Quantum lattice fluctuations in a frustrated Heisenberg spin-Peierls chain

9 Pages·0.19 MB·English

by  A. Weisse

Quantum lattice fluctuations in a frustrated Heisenberg spin-Peierls chain∗
A. Weißea, G. Welleinb and H. Fehskec
aPhysikalisches Institut, Universit¨at Bayreuth, 95440 Bayreuth, Germany
bRegionales Rechenzentrum Erlangen, Universit¨at Erlangen, 91058 Erlangen, Germany
cTheoretische Physik, Universit¨at des Saarlandes, 66041 Saarbru¨cken, Germany
(March 3, 2009)
9 As a simple model for spin-Peierls systems we study a frustrated Heisenberg chain coupled to
9 opticalphonons. Inviewoftheanorganicspin-PeierlscompoundCuGeO3 weconsidertwodifferent
9 mechanisms of spin-phonon coupling. Combining variational concepts in the adiabatic regime and
1
perturbation theory in theanti-adiabatic regime we derive effectivespin Hamiltonians which cover

n the dynamical effect of phonons in an approximate way. Ground-state phase diagrams of these
a models are determined, and the effect of frustration is discussed. Comparing the properties of the
J groundstateandoflow-lyingexcitationswithexactdiagonalization dataforthefullquantumspin

5 phonon models, good agreement is found especially in theanti-adiabatic regime.
2
PACS numbers: 75.10.Jm, 63.20.Kr

]
l
e
- I. INTRODUCTION spinchain, whichis basedonthe unitary transformation
r
t method. It works well in the adiabatic and in the anti-
s
. The effect of a Peierls instability in quasi-one- adiabatic regime. In the latter case there are also some
t
a dimensional spin systems, i.e. the instability of a uni- approachestotheHeisenbergspinchaininteractingwith
m form spin chain towards dimerization induced by the in- optical phonons: Kuboki and Fukuyama [12] used per-
- teraction with lattice degrees of freedom, has attracted turbationtheory toderive aneffective spinHamiltonian,
d considerable attention over the last decades. Starting in while Uhrig [13] applied the flow-equation method [14].
n
the seventies with organic compounds of the TTF and As a simple model which contains all important fea-
o
c TCNQ family [1], the interest in the Peierls instability tures of a SP system in the following we consider an
[ was renewed with the discovery of a spin-Peierls (SP) antiferromagnetic Heisenberg chain coupled to a set of

transition in the anorganic compound CuGeO in 1993 Einstein oscillators,
1 3
v by Hase et al. [2]. The most significant feature distin- H =H +H +H (1)
2 guishing CuGeO3 from other SP-compounds is the high s p sp
6 energy of the involved optical phonons, which is compa- with
2 rabletothemagneticexchangeintegralJ. Incontrastto
1 theorganicmaterialsnosofteningofthesephononmodes H =J (S ·S +αS ·S ), (2)
s i i+1 i i+2
0
is observed near the transition. Therefore the adiabatic i
9 X
9 treatment of the phonon subsystem used in the works of H =ω b+b . (3)
/ Pytte [3] or Cross and Fisher [4], does not seem appro- p 0 i i
at priatetodescribetheSPtransitioninCuGeO3,although Xi
m there are recent efforts in this direction [5]. Rather one The interaction of spins and phonons, Hsp, can be mod-
hastotakeintoaccounttheeffectofquantumlatticefluc- elled in two different ways,
-
d tuations which tend to decrease the SP transition tem-
n peraturerespectivelytheenergygapbetweentheground Hslopc =g¯ (b+i +bi)Si·Si+1, (4)
o
state and lowestexcitations in the dimerized phase. Un- i
c X
v: fhoarntudnleacteoluyptlehderseysatreempsraocftsicpainllsy(neloecatnraolnyst)icanmdetphhoodnsontos Hsdpiff =g¯ (b+i +bi)(Si·Si+1−Si·Si−1), (5)
i
i X
X when all energy scales and coupling strengths are of the
r same order of magnitude. This is why many studies in- where Si denote spin-21 operators at lattice site i, while
a volving dynamical phonons rely on numerical methods, b+andb arephononcreationandannihilationoperators,
i i
such as exact diagonalization (ED) [6,7], density matrix respectively. HlocandHdiff differinthemechanism,how
sp sp
renormalizationgroup(DMRG)[8,9]orMonteCarlosim- thelatticeinfluencestheexchangeintegral. ForHloc,the
sp
ulation(MC)[10]. OnlyrecentlyZheng[11]developedan local coupling, one can think of a single harmonic degree
analyticalapproachtodescribetheSPinstabilityofaXY of freedom directly modifying the magnetic interaction.
