H. Zheng- Quantum Lattice Fluctuations in the Ground State of an XY Spin-Peierls Chain

8/3/2019 H. Zheng- Quantum Lattice Fluctuations in the Ground State of an XY Spin-Peierls Chain

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I IntroductionRecently the physics of quasi- one- dimensional spin- Peierls systems has attracted consid-erable interests of both theoretists an d experimentalists because of the discovery of aspin- Peierls transition at TSP ~ 14K in the cuprate compound CuGeO3[?]. BelowTSPt h e lattice is dimerized[?] and a spin- gap has been observed[?].

From the theoretical point of view, th e quasi- one- dimensional spin- Peierls system canbe described by the Heisenberg antiferromagnetic chain coupled with lattice phonons.Within th e adiabatic approximation, that is, treating th e phonon degrees of freedomclassically, Bray et al.[?] and Cross and Fisher[?] treated this type of model system anddiscovered some interesting physics. But the nonadiabatic effect related to the finitephonon frequency, which tends to decrease th e Peierls transition temperature and theorder parameter, was not taken into account. Very recently, Caron an d Moukouri[?]suggested to study the effect of quan tum latt ice fluctuations in th e spin- Peierls system bystarting from an XY spin chain in which the phonons interact with the spins by modifyingt h e magnetic interaction,

l

{ ^ 2 f 2 } (i)Here , SlX an d S L Y are the spin-12 operators on site l, J > 0 is the usual antiferromagneticexchange energy, is the spin- phonon coupling constant, ul an d pl are the displacementand momentum operators of the magnetic ion on site l. M is the mass of the magneticio n an d K the spring constant. By using the Jordan- Wigner transformation[?]

12 , (2)n


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As pointed out by Caron and Moukouri, this may be the simplest model but contains theessential elements for a spin- Peierls system. In th is work we start from this model.

Within the adiabatic approximation the model can be solved easily. In th e half- filledcase the system undergoes a Peierls instability and the ground stat e is dimerized with anenergy gap 2A at the Fermi points k = / 2 [ ? ] . The theoretical analysis becomes muchmore difficult when th e quantum lattice fluctuations are taken into account. Fradkin andHirsch[?] have calculated the electronic and lattice structure of th e half- filled SSH modelby the Monte Carlo simulations, and concluded that for spinless fermions quantum latticefluctuations destroy the Peierls dimerization for small coupling constant if the ionic massis finite. Caron and Moukouri[?] calculated the T = 0K phase diagram of the system byusing the density matrix renormalization group ( DMRG ) method. Their results showeda power- law dependence of the critical spin- phonon coupling on the phonon frequencyfor the onset of a spin gap. They also observed a classical- quantum crossover when th espin- Peierls gap 2 A is of order .

In this work, we use th e unitary transformation to take into account th e fermion-pho n o n correlation[?] and show that when > 0 there may exist a static dimerizationof the lattice but the quantum lattice fluctuations play a very important role. When thespin- phonon coupling constant 2 / 4K decreases or the phonon frequency increasest h e lattice dimerization and the spin gap decrease gradually. At some critical value( 2/4K)c or c, the dimerization disappears and the system becomes gapless. This canbe attributed to the fact that the ground state fails to develop a spin- Peierls long- rangeorder because of the quant um lattice fluctuations. Throughout this paper we pu t h = 1and kB = 1.

II Theoretical analysisIn Hamiltonian (3) th e operators of the lattice modes, ul and pl, can be expanded byusing the phonon creation and annihilation operators,

( 4 )

(5)

N is the total number of sites. Q 2 = ^- sin 2(q/ 2). Then H becomes(6)

k,q


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where tk = Jcos(k) is the band function. The coupling function g(k + q, k) isg(k + q,k) = - iaJ1(sink - sin(k + q)). (7)

y 2MujqIn order to take into account the fermion-phonon correlation a unitary transformation

is applied to H,H' = exp(5)Jffexp(- 5), (8)

where the generator S is E 9{k + " ^ ^ t . (9)

Here we introduce a function 6(k', k) which is a function of the energies of the incomingand outgoing fermions in the fermion-phonon scattering process. The form of 5(k ' , k )will be defined later. We divide the original Hamiltonian into H = H + H1, where Hcontains the first two terms and H1 the last term. Then the transformation can proceedorder by order,

