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arXiv:cond-mat/0004426v1 [cond-mat.str-el] 26 Apr 2000

Lattice dynamical e?ects on the Peierls transition in one-dimensional metals and spin chains

Holger Fehske, Michael Holicki, Alexander Wei?e

Physikalisches Institut, Universit¨t Bayreuth, D-95440 Bayreuth a

Summary: The interplay of charge, spin and lattice degrees of freedom is studied for quasi-one-dimensional electron and spin systems coupled to quantum phonons. Special emphasis is put on the in?uence of the lattice dynamics on the Peierls transition. Using exact diagonalization techniques the ground-state and spectral properties of the Holstein model of spinless fermions and of a frustrated Heisenberg model with magneto-elastic coupling are analyzed on ?nite chains. In the non-adiabatic regime a (T = 0) quantum phase transition from a gapless Luttinger-liquid/spin-?uid state to a gapped dimerized phase occurs at a nonzero critical value of the electron/spin-phonon interaction. To study the nature of the spin-Peierls transition at ?nite temperatures for the in?nite system, an alternative Green’s function approach is applied to the magnetostrictive XY model. With increasing phonon frequency the structure factor shows a remarkable crossover from soft-mode to central-peak behaviour. The results are discussed in relation to recent experiments on CuGeO3 .

1

Introduction

Low dimensional electronic materials are known to be very susceptible to structural distortions driven by the electron-phonon interaction. Probably the most famous one is the Peierls instability [1] of one-dimensional (1D) metals: as the temperature is lowered the system creates a periodic variation in charge density, called a “charge-density-wave” (CDW), by shifting the electrons and ions from their symmetric positions. For the half-?lled band case the CDW is commensurate with the lattice and cannot slide as a whole. As a result the unit cell is doubled and the system has a broken-symmetry ground state. Since the dimerization of the lattice opens a gap at the Fermi surface the Peierls process transforms a metal into an insulator. Spontaneous dimerization transitions to a less symmetric but lower-energy con?guration like those shown in Fig. 1 (left panel) have been found in many quasi-1D materials, such as the organic conjugated polymers [e.g., (CH)x ] and charge transfer salts [e.g., TTF(TCNQ)] or the inorganic blue bronzes [e.g., K0.3 MoO3 ] and MX-chains [2].

2

Fehske et al.

As a generic theoretical model for such systems the 1D Hubbard model is frequently considered, He = ?

i,σ

tii niσ ?

i,σ

tii+1 (c? ci+1σ + H.c.) + U iσ

i

ni↑ ni↓ ,

(1.1)

supplemented by a coupling to the phonon system Hp =

i

K 2 p2 i + qi 2M 2

(1.2)

according to SSH-type [3]: Holstein-type [4]: tii = 0 tii → λqi tii+1 → t(1 + λqi ) tii+1 = t qi = ui ? ui+1 qi = ui ,

(1.3) i.e., the lattice vibrations qi interact with the electrons by modifying the electron hopping matrix element tij and on–site potential tii , respectively. In Eq. (1.1), c? iσ (ciσ ) creates (annihilates) a spin-σ electron at Wannier site i, and niσ = c? ciσ . iσ If there is one electron per site and the Coulomb repulsion is strong, U ? t, we are in the limit of localized electrons interacting via an e?ective antiferromagnetic (AF) exchange interaction J (∝ t2 /U ) and the system can be described by an Heisenberg Hamiltonian with magneto-elastic coupling [5] H = ?J

i

(1 + λqi )Si Si+1 + Hp ,

(?) (?)

(1.4)

?iσ where Si = σσ′ c? τσσ′ ciσ′ with ciσ = ciσ (1 ? ni,?σ ). In analogy to the Peierls ? ? ? instability in 1D metals, the dependence of J on the distance between neighbouring spins again gives rise to an instability, where the energy of the spin chain is lowered by dimerizing into an alternating pattern of weak and strong bonds. This so-called “spin-Peierls” (SP) transition leads to the formation of a singlet (dimer) ground state and there is an energy gap to elementary (massive) spin triplet excitations being well-separated from the continuum [see Fig. 1 (right panel)]. Experimentally, the SP phenomenon was observed in a number of organic compounds, such as (TTF)(CuBDT) or MEM(TCNQ)2 [6]. Most theoretical treatments of the Peierls instability describe the lattice degrees of freedom classically. In a wide range quasi-1D metals, however, the lattice zero-point motion is comparable to the Peierls lattice distortion, which makes the rigid lattice approximation questionable [7]. By any means lattice dynamical (quantum phonon) e?ects should be included in a theoretical analysis of the extraordinary transport and optical phenomena observed in Peierls-distorted systems [8, 9]. Likewise the interest in models of spins coupled dynamically to phonons has increased signi?cantly since it was recognized that the ?rst inorganic

