Adiabatic-antiadiabatic crossover in a spin-Peierls chain

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Adiabatic-antiadiabatic crossover in a spin-Peierls chain

CITRO, R., ORIGNAC, E., GIAMARCHI, Thierry

Abstract

We consider an XXZ spin-1/2 chain coupled to optical phonons with nonzero frequency ω0. In the adiabatic limit (small ω0), the chain is expected to spontaneously dimerize and open a spin gap, while the phonons become static. In the antiadiabatic limit (large ω0), phonons are expected to give rise to frustration, so that dimerization and formation of spin gap are obtained only when the spin-phonon interaction is large enough. We study this crossover using bosonization technique. The effective action is solved both by the self-consistent harmonic approximation (SCHA) and by renormalization group (RG) approach starting from a bosonized description. The SCHA allows to analyze the low-frequency regime and determine the coupling constant associated with the spin-Peierls transition. However, it fails to describe the SU(2) invariant limit. This limit is tackled by the RG. Three regimes are found. For ω0⪡Δs, where Δs is the gap in the static limit ω0→0, the system is in the adiabatic regime, and the gap remains of order Δs. For ω0>Δs, the system enters the antiadiabatic regime, and the gap decreases rapidly as ω0 [...]

CITRO, R., ORIGNAC, E., GIAMARCHI, Thierry. Adiabatic-antiadiabatic crossover in a spin-Peierls chain. Physical Review. B, Condensed Matter , 2005, vol. 72, no. 2

DOI : 10.1103/PhysRevB.72.024434

Available at:

http://archive-ouverte.unige.ch/unige:36135

Disclaimer: layout of this document may differ from the published version.

Adiabatic-antiadiabatic crossover in a spin-Peierls chain

R. Citro

Dipartimento di Fisica “E. R. Caianiello” and Laboratorio Reg. Supermat, I.N.F.M. di Salerno, Università degli Studi di Salerno, Via S. Allende, I-84081 Baronissi (Sa), Italy

E. Orignac

Laboratoire de Physique Théorique de l’École Normale Supérieure CNRS-UMR8549, 24, Rue Lhomond F-75231 Paris Cedex 05 France

T. Giamarchi

Département de Physique de la Matière Condensée, Université de Genève, 24, quai Ernest-Ansermet CH-1211 Genève, Switzerland 共Received 11 November 2004; revised manuscript received 12 April 2005; published 15 July 2005兲

We consider anXXZspin-1 / 2 chain coupled to optical phonons with nonzero frequency␻0. In the adiabatic limit 共small ␻0兲, the chain is expected to spontaneously dimerize and open a spin gap, while the phonons become static. In the antiadiabatic limit 共large␻0兲, phonons are expected to give rise to frustration, so that dimerization and formation of spin gap are obtained only when the spin-phonon interaction is large enough. We study this crossover using bosonization technique. The effective action is solved both by the self-consistent harmonic approximation 共SCHA兲 and by renormalization group 共RG兲 approach starting from a bosonized description. The SCHA allows to analyze the low-frequency regime and determine the coupling constant associated with the spin-Peierls transition. However, it fails to describe the SU共2兲invariant limit. This limit is tackled by the RG. Three regimes are found. For␻0Ⰶ⌬s, where⌬sis the gap in the static limit␻0→0, the system is in the adiabatic regime, and the gap remains of order ⌬s. For ␻0⬎⌬s, the system enters the antiadiabatic regime, and the gap decreases rapidly as␻0increases. Finally, for␻0⬎␻BKT, where␻BKTis an increasing function of the spin-phonon coupling, the spin gap vanishes via a Berezinskii-Kosterlitz-Thouless transition. Our results are discussed in relation with numerical and experimental studies of spin-Peierls systems.

DOI:10.1103/PhysRevB.72.024434 PACS number共s兲: 75.10.Pq, 71.10.Pm, 63.70.⫹h, 05.10.Cc I. INTRODUCTION

The properties of the spin-Peierls共SP兲state in quasi-one- dimensional materials has attracted considerable attention over the last decades since its discovery in the organic com- pounds of the TTF and TCNQ series,^1–3and more recently in the inorganic compound CuGeO₃.^4,5In analogy to the Peierls instability in quasi-one-dimensional metals,⁶a spin chain un- dergoes a SP transition by dimerizing into an alternating pat- tern of weak and strong bonds,^7–9with the magnetic energy gain compensating the energy loss from the lattice deforma- tion. Although this physical picture gives a good qualitative understanding of the SP phenomenon, the real SP transition is in fact much more complicated to describe. In particular, the above picture of SP transition is only valid in the adia- batic regime, in which the frequency of the phonons is neg- ligible compared to the magnetic energy scales in the system, such as the spin gap or the exchange interactionJ. The va- lidity of this approximation is clearly dependent on the sys- tem at hand. Recently, it was pointed out that the difference between CuGeO₃and the other SP compounds consists in the high energy of the optical phonons involved in the transition, which is of the order of the exchange integralJ.^10–12Another feature that distinguishes CuGeO₃from the other organic SP compounds is that no softening of the phonon modes is ob- served near the transition. All these findings stem from the fact that an adiabatic treatment of the phonon subsystem^8,9is inadequate to describe the SP transition in CuGeO₃, and an appropriate treatment of phonons in the antiadiabatic regime is required.^13,14

