spin chain segments with open boundary conditions. As long
as we do not know in detail how these defects are distributed it seems reasonable to assume a
Poisson distribution, i. e., the probability of having an impurity at site j is not effected by the
other impurities. We might, however, expect this assumption to break down for large impurity
concentrations where some sort of impurity order might set in. Using a Poisson distribution we
have a normalized probability P (N) = p(1−p)N to find a chain segment withN sites if the impurity
concentration is p. Any measurement corresponds to taking an average over this ensemble of chain
segments. For the susceptibility, for example, we find
χp = p2
∑
N
N(1 − p)Nχ(N) . (4.1)
Note that in a bulk measurement only the uniform part of the susceptibility contributes. The
staggered part cancels out. Using the formula (3.18) we can immediately calculate the average
susceptibility χp. It is, however, very useful and instructive to derive a much simpler approximate
formula. To this end, we notice that we have two different regimes for a finite spin chain. If
the temperature is larger than the finite size gap, T/J > 1/N , we can approximate χ(N) ≈
χbulk + χB/N . If, on the other hand, T/J < 1/N we expect to see more or less the ground
state properties of the finite chain. This means, according to equation (3.6), that there will be
no contribution when the chain length is even while χ(N) ≈ 1/(4TN) if the chain length is odd.
We can, therefore, introduce a length Nc = γJ/T where this crossover occurs. Here γ is a parameter
which we expect to be of the order of 1. For the average susceptibility we can, therefore, write
χp ≈ p2
4T
Nc∑
N odd
(1 − p)N + p2
∞∑
N=Nc
(Nχbulk + χB)(1 − p)N
=
p
4T
1 − p
2 − p
(
1 − (1 − p)γ/T
)
+ (1 − p)γ/T
[(
1 − p+
pγ
T
)
χbulk + pχB
]
, (4.2)
where χbulk is given by equation (3.19) and χB by equation (3.20).
An interesting experiment has been performed by Kojima et al. [5] where Palladium (Pd) has
been doped into the spin chain compound Sr2CuO3. The Pd ions replace the Cu ions, act as non-
magnetic impurities, and cut the chain into finite segments. A complication in the analysis of this
experiment arises because even the undoped Sr2CuO3 samples already have a substantial amount
of chain breaks. It is believed that this is mainly a consequence of excess oxygen, i. e., we are in
reality dealing with Sr2CuO3+δ. In a simple picture, an excess oxygen ion pulls two electrons out
of the copper chain. If these holes are relatively immobile this also corresponds to a chain break.
457
J. Sirker
In the comparison of the experimental data for Sr2Cu1−xPdxO3+δ and formula (4.2) shown in
figure 6 we, therefore, use p as an effective impurity concentration incorporating both the chain
breaks due to excess oxygen and due to the non-magnetic Pd ions. Furthermore, we have subtracted
a constant contribution χconst from the experimental data which is expected to be present due to
core diamagnetism and Van Vleck paramagnetism. The values for p and χconst which yield the
best fit of the experimental data are shown in table 1. For Pd concentrations of x = 0.5% and
10 100
T[K]
10-3
10-2
χT
[K
e
m
u/
m
ol
]
Figure 6. Measured susceptibility T (χ − χconst) for Sr2Cu1−xPdxO3+δ with impurity concen-
trations x = 0%, 0.5%, 1%, 3% (crosses from bottom to top) from [5]. Here a constant χconst as
given in table 1 has been subtracted from the experimental data. Subsequent curves are shifted
by 5 × 10−3. For comparison theoretical results according to (4.2) with γ = 1 are shown with
an effective impurity concentration p as given in table 1.
Table 1. Concentration x of Pd ions in experiment compared to chain break concentration p
and constant contribution χconst yielding the best theoretical fit. The first line corresponds to
the “as grown” sample of Sr2CuO3+δ from [1].
x (Exp.) p (Theory) χconst [emu/mol]
0.0 0.006 −7.42× 10−5
0.005 0.012 −7.7× 10−5
0.01 0.014 −7.5× 10−5
0.03 0.024 −7.5× 10−5
x = 1% the obtained values for p are consistent with the picture of having a certain amount of
chain breaks due to excess oxygen already in the undoped compound. For x = 3%, however, this
picture seems to fail. Reasons for this could be either on the experimental side (perhaps not all
Pd ions really go into the sample, or some go in interstitially) or in the theoretical description. If,
for example, the Pd ions tend to cluster above a certain concentration, then our assumption of a
Poisson distribution becomes incorrect.