In the context of CuGeO this could correspond to side
3
group effects (by the Germanium atoms) as discussed
in Ref. [15]. In the case of Hdiff, the difference coupling,
sp
∗Dedicated to H. Bu¨ttner on the occasion of his 60th theexchangedependsdirectlyonthespatialdistancebe-
birthday. tweenneighbouring spins. Note that it is not possible to
1
uniformlydecreaseorincreaseallexchangeintegralswith |nihn|H˜sp|mihm|
S = , (11)
this type of spin phonon interaction. 2 E −E
n m
Although Hloc seems to be more appropriate for
sp
where|niand|midenoteeigenstatesofthe unperturbed
CuGeO we will consider both variants and compare
3
Hamiltonian H +H with the corresponding eigenener-
their properties. Inadditionwe takeinto accounta frus- s p
gies E and E . Since we do not know the exact eigen-
trating next-nearest neighbour interaction Jα, which in n m
statesofH ,equation(11)cannotbewritteninasimple
viewofCuGeO wasintroducedtoexplainsusceptibility s
3
form. Onefirstguesswhichdescribesthe phonon-partof
data [29]. As wewill see below,the spin-phononinterac-
S in a correct way and ensures S to be antihermitean
tion is able to induce this kind of long ranged exchange 2 2
is
as well.
Motivated by the success of methods combining uni- g¯
Sloc =f (b+−b )S ·S , (12)
tary transformations with variational and numerical 2 ω i i i i+1
0
techniques, which we used to study the Peierls transi- Xi
g¯
tiniosnpirinedthbeyHtohlestewinormkoodfeZl hofensgpin[1le1s]s, fienrmthioisnsar[1ti6c]l,eawnde S2diff =fω0 (b+i −bi)(Si·Si+1−Si·Si−1). (13)
i
X
analyse the model (1) within the same framework. In
particular we focus on the ground-state phase diagram Comparing with (11) ω0 gives the appropriate contribu-
as a function of spin-phonon coupling, frustration and tion to the energy denominator, since Hsp connects only
phonon frequency, and compare our results with exact phonon states differing by one in their occupation num-
diagonalization data. ber. What is missing is the contribution of spin states
whichhaveoverlapviaH . Therefore,usinganumerical
sp
procedure we fix the free parameter f by the condition,
II. EFFECTIVE SPIN MODELS that the amplitude of the state resulting from the appli-
cation of H˜ +[S ,H˜ +H˜ ]| to the ground state
sp 2 s p ∆π=0
of H +H is minimal. Figure 1 illustrates the variation
To describe a static lattice dimerization in the adia- s p
of f with varying phonon frequency ω . We find that
batic case of small phonon frequency ω we start with a 0
0
unitarytransformationofH whichshiftstheequilibrium
positionofeachoscillatorbyaconstantamountalternat-
ing from site to site, H˜ =exp(S )Hexp(−S ) with
1 1 1.0
∆
S = π (−1)i(b+−b ). (6)
1 2g¯ i i 0.8
i
X
For the terms involving phonons this yields
0.6
H˜ =H −ω ∆π (−1)i(b++b ) f
p p 0 2g¯ i i 0.4
i
X
∆π 2 α = 0.0
+Nω0 2g¯ , (7) 0.2 α = 0.2
(cid:18) (cid:19) α = 0.4
H˜loc =Hloc−∆ (−1)iS ·S , (8)
sp sp π i i+1 0.0
Xi 10−2 10−1 100 101 102
H˜sdpiff =Hsdpiff −2∆π (−1)iSi·Si+1. (9) ω0/J
i
X
∆ will act as the variational parameter describing the
π
dimerization of the system. FIG. 1.: Variation of f in the case of local coupling and
Next we want to decouple spin and phonon degrees lattice size N =16.