Unfortunately, not many analytical methods to study the system of coupled spin and phonons in the full frequency range are available. The main difficulty relies in the fact that when the phonon frequency becomes comparable to the en- ergy gap in the spin-excitation spectrum, one is entering a quantum regime in which quantum fluctuations completely impregnate the ground state. This is why many of the known studies involving dynamical phonons rely on numerical methods such as exact diagonalization共ED兲,^15–18strong cou- pling expansions,¹⁹ density matrix renormalization group^20–23 共DMRG兲, or quantum Monte Carlo simulations.^24–28 From the analytical point of view, various approaches have been developed, but they work well either in adiabatic or in the antiadiabatic regime. In the former case, most approaches are based on the mean-field approximation.^9,29,30 In the latter case, various perturbation studies were performed to derive an effective spin Hamiltonian.^31,32Another approach was developed, based on integrating out the phonon modes,³³ in order to map the model onto the Gross-Neveu model,³⁴for which various ex- act results are available.^35,36Other approaches are based on the flow-equation method¹³ that works well in the antiadia- batic regime, or on the unitary transformation method for the XYspin chain.³⁷Since various compounds are rather close to the border between the two regimes,¹² it would be thus highly desirable to have a good method to tackle the adiabatic-antiadiabatic crossover. In this paper we provide such a method. We combine the renormalization group共RG兲 method and the self-consistent harmonic approximation

共SCHA兲 to study the adiabatic-antiadiabatic crossover in a spin-1 / 2 Heisenberg chain coupled to dynamical phonons. A previous attempt to use the RG to study this adiabatic- antiadiabatic crossover has been published.³⁸ We will com- ment on the differences of the two approaches.

The plan of the paper is as follows. In Sec. II we intro- duce the model of spin chain coupled to dynamical phonons, and write it in the continuum limit using the bosonization representation. In Sec. III we describe a variational approach, inspired from the self-consistent harmonic approximation, and use it to describe the crossover. In the adiabatic regime we find an expression for the spin-Peierls gap consistent with the mean-field treatment of Cross and Fisher.⁹ In the antia- diabatic regime, the gap is essentially described by a sine- Gordon model. In Sec. IV we study the crossover using a renormalization group method. The RG is specially well adapted around the Heisenberg isotropic point. This allows one to extract the phase diagram as a function of the strength of the electron-phonon coupling and the phonon frequency.

In Sec. V we discuss the findings of the two methods, both relative to each other and in connection with experiments.

Conclusions can be found in Sec. VI, and some technical details have been put in the appendices.

II. MODEL AND CONTINUUM LIMIT

As a simple model which describes a SP system in the following we consider an antiferromagnetic spin-1 / 2 chain coupled to a set of Einstein oscillators, given by

H=Hs+Hp+Hsp, 共1兲 with

H_s=J

兺

_n ^{Sn·Sn+1, 共2兲}

Hp=

兺

n

冋

^{2mpn2 +m2␻02qn2}

册

^{, 共3兲}

whereSnare spin-1 / 2 operators, J⬎0,关qn,pn⬘兴=i␦n,n⬘. The quantitym␻02=keis the stiffness of the Einstein phonon. The interaction of spins with phonons can be modeled by

H_sp=g

兺

_n ^{qnSn·Sn+1. 共4兲}

The coupling to optical phonons described by共4兲is adequate for the CuGeO₃ since it would correspond to a side group effect by Germanium atoms as discussed in Refs. 39 and 40.

Acoustic phonons could of course be treated in a very similar way, but one would have to replaceqnwith共qn+1−qn兲. Note that some authors^13,18 prefer to diagonalize the phonon Hamiltonian in 共2兲 using the boson operator b=

冑

m␻0/ 2q +ip/

冑

2m␻0, and write the interaction˜g兺n共bn†+bn兲Sn·S_n+1. It is obvious that one has˜g=g/

冑

2m␻0. This remark will be useful when we will compare the results of the different ap- proaches. The adiabatic limit is␻0→0,m→⬁withkefixed.

In that limit, the phonons become classical, i.e., the q_n’s commute with the Hamiltonian共1兲, and one can simply mini-

mize the ground state energy with respect to their expecta- tion value. In that limit, the results of Ref. 9 are recovered.

The opposite antiadiabatic limit is␻0→⬁. In that limit, one can integrate out the phonons, and one is left with the Hamil- tonian of a frustrated spin-1 / 2 chain.³¹For a frustration large enough,^41–43i.e., for large enough spin-phonon coupling, a spontaneous dimerization of the spins takes place, and the system presents a spin gap. Our purpose is to provide a uni- fied treatment of these two limits.

To solve the spin-phonon problem, we use first the well- known Jordan-Wigner transformation to express the spin op- erators in terms of spinless fermions. Thus, the Hamiltonian Hsbecomes

Hf= −t

兺

_n ^{关cn+1† cn+ H.c.兴+V}

兺

_n

冉

^{cn+1† cn+1−1}2

冊冉

^cn†cn−12

冊

^,

共5兲 witht=J/ 2 and V=J. The spin-phonon Hamiltonian Hsp is transformed into

H_{f p}=g

兺

_n ^qn

冋

^{12共cn+1† cn+cn†cn+1兲}

+

冉

^{cn+1† cn+1−1}2

冊冉

^cn†cn−12

冊 册

^{. 共6兲}

We now proceed in the standard way to take the continuum limit共see, e.g., Ref. 44, Chap. 6兲. In the continuous approxi- mation, 共6兲 generates a coupling between the lattice defor- mation共phonon mode兲and theq= 2kF=␲component of the charge density,␳共2k_F,x兲.