H' = H + H1 + [S , H] + [S , H1] + 1[[H, S],S] + O( 3).The first- order terms in H' are

H 1 + [ S , H ] = ^J29(k + q , k)(blq + bq)fl+qfkJ2g(k + q , k ) ( k + q , k)(blq + bq)fl+qfkk,qJ2g{k + q'k)(e k - e k+ q)5(k + q , k)(blq - bq)fl+qfk. (10)k ,q q

Now we can choose the functional form of (k + q, k) to make the contribution of thesefirst- order terms as small as possible. Note that the ground state \go) of H, the non-interacting system, is a direct product of a filled Fermi-sea \FS) and a phonon vacuumstate |ph, 0):

\go) = \ FS)\ph,0). (11)Applying the first- order terms on \g0) we get

q , k ) b l q f k + q f k [ l - 5 ( k + q,k) + q ( k + q , k ) | \g0), ( 1 2 )


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since bq|ph, 0) = 0. As the band is half- filled the Fermi energy tp = 0. Thus fk+ qf k \F S )0 only if tk+q > 0 and tk < 0. So, if we choose

q,k) = 1/ (1 + \ ek+g- ek\ / ug), (13)we have

(H1 + [S,H])|g0>=0. (14)We believe that this form of 6(k', k) makes the contribution of first- order terms as smallas possible. The second- order terms in H' can be collected as follows:

q,k)g(k' + q',k') tLOqLOqi

q , k)5(k' + q', k ')[fL qfk'8 k'+ q'tk - fl+qjk(UJq

- 5 ( k + g, k)5(k' + g', c / ) ] [/ fc+ o / f c '^ fc '+ ')fc fi+a'fk$k+q,k'q, k)

, + q,

')]flfkfl, k)5(k> - q , k')]fl+qfkfl_qfk, (15)$k'+ q', k is the Kronecker symbol. All terms of order higher t h a n 2 will be omitted inth e following treatment.

Th en we make a displacement transformation to H' to take into account th e staticphonon- staggered ordering,

H = exp(JR)Jff/exp(- JR). (16)Here

and exp(R) is a displacement operator:

^ ( b- q + bq) ^M^)- (18)If the ground state of H is \g), then t he ground state of H is l f'): \g) = exp(S) exp(R)\g')We assume that for \g') the fermions and phonon s can be decoupled: \ g') ~ \ fe)\ ph, 0),


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where |/ e) is the ground state for fermions. After averaging H over the phonon vacuumstate we get an effective Hamiltonian for the fermions,

Heff = {ph,0\H\ph,0)= 2Ku20N + E0(k)ftfk + ]T zA0(k)(flfk.n - fl_Jk)

k k>0_ i g(fc + g , % ( f c - g , f c ) j ( f c + g> fc)[2 _ _g> w t / f c / j t / fc,. (1

7 V Uy q k,k' q

where$ 2(k',k), (20))]. (21)

We find bymeans ofthe variational principleno = - - ^- y:2aSm(k)[l-5(k,k-n)](fe\ftfk^- fljk\fe). (22)

We note that in the adiabatic limit where =0one has 5(k', k) =0 and Heff goesback to the adiabatic mean- field H amilton ian,

k- 7T- fljk). (23)k k>0On the other hand, in the ant iadiabatic limit where > oo, we have u0 =0, 5(k', k) = 1,and H eff becomes

- g , f c ) f t f f t f; Jk+qJkJk'-qJkk kN q k,k> q

Returning to the real space, this Hamiltonian is

Heff(uv - oo) = - ^ XXtf/ j+i + flji) + ^ Y.UUiflifi+1 - fUi)- (25)This is the antiferromagnetic XXZ model (through th e Jordan- Wigner transformation) [?].I t can be solved exactly and there exists a transition point at 2 / 4K = J[?]. Thus oureffective Hamiltonian works well in these two limits. When 0 < < oo, we have0 < 5(k', k)


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III < 2 JI n this case the four-fermion term can be treated as a perturbation and the unperturbedH amiltonian is

Heff = 2KulN + YE0(k)ftfk + y iAQ(k)(flfk- n - f! Jk). (26)k k>0The four- fermion term can be re-written as

I T i _ y ye/ / AT 2-^i 2- ^i

k+q-TTJk-TTjKJk'-qJk' ~ JU'-q- itlk'-n)q,k- 7v)g(k' - q,k' - ) , ,(fc + g,fc- 7r)[2- ^(fc - g,fc - T T ) ]

q, k- ir)g(k' - q, k'- ) , ,^j (fc + g,fc- 7r)[2- (fc - 9,fc - T T ) ]

X [fk+qfk- Trfk'- qfk'-TT + fk+g- Trfkfk'- g- Trfk'J (27)

I n these terms we have the constraintsk + q>0, k>0, k ' -q>0, k' > 0.