of freedom. The idea is to apply another unitary trans-
formation H¯ =exp(S )H˜ exp(−S ), where S should be the general shape of f = f(ω ) depends only weakly on
2 2 2 0
chosen such that all contributions of first order in the both, system size N and frustration α. While f → 1 in
couplingconstantg¯disappearinthetransformedHamil- the anti-adiabatic frequency range (ω ≫ J), the trans-
0
tonian. This gives the usual condition for a Schrieffer- formation exp(S ) vanishes completely as the frequency
2
Wolff transformation [17], becomes small (ω ≪J).
0
In contrast to electron-phonon systems with Holstein
H˜sp+[S2,H˜s+H˜p] =! 0 (10) coupling,whereatransformationsimilartoexp(S2)com-
∆π=0 pletelyremovestheelectron-phononinteractionterm,ap-
(cid:12)
or (cid:12) plying the unitary transformation exp(S ) to H˜, we ob-
(cid:12) 2
2
tainaninfiniteseriesofterms,whichcannotbesummed Jdiff =1+2λf(1− f)+ 3f2g2Y(1−α) (24)
up to a simple expression, i.e. 1 2 2
3λg2Yf3 f f4g4Y2
H¯ = [S ,H˜] /k!, (14) + (1− )+ (59−75α),
2 k 2 4 24
Xk f f2g2Y(3−5α)
Jdiff =α+λf(1− )− (25)
where [S ,H˜] denotes the iterated commutator 2 2 2
2 k
[S ,H˜] =[S ,[S ,H˜] ] with [S ,H˜] =H˜. λg2Yf3 f f4g4Y2
2 k+1 2 2 k 2 0 − (1− )− (75−124α),
To get an effective spin model we consider only con- 12 4 24
tributions up to fourth order in g¯ and average over 5λg2Yf3 f
the phonon subsystem. The resulting spin Hamiltonian J3diff =−f2g2Yα− 6 (1− 4) (26)
H =hH¯i contains long rangedHeisenberg interactions
eff f4g4Y2
as well as numerous four- and six-spin couplings of the + (32−119α),
form (S ·S )(S ·S)...(S ·S ). To a good approxima- 48
i j k l m n
21f4g4Y2α
tion we can neglect them and obtain Jdiff = . (27)
4 48
H /J =J + (J +(-1)iδ)S ·S (15)
eff 0 1 i i+1
To point out the relevant model parameters we intro-
i
X ducedthedimensionlesscouplingconstantsλ=g¯2/(Jω )
4 0
+ J S ·S . (cf. Refs. [3,4]) and g =g¯/ω0. Besides we used
k i i+k
i k=2
XX −1 if T =0
Y :=h(b+−b )2i= , (28)
Note that all phonon dynamics disappeared from the i i −coth(ω0) if T >0
(cid:26) 2T
Hamiltonian H , but the effect of the spin-phonon in-
eff
teractionentersthroughboth,thestaticdimerizationpa- asashorthandnotation. Tocompareourresultwiththat
rameter∆ andthedifferentlongrangespininteractions. of Uhrig [13], in Hdiff we have to set f =1, which corre-
π eff
For the local coupling the corresponding interaction spondstotheanti-adiabaticregime,and∆ =0. Indeed
π
strengths are we recover (except for a prefactor 1/2 which in Ref. [13]
enters erroneously going from eq. (11c) to eq. (13), and
∆ f2g2Y f
δloc =− π (1−f)+ 1− , (16) which consequently is also wrong in Ref. [9, eq. (3-4)])
J 2 3 all second order terms derived with the flow-equation
(cid:20) (cid:21)
(cid:0) 2 (cid:1) method, supplemented by some new fourth order con-
ω 1 ∆
J0loc =N J0hb+i bii+ 4λ Jπ tributions.