In order to get a continuous description we separate fast and slow components of the phonon field and similarly for the fermion fields, and we get the interaction⁴⁵

Hf p=i

冕

^{dx关q共x兲␳共2kF,x兲− H.c.兴. 共7兲}

We now use the boson representation of one dimensional fermion operators. In this representation the HamiltonianHf

becomes H_f= 1

2␲

冕

^{dx uK关␲⌸共x兲兴2+Ku关ⵜ␾共x兲兴2, 共8兲}

where the field ␾共x兲 is related to the density of fermions⁴⁴ and 关␾共x兲,⌸共x⬘^兲兴=i␦共x−x⬘^{兲. We have u=共}␲/ 2兲J␣^{, with} ␣ the lattice spacing, K= 1 / 2,qn=q共x=n␣兲and we have kept only the most relevant terms. Changing the parameterKal- lows one to explore the more general case of XXZ spin chains with an easy plane anisotropy.⁴⁴The long-wavelength part of the fermion density is ␳q⬃0共x兲= −共1 /␲兲ⵜ␾共x兲, whereas the higher Fourier components are^46–48

␳2k_F共x兲= 3

␲²␣

冉

^␲2

冊

^{1/4cos关2␾共x兲兴, 共9兲}

where we have specialized to an isotropic spin chain. The prefactor in Eq. 共9兲 has been shown in Ref. 48 to yield a good agreement of the gap calculated within bosonization

and numerical calculations.⁴⁹ The Matsubara action for the phonon field has a standard quadratic form

Sp=␳

2

冕

^dx

冕

0

␤

d␶关共⳵␶q兲²+␻02

q²兴, 共10兲 where ␳^=m/␣ is the mass density of the optical phonon mode, and q共x=n␣兲=qn is the lattice deformation field. In the first approximation 共2兲, we have neglected the fact that the phonon disperses. It can be shown that the dispersion along the chain leads to insignificant corrections. On the con- trary, if the phonons are three-dimensional; i.e., if ␻ ^dis- perses with the transverse momentum, then significant changes can occur. Indeed in that case, since the phonon are three-dimensional they couple the different spin chains and can induce a three-dimensional transition at low tempera- tures. We will come back to the case of three-dimensional phonons in Sec. V.

Since the total action is quadratic in the phonon fields, we can integrate them out to obtain the following bosonized action with a retarded interaction between the electronic den- sities:

S=

冕

^dx

冕

0

␤ d␶

2␲^K

冋

^{u共⳵x␾兲2+1u共⳵␶␾兲2}

册

− g²

2共␲␣兲²␳␻0 2

冕

^dx

冕

0

␤

d␶

冕

0

␤

d␶⬘^{cos 2}␾共x,␶兲

⫻D_␻0,␤共␶⁻␶⬘^{兲cos 2}␾共x,␶⬘^{兲, 共11兲} whereD_␻0,␤共␶兲 is the propagator of an Einstein phonon of frequency␻0corresponding to the action共10兲:

D_␻0,␤共␶⁻␶⬘^兲=具q共␶兲q共␶⬘^兲典

=␻0

2

冋

^{e−␻0兩␶−␶⬘兩+2 coshe␤␻共␻00共− 1␶−␶⬘兲兲}

册

^.

共12兲 The action共11兲fully describes a one-dimensional spin chain coupled to phonons, and does not rely on adiabatic or antia- diabatic limit. However, one has to note that because of the cutoff, the action共11兲is valid only for␻0Ⰶu/␣. For higher values of the phonon frequency ␻0, the phonon propagator 共12兲must be replaced with␦共␶⁻␶⬘兲. In that case the action 共11兲 is simply the continuum action of a frustrated spin chain,⁴¹ in agreement with the canonical transformation approach.^13,31.

The action共11兲is of course impossible to solve exactly. In order to obtain the physical properties of the system, we will analyze it using two different techniques in the next sections.

The first technique is a self-consistent approximation. Such an approximation will be very useful to define the various phases of the system as well as the relevant parameters. As any variational approximation, although it can be very effi- cient in describing the various phases, it can only describe the transitions between these phases approximately. There- fore, in order to study the critical points we use a renormal- ization group method, building on the knowledge of the rel- evant parameters extracted from the SCHA.

III. SELF-CONSISTENT HARMONIC APPROXIMATION To study the action共11兲, we apply first the self- consistent harmonic approximation or Gaussian variational method.^50–53 The idea is that the action 共11兲would be clas- sically minimized by ␾= 0. One can thus expect that the physics will be dominated by small deviations around this minimum and approximate the action 共11兲 by a quadratic action. We thus consider as a trial action

S₀= 1 2␤⍀

兺

q,␻

G⁻¹共q,␻n兲␾^*共q,␻n兲␾共q,␻n兲. 共13兲 We have to find the propagatorG共q,␻n兲 so that 共13兲 is the best approximation for 共11兲. For that we define the varia- tional free energy

F_var=F₀+具S−S₀典0, 共14兲 where

F₀= − 1

␤^{LlnZ0= −} 1

␤

兺

q,␻n

lnG共␻n兲, 共15兲

Z₀=

冕

^{D␾e−S0关␾兴, 共16兲}

and具¯典0represents an average with respect to the actionS₀. The second term in the action共11兲can be rewritten as

− g²

2共␲␣兲²␳␻0

2

冕

^dx

冕

0

␤

d␶

冕

0

␤

d␶⬘D_␻0,␤共␶⁻␶⬘兲

⫻兵cos关2␾共x,␶兲+ 2␾共x,␶⬘^{兲兴+ cos关2}␾共x,␶兲− 2␾共x,␶⬘^兲兴其.

共17兲 Given that the phonon propagator 共12兲 decays for a large time difference共␶⁻␶⬘兲, one can see from共17兲that the cosine of the sum is roughly equivalent to⬃cos关4␾共x,␶兲兴and can be responsible for the opening of a gap in the spectrum, while the cosine of the difference is ⬃共␶−␶⬘^兲2关ⵜ␶␾共x,␶兲兴² and thus will modify the quadratic part of the action.

In the following, we consider the gapless共⌬= 0兲 and the gapful共⌬⫽0兲case separately, at zero temperature. The gap- less case is interesting in connection with systems of elec- trons at an incommensurate filling interacting with phonons.^54–56 In these systems, the term cos关2␾共x,␶兲 + 2␾共x,␶⬘^兲兴 does not appear in 共17兲, leading to ⌬= 0. The spin-Peierls problem corresponds to the half-filling case for the fermions.