We can distinguish between different physical processes. The first term in (27) is theforward scattering one (g2 and g4 terms in the g- ology language[?]), the second is theback scattering one (g1 term), and the last is the Umklapp scattering one (g3 term).q 7^ 0 in the first term means that there are no phonons of q = 0.

We use the Green's function method to implement the perturbation t rea tment [ ? ] . Itis more convenient to work within a two-component representation,

* f c = ( fk ) , (28)in which k > 0. Thus the Hamiltonian becomes

H eff 0 = 2Ku20N + ]T E0(k)VlazVk + ]T iA0(k)^liay^k, (29)k>0 k>0

, , = 4N 1N k,k'

-

, k- ir)g(k' - q, k'- n) . u . , , ( k + q , k - n ) [ 2 - d ( k - q , kk , k >


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g(k + q, k ir)g(k' q, k vr) ) . . . .f S ( k + q , k - K ) [ 2 - 6(k' - q , k ' - T T )y /v,/v y

x (*l+ ,^ *fc*l- ,^ *fc ' + *l+ g^*fc*L^x*fc'). (30)(T/j ( = x,y,z) is th e Pauli matrix. The matrix Green 's function is defined as (thetemperature Green's function is used and at the end T > 0)

G(k,r) = - (T T * f c (r)*I (0)) = T^exp(- zcc; r ir)G(fc, wn). (31)

T h e Dyson equation is

G ( k , n ) = G0(k, n) + G0(k, n) (k, n)G(k, n), (32)where

G 0 ( k, n ) = {iun - E0(k) z + Aoi^ay}'1 (33)is the un per turbed G reen's function. The self- energy (k, n) can be calculated by theperturbation theory[?],

i V fc'>0x J T [ i G ( f c /

^ V fc'>0,k)[2 - 5(k,k')}{G0(k',ujm)fc'>0 " i ^fc'- fc

Tk ' > 0

g ( k ,N k'

x i iayGo(k', m)i y axGo(k', m) x . (34)

Here (k) = (k,k ) and Tr[...] is the trace of .... When making the pertu rbat ioncalculation we have taken into account th e fact that the forward and back scatterin gterm s do not con tribute to the "charge" gap[?]. Then, by using the Dyson equation wec a n get

G(k, n) = i n E(k) z + A(k)(jy > . (35)F rom G(k, n) the fermionic spectrum in the gapped state can be derived

W ( k ) = E2(k) + A 2(k) . (36)


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The renormalized band function isE(k) = E0(k)-^- j: {i V fc'>o z n

k' - I T , k)[2 - 5{k' - I T , fc)]}- _^22_ (37)The gap function is

k) = 2auosm(k)[c- d5(k)}, (38)where

c = 1 + l - s i n ( fcW f c ) . A( f c ) (39)N t2K J2 / E ( k ) ^(k)'d = l - ly; - ^s in(fc)[l - 5(fc)] . A( f c ) (40)

The equation to determine U0 is

In the nonadiabatic case u0 is a variational parameter and cannot be measured by ex-periment or Monte Carlo simulations. The quant ity which can be measured is mp,

mp = I E = ^ E ( - l ) 1 E \ l^r{90- q + bq)e\ g)= ^E(- i )'E

k>0A(fc)

) . (42)These are basic equations for the


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This is th e true gap in the fermionic spectrum. Fig.1 shows the density of stat es (DOS)of fermions,

for some and g2 = 2 /4KJ values. One can see that a nonzero DOS star ts fromt h e gap edge and, for smaller values of / J , there is a peak above the gap edge with asignificant tail between it and the true gap edge. The inverse- square- root singularity att h e gap edge in the adiabatic case[?] disappears.