5. The Knight shift and the role of interchain couplings
As already mentioned, the staggered part of the susceptibility cannot be observed in a bulk
measurement. This, however, is possible by NMR because here the resonance frequency gets shifted
by the local magnetic field. This so-called Knight shift Kj can, therefore, be directly related to the
458
Spin-1/2 chain compounds
local susceptibility. For a chain of length N it is given by
K
(N)
j = (γe/γn)
∑
j′
Aj−j′χ
(N)
j′ , (5.1)
where γe (γn) is the electron (nuclear) gyromagnetic ratio, respectively and Aj−j′ the hyperfine
coupling tensor. It is usually sufficient to take only A0 and A±1 into account because of the short-
range nature of the hyperfine interaction.
7.5 7.55 7.6 7.65 7.7
h [T]
In
te
ns
ity
[A
rb
. u
ni
ts
] T = 30 K
h || a
theory I
experiment theory II
∆h
Dh
δh
Figure 7. NMR spectrum for Sr2CuO3 at T = 30 K taken from [6]. In comparison the theoretical
results for a single chain with a Poisson distribution of chain breaks (theory I) and for the case
where also interchain couplings (J⊥ = 5 K) are taken into account (theory II) are shown. In
both cases p = 5× 10−4, Γ = 4× 10−4, and h0
res = 7.598 T. The material-dependent parameters
are given in the text.
To compare with experiment, we assume again a Poisson distribution of chain breaks so that
the measured Knight shift is given by an average over all possible chain lengths. Furthermore, each
site in a chain of length N gives a different Knight shift according to (5.1). If we assume that each
of these Knight shifts has a Lorentzian lineshape with width Γ we find for the NMR spectrum
P (K) =
Γ
π
∞∑
N=1
p(1 − p)N−1
N
N∑
j=1
1
(K −K
(N)
j )2 + Γ2
. (5.2)
Using the results for the uniform (3.18) and the staggered part (3.21) we can now immediately
calculate the NMR spectrum for an ensemble of isotropic Heisenberg chain segments and compare
to experimental data for Sr2CuO3 obtained by Takigawa et al. [6,7]. We use J = 2200 K as exchange
constant in (1.1) and hyperfine coupling constants A0
c/(2~γn) ≈ −13 T, A0
ab/(2~γn) ≈ 2 T, and
A1/(2~γn) ≈ 4 T [39]. Here the index denotes the magnetic field direction. We calculate the
spectrum as a function of h = (1 + K)h0
res and use h0
res = 7.598 T. For the resonance field
ν = 86 MHz used in the experiment [6] this is consistent with h0
res = ν/γn where γn ≈ 11.3 MHz/T
[40]. A comparison of formula (5.2) for this set of material-dependent parameters with experiment
is shown in figure 7. Here the impurity concentration p and the Lorentzian linewidth Γ have been
used as fitting parameters. Note that the sample used for the NMR experiment has been annealed,
thus dramatically reducing the amount of excess oxygen and the associated chain breaks compared
to the sample used for the susceptibility measurements shown in figure 6. The theory predicts
a central peak corresponding to the bulk susceptibility value and two shoulders with separation
∆h (see curve ‘theory I’ in figure 7) which are caused by the maxima of the local susceptibility
(see, for example, figure 2). Theoretically, we find that ∆h ∼ h0
res
√
v/T ln1/4(v/T ), i. e., the
separation of the shoulders increases ∼ 1/
√
T with decreasing temperature. This is in agreement
459
J. Sirker
with experimental findings [6]. The central peak, however, has a much more complex structure
than predicted. The peak is split (feature δh in figure 7) and has prominent shoulders (feature Dh
in figure 7). These features are only observed in experiment at temperatures T . 60 K, whereas
at higher temperatures the theoretical prediction agrees well with experimental data (not shown).