A. Incommensurate case

Using共13兲, the variational free energy is given by F_var=1

2

冕

^2dq␲

冕

^d2␻␲^{兵关G0−1共q,}␻兲−G⁻¹共q,␻兲兴G共q,␻兲

− lnG共q,␻兲其− g²

2共␲␣兲²␳␻02

冕

_−⬁^{⬁ d␶␻20e−␻0兩␶兩}

⫻具cos关2␾共0,␶兲−␾共0,0兲兴典, 共18兲

whereG₀⁻¹共q,␻兲=共1 /␲^K兲共uq²+␻^2/u兲 and we used共12兲for

␤⁼⬁. Introducing the propagator for the field ␾^{, G}共x,␶兲

=具␾共x,␶兲␾共0 , 0兲典, we can rewrite 具cos 2␾共0,␶兲cos 2␾共0,0兲典

=1

2exp

再

^{− 4}

^冕

^2dq␲

^冕

^{d2␻␲G共q,␻兲关1 − cos共␻␶兲兴}

冎

^.

共19兲 Using this expression, we obtain

F_var=1

2

冕

^2dq␲

冕

^d2␻␲^{关G0−1共q,}␻兲G共q,␻兲− lnG共q,␻兲兴

− g²

4共␲␣兲²␳␻02

冕

−⬁

⬁

d␶␻0

2 e^{−␻0兩␶兩}

⫻exp

再

^{− 4}

^冕

^2dq␲

^冕

^{d2␻␲G共q,␻兲关1 − cos共␻␶兲兴}

冎

^.

共20兲 Minimizing the action共20兲with respect toG共q,␻兲and using the fact that G⁻¹共q,␻兲=G₀⁻¹共q,␻兲−⌺共q,␻兲, where ⌺ is the self-energy, we get

␦^Fvar

␦G共q,␻兲= 0 =1

2

冕

^2dq␲

冕

^d2␻␲

冠

^{⌺共q,␻兲+共␲␣g兲22␳␻}0 2

⫻

冕

_−⬁^{⬁ d␶␻20e−␻0兩␶兩关1 − cos共␻␶兲兴exp}

再

^{− 4}

^冕

^2dq␲

⫻

冕

^d2␻␲^G共q,␻兲关1 − cos共␻␶兲兴

冎 冡

^{. 共21兲}

As is obvious from the above equation, the low-frequency behavior of G⁻¹共q,␻兲=G₀⁻¹共q,␻兲 is similar to the one of G₀⁻¹共q,␻兲, and corresponds to a variational action of the form

S₀=

冕

^dx

冕

0

␤ d␶

2␲^K¯

冋

^{¯u共⳵x␾兲2+¯1u共⳵␶␾兲2}

册

^{; 共22兲}

thus,G⁻¹共q,␻兲=共1 /␲K兲共u¯q²+␻^2/¯u兲. In the equation above we thus have

冕

^2dq␲

冕

^d2␻␲^G共q,␻兲关1 − cos共␻␶兲兴

=␲^K¯

冕

^2dq␲

冕

^d2␻␲

关1 − cos共␻␶兲兴 共u¯q²+␻^2/u¯兲 =K¯

2 ln共␻c␶兲, 共23兲 where ␻c=u/␣, is a frequency cutoff. Using 共23兲 we can write our variational equation as

⌺共q,␻兲+ g² 共␲␣兲²␳␻02

冕

−⬁

⬁

d共␻0␶兲

2 e^{−␻0兩␶兩}关1 − cos共␻␶兲兴 共␻c␶兲^2K¯ = 0.

共24兲 If we confine to an expansion up to order ␻^{2 in} 关1

− cos共␻␶兲兴, as requested by the analytical behavior of the Green’s function for␻→0, we obtain

⌺共q,␻兲= − g²

共␲␣兲²␳␻02

冕

_−⬁^{⬁ d共␻20␶兲e−␻0兩␶兩}_共␻^␻_c²_␶^␶_兲^22K¯

= − g²

共␲␣兲²␳␻02

冉

^␻␻⁰c

冊

^{2K¯⌫共3 − 2K兲}␻02 ␻^2, 共25兲 where ⌫ is the gamma function. As we see the integral is convergent whenK⬍3 / 2. Going back to the definition of the self-energy, we have

⌺共q,␻兲=共G₀⁻¹−G⁻¹兲共q,␻兲=

冉

^2␲1uK−2␲^1¯Ku¯

冊

^␻2.

共26兲 Equating共25兲 with共26兲, and using the fact that u/K=¯u/K¯, we obtain the following value of the parameterK:

K²=K¯²

冋

^{1 +␲2Kgu␳␻2}0

2

冉

␣␻^u0

冊

^{2−2K¯⌫共3 − 2K兲}

册

^{. 共27兲}

Expanding aroundK, we obtain the renormalized value ofK¯: K¯²⯝K²

冋

^{1 −␲2Kgu␳␻2}0

2

冉

␣␻^u0

冊

^{2−2K⌫共3 − 2K兲}

册

^{. 共28兲}

One thus recovers a Luttinger liquid but with a renormalized value of the Luttinger parameter K. Equation 共28兲 implies thatK¯⬍K. A similar result can be obtained via the renormal- ization group analysis共see the next section兲. In a RG analy- sis, the result共28兲 would correspond to integrating the RG equation for the coupling constantg, assuming that Kis not renormalized and then computing the lowest-order correction toK with the renormalized coupling constant. Our method thus reproduces in a crude way the renormalization of K downwards. As in Refs. 55–57, we find that the tendency of the system to form charge density waves is increased.