T h e adiabatic theory predicts a ratio A/amp = 2[?, ?, ?]. But our calculations showt h a t A/amp is around or smaller than 1 when > 0. The DMRG results of Caronan d Moukouri[?] is similar to ours. This fact seems to indicate that there might be adiscontinuous transition of the ratio between = 0 and > 0. But if we use the peakposition in the DOS, peak, to calculate th e ratio peak/ mp there will be no discontinuitybecause, for smaller , peak ~ 2A. From an experimental point of view, when / J the behavior of the dimerization mp, as a function of g2, can be described byt h e form of an adiabatic mean- field solution,

1 _ 4 / 4ci!2mp:\ ~ 1g12 = A 41 ~ J2B2 p

but the mean- field parameters are renormalized by th e quan tum lattice fluctuations. In(46) E[m] and K[m] are first- and second- type complete elliptic functions. A and B are


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fitting parameters and in the adiabatic case A = 1 and B = 1. In Fig.4 we show the fittedvalues of A an d B for < 0.5 J . Both A an d B increase with increasing , however, inour case A < B but in Caron and Moukouri's work A > B.

F or the quantum region where 2 mp < we use the formgc ( / - ^ r e xp[ - 6 ( , 2 - , c 2 )- 0 - 5 ] (47)

t o fit our calculations, where b is a fitting parameter. We find th at the form used byCa ron and Moukouri[?],

^ ex (/ - ^ )- 0 - 5exp[ - 6(/ - ^ 2)- 0 - 5 ] , (48)is not good for fitting our calculations compared with the form of Eq. (47) . We believeth is is because of th e retarda tion effect of the spin- phonon interact ion.

Fig.5 shows th e phase diagram. We use / ( + 2J), instead of , as the variablebecause it goes to 1 when > . Th e solid line is th e result of th is section . The dashedline is

gc2 = 0.857125 \] . (49)We can see that a power- law is quite good to fit our calcu lation s. The power- law rela-tionship gc 2 CJ'4 is the result of Caron and Moukouri[?] (they showed (

cm2)

c~ CM1.4 and

we have the relation cm2 /J2 = g2 / 2J an d cm = ) .

IV > 2 JIn this case Heff can be re- written as

k

^ ^ 2 i f c o s { k + q / 2 ) cos{k' ~ qwti+ qhfi ' - qfv< ? k , koik) - pQek)fkfk + 2^ *Ao(fc)(/ fc/fc-^ - fk- nfk)

k>0

11 i / n \ 11 Ii V q k,k'2K

{6(k + q, k)[2 - 6(k ' - q, k')] - V}ft+qfkfl_qfk,, (50)where

9 ( k), (51)


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^ 3 - q , k / ) } . (52)N 3 q k,k>

One can show that 0 . Theunperturbed Hamiltonian is

" ^ E E ^ cos(fc + 9/2) cos(fc' - q/ 2)fl+qfkfl_qfk,1 k,k'I I

I t is the antiferromagnetic XXZ model with JX = JY = J 0 and JZ = 2V/4K. Thismeans that, because of the spin- phonon coupling, when > 2Jwe can get an effectiveXXZ model as the unperturbed H amiltonian with a phonon - i n duced Z interaction 2V/4Kand the XYmagnetic interaction being renormalized by a factor 0. The result ofYangand Yang[?] shows that there exists a transition point at JX = JY = JZ, that is, at

2V/4K = J 0. (54)The tr ansition po ints determined by this equation are shown in Fig.5 by the dashed- dottedline. The dotted line is a fit by

c 2 =One can see that, although the formula isvery simple, the interpolated result is, at least,qualitatively correct. No t e th at the power- law of Eq.(49) cannot beused for the wholerange 0 < oo when > oo.

V Conclusions

An analytical approach has been developed to study the effect ofquantum lattice fluc-tuations on the ground state of an XY spin- Peierls chain, which is equivalent to thespinless Su- Schrieffer- Heeger model in half- filling after the Jordan - Wigner transforma-t i on . We have shown that when th e spin- phonon coupling constant g2 decreases or thepho n o n frequency increases the lattice dimerization mp and the gap function A(fc) int h e fermionic spectrum decrease gradually. At some critical value gc2 or c, the systembecomes gapless and the lattice dimerization disappears. This can beattributed to thefact that the ground state fails to develop th e spin- Peierls long- range order because of th equ a n t um lattice fluctuations. Aphase diagram in the g2 ~ plane is derived.


