Using a new version of the density-matrix renormalization group we determine
the phase diagram of a model of an antiferromagnetic Heisenberg spin chain
where the spins interact with quantum phonons. A quantum phase transition from
a gapless spin-fluid state to a gapped dimerized phase occurs at a non-zero
value of the spin-phonon coupling. The transition is in the same universality
class as that of a frustrated spin chain, which the model maps to in the
anti-adiabatic limit. We argue that realistic modeling of known spin-Peierls
materials should include the effects of quantum phonons.Comment: RevTeX, 5 pages, 3 eps figures included using epsf. Improved theories
  in adiabatic and non-adiabatic regimes give better agreement with DMRG. This
  version accepted in Physical Review Letter

A. Auerbach

A. Sandvik

B. van Bodegom

C. Gros

C. Zhang

Chris J. Hamer

D. Augier

E. Jeckelmann

G. Castilla

G. S. Uhrig

G. Wellein

I. Affleck

J. P. Boucher

J. S. Kasper

J. W. Bray

K. Fabricius

K. Kuboki

K. Nomura

K. Okamoto

L. G. Caron

M. Braden

M. C. Cross

M. Hase

R. J. Bursill

Robert J. Bursill

Ross H. McKenzie

S. Eggert

S. Huizinga

S. R. White

English

arXiv

Crossref

Phase Diagram of a Heisenberg Spin-Peierls Model with Quantum Phonons

Using a new version of the density-matrix renormalization group we determine the phase diagram of a model of an antiferromagnetic Heisenberg spin chain where the spins interact with quantum phonons. A quantum phase transition from a gapless spin-fluid state to a gapped dimerized phase occurs at a nonzero value of the spin-phonon coupling. The transition is in the same universality class as that of a frustrated spin chain, to which the model maps in the diabatic limit. We argue that realistic modeling of known spin-Peierls materials should include the effects of quantum phonons