B. Commensurate case

In the commensurate case the derivation of the variational free energy from共14兲–共21兲remains the same. Let us rewrite 共20兲in slightly different way:

F_var= 1

2

冕

^2dq␲

冕

^d2␻␲^{关G0−1共q,}␻兲G共q,␻兲− lnG共q,␻兲兴

− g²

4共␲␣兲²␳␻0

2

冕

_−⬁^{⬁ d␶␻20e−␻0兩␶兩}

冠

^{12 exp}

再

^{− 4}

^冕

^2dq␲

⫻

冕

^d2␻␲^G共q,␻兲关1 − cos共␻␶兲兴

冎

+1

2 exp

再

^{− 4}

^冕

^2dq␲

^冕

^{d2␻␲G共q,␻兲关1 + cos共␻␶兲兴}

冎 ^冡

^.

共29兲 Minimizing this action with respect toG共q,␻兲, we obtain the following expression for the self-energy:

⌺共q,␻兲= − g²

共␲␣兲²␳␻02

冕

_−⬁^{⬁ d␶␻20e−␻0兩␶兩}

冠

^{关1 − cos共␻␶兲兴}

⫻exp

再

^{− 4}

^冕

^2dq␲

^冕

^{d2␻␲G共q,␻兲关1 − cos共␻␶兲兴}

冎

+关1 + cos共␻␶兲兴

⫻exp

再

^{− 4}

^冕

^2dq␲

^冕

^{d2␻␲G共q,␻兲关1 + cos共␻␶兲兴}

冎 ^冡

^.

共30兲 As is obvious from Eq.共30兲,⌺共q,␻兲 is in fact independent ofq. Moreover, we can use the following expansion for the self-energy:

⌺共q,␻兲= − 1

␲^¯Ku¯共⌬

2+␥␻²兲. 共31兲

In Eq.共31兲, the variational parameter ⌬ stands for the gap caused by the commensurability, and the variational param- eter ␥ stands for the renormalization of the bare Luttinger exponentK. Such a restricted ansatz is justified by the fact that higher powers of ␻ ⁱⁿ ⌺共␻兲 are associated with irrel- evant operators in the action, whereas⌬ and␥ correspond, respectively, to a relevant and a marginal operator. Keeping only⌬amounts to neglect any renormalization of Kby the spin-phonon interaction.

The self-energy共31兲leads to a Green’s functionG共q,␻兲:

G共q,␻兲= ␲^K¯

¯qu ²+␻^2/u¯+u¯⌬², 共32兲 where⌬is the mass term. The integral of the Green’s func- tion is:

冕

^2dq␲

冕

^d2␻␲^G共q,␻兲e^i␻␶=K¯

2K₀共⌬¯u␶兲, 共33兲 where K₀ is the Bessel function. The corresponding varia- tional action is

S₀=

冕

^dx

冕

0

␤ d␶

2␲^K¯

冋

^{¯u共⳵x␾兲2+¯1u共⳵␶␾兲2+␰¯u2␾2}

册

^{, 共34兲}

whereu/␰⁼⌬is the gap andu/K=¯u/K¯ as no term共⳵x␾兲²is generated from共17兲.

Equating the coefficient of共31兲with that coming from the expansion for small␻^of 共30兲, we obtain the following two equations:

¯u⌬²

␲^{K¯ =} 2g² 共␲␣兲²␳␻0

2

冕

−⬁

⬁

d␶␻0

2 e^{−␻0兩␶兩}exp

再

^{− 4}

^冕

^2dq␲

⫻

冕

^d2␻␲^G共q,␻兲共1 + cos␻␶兲

冎

^{, 共35兲}

␥

␲^{u¯K¯ =} 2g²

共␲␣兲²␳␻02

冕

_−⬁^{⬁ d␶␻20e−␻0兩␶兩}

⫻␶²

2关e^{−4兰共dq/2␲兲兰共d␻/2␲兲G共q,␻兲关1−cos共␻␶兲兴}

−e^{−4兰共dq/2␲兲兰共d␻/2␲兲G共}q,␻兲关1+cos共␻␶兲兴兴. 共36兲 Using thatu/K=¯u/K¯, the left-hand side 共l.h.s.兲of 共36兲 can also be rewritten as

␥

␲^{¯Ku¯ = −} 1 2␲^uK+

1

2␲^{¯Ku¯. 共37兲} The two self-consistent equations 共35兲 and 共36兲 can be solved analytically in the antiadiabatic limit共␻0Ⰷ⌬兲. Using 共33兲, and after a straightforward but lengthy calculation, we obtain

K¯²⯝K²

冋

^{1 −␲Kgu␳␻2}02

冉

␣␻^u0

冊

^{2−2K⌫共3 − 2K兲}

册

^共38兲

which is the same change ofKthan in共28兲. The system also develops a gap given by

⌬= u

␣

冋

^{␲Kgu␳␻2}02

冉

␣␻^u0

冊

^{2K⌫共1 + 2K兲}

册

^{1/共2−4K¯兲. 共39兲}

As we can see from 共38兲, for K⬎1 / 2, we can have K¯

⬍1 / 2, so that共39兲can still lead to a gap provided thatg is large enough. Combining the two Eqs. 共38兲 and 共39兲, we finally have

K¯²⯝K²

冋

^{1 −共⌬␣兲2−4K¯⌫共3 − 2K兲⌫共1 + 2K兲}

册

^{. 共40兲}

The SCHA thus correctly describes the formation of a gap in the antiadiabatic limit. As for the incommensurate case, the SCHA captures part of the effects of the renormalization of the parameters. Note that the SCHA, as any variational method, is efficient in capturing the nature of the ordered phases, but in order to determine the nature of transition one needs the full RG analysis. Such an analysis will be dis- cussed in Sec. IV.