To explain the additional features in the NMR spectra at T . 60 K we have to take the
interchain coupling into account. Sr2CuO3 shows three-dimensional Néel ordering at TN ∼ 5 K.
According to the usual chain mean field argument the interchain coupling should, therefore, also
be of the order of J⊥ ∼ 5 K. This estimate is consistent with band structure calculations [8]. As we
have already demonstrated numerically in figure 4, a ladder-like interchain coupling as present in
Sr2CuO3 leads to a reflection of susceptibility oscillations. Therefore, another typical Knight shift
is expected to be present in the NMR spectra, related to the maxima of the reflected oscillations.
As long as J⊥/T � 1 we can calculate this effect perturbatively. Starting from the definition of
the local susceptibility given in equation (1.2) we obtain to the first order
χ
(1)
j,1 = −J⊥
T 2
∑
k
〈Sz
j,1S
z
k,1〉︸ ︷︷ ︸
Gzz
1 (j−k)
〈Sz
tot,2S
z
k,2〉︸ ︷︷ ︸
χk,2
. (5.3)
Here, the lower index 1 stands for the infinitely long chain without chain breaks whereas the lower
index 2 denotes the chain with a chain break at site k = 0. Both, the two-point correlationGzz
1 (j−k)
and the local susceptibility χk,2 have a uniform and a staggered part in the low temperature limit.
We, therefore, find χ
(1)
j,1 ≈ χ
uni(1)
j,1 + (−1)jχ
st(1)
j,1 with χ
uni(1)
j,1 = −J⊥χuni(0)
1 χ
uni(0)
2 /T and
χ
st(1)
j,1 = −J⊥
T
(−1)j
∑
k
χst
k,2G
zz,st
1 (j − k) . (5.4)
Here equation (3.8) has to be used for χst
k,2 whileGzz,st
1 (j−k) = 〈Sz
j S
z
k〉st = c2/
[
v
πT sinh(πT
v |j − k|)
]K
is the staggered part of the bulk two-point correlation function. In figure 4, formula (5.4) is com-
pared with the numerical result and good agreement is found.
If we take these reflections into account, then also the additional features in the NMR spectra
are explained as shown in figure 7 (curve ‘theory II’). In particular, the shoulders with separation
Dh directly correspond to the maxima of the reflected susceptibility oscillations. The splitting of
the peak δh is of different origin. It is in fact not a splitting but rather a loss of intensity at the value
which corresponds to the bulk susceptibility. The oscillations and reflected oscillations spread over
the entire crystal at low temperatures so that there are simply no sites left which show bulk behav-
ior. Rather interestingly, the peak usually associated with the bulk susceptibility value, therefore,
turns into a dip at low temperatures due to the presence of chain breaks and interchain couplings.
Finally, we also want to shed some light on the role played by the geometry of the interchain
couplings. To this end, we consider an interchain coupling as shown in figure 1c. Such a coupling
is realized in SrCuO2 with a ferromagnetic J⊥ ∼ [−0.1 J,−0.3 J ] [1,8]. Using again first order
perturbation theory we find in this case
χ
(1)
j,1 = −J⊥
T 2
∑
k
〈Sz
j,1S
z
k,1〉︸ ︷︷ ︸
Gzz
1 (j−k)
〈Sz
tot,2(S
z
k−1,2 + Sz
k,2)〉︸ ︷︷ ︸
χk−1,2+χk,2
. (5.5)
Separating this into a uniform and a staggered part we find χ
uni(1)
j,1 = −2J⊥χ
uni(0)
1 χ
uni(0)
2 /T and
χ
st(1)
j,1 = −J⊥
T
(−1)j
∑
k
(χst
k,2 − χst
k−1,2)︸ ︷︷ ︸
≈0
Gzz,st
1 (j − k) . (5.6)
Therefore, no reflections will occur in this case to first order in perturbation theory consistent with
the numerical results shown in figure 5. This also means that an NMR spectrum for SrCuO2 would
not show any shoulders associated with the coupling to the nearest neighbor chain. Reflections in
chains further away might, however, be still possible which then would again lead to additional
structures in the NMR spectra at low temperatures.