" (cid:18) (cid:19) However, in contrast to Uhrig [13], who proposed to
3 f λg2Yf3 f takeathermalaveragewithinthephononsubsystem,we
− λf(1− )− (1− ) , (17) believe,thataveragingoverthephononvacuum,i.e. set-
8 2 16 4
# ting T = 0, is a more natural choice, especially if we
f f2g2Y(1−α) are dealing with phonon-energies in the anti-adiabatic
Jloc =1+λf(1− )+ (18)
1 2 2 regime, where the interesting energy- and temperature-
λg2Yf3 f f4g4Y2 scalesaresmallinrespectofω0. Thereforeinthe follow-
+ (1− )+ (28−37α), ing we use Y =−1 and hb+b i=0 exclusively.
2 4 96 i i
f2g2Y(1−2α)
Jloc =α− (19)
2 2 III. TRANSITION TO A GAPPED PHASE
λg2Yf3 f 37
− (1− )−f4g4Y2 (1−2α),
4 4 96 AprominentfeatureassociatedwiththeSPinstability
Jloc =−f2g2Yα + f4g4Y2(9−46α), (20) is of course the existence of an energy gap between the
3 2 96 ground state and lowest excitations. Considering, in a
Jloc = 9f4g4Y2α, (21) first step, the pure spin model Hs, it is known that the
4 96 spectrum is gaplessfor the Heisenberg chainwith α=0,
where the lowest spinon excitations (triplet and singlet)
while for the difference coupling we find
are degenerate with the ground state at momenta q = 0
δdiff =2 δloc, (22) and π [18–20]. In contrast the system has a two-fold
degenerategroundstate anda gapto lowesttriplet exci-
2
ω 1 ∆
Jdiff =N 0hb+b i+ π tations at α = 0.5, the Majumdar-Ghosh point [21]. At
0 " J i i 4λ(cid:18) J (cid:19) some intermediate frustration αc the model undergoes a
transition from the gapless to the gapped phase, which
3 f 3λg2Yf3 f
− λf(1− )− (1− ) , (23) is of Kosterlitz-Thouless type [22–24]. Using arguments
4 2 16 4 # of conformal field theory one can show that the lowest
3
singletandtripletexcitationsofafinite systemofsizeN
become degenerate at α (N), where the dependence on
c
N isonlyweakandα (N)−α (∞)∼N−2 [25–27]. This 0.4
c c
waLsouoskeidnginatRoefusr. e[2ff8e,c2t9iv]etospdientemrmodineelsαHc =w0.e24fi1n1d. that (a) HHHee cffff rcαoresoffs=s.αs (.cN (N=8=)16)
theinteractionwiththeopticalphononsinedffucesthesame 0.3 HHHeeeffffff dddiiimmm... (((NNN===811)26))
kind of frustrating next-nearest neighbour interaction.
ω/J=10
Therefore, without any explicit frustration α, the effec- 0
tive frustration α := J /J due to the phonons can α 0.2
eff 2 1
leadtoagapinthe energyspectrumandtospontaneous
gapless
dimerization, as was already discussed in Ref. [12]. This
effect is most important in the anti-adiabatic frequency 0.1 gapped
range.
Another mechanismproducinga gapis (static)dimer-
ization, i.e. an alternation of the nearest neighbour ex- 0.0
0.0 0.5 1.0 1.5
change integral. Taking the adiabatic limit of our effec- g
tive model, f →0 and δ →∆ , the ground-state energy
π
of H + δ (-1)iS ·S is known to deviate from its
s i i i+1
value at δ = 0 like δ4/3 [4], while the elastic energy in- 0.4
cstraeateseesnweritgPhyδo2f.HThe(rfef=ore0f)oirsamllicnoiumpalli,ngifsδλitshfiengirtoeu.nAdt- (ωb)/J=1 HHHee cffff rcαoresoffs=s.αs (.cN (N=8=)16)
eff 0 H cross. (N=10)
the same time proportional to δ2/3 a gap opens in the 0.3 Heff dim. (N=16)
spectrum.