C. Adiabatic-antiadiabatic crossover in the SCHA Using the SCHA we are now in a position to describe the crossover from adiabatic to antiadiabatic regime. We will assume that we are far from the pointK= 1 / 2 and in fact that we haveK⬍1 / 2. For the spin chain this would correspond in being in the Ising limit. In that regime, we can neglect the renormalization ofKand takeK=K¯ in our variational action.

The variational free energy共29兲can be written F=F₀− u

2␲^K␰^{2G共0,0兲−} g²

4共␲␣兲²␳␻02

冉

^e2^␥␰^␣

冊

^2K

冕

0

⬁

d␶D共␶兲

⫻关e^{−4G共0,␶兲}+e^{4G共0,␶兲}兴, 共41兲

whereG共0 ,␶兲is given by Eq.共33兲. Using the expansion for

the Bessel function we obtain the following approximate ex- pression:

G共0,␶兲= −K

2 ln

冉 ^冑

^共u␶兲␰^2+␣2 e^␥

2

冊

^{, ifu␶Ⰶ2e−␥␰,}

G共0,␶兲= 0, ifu␶Ⰷ2e^−␥␰^, 共42兲 where␥is the Euler-Mascheroni constant.⁵⁸Using the above expression for the Green’s function, we are able to calculate the variational free energy共41兲, and minimizing it with re- spect to␰we obtain the following variational equation:

u 4␲^K␰²⁼

g²

共␲␣兲²␳␻02

冋 ^冉

^e2␥␰␣

^冊

^{2Ke−2e−␥␻0␰/u}

+

冉

^e2^␥␰^␣

冊

^4K

冉

␣␻^u0

冊

^2K␥

冉

^{1 + 2K,␻}u^0␰

冊 册

^{, 共43兲}

where␥共· , ·兲is the incomplete gamma function.⁵⁸

Two interesting limits in Eq. 共43兲 must be discussed. If

␻0␰^/u→0, one is in the adiabatic limit, whereas the antia- diabatic one corresponds to ␻0␰^/u→⬁. In the adiabatic limit, one sees that the term on the right hand side of共43兲 reduces to a contribution⬃共a/␰兲^2K, the term depending on the incomplete Gamma function being zero in that limit. As a result, the Cross-Fisher prediction for the gap,⁹

⌬= u

␣

冉

2k^g2e

冊

^{1/共2−2K兲 共44兲}

is recovered. In the antiadiabatic limit, the exponential term in共43兲disappears, and the incomplete Gamma function can be replaced by a gamma function, leading to the result for the gap we have found in Sec. III B, in共39兲. In this limit, the gap can be understood as resulting from a cos 4␾interaction induced by integrating out the phonon modes.

To perform a general study for any␻0␰^/u, we rewrite共43兲 for␰^as

冉

^␣␰

冊

^2−2K

e^{−2e−␥共␻0␰/u兲}+共ue^␥/2␻0␰兲^2K␥

冉

^{1 + 2K,2e−␥␻u0␰}

冊

= 4Kg²

␲^u␳␻02

冉

^e2^␥

冊

^{2K. 共45兲}

In terms of the gap, this equation reads

f

冉

␻^⌬0

冊

⁼_␲^4Kgu␳␻²02

冉

␻^u0␣

冊

^2−2K

冉

^e2^␥

冊

^{2K, 共46兲}

where

f共x兲= x^2−2K

e^{−2共e−␥兲/x}+

冉

^xe2^␥

冊

^2K␥

冉

^{1 + 2K,2ex−␥}

冊

^{. 共47兲}

The graph of the function f共x兲 is represented on Fig. 1. In this figure, the crossover from the adiabatic to the antiadia- batic regime is easily observed, with the two limiting forms

of the gap given, respectively, by Eqs. 共44兲 and 共39兲. The SCHA allows one to obtain the full interpolating function between the two regimes, and thus to obtain precisely the crossover scale. We obtain that the limit between the adia- batic and the antiadiabatic regime is given by␻0⬃⌬and not by␻0⬃J. This point will be further discussed in the forth- coming Sec. IV.

The SCHA also yields the expectation value of the nearest-neighbor correlationsS_n·S_n+1, as it is proportional to 共−1兲ⁿ具cos 2␾典. One finds

具Sn·S_n+1典 ⬃

冉

^␣␰

冊

^{K. 共48兲}

For␻0Ⰶ⌬, i.e., in the adiabatic regime, one has

具Sn·S_n+1典 ⬃

冉

␲␳^gu2␻02

冊

^{K/共2−K兲. 共49兲}

In the antiadiabatic regime, forK⬍1 / 2, we find

具Sn·S_n+1典 ⬃

冋

^␲␳gu2␻02

冉

␣␻^u0

冊

^2K

册

^{K/共2−4K兲. 共50兲}

IV. RG ANALYSIS

As we have discussed in the previous section, the SCHA describes only approximately the renormalization of the qua- dratic part by the phonon coupling term. Such a renormal- ization of the parameterK is of course especially crucial to take into account precisely close to the isotropic Heisenberg pointK= 1 / 2. In this section, we thus apply an RG method to analyze the adiabatic-antiadiabatic crossover.

Attempts to an RG analysis of such a problem or of di- rectly related fermionic problems have been described in the literature. In particular, an RG analysis was performed¹⁸ at FIG. 1. The graph of the functionf共x兲 共solid line兲defined in Eq.

共47兲 with K= 1 / 3. Two regimes are visible. For ⌬Ⰷ␻0, f共x兲

⬃x^2−2K 共dashed curve兲. In that regime, the gap is given by the adiabatic formula Eq.共44兲. For⌬Ⰶ␻0, f共x兲⬃x^2−4K共dotted curve兲 and the gap is given by the antiadiabatic formula Eq. 共39兲. The crossover regime is observed for 0.3⬍⌬/␻0⬍3.