Lattice dynamical e?ects on the Peierls transition

3

Peierls Instability

t1 t

a

Spin-Peierls Instability

t2 J1 J2

E(k)

ω(q)

C

? ? ? k ?

π 2a

T

S

π 2a 0

π 2a

0

π 2a

q

Figure 1 Schematic representation of the Peierls- and spin-Peierls scenarios. The left panel shows the opening of a gap ? in the electronic band structure E(k) of an 1D metal at the Fermi surface if, according to an SSH- or Holsteintype of electron-phonon coupling, a static lattice distortion with the new lattice period 2a occurs. The right panel represents the main features of the excitation spectrum of a Peierls-distorted 1D quantum spin chain. Above the singlet (S) ground state at least one elementary excitation, corresponding e.g. to a triplet (T) bound state, is split from the continuum (C).

SP compound CuGeO3 [10] shows no clear separation between the magnetic and phononic energy scales: the two (weakly dispersive optical) Peierls-active T+ phonon modes have frequencies ω0,1 ? J and ω0,2 ? 2J [11, 12]. Moreover, 2 in contrast to the organic SP materials, no phonon-softening is observed at the SP transition [13]. The SP physics in CuGeO3 is therefore in the non-adiabatic regime. Motivated by this situation, in this report, we study the perhaps minimal microscopic models capable of describing the Peierls and spin-Peierls transitions in 1D systems by the use of numerical techniques allowing an essentially exact treatment of both subsystems, electrons/spins and phonons, at a fully quantum-mechanically level. In the weak-coupling regime, the random-phaseapproximation (RPA) approach is shown to be consistent not only with phonon softening but also with phonon hardening at the SP transition, as observed, e.g., in (TTF)(CuBDT) and CuGeO3 , respectively.

4

Fehske et al.

2

Luttinger-liquid vs. charge-density-wave behaviour

First we consider the 1D Hubbard model, Eq. (1.1), at quarter ?lling and con?ne ourselves to the case of spinless fermions for simplicity. This model is of physical relevance in the strong interaction limit U → ∞ and is particularly interesting because a quantum phase transition from a Luttinger liquid (LL) to a CDW phase occurs at a ?nite electron-phonon interaction, demonstrating that the quantum phonon ?uctuations destroy the dimerized ground state for weak electron-phonon couplings [14, 15]. ? ? ? Rescaling H → H/t = He + Hp + Hep and setting λ=g 2Kω0 with

2 ω0 = K/M

(2.5)

the Holstein Hamiltonian in a particle-hole symmetric notation is given by ? He = ?

i

? (c? ci+1 + c? ci ) , Hp = ω0 i i+1

i

1 (b? bi + 2 ) , i

(2.6)

and ? He?p = ?gω0

i (?)

1 (b? + bi ) (ni ? 2 ) . i

(2.7)

In Eqs. (2.6) and (2.7), bi annihilates (creates) a dispersionsless Einstein phonon of frequency ω0 coupled to the local electron density ni = c? ci (? = 1, and all h i energies are measured in units of t). Note that for the Holstein model the dimensionless electron-phonon coupling constant g = εp /ω0 is directly related to the familiar polaron shift εp being a second natural measure of the strengths of the electron-phonon interaction. Both parameters are necessary in order to characterize the weak (εp ? 1) and strong coupling (εp ? 1 and g ? 1) situations in the adiabatic (ω0 ? 1) and anti-adiabatic regimes (ω0 ? 1), respectively. Previous results for the ground-state phase diagram of the Holstein model at half-?lling obtained by WL QMC [14] and GF MC [16] simulations showed signi?cant discrepancies in the region of small ω0 (0 < ω0 < 1). Only very recently ? Bursill et al. [15] provided more reliable information from level crossings in their DMRG data. Applying a new optimized phonon approach for the diagonalization of coupled electron/spin-phonon systems [17], we consider the Holstein model on chains of even length with up to 10 sites and periodic (antiperiodic) boundary conditions if there is an odd (even) number of fermions in the system. The resulting phase diagram is shown in Fig. 2 over a wide range of frequencies and coupling strengths. For small g the system is a metal, more precisely a Luttinger liquid with parameters that vary with the coupling (see below). For large g the system has an energy gap and develops true long-range CDW order in the thermodynamic limit. The phase boundary obtained with our optimized