460
Spin-1/2 chain compounds
6. Summary and conclusions
We have investigated here how chain breaks and interchain couplings affect the physical prop-
erties of spin chain compounds. A weak coupling of two Heisenberg chains is an irrelevant per-
turbation in the renormalization group sense whereas the weakening of a bond in an otherwise
homogenous chain is relevant. Open boundary conditions are, therefore, the stable fix point. Due
to this reasoning it is expected that a wide class of perturbations like impurities or dislocations
present in any real compound can be effectively described at low energies as a chain break. An
experimental measurement then corresponds to taking an average over an ensemble of finite chain
segments with open boundaries.
By combining a low-energy effective field theory with Bethe ansatz we have derived parameter-
free formulas for the thermodynamics of finite spin-1/2 Heisenberg chains with open boundary con-
ditions. Particular emphasis was put on a calculation of the susceptibility. Due to the broken trans-
lational invariance there exists a site-dependent staggered susceptibility in addition to the uniform
site-independent part. However, even the uniform part is affected by the open boundary conditions
in the sense that a surface contribution arises which is not present for periodic boundary conditions.
We have shown that susceptibility measurements on Sr2CuO3 doped with non-magnetic Pd
ions are well described by the theory presented here. One of the complications arising for this
compound is, however, that even the undoped sample apparently already has a relatively large
amount of chain breaks which are believed to be caused by excess oxygen. The impurity concen-
tration used in the theory to fit the experimental data, therefore, differs significantly from the
nominal Pd concentration. The emerging picture nevertheless seems to be consistent – at least at
low impurity concentrations – with a fixed concentration of additional chain breaks already present
in the undoped sample. It would certainly be of some value to obtain experimental data for this or
some other spin-chain compound where the impurity concentration is well controlled and a direct
comparison with theory is, therefore, possible.
Furthermore, we used the field theory to calculate NMR spectra. Importantly, the Knight
shift is proportional to the local susceptibility so that the staggered site-dependent part of the
susceptibility, which cannot be observed in a bulk susceptibility measurement, becomes observable.
As has already been shown previously [6], the staggered susceptibility leads to a broad background
in the NMR spectra with edges caused by the maxima of the staggered susceptibility. For Sr2CuO3
it has, however, been found that the NMR spectra at low temperatures show puzzling additional
features which have been ascribed in [7] to a coupling to phonons. Here we have shown that these
features quite naturally arise in a spin-only model if the known interchain couplings are taken into
account. It is also important to note that the geometry of the interchain couplings is crucial. As a
specific example we also considered, in addition to the ladder-like interchain coupling relevant for
Sr2CuO3, the zigzag-like interchain coupling realized in SrCuO2. In the latter case no additional
structures in the NMR spectra related to this interchain coupling will occur. NMR experiments on
spin chain compounds are, therefore, not only helpful to study of impurity effects. They can also
be used to investigate the geometry and strength of interchain couplings.
Acknowledgements
I want to thank all my collaborators on this and related topics, in particular, Ian Affleck,
Michael Bortz, Sebastian Eggert, Andreas Klümper, and Nicolas Laflorencie.
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Наслiдки недосконалостей гратки i мiжланцюжкових
взаємодiй для критичних властивостей спiн-1/2
ланцюжкових сполук
Є. Сiркер
1 Факультет фiзики i дослiдницький центр OPTIMAS, Унiверситет Кайзершлаутена, Нiмеччина
2 Iнститут дослiджень твердого стану Макса Планка, вул. Гайзенберга 1, D–70569 Штутгарт,
Нiмеччина
Отримано 7 травня 2009 р.
Щоб мати можливiсть порiвняти теоретичнi передбачення для спiнових ланцюжкiв з експеримен-
тальними даними, часто важливо взяти до уваги ефекти домiшок, так само, як мiжланцюжковi взає-
модiї. Тут ми представляємо теорiю поля для скiнчених спiнових ланцюжкiв при скiнченiй темпера-
турi i обчислюємо експериментально вимiрюванi величини як сприйнятливостi i спектри ядерного
магнiтного резонансу. Для мiжланцюжкових взаємодiй ми зосереджуємося на геометрiях, iстотних
для купратних спiнових ланцюжкiв як от Sr2CuO3 i SrCuO2. Теоретико-польовi результати порiвнюю-
ться з експериментальними i чисельними даними, отриманими з використанням ренормалiзацiйної
групи для матрицi густини.