gapped
By taking into account both mechanisms we can now
determine the transition from the gapless phase to the α 0.2
gapped one. As the SP-system behaves different for the
two couplings, we treat them separately, starting with
the local coupling case. 0.1 gapless
0.0
A. Local coupling
0.0 0.5 1.0 1.5
g
Inafirststepweset∆ =0andusethe level-crossing
π
criterium[28,29,9]tocalculatethecriticallineintheα-g-
0.4
planefordifferentphononfrequenciesω andsystemsizes
0 (c)
aNs.SSi·inSci+e3HaelnoffcdcSoin·tSaii+n4s,lothnigserlinreansgliegdhtilnytedreavcitaiotenssfsruocmh 0.3 ω0/J=0.1 HHHHeeeeffffffff ddddiiiimmmm.... ((((NNNN====8112)260)))
the line α =α , andwe haveto calculateitseparately.
eff c
Applying the Lanczos algorithm to the effective model
we obtain the critical line with high accuracy on local
workstations(N ≤20). Ontheotherhandwedetermine α 0.2 4
the level-crossing in the original model (1) by using the α=0.36 (/δω
methodsdescribedinRef.[6]. Inthecaseω /J =0.1,the 0)2
0
0.1
latter is complicated for the small spin phonon systems
we can handle with our Lanczos diagonalizationcode on 0
0 1 2
present day parallel computers, since the finite size gap g
0.0
tothefirsttripletexcitationisafewtimeslargerthanω0. 0.0 0.5 1.0 1.5 2.0
If g is small enough, each eigenvalues of the spin system g
is accompanied by a ladder of phonon states. Therefore
several ’copies’ of the singlet ground state lie within the
singlet-tripletgapif ω0 ≪J, andthe singletwe consider FIG.2.: Singlet-tripletlevel-crossing(solidlines)andon-
for the level-crossing is not easy to determine unless we setofdimerization(dashedwithsymbols)intheeffective
go to very large systems (see Ref. [9, Tab.II]). model in comparison to the level-crossing in the original
Figure2(a)and2(b)showthecriticallinesintheeffec- model (bold dashed lines) at ω /J =10, 1 and 0.1.
0
tive (bold solid) andthe originalmodel (bold dashed)as
well as the lines α =α (thin solid). As in the case of
eff c
thepurespinmodel,thecriticallinesdependonlyweakly onN. Wecanthereforecompareexactdataforthe orig-
4
inal model and N = 8 with data for the effective model In the case small phonon frequency, ω ≪ J, the sec-
0
and N = 16. While the results differ noticeable for in- ondtransformationexp(S )loosestheirimportance,and
2
termediate phonon frequency ω ∼ 1, the agreement is the effective frustration due to the spin phonon interac-
0
excellent in the anti-adiabatic frequency range ω ≫ J. tionisreplacedbythedimerizationastherelevantmech-
0
With increasing ω the critical curve exhibits a remark- anismleadingtoanenergygap. Weaccountforthiseffect
0
able upturn before crossingthe abscissa,i.e. the frustra- by allowing for a finite ∆ in our approximation. Using
π
tion is suppressed for small spin phonon coupling, but theHellmann-Feynmantheoremandnumericaldiagonal-
overcritical for strong coupling. It is this feature which ization of finite spin systems we determine ∆ such that
π
makes it necessary to expand (14) up to fourth order to the ground-state energy of Hloc is minimal. Depend-
eff
approximate Hloc in a correct way. A second order the- ing on coupling strength g and frustration α the sys-
ory is not capable to describe the observed critical line. tem prefers to remain in the undimerized, gapless phase
Anotherpointwecanstudywithinoureffectivemodel (∆ = 0) or to develop a non-zero dimerization leading
π
is the behaviour of the critical spin phonon coupling g to a gap. In Figs. 2(a) – (c) we plotted these transi-
c
at α = 0 in the limit ω /J → ∞, i.e. the limit of the tionlines(dashedlinewithsymbols)inadditiontothose
0
crossingpointofthecriticallineandtheabscissa. Asthe obtained by level-crossing. As we already found in our
effects ofthe longerrangedinteractionsarerathersmall, study of the Holstein model of spinless fermions [16], for
we can solve the equation αloc = α . Setting f = 1 small ω the transition to the dimerized phase depends
eff c 0
(compare Fig. 1) and α=0 we find noticeable on the system size N (see Fig. 2(c)), while
the finite-size dependence is weak in the anti-adiabatic
2 regime (cf. Fig. 2(a)). In addition, for ω /J = 10 the
P P α 0
g2 = + + c with (29) transition is consistent with the critical line determined
c 2Q s 2Q Q
(cid:18) (cid:19) via level-crossing.