Bursill, RJ

McKenzie, RH

Hamer, CJ

University of Queensland eSpace

VOLUME 83, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R S 12 JULY 1999
408Phase Diagram of a Heisenberg Spin-Peierls Model with Quantum Phonons
Robert J. Bursill,* Ross H. McKenzie, and Chris J. Hamer
School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia
(Received 24 December 1998)
Using a new version of the density-matrix renormalization group we determine the phase diagram of
a model of an antiferromagnetic Heisenberg spin chain where the spins interact with quantum phonons.
A quantum phase transition from a gapless spin-fluid state to a gapped dimerized phase occurs at a
nonzero value of the spin-phonon coupling. The transition is in the same universality class as that of a
frustrated spin chain, to which the model maps in the diabatic limit. We argue that realistic modeling
of known spin-Peierls materials should include the effects of quantum phonons.
PACS numbers: 75.10.Jm, 71.38.+ i, 71.45.Lr, 75.50.EeChallenged by the discovery of high-temperature super-
conductivity in doped antiferromagnets, our understanding
of quantum magnetism in low dimensions has increased
significantly over the past decade [1]. However, the ef-
fect of the interaction of quantum spin systems with further
degrees of freedom such as disorder, phonons, and holes
produced by doping is still poorly understood. Interest in
models of spins interacting with phonons has increased sig-
nificantly since the discovery of a spin-Peierls transition in
the inorganic compound CuGeO3 [2]. The availability of
large, high-quality single crystals has led to much more ex-
tensive experimental studies [3] than on the organic spin-
Peierls materials studied in the 1970s [4].
The fact that a spin-12 antiferromagnetic Heisenberg
chain is unstable to a static uniform dimerization [4,5]
is known as the spin-Peierls instability. This occurs
because dimerization opens a gap D in the spin excitation
spectrum and lowers the total magnetic energy by a
greater amount than the increase in elastic energy due
to the dimerization. Until very recently, almost all
theoretical treatments have used this static picture which
assumes that the frequency v of the phonon associated
with the dimerization is much smaller than D and the
antiferromagnetic exchange integral J . It has recently
been pointed out that CuGeO3 is not in this adiabatic
regime [6–8], stimulating several numerical studies of
models with dynamical phonons [8,9].
In this Letter, we study a model of a spin-12 anti-
ferromagnetic Heisenberg chain interacting with quantum
phonons using a powerful new numerical technique that
allows an essentially exact treatment of both the spins and
the phonons at a fully quantum-mechanical level. Our
main result is the phase diagram in Fig. 1 in which the
adiabaticity parameter Jv varies over several decades.
We find that the spin-phonon coupling must be larger
than some nonzero critical value for the spin-Peierls in-
stability to occur. This is in contrast to the static case
(vJ ! 0) for which dimerization occurs for any value
of the coupling. Hence, quantum lattice fluctuations can
destroy Heisenberg spin-Peierls order. We find that the
quantum phase transition from the spin-fluid state to the
gapped state is in the same universality class as the dimer-0031-90079983(2)408(4)$15.00ization transition of the J1-J2 frustrated spin chain. Our
results have important implications for the modeling of
spin-Peierls materials.
The model we study is one of the simplest possible. It
consists of a local phonon on each site and the antifer-
romagnetic exchange on neighboring sites varies linearly
with the difference between the phonon amplitudes on the
two sites. The Hamiltonian is
H 
NX
i1
J 1 gbi11 1 b
y
i11 2 bi 2 b
y
i  Si ? Si11
1 v
NX
i1
b
y
i bi . (1)
Here Si is the S  12 spin operator on site i and bi
destroys a phonon of frequency v on site i. We assume
a periodic chain of N sites.
Insight into this model can be obtained by considering
the diabatic limit (v ¿ J). One can then integrate out
the phonon degrees of freedom to obtain the following
effective Hamiltonian for the spin degrees of freedom [10:
Heff  J1
NX
i1
Si ? Si11 1 J2
NX
i1
Si ? Si12 , (2)
FIG. 1. Zero temperature phase diagram of the spin-Peierls
antiferromagnetic chain of spins interacting with quantum
phonons [Eq. (1)]. For small spin-phonon coupling g the
system is a gapless spin-fluid. For large g the system is
dimerized and has an energy gap. The diamonds with error
bars denote the phase boundary from this DMRG study. The
dotted line is [Eq. (5)] the phase boundary which results from
an approximate mapping onto the J1-J2 model [frustrated
antiferromagnetic chain, Eq. (2)] which becomes exact in the
diabatic limit Jv ! 0.© 1999 The American Physical Society
VOLUME 83, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R S 12 JULY 1999where J1  J 1 g2v and J2  g22v. Uhrig [6]
recently obtained the same Hamiltonian, calculating J1
and J2 to next order in Jv.
J1  J 1 g
2v 2 3g2J2v2 1 · · · , (3)
J2  g
22v 1 3g2J2v2 1 · · · . (4)
The frustrated spin chain Eq. (2) or J1-J2 model has been
extensively studied and is well understood. If a  J2J1
then at a critical value of a  ac  0.241 1675 the
model undergoes a quantum phase transition from a gap-
less spin-fluid state with quasi-long-range antiferromag-
netic order to a gapped phase with long-range dimer order
[11,12]. Uhrig pointed out that this implies that in the