Ключовi слова: спiновi ланцюжки, домiшки, термодинамiка, бозонiзацiя, ренормалiзацiйна група
для матрицi густини
PACS: 75.10.Pq, 75.10.Jm, 11.10.Wx, 0230.Ik
462
|
| id | nasplib_isofts_kiev_ua-123456789-120021 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T18:39:21Z |
| publishDate | 2009 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Sirker, J. 2017-06-10T18:30:46Z 2017-06-10T18:30:46Z 2009 Consequences of lattice imperfections and interchain couplings for the critical properties of spin-$1/2$ chain compounds / J. Sirker // Condensed Matter Physics. — 2009. — Т. 12, № 3. — С. 449-462. — Бібліогр.: 40 назв. — англ. 1607-324X PACS: 75.10.Pq, 75.10.Jm, 11.10.Wx, 02.30.Ik DOI:10.5488/CMP.12.3.449 https://nasplib.isofts.kiev.ua/handle/123456789/120021 To allow for a comparison of theoretical predictions for spin chains with experimental data, it is often important
 to take impurity effects as well as interchain couplings into account. Here we present the field theory for finite
 spin chains at finite temperature and calculate experimentally measurable quantities like susceptibilities and
 nuclear magnetic resonance spectra. For the interchain couplings we concentrate on geometries relevant for
 cuprate spin chains like Sr₂CuO₃ and SrCuO₂. The field theoretical results are compared to experimental as
 well as numerical data obtained by the density matrix renormalization group. Щоб мати можливiсть порiвняти теоретичнi передбачення для спiнових ланцюжкiв з експериментальними даними, часто важливо взяти до уваги ефекти домiшок, так само, як мiжланцюжковi взаємодiї. Тут ми представляємо теорiю поля для скiнчених спiнових ланцюжкiв при скiнченiй температурi i обчислюємо експериментально вимiрюванi величини як сприйнятливостi i спектри ядерного магнiтного резонансу. Для мiжланцюжкових взаємодiй ми зосереджуємося на геометрiях, iстотних для купратних спiнових ланцюжкiв як от Sr₂CuO₃ і SrCuO₂. Теоретико-польовi результати порiвнюються з експериментальними i чисельними даними, отриманими з використанням ренормалiзацiйної групи для матрицi густини. I want to thank all my collaborators on this and related topics, in particular, Ian A eck,
 Michael Bortz, Sebastian Eggert, Andreas Kl umper, and Nicolas La
 orencie. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Consequences of lattice imperfections and interchain couplings for the critical properties of spin-$1/2$ chain compounds Наслiдки недосконалостей гратки i мiжланцюжкових взаємодiй для критичних властивостей спiн-1/2 ланцюжкових сполук Article published earlier |
| spellingShingle | Consequences of lattice imperfections and interchain couplings for the critical properties of spin-$1/2$ chain compounds Sirker, J. |
| title | Consequences of lattice imperfections and interchain couplings for the critical properties of spin-$1/2$ chain compounds |
| title_alt | Наслiдки недосконалостей гратки i мiжланцюжкових взаємодiй для критичних властивостей спiн-1/2 ланцюжкових сполук |
| title_full | Consequences of lattice imperfections and interchain couplings for the critical properties of spin-$1/2$ chain compounds |
| title_fullStr | Consequences of lattice imperfections and interchain couplings for the critical properties of spin-$1/2$ chain compounds |
| title_full_unstemmed | Consequences of lattice imperfections and interchain couplings for the critical properties of spin-$1/2$ chain compounds |
| title_short | Consequences of lattice imperfections and interchain couplings for the critical properties of spin-$1/2$ chain compounds |
| title_sort | consequences of lattice imperfections and interchain couplings for the critical properties of spin-$1/2$ chain compounds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120021 |
| work_keys_str_mv | AT sirkerj consequencesoflatticeimperfectionsandinterchaincouplingsforthecriticalpropertiesofspin12chaincompounds AT sirkerj naslidkinedoskonalosteigratkiimižlancûžkovihvzaêmodiidlâkritičnihvlastivosteispin12lancûžkovihspoluk |