ω α 1
0 c
P = − (1+α ), (30)
c
J 2 2
Q= ω03(1 +α )−(37 + 7 α ), (31) 102
c c
J 8 2 96 24 local model
and in the limit of infinite phonon frequency g ap-
c
proaches a finite value, 101
gapped
8α
c s
ω0/liJm→∞gc =s3(1+2αc) ≈0.66, (32) ωJ/0 100 aples
g
for the model with local coupling.
N=8
10−1 N=12
N=16
N=20
1.5 α = α
eff c
ωω==01.,3 g, =g0=.05.37 10−2
ω=2, g=0.5 0 1 2 3 4
g
α=0.36
1.0
E FIG. 4.: Critical coupling gc versus frequency for Heloffc
with α=0.
0.5
To compare properties of the original and the effec-
tive model also in the case of small phonon frequency,
we consider the dimerization. As a quantity which cor-
local model
responds to δloc/diff we take the static (lattice) structure
0.0 factor [6,7],
π/2 3π/4 π
q
g¯2
δ2 = hu u ieiπ(j−k), (33)
N2 j k
j,k
X
FIG.3.: Low-lyingexcitationsintheeffective(filledsym-
calculated in the ground state of (1), where u = b+ +
bols)andthe originalmodel(opensymbols)fordifferent j j
b . The inset of Fig. 2(c) displays (δ/ω )2 for α = 0.36,
frequency and local coupling g. j 0
ω /J = 0.1 (solid line), 0.316 (dashed line), and N = 8.
0
5
The results for the effective model (bold lines) and the
originalmodel(thinlineswithsymbols)agreeratherwell,
especially for ω /J =0.1. 0.4
0
lowA-nlyoitnhgerexfceiattautrioenws.e Fcaignurceom3psahroewisstthheeedniesrpgeyrsoiofnthoef (ωa)0/J=10 HHHee cffff rcαoresoffs=s.αs (.cN (N=8=)16)
lowest triplet excitations, calculated exactly and within 0.3
our approximation. Clearly for ω /J = 0.3 the correct
0
dispersionisflattenedatmomentanearq =π/2. Thisre-
sultsfromtheenergyofthedispersionlessphonons,which α 0.2 gapped
is added to the lowest triplet at q =π. Of course our ef-
fective model does not contain these low-lying phonon
excitations. However, as soon as ω & J the lowest ex- 0.1 gapless
0
citations are due to renormalized spin interactions and
well approximated by the effective model.