T= 0 based on a previous work on spinful fermions coupled to phonons.^59–61In this work, the interaction of the spinful fermions with the electrons is viewed as a retarded back- scattering interaction. However, although this description is appropriate for fermions, in the case of the spin chain it neglects the fact that the staggered dimer operator gives rise to more relevant interactions than current-current ones. As a result, this fermionic description underestimates the size of the dimerization gap. Our analysis, directly based on the bo- son representation of the spin chain does not suffer from such a limitation. In addition to providing us with a better description of the dimerization gap, the use of the boson representation also allows us to tackle the case of a finite frequency␻0and nonzero temperature.

Another closely related problem is the one of fermions in a random potential,⁶² which has an action quite similar to 共11兲but with a constantD. One could be tempted to simply reuse the RG equations derived for this system. However, here, the situation is more subtle. In 共12兲, for ␻0/T→0, D共␶兲→T. Therefore, we see that the rescaling of the tem- perature is going to modify the RG equations with respect to the case of disordered fermions. Moreover, for ␻0/T→⬁, D共␶兲=共␻0/ 2兲e^{−␻0兩␶兩}. As a result, the limit ofT→0 is delicate to handle properly. In particular, the definition of the spin- phonon coupling constant becomes ambiguous in this limit.

However the variational analysis performed in the previ- ous section allows us to build the correct RG procedure.

First, the variational approach shows that in order to obtain the correct results, it is important to first perform the calcu- lation of the ground state free energy for 0⬍TⰆ⌬, where⌬ is the spin-Peierls gap, and then take the limit of T→0.

Second, it gives us that the proper dimensionless coupling constant measuring the strength of the electron-phonon inter- action is

G= g²

␲^u␳␻02. 共51兲 We now proceed with the RG. We start from the following action:

S=

冕

^dx

冕

0

␤ d␶

2␲^K

冋

^{u共⳵x␾兲2+1u共⳵␶␾兲2}

册

− 1

2␳␻02

冉

␲^ga

冊

²

冕

^dx

冕

0

␤

d␶

冕

0

␤

d␶⬘^{cos 2}␾共x,␶兲

⫻D_␻0,␤共␶−␶⬘兲cos 2␾共x,␶⬘兲

− 2g_⬜

共2␲a兲²

冕

^dx

冕

0

␤

d␶cos 4␾. 共52兲

The cos共4␾兲operator is the marginal operator needed to de- scribe an spin-isotropic spin chain. The derivation of the equations is given in Appendix C. They read

d

dl

冉

^K1

冊

⁼

冉

␲^g⬜u

冊

²⁺␲^ug␳␻² 0 2

␣

uD_␻0共l兲

冉

^␣u

冊

^{, 共53a兲}

d

dl

冉

^g␲^⬜u

冊

^{=共2 − 4K兲}␲^g⬜u+ g²

␲^u␳␻0 2

␣

uD_␻0共l兲

冉

^␣u

冊

^{, 共53b兲}

d

dl

冉

␲^ug␳␻² 0

2

冊

⁼

冋

^{2共1 −K兲+}␲^g⬜u

册

␲^ug␳␻² 0

2, 共53c兲

d␻0

dl =␻0. 共53d兲

These RG equations are conveniently expressed using the coupling constant G defined in Eq. 共51兲. At this one-loop order, we find no corrections to the phonon frequency as can be seen in共53d兲. However, we expect such corrections to be obtained in a higher-loop-order calculation.

A. Anisotropic case

Since the action 共11兲 also describes spinless fermions coupled to phonons, our equations have similarities with the RG equations that have been derived for the electron-phonon problem.55–57,60,63 There are however important differences.

First, for a spin chain the equivalent fermionic band is auto- matically half-filled共in the absence of an external magnetic field兲. Thus, in addition to the standard terms that were con- sidered for the electron-phonon problem with incommensu- rate filling, one has here to take into account the marginal umklapp operator cos共4␾兲 as in Ref. 63. Second, in the electron-phonon problem a different coupling constant is used,^55–57 namely, Y_sp² =G关␻0共l兲␣^/u兴. Such a definition ap- pears natural when looking at the RG Eqs.共53a兲and共53b兲, sinceY_spseems to be the amount by whichKis renormalized in the limitT= 0. However, such definition would be at odds with the calculations performed with the SCHA. In fact, the integral兰0⬁dl共␣^/u兲D_␻0e^l共␣^/u兲= 1 for all␻0. As a result, if we neglect g_⬜ in共53a兲, and the renormalization ofK in 共53c兲, we find the following approximate RG equations for GandK:

G共l兲=G共0兲e^{共2−2K兲l}, 共54兲 d

dl共K⁻¹兲=G共0兲e^{共2−2K兲l}␻0␣

u e^lexp

冉

^{−␻u0␣el}

冊

^{, 共55兲}

and by a variable change toV=共␻0␣/u兲e^l, we easily obtain that

K⁻¹共⬁兲−K⁻¹共0兲=G共0兲

冉

␻^u0␣

冊

^{2−2K⌫共3 − 2K兲. 共56兲}

This equation is easily understood:␻0gives an energy cutoff that stops the RG flow ofKinduced byGat an energy scale of order ␻0=u/␣^e−l*. We note that it is identical to the SCHA result共28兲. We thus see that at that scale,Kis renor- malized by an amount proportional to G共l^*兲 and not G共l兲

⫻关␻0共l兲␣/u兴 as a result of the exponential factor in 共55兲.

This confirms that the right coupling constant in this theory is G and notG␻0共l兲␣^/u. In Ref. 63, the same prescription was used to define the coupling constant, whereas in Ref. 38, the incorrect rescaling of Ref. 55 was used. As a result, we

expect the conclusions of Ref. 38 to be incorrect in the adia- batic regime.