diabatic regime (1) possesses a nonzero critical coupling
gc. To second order in Jv,
g2cv 
acJ
12 2 ac 1 31 1 acJ2v
. (5)
We have confirmed this result numerically (see Fig. 1).
Furthermore, this nonzero critical coupling gc still occurs
well into the adiabatic regime. It is interesting that
although (5) is valid only to second order in Jv it gives
a good description of gc up to Jv  1.
Models such as (1), which involve bosons are a chal-
lenge to study numerically due to the large number of
degrees of freedom per site. The density matrix renor-
malization group (DMRG) method [13] has the potential
for obtaining definitive results for these models by study-
ing very large systems. Several schemes based on the
DMRG have recently been developed to treat models in-
volving phonons [14–16]. We employ a new “four-block”
DMRG method [16] which allows us to treat the phonons
and spins on an equal footing and to study systems as large
as 256 sites. This is in contrast to some recent exact diago-
nalization studies of spin-phonon models that were limited
to small systems and/or used uncontrolled truncations of
the phonon degrees of freedom [8,9]. We previously used
this method to obtain the phase diagram of the Holstein
model with spinless fermions [17].
The four-block method can be used to calculate the
ground state energy E0 and the singlet and triplet gaps
Dss and Dst for periodic systems [16]. Table I shows the
DMRG convergence of the gaps with the single truncation
parameter e [18] for a representative parameter set. It can
be seen that the gaps are sufficiently well resolved to be
useful for finite-size scaling analyses. The error of around
0.1% in the N  128 site system is typical of the error in
the largest systems studied for a given set of parameters.
We determine the critical coupling using the gap-
crossing method used by Okamoto and Nomura [12]
to determine the critical coupling ac in the frustrated
Heisenberg model (2). The convergence of the crossover
coupling acN with N is rapid due to the absence of
logarithmic corrections at the critical point [11,12,19]. If
the system is gapless with quasi-long-range Néel order for
0 # g # gc, the lowest excitation is the triplet state, i.e.,TABLE I. Four-block DMRG convergence of the singlet and
triplet gaps Dss and Dst of the spin-Peierls model (1) with the
truncation parameter e for various periodic lattices of size N ,
where Jv  1 and gv  0.4.
N e Dssv Dstv
8 10215 0.313 749 61 0.518 325 1
8 10220 0.313 728 89 0.518 325 4
8 10222 0.313 728 70 0.518 325 4
32 10213 0.076 478 2 0.133 925
32 10215 0.076 595 8 0.133 785
32 10216 0.076 593 3 0.133 778
128 10210 0.014 909 0.040 09
128 10211 0.014 817 0.038 56
128 10213 0.014 619 0.037 90
128 10214 0.014 648 0.037 75
Dst , Dss (for sufficiently large N) and Dst,Dss ! 0 as
N ! `. If for g . gc the system has a nonzero gap D
and is dimerized with a doubly degenerate ground state,
then the first excited singlet state becomes degenerate with
the ground state in the bulk limit [19]. That is, Dss , Dst
(for sufficiently large N), Dss ! 0, and Dst ! D . 0
as N ! `. A finite lattice crossover coupling gcN
is defined by Dst  Dss. As shown in Table II, gcN
rapidly approaches a limit as N ! `. This limit is the
critical coupling gc separating gapless and gapped phases.
For the Jv . 1 cases, where the N dependence is
substantial, gcN is well described by the functional form
gcN  gc 2 A exp2BN and nonlinear fitting is used
to determine gc [20]. The resulting phase boundary is
plotted in Fig. 1. The DMRG, discretization, and fitting
errors in gc are estimated to be no greater than a few
percent.
From conformal invariance the finite-size energies of the
spin-fluid should satisfy [12]:
E0  Ne` 1
py0
6N
1 · · · , (6)
1
4
3Dst 1 Dss 
py1
N
1 1 · · · , (7)
TABLE II. Convergence of the crossover coupling gcNv
with lattice size N for various values of the adiabaticity param-
eter Jv. gcN is defined by Dss  Dst and converges to the
critical coupling gc as N ! `.
Jv
N 0.005 0.1 1.0 2.0 10.0
4 0.0692 0.237 0.1201 · · · · · ·
8 0.0681 0.228 0.2735 0.092 · · ·
16 0.0671 0.225 0.3021 0.274 · · ·
32 · · · 0.223 0.3087 0.310 · · ·
64 · · · · · · 0.3092 0.318 0.249
128 · · · · · · · · · · · · 0.318
256 · · · · · · · · · · · · 0.339409
VOLUME 83, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R S 12 JULY 1999where e` is the bulk ground state energy density and y0 
y1  ys is the spin wave velocity. The combination of
the gaps in Eq. (7) is chosen to cancel the logarithmic
corrections.
We have performed a number of consistency checks
on our results. First, ys as determined from Eq. (6) and
our DMRG calculations for Jv  0.005 and gv ,
0.05 agrees with results for the same quantity determined
for the corresponding J1-J2 model, again using DMRG
techniques. This confirms the mapping between the two
models in the diabatic regime. Second, we note that the
DMRG results for the phase boundary agree well with
the result (5) from the mapping in the diabatic limit (see
the dotted line in Fig. 1). Third, for general phonon
frequencies, we calculate the ratio y0y1 which should
equal unity. At g  gc it is one within errors expected
from corrections to scaling and DMRG truncation, over
the range of frequencies studied. Values vary from 0.98 6
0.04 for Jv  0.005 to 1.07 6 0.10 for Jv  10.
For a Kosterlitz-Thouless (KT) transition, the gap D 
limN!` Dst is expected to have an essential singularity at
g  gc. In Fig. 2,Dst is plotted as a function of g for vari-