Lattice dynamical e?ects on the Peierls transition

5

10.0

uρ/2 Kρ ω0=0.1 1.2

DMRG XXZ ED 0.8

reg

1.0

0.6 g=4.47 0.4 0.2 0.0 0 2 ω0=0.1

0

1 g

2

3

0.8

reg

attractive

σ ,S

1/ω0

1.0

0

LL

g 1 2 1.0

CDW

0 50 0.04

reg

ω ω

4

6

8

100

150

200

repulsive

density x 10 g=2.45

σ ,S

reg

ω0=10

0.5

0.02 0.00

ω0=10

0.1

ED vs. XXZ 0.0

0.0

1.0

2.0 g

3.0

4.0

Figure 2 Ground-state phase diagram of the 1D Holstein model of spinless fermions at half-?lling (Ne = N/2), showing the boundary between the Luttinger liquid (LL) and charge-density-wave (CDW) states obtained by exact diagonalization (ED) and density matrix renormalization group (DMRG) [15] approaches. The dashed line gives the asymptotic result for the XXZ model. Left insets show the LL parameters uρ and Kρ as a function of the electron-phonon g in the metallic regime; right insets display for a six-site chain the regular part of the optical conductivity σ reg (ω) (dotted lines) and the integrated spectral weight ω S reg (ω) = 0 dω ′ σ reg (ω ′ ) (solid lines) in the CDW region.

phonon diagonalization methods agrees with recent DMRG results [15]. In the adiabatic limit ω0 = 0, the critical coupling converges to zero, as expected for the Holstein Hamiltonian, Eqs. (1.1)-(1.3), with M → ∞. For 0 < ω0 < 1 the accurate ? determination of gc is somewhat di?cult. On the other hand, in the strongcoupling anti-adiabatic regime, the half-?lled Holstein model can be transformed to the exactly soluble XXZ (small polaron) model [14, 18]

2 N ? HXXZ = (2ω0 ? g 2 ω0 ? V2 ) ? e?g 4

+ ? ? + z z (Si Si+1 + Si Si+1 ) ? V2 eg Si Si+1 i

2

(2.8)

6

Fehske et al.

using second order perturbation theory with respect to t. Here V2 (g 2 , ω0 ) = 2 ?1 2 s 2e?2g ω0 s=0 (2g ) /(ss!), and the (Kosterlitz-Thouless) phase transition line is given by the condition V2 (α, g 2 )eg /2 = 1 (long-dashed curve in Fig. 2). Let us now characterize the LL and CDW phases in some more detail. According to Haldane’s Luttinger liquid conjecture [19], 1D gapless systems of interacting fermions should belong to the same universality class as the TomonagaLuttinger model. As stated above, the Holstein system is gapless for small enough coupling g. Thus it is obvious to prove, following the lines of approach to the problem by McKenzie et al. [16], whether our Lanczos data shows a ?nite-size scaling like a Luttinger liquid. For a LL of spinless fermions, the ground-state energy E0 (N ) of a ?nite system of N sites scales to leading order as πuρ E0 (N ) , = ?∞ ? N 6N 2 (2.9)

2

where ?∞ denotes the ground-state energy per site for the in?nite system and uρ is the velocity of the charge excitations. If E±1 (N ) is the ground-state energy with ±1 fermions away from half ?lling, to leading order the scaling should be E±1 (N ) ? E0 (N ) = πuρ . 2Kρ N (2.10)