Until now, we have assumed that at the scale l^*

= ln关u/共␣␻0兲兴, the coupling constant G共l^*兲Ⰶ1. If this as- sumption breaks down, since the coupling constant G共l兲

=e^{共2−2K兲l}G共0兲, one finds a gap

⌬= u

␣^{G共0兲1/共2−2K兲⬎}␻0. 共57兲

This gap is in agreement with the SCHA result and with the mean-field theory treatment of Cross and Fisher.⁹ For K

⬍1 / 2, in the antiadiabatic limit␻0Ⰷu/␣, we know from the SCHA that the phonons can generate a relevant perturbation cos 4␾and thus induce a gap.³³This effect is also captured in the RG by共53b兲. This can be seen by a two step renor- malization procedure. In the first step, forl⬍l^*= ln共u/␣␻0兲, a termg_⬜is induced by the RG flow. This term is found to be of order

y共l^*兲=g_⬜共l^*兲

␲^{u =G共0兲}

冉

␣␻^u0

冊

^2−2K

冋

^{␥共2K+ 1,1兲}

−␥

冉

^{2K+ 1,␣␻u0}

冊 册

^{. 共58兲}

Since␻0Ⰶu/␣, we can actually neglect␥共2K+ 1 ,␣␻0/u兲in Eq.共58兲. Forl⬎l^*,D_␻0共l兲共␣/u兲→0, and we can dropGfrom the RG equations. We then have a simple Kosterlitz-Thouless RG flow, which leads to a gap of the form

⌬=u

␣

冋

^G共0兲

冉

␣␻^u0

冊

^{2K␥共2K+ 1,1兲}

册

^{1/共2−4K兲. 共59兲}

This gap is in agreement with the SCHA prediction in the antiadiabatic limit 共38兲. Therefore, we see that SCHA and RG methods agree perfectly, far from the isotropic point, once the proper coupling constant is used in the RG.

Using our RG equations, we can now study the SU共2兲 invariant limit for which the SCHA cannot be used, due to importance at that point of the marginally irrelevant operator cos共4␾兲.

B. SU(2) invariant case In the isotropic limit, we have

K=1

2

冉

^{1 −2g}␲^⬜u

冊

^{. 共60兲}

This ensures that, in the absence of spin-phonon coupling, the flow will renormalize to the fixed point K^*= 1 / 2 and g_⬜^* = 0. It is then easily seen that the Eqs. 共53a兲 and 共53b兲 reduce to a single equation fory=g_⬜/␲u. This leads to the following RG flow:

dy

dl =y²+G共l兲␻0␣

2u e^le^{−共␻0␣/u兲el}, 共61兲

dG

dl =

冉

^{1 +32y}

冊

^{G. 共62兲}

These RG equations allow for the full interpolation between the adiabatic and antiadiabatic limit.

The simple analysis of the previous section showed that the gap should behave as ⌬=共u/␣兲G共0兲 in the adiabatic limit. For the isotropic case, using共61兲and共62兲, we obtain logarithmic corrections to the gap ⌬=共u/␣兲G兩lnG兩^−3/2 re- sulting from the marginal flow of y共l兲. These logarithmic corrections共for details see the Appendix D兲are identical to those obtained by incorporating the logarithmic corrections to the gap of the dimerized spin-1 / 2 chain^42,49,64 into the Cross-Fisher mean-field theory. This confirms thatG is the right coupling constant to study the formation of the spin- Peierls gap in the adiabatic limit. On the other hand, as dis- cussed in the previous section, in the antiadiabatic limit, it is the flow of y共l兲 that determines whether or not the gap is formed. To analyze the flow in the antiadiabatic regime, we can use the approximationG共l兲=G共0兲e^l; i.e., we neglect the logarithmic corrections to the flow ofG. We have checked that this approximation leads to a good agreement with the numerical study of the RG flow using the fourth-order Runge-Kutta algorithm. Using the previous approximation, the RG flow 关共61兲 and 共62兲兴 can be reduced to a Ricatti differential equation共cf. Appendix E兲leading to the follow- ing dependence of the gap onG:

⌬=␻0e^␥−1exp

冋

^{−uG共0兲2␻0␣}

册

^{, 共63兲}

for the case ofy共0兲= 0. Wheny共0兲⬍0, it is found that a gap exists only if

uG共0兲 2␻0␣ ^⬎

兩y共0兲兩

1 +兩y共0兲兩ln共ue^1−␥/␣␻0兲. 共64兲 The physical content of this equation is transparent. At the scalel^*such that␻0e^l*=u/␣,G共l^*兲is equal to the l.h.s. of the inequality, whereas 兩y共l^*兲兩is equal to the right-hand side of the inequality. The gap can form only if the renormalized spin-phonon interaction is stronger than the renormalized marginal coupling at the energy scale ␻0. This is in agree- ment with the two-step RG approach⁶⁰of the preceding sec- tion. When the condition共64兲is satisfied, the gap behaves as

⌬=␻0e^{−共1−␥兲}exp

冤

^{uG共0兲2␻0␣ +1 −− 1y共0y共0兲兲ln}

^冉

^{ue␻01−␥␣}

^冊 冥

^{. 共65兲}

This expression shows that the gap vanishes as exp关−Ct./共G共0兲−G^c兲兴 when the spin-phonon coupling con- stant goes to the critical value, indicating that the phase tran- sition between the gapped phase and the gapless phase in the antiadiabatic regime is in the Berezinskii-Kosterlitz-Thouless 共BKT兲universality class. For fixedG共0兲, Eq.共65兲also indi- cates that there exists␻BKT such that for␻0⬎␻BKT the gap vanishes via a BKT transition. The implicit equation giving

␻BKTreads