ous N in a case of intermediate coupling Jv  1. Two-
point linear extrapolations (in 1N) to N  ` are included
in the plot. These estimates of D are shown to be well
fitted by the KT form [12] D  Afg exp2B fg2	
where fg  g 2 gc212. Note that the gap crossover
method (Table II) is substantially more accurate than this
fitting procedure for determining gc, the latter tending to
overestimate gc [16].
FIG. 2. The singlet-triplet gap Dst of the spin-Peierls model
as a function of the coupling g for various lattice sizes N for
an intermediate phonon frequency Jv  1. Extrapolations
(in 1N , using the two largest values of N) to N  ` are
given by the solid diamonds. These are fitted to the KT form
Afg exp2Bfg2, where fg  g 2 gc212 (solid line).
The critical coupling gc is not obtained from this fit. It is
substantially more accurate to use the gap crossover method
(see Table III). The inset shows the extrapolated gap (using
N  32 and 64) for a small phonon frequency (adiabatic
regime) Jv  10. The dashed line is the result for the static
limit where the quantum phonon fluctuations are neglected [21].410In the adiabatic regime (v ø J) there is strong mixing
between spin singlet and phonon excitations. An analo-
gous effect was observed for the Holstein model [17]. In
the case of (1) this is manifested in nonlinear corrections
to the scaling of Dss. That is, Dss is found to be phonon
like (flat in 1N) until the characteristic spin energy
2pJN decreases below the bare phonon frequency v, at
which point Dss begins to vanish, as 1N 0 # g # gc,
or exponentially g . gc. This can be seen in Table II
from the slow convergence of gcN with N for the
Jv  10 case.
Next, we consider the validity of the static approxima-
tion in the adiabatic regime, where the phonon operators
bi in (1) are replaced by the constant dimerization 21id,
the total energy is minimized as a function of d then the
gap is calculated for this optimal value of d. This cal-
culation was performed by using the four-block DMRG
method to solve for the ground state energy and gap
in the dimerized Heisenberg model [21]. The resulting
adiabatic curve is compared in Fig. 2 to the extrapolated
gap D found from our numerical results for Jv  10.
We see that even in this adiabatic region treating the
phonons in the mean-field approximation is not fully re-
liable, particularly for the purposes of quantitatively ex-
tracting the coupling g from the experimental triplet gap.
The situation is far worse for phonon frequencies relevant
to CuGeO3. For example, for the Jv  1 case in Fig. 2,
the adiabatic curve would not fit on the same scale as the
curve from the fully dynamical model.
To consider our results in the context of experiment,
estimates of a number of parameters for various spin-
Peierls compounds are listed in Table III. It can be
seen from these estimates and our results that the static
approximation is highly questionable for CuGeO3, and
may not be valid for the organic spin-Peierls materials.
A related question is the use of an explicit next-neighbor
(J2, frustration) term in adiabatic spin-phonon models of
CuGeO3 [27,28]. The value of J2 required to achieve
agreement with susceptibility and magnetic specific heat
data is generally very large (J2J1 
 0.3. Attempts
have been made to justify the inclusion of a J2 term
on the basis of Cu-O-O-Cu superexchange paths [28].
However, Ref. [6] and the present analysis suggests that
an explicit J2 term may not be required in order to
describe experimental results if the phonons are treated
TABLE III. Estimates of the exchange J, phonon frequency
v, and energy gap D for various spin-Peierls materials. All are
given in units of degrees kelvin. (We are unaware of any other
measurements of the frequencies of the dimerization phonon in
organic materials.)
Material J v D Ref.
CuGeO3 100 150, 300 20 [2,22]
TTFCuS4C4C3F4 70 ? [24] 20 [4,23]
MEM TCNQ2 50 100 60 [25,26]
VOLUME 83, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R S 12 JULY 1999quantum mechanically since the phonons induce a next-
nearest neighbor interaction.
To conclude, we have numerically determined the
phase diagram of a spin-Peierls model (1) with high ac-
curacy. Our results are consistent with a mapping of
the model to the frustrated spin chain (2) in the dia-
batic limit (large phonon frequency). For a wide range
of phonon frequencies compared to the exchange there is
a phase transition at a nonzero value of the coupling g
from a gapless spin-fluid state to a gapped dimer phase
[29]. The transition is in the same universality class as
the Kosterlitz-Thouless transition in the frustrated antifer-
romagnetic chain (2). Quantum phonon fluctuations are
important in known spin-Peierls materials.
This work was supported by the Australian Research
Council. We thank I. Affleck, A. Sandvik, O. Sushkov,
J. Voit, V. Kotov, H. Fehske, A. Weisse, and G. Uhrig
for discussions and A. Sandvik for showing us results
prior to publication. Calculations were performed at the
New South Wales Centre for Parallel Computing and the
Australian National University Supercomputing Facility.
*Email address: ph1rb@newt.phys.unsw.edu.au
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9905337]. A nonzero critical spin-phonon coupling was
also found.411


Phase diagram of a Heisenberg spin-Peierls model with quantum phonons

Phase diagram of a Heisenberg spin-Peierls model with quantum phonons

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