Kρ is the renormalized e?ective coupling (sti?ness) constant. The left insets of Fig. 2 show the LL parameters as a function of g in the region, where the scaling relations Eqs. (2.9) and (2.10) hold. A very interesting result is the changing character of the interaction below ω0 ? 1. For small frequencies the e?ective fermion-fermion interaction is attractive, while it is repulsive for large frequencies, where the system forms a polaronic metal with strongly reduced kinetic energy [17]. Self-evidently there is a transition line in between, where the model describes “free” particles in lowest order. In the CDW state extremely valuable information about the low-energy excitations can be obtained from the behaviour of the optical conductivity. The real part of σ(ω) contains two contributions, the (coherent) Drude part at ω = 0 and a so-called “regular term”, σ reg (ω), due to ?nite-frequency dissipative optical transitions to excited quasiparticle states. In spectral representation (T = 0), the regular part takes the form σ reg (ω) =

m>0

| Ψ0 |i

? j (cj cj+1

? c? cj )|Ψm |2 j+1

Em ? E0

δ[ω ? (Em ? E0 )] ,

(2.11)

where σ reg (ω) is given in units of πe2 and we have omitted an 1/N prefactor. The evaluation of dynamical correlation functions, such as Eq. (2.11), can be carried out by means of very e?cient and numerically stable Chebyshev recursion and maximum entropy algorithms [20]. Clearly the optical absorption

Lattice dynamical e?ects on the Peierls transition

7

spectrum in the strong EP coupling regime is quite di?erent from that in the LL phase (cf. Ref. [21]). It can be interpreted in terms of strong electron-phonon correlations and corroborates the CDW picture. Since for g > gc the electronic band structure is gapped we expect that the low-energy gap feature survives in the thermodynamic limit. In the adiabatic region (upper right inset), the broad optical absorption band is produced by a single-particle excitation accompanied by multi-phonon absorptions and is basically related to the lowest unoccupied state of the upper band of the CDW insulator. The lineshape of σ reg (ω) re?ects the phonon distribution in the ground state. The most striking feature is the large spectral weight contained in the incoherent part of optical conductivity. Moreover, employing the f-sum rule for the optical conductivity [22] and taking into account the behaviour of the kinetic energy (∝ uρ ) as function of g, we found that in the metallic LL and insulating CDW phases nearly all the spectral weight is contained in the coherent (Drude) and incoherent (regular) part of Re σ(ω), respectively. As stated above, in the anti-adiabatic regime the LL phase is basically a polaronic metal, i.e., the electrons will be heavily dressed by phonons. Since the renormalized coherent bandwidth of the polaron band is extremely small, the ?nite-size gaps in the band structure are reduced as well. Therefore, the gap occurring in the CDW state (?CDW ? εp ) may be identi?ed with the optical absorption threshold (see lower right inset).

3

3.1

Non-adiabatic approach to the spin-Peierls transition

Exact diagonalization results for T=0

In spite of the experimental fact that a realistic modeling of the inorganic SP compound CuGeO3 should include the phonon dynamics, previous theoretical studies have commonly adopted an alternating and frustrated AF Heisenberg spin chain model [23] ? Hstatic =

i

(1 + δ(?1)i )Si Si+1 + α Si Si+2

(3.12)

with a static dimerization parameter δ, thus representing the extreme adiabatic limit of a SP chain (in this section all energies are given in units of J). α determines the strength of the frustrating AF next-nearest-neighbour coupling. The spin model (3.12) contains two independent mechanisms for spin gap formation. At δ = 0 and for α < αc the ground state is a spin liquid and the elementary excitations are massless spinons [24]. The critical value of frustration αc = 0.241 was accurately determined by numerical studies [23, 25]. For α > αc the ground state is spontaneously dimerized, the spectrum acquires a gap, and the elementary excitations are massive spinons [26, 27]. On the other hand for any ?nite δ,

8

Fehske et al.