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From our work we can see that t he main effect of quan tum lattice fluctuations istwofold. One is to lower the effective dimerization potent ial seen by Jordan- Wignerfermions, as is represented by the factor 1 (k) in Eq.(41). The other is to inducea four-fermion interaction term. In this work the standard perturbation approach is usedt o treat the interaction term.

In our model system, quantum lattice fluctuations compete with the long- range dimer-ization order and the physical properties of the system should be determined by thiscompetition. When 2 mp > the long- range dimerization order dominates and thesystem is in th e classical region. In this region the behavior of th e dimerizat ion mp canbe described by the form of an adiabatic mean- field solution but the mean- field parame-ters are renormalized by the quantum latt ice fluctuations. For the quant um region where2 mp < the behavior of mp can be described by an exponential function of (g2 g2) '0 '5which is similar t o, but different from the fitting formula of Caron and Moukouri[?]. Webelieve that this type of difference comes from the reta rdat ion effect of th e spin- phononcoupling.

In th is work we concen trated mainly on the long- range ordering phase, where mp >0 and there is a gap in the fermionic spectrum, and the phase transition point. Theproperties of the disordered phase with gapless fermions are also of interest because thedisordered phase should be a Lut tinger liquid with fermion- phonon interaction. This willbe th e topic of a future work.

AcknowledgmentsThe author would like to thank Prof. Yu Lu for stimulating discussions on these and

related topics. This work was supported part ly by the China National N atu ral ScienceFounda t i on and the China State Committee of Education.


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References[1] M. Hase, I. Terasaki, and K. Uchinokura, Phys. Rev. Lett. 70, 3651 (1993).[2] J.P.Pouget et al., Phys. Rev. Lett. 72, 4037 (1994); K.Hirota et al., Phys. Rev. Lett.

73 , 736 (1994).[3] M.Nishi, O.Fujita, and J.Akimitsu, Phys. Rev. B50, 6508 (1994); M.C.Martin et al.

Phys. Rev. B53, 14713 (1996).[4] J.W.Bray et al., Phys. Rev. Lett. 35, 744 (1975).[5] M.Cross and D.S.Fisher, Phys. Rev. B19, 402 (1979).[6] L. G. Caron and S. Moukouri, Phys. Rev. Lett. 76, 4050 (1996).[7] P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928).[8] W.P.Su, J.R.Schrieffer, and A.J.Heeger, Phys. Rev. Lett. 42, 1698 (1979); Phys. Rev.

B22 , 2099 (1980).[9] E.Fradkin and J.E.Hirsch, Phys. Rev. B27, 1680 (1983).

[10] H.Zheng, Phys. Rev. B50, 6717 (1994); H.Zheng and S.Y.Zhu, Phys. Rev. B53, 3107(1996).

[11] C.N.Yang and C.P.Yang, Phys. Rev. 150, 321, 327 (1966).[12] V.J.Emery, in Highly Conducting One-Dimensional Solids, edited by J.T.Devreese,

R.P.Evrard and V.E.van Doren (Plenum, New York, 1979).[13] See, for example, A.L.Fetter and J.D.Walecka, Quantum Theory of Many-Particle

Systems, McGraw-Hill, New York, 1971.[14] J.Solyom, Adv. Phys. 28, 209 (1979); J.Voit, Rep. Prog. Phys. 58, 929 (1995).


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Figure CaptionsF ig.1 The density of states of fermions for (a) g2 gc2 = 0.15 with = 0.002J,

0.025J, and 0.3J, and (b) = 0.025J with g2 - gc2 = 0.03, 0.15, and 0.3.F ig.2 The normalized dimerization param eter mp/ mp0 as functions of the normalized

pho n o n frequency / C for g2 = 0.2 (curve 1), 0.3 (2), 0.4 (3), and 0.5 (4).F ig.3 The dimerization parameter 2 mp/ J as functions of the coupling constant

g2 in th e cases of (a) = 0.025J, (b) 0.3J an d (c) J . The solid lines are results ofour theory, the dashed lines are fitted results of Eq.(46), and th e dashed- dotted lines arefitted results of Eq.(47).

F ig.4 The values of fitting param eters A an d B in Eq.(46).F ig.5 The phase diagram. See text for details.


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H. Zheng- Quantum Lattice Fluctuations in the Ground State of an XY Spin-Peierls Chain

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