To collect the results of this subsection we show in 0.0
0.0 0.1 0.2 0.3 0.4
Fig. 4 the critical coupling gc(α = 0) over a wide range g
of phonon frequencies, using both criteria for the phase
transition. Symbols stand for the onset of dimerization,
while the bold line corresponds to α = α . As ex- 0.4
eff c
preelciatebdle,fwoerifinntdermtheadtiaotuerpahpopnroonxifmreaqtuioennciys.sTomheewsihnagtuluanr-- (ωb)/J=1 HHHee cffff rcαoresoffs=s.αs (.cN (N=8=)16)
Tityheofcogrcraetctωc0r/iJtic=al1liinseawmilalnciofensnteacttioandoiafbthatisicdaenfidcieannctyi-. 0.3 0 HHHeeeffffff dddiiimmm... (((NNN===811)26))
adiabatic behaviour in a continuous way (compare also
next subsection and [9]). α 0.2 gapped
B. Difference coupling 0.1
gapless
The procedure to determine the phase transition in
the SP-system with difference coupling is the same as 0.0
0.0 0.2 0.4 0.6
described before. In the anti-adiabatic regime we set g
∆ = 0 and calculate the position of the crossing of
π
the first triplet and the first singlet excitation for both,
the original and the effective model. The results for 0.4
tωi0v/eJly.=In10caonndtra1satrteoshthoewnloicnalFcigosu.p5li(nag)atnhde 5st(rbu)crteusrpeeoc-f (ωc)/J=0.1 HHHHeeeeffffffff ddddiiiimmmm.... ((((NNNN====8112)260)))
the critical line for high phonon frequency (ω /J = 10) 0.3 0
0
is much simpler. Itappears that one wouldgetthe same
shapealsoforasecondordertheory. However,toenlarge
theapplicationareaofourapproximationtakingintoac- α 0.2 8
cboeufonrte,hitghheeragorredeemrecnotntbreibtwuteieonnsthisesotirlilgainpaplroapnrdiatthee. Aefs- α=0.36 0(/)δω2
fective model is excellent in the anti-adiabatic regime, 0.1
while the deviations increase with approaching interme- 0
0 1 2
diate frequencies. g
Calculating the behaviour of the critical coupling, 0.0
0.0 0.2 0.4 0.6 0.8
gc(α = 0) in the limit of infinite phonon frequency, g
ω /J →∞, we now find
0
2
P P α
g2 =− + + c with (34) FIG. 5.: Level-crossing (solid lines) and onset of dimer-
c 2Q s 2Q Q ization (dashed with symbols) in the effective model in
(cid:18) (cid:19)
ω 1 3 comparison to the level-crossing in the original model
0
P = ( −α )+ (1+α ), (35)
J 2 c 2 c (bold dashed lines) at ω0/J =10, 1 and 0.1.
ω 1 9 25 59
0
Q= ( + α )−( + α ), (36)
c c
J 16 8 8 24
lim g =0. (37)
c
and,differenttothelocalcouplingcase,gc tendstozero, ω0/J→∞
6
While the q =0 and the q =π phonon mode compete in IV. GROUND-STATE PHONON DISTRIBUTION
thecaseoflocalspinphononcoupling,allowingforasta-
blegaplessphaseuptoacriticalg,thereisnointeraction In the course of exact diagonalizations of the phonon
with the q =0 mode in Hsdpiff. Therefore the q =π mode dynamical model (1) we observed another interest-
induces longrangedexchangemore efficiently,leading to ing feature distinguishing the two mechanisms of spin
a vanishing gc for ω0/J →∞. phonon interaction. Turning our attention to the
For small phonon frequencies ω0 ≪ J again we deter- phonon distribution in the ground state of (1), we find
mine the optimal dimerization ∆ and the critical line
π
beyond which ∆ starts to be non-zero. The results
π
are shown in Figs. 5(b) and 5(c). For large phonon fre-
quency we find an unstable behaviour of ∆ , it is finite 0.2 (a) difference model
π
for some g, but vanishes before growing to substantial 0.1 ω0/J=α0.=10
values again. We therefore make no attempt to fix the
onset of ∆ 6=0 for ω /J &1.
π 0 0.0
Asinthe caseoflocalcouplingthe dimerizationinthe
original and the effective model agrees well for ω /J = 0.2 (b) local model
0
0.1 (inset Fig. 5(c)). ω0/J=0.1
0.1 α=0
102 m2c| 0.0
difference model | 0.2 (c) difference model
ω/J=0.1
0
0.1 α=0.36
101
0.0
gapped
s 0.2 (d) difference model
ωJ/0 100 aples 0.1 ω0/J=α1=00
g
N=8 0.0
10−1 N=12 0 2 4 6 8 10 12 14 16 18 20
N=16 m
N=20
α = α
eff c
10−2
0 1 2
FIG. 7.: Even/odd-phonon distribution in the ground
g
state of the model with difference coupling (g = 1.4,
ω /J = 0.1) and mechanisms that suppress the effect:
0
local coupling, frustration and anti-adiabaticity.