the singlet ground state of the model (3.12) is also dimerized, but the elementary excitation is a massive magnon [24, 28]. A comprehensive study of the spectral properties of the model (3.12) in terms of the spin dynamical structure factor has been carried out by Yokoyama and Saiga [29]. From the magnetic properties of the uniform phase J ? 160 K and α = 0.36 have been estimated for CuGeO3 [30]. However, if one attempts to reproduce the observed spin gap ?ST ? 2.1 meV within the static model (3.12), a very small value of δ ? 1.2% results. From the uniaxial pressure derivatives of the exchange coupling J [31] and the structural distortion in the dimerized phase [11] a minimum magnetic dimerization of about 4% is obtained, incompatible with an adiabatic approach to the SP transition. The simplest model that maintains the full quantum dynamics of the lattice vibrations may be obtained from Eq. (3.12) by replacing (?1)i δ → gω0 (b? + bi ): i ? Hs =

i

? (Si Si+1 + αSi Si+2 ) , Hp = ω0

i

b? bi , i

(3.13)

?I Hsp = gω0

i

(b? + bi )Si Si+1 . i

(3.14)

Recently it was shown that such a dynamical spin-phonon model describes the general features of the magnetic excitation spectrum of CuGeO3 [32, 33, 34]. Here we focus on the behaviour of the lattice dimerization which can be found from the displacement structure factor at wave number q = π δ2 = gω0 N

2 i,j uu Cij eiπ(Ri ?Rj ) uu with Cij = (bi + b? )(bj + b? ) . i j

(3.15)

The alternating structure of the correlation function C1i as shown in the inset Fig. 3 (a) implies the Peierls formation of short and long bonds and thus alternating strong and weak AF exchange interactions, i.e., a dimerized ground state. The structure is enhanced (weakened) increasing the spin–phonon coupling (phonon frequency) [32]. As in the ordinary Peierls phenomenon, a ?nite dimerization δ > 0 necessarily leads to a gap ?ST in the magnetic excitation spectrum. For evaluating the relation between the dimerization and the resulting magnitude of the spin triplet excitation gap, we keep the phonon frequency ?xed, vary T S the coupling strength g and calculate for each parameter set ?ST = E0 ?E0 and ST the dimerization δ from Eq. (3.15). ED results for ? obtained for the static spin-only model (3.13) and the quantum phonon model (3.14) are compared in the main part of Fig. 3 (a). For vanishing dimerization, i.e. in the absence of any spin–phonon coupling (g = 0) the results for the static and the dynamic model naturally agree (note that for δ = 0 there remains a spin excitation gap due to the frustration driven singlet dimer ordering). The dynamic model (3.14) partially resolves the ?ST ? δ con?ict we are faced within the static approach,

Lattice dynamical e?ects on the Peierls transition

0.6

N=12 M=N 0.3

9

ω0=0.05 ω0=0.1 ω0=0.3 ω0=1 ω0=2 ω0=10

g=0.75 g=1.25

5.0

C1,j ?δ1,j

ω0=1.0 g=0.5 δ =0.0514

2

0.0

δ (N)?δ (10)

4.0 3.0

0.02

ω0=1.0

uu

2 2

?0.02

7 8 9 10 11 12 1 2 3 4 5 6 7

?

0.5

static model ω0=0.1 ω0=0.3 ω0=1.0 ω0=2.0

(δ/ω0)

j

ST

2

2.0 1.0

?0.06

8

10

12

N

α=0.36 N=8 M=20 0.0 0.5 1.0 g 1.5

(b)

α=0.36 N=8

(a)

0.4 0.00

0.0

0.05 δ 0.10

2.0

Figure 3 Spin gap (a) and dimerization (b) in the frustrated Heisenberg spin chain with dynamical spin-phonon coupling.

because the dimerization δ grows with the phonon frequency at ?xed ?ST . Thus matching ?ST to the neutron scattering data, a larger δ ? 5% may result. Figure 3 (b) shows the dimerization as a function of the spin-phonon interaction strength. As for the Holstein model the precise determination of the critical coupling gc is di?cult for phonons in the adiabatic regime ω0 < 0.5. On the contrary, in the non-adiabatic regime the SP transition takes place at gc ? 1 nearly irrespective of ω0 . Most remarkably, below (above) gc the dimerization decreases (increases) with increasing lattice size N (see inset), indicating that the in?nite system exhibits a true phase transition. 3.2 Anti-adiabatic limit: mapping to an e?ective magnetic problem