FIG. 6.: Critical coupling g versus frequency for Hdiff
c eff
with α=0. for the model with difference coupling in the adia-
batic regime, that the system prefers states with even
Finally in Fig. 6 we combine the above results to ob- phonon occupation numbers (cf. Fig. 7(a)). This be-
tain the phase diagram in the J/ω0-g plane. Except for haviour reminds of a simple two-level system, more pre-
ω0/J →∞,wheregc →0,thebehaviourissimilartothe cisely the Rabi (pseudo-Jahn-Teller) Hamiltonian [30]
case of local coupling. Again at ω0/J ∼ 1 the effective H =∆σz +g¯(b++b)σx+ω0b+b σ0, where an interac-
model gives a singularity in gc, while the correct result tion with a bosonic degree of freedom connects the lev-
shouldbecontinuous. Comparingthetransitionlinewith els. Solving this model in the adiabatic strong-coupling
the very recent DMRG data of Bursill et al. [9], we find case, ω /∆≪1 and g =g¯/ω >1, we obtain a similar
0 0
very good agreement in the anti-adiabatic regime (the even/odd oscillation in the ground-state phonon distri-
phase transition does not change, going from the second bution. One can understand this effect within standard
to the fourth order effective theory). For small phonon perturbationtheory. Startingwiththetwolevels±∆and
frequencyω0/J →0theresultsofBursilletal. suggesta the corresponding eigenstates 10 and 01 , it is obvious,
finitelimitforgc/ω0,whileourmodelgivesanincreasing that adding only one phonon requires an extra amount
(cid:0) (cid:1) (cid:0) (cid:1)
ratioofgc/ω0. However,asthefinitesizeeffectsarelarge of 2∆ exchange energy, because the system is excited to
for ω0 ≪J, determining the correct value of gc is a del- the upper level. Therefore the system prefers an even
icate procedure and the exact gc might be much smaller number of phonons for the ground-state wave function.
than depicted in Fig. 6. In the anti-adiabatic case, ω ≫ ∆, to a good approxi-
0
mationwecanomitthe∆σ termandsolvethemodelby
z
7
unitarytransformation,leadingtoaPoissondistribution phase transitionby means ofthe static dimerization∆ ,
π
of the phonons (coherent state). which changes from zero at small spin phonon coupling
In view of the SP system with difference coupling, the to a finite value beyond a critical coupling.
q =π phonon is the most relevant(in fact in the ground Atintermediatefrequencythesituationremainsunsat-
state almost all phonons occupy this mode), and adding isfactoryasthecriticalcouplingbehavesdiscontinuously.
one phonon changes the momentum of the spin system Here numerical methods, including the full phonon dy-
byπ. Consequentlythegapbetweenthe(singlet)ground namics, still provide the only reliable tool to study the
stateatq =0(N =4k, k ∈N)orπ(N =4k+2)andthe transition. Nevertheless the proposed effective models
lowest singlet state at q =π, or 0 respectively, plays the help to understand the physical mechanisms leading to
roleof2∆inthe two-levelsystem. Asthissinglet-singlet spontaneous dimerizationof the interacting spin-phonon
gap is only due to the finite size of the considered spin system.
chain, we believe that the even/odd oscillations will dis-
appear in the thermodynamic limit. The reduction (dis-
appearance) of the even/odd imbalance resulting from a VI. ACKNOWLEDGEMENT
finite frustration with a much smaller singlet-singlet gap
(zero at α = 0.5), cf. Fig. 7(c), can be taken as a first WethankJ.Schliemannforvaluablediscussions. Some
indication. Of course we observe a smooth phonon dis- computationsweredoneatLRZMu¨nchen,HLRZJu¨lich,
tribution in the anti-adiabatic case (Fig. 7(d)). HLRS Stuttgart, and GMD Bonn. H.F. acknowledges
The model with local spin phonon coupling exhibits financial support from the Graduiertenkolleg ’Nichtlin-
the usual Poisson distribution for all frequencies, since eare Spektroskopie und Dynamik’ at the University of
the interactionis due to the q =π and the q =0 phonon Bayreuth.
mode (Fig. 7(b)).
V. CONCLUSION
In summary, we have studied the spin-Peierls insta-
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8
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9