Figure 3 (b) implies that more insight into the dynamic spin-phonon model can be obtained by considering the limit of large phonon frequencies. One can then integrate out the lattice degrees of freedom in order to derive an e?ective spin Hamiltonian which cover the dynamical e?ect of phonons in a approximate way. Technically this can be achieved by a variety of methods, such as standard perturbation theory [35], continuous (?ow-equation based) [36] or variational unitary transformations [37]. In what follows we consider besides the magneto-elastic interaction, Eq. (3.14), another type of spin-phonon coupling [35, 38], ? II Hs?p = gω0

i

(b? + bi )(Si Si+1 ? Si Si?1 ) , i

(3.16)

where the AF exchange integral varies linearly with the di?erence between the

10

Fehske et al.

phonon amplitudes on neighbouring sites, and perform a Schrie?er-Wol? trans? ? ? formation [39], H = exp(S)H exp(?S), with SI = g

i

(b? ? bi )Si Si+1 i

and S II = g

i

(b? ? bi )(Si Si+1 ? Si Si?1 ) , (3.17) i

respectively. In contrast to electron-phonon systems with Holstein coupling, where the (Lang-Firsov) transformation similar to exp(S) completely removes ? the electron-phonon interaction term (He?p ), applying the unitary transforma? with Eqs. (3.14) and (3.16), we obtain an in?nite series of tion exp(S) to H terms, which can not be summed up to a simple expression. To derive an e?ective spin model we now take the average over the (transformed) phonon vacuum. ? ? The resulting spin Hamiltonian He?,s = ?p |H|?p contains longer than next0 ? 0 nearest-neighbour ranged Heisenberg interactions as well as numerous multi-spin couplings. To a good approximation we can neglect them and obtain (cf. [37]) ? I/II He?,s =

i

(Si Si+1 + αe? Si Si+2 )

I/II

(3.18)

with αI = e? α + g 2 (1 ? 2α)/2 + 3g 4 ω0 /16 ? 37g 4 (1 ? 2α)/96 , 1 + g 2 ω0 /2 ? g 2 (1 ? α)/2 ? 3g 4 ω0 /8 + g 4 (28 ? 37α)/96 (3.19)

αII = e?

α + g 2 ω0 /2 + g 2 (3 ? 5α)/2 + 3g 4 ω0 /48 ? g 4 (75 ? 124α)/24 . 1 + g 2 ω0 ? 3g 2 (1 ? α)/2 ? 9g 4 ω0 /8 + g 4 (59 ? 75α)/24

(3.20)

That is the integration over the phonon subsystem yields an additional frustrating next-nearest-neighbour exchange interaction. Therefore, without any explicit frustration α, the e?ective frustration αe? due to the phonons can lead to a gap in the energy spectrum and to spontaneous dimerization. This e?ect is most important in the anti-adiabatic frequency range [37]. In order to determine the critical line in the the α ? g plane indicating the transition from the gapless to the gapped ground state, we use the level crossing criterion [25, 23, 37, 38] for the lowest singlet and triplet excitations which become degenerate at αc (N ). At αc the ?nite-size corrections αc (N ) ? αc (∞) ? N ?2 are small. The resulting phase diagrams are displayed in Fig. 4. We ?nd that the phase boundary of the original quantum spin-phonon and e?ective spin models agree surprisingly well, where for the e?ective model (3.18) αc (g) can be obtained with high accuracy on local workstations if N ≤ 20. For spinphonon coupling of type I the critical curve exhibits a remarkable upturn before crossing the abscissa; i.e., the frustration is suppressed for small spin-phonon coupling, but over-critical for strong coupling. It is this feature which makes it ? ? ? necessary to expand H up to fourth order in g to approximate H in a correct

Lattice dynamical e?ects on the Peierls transition

0.4

coupling I

α=0 10

0

11

10

0

0.4

gapless

coupling II

1/ω0

α=0

gapped

αeff = αc

10

?1

gapped

αeff = αc

10

?1

α

α

0.2

ω0/J=10

0 1 2

10

?2

0.2 gapless ω0/J=10

10 0 1 2

?2

g

g

gapless 0.1

H cross. (N=8) Heff αeff=αc

gapped (a)

0.1

H cross. (N=8) Heff αeff= αc

gapped (b) 0.2 0.3 0.4

0.0 0.0

0.5

1.0

1.5

0.0 0.0

0.1

g

g

Figure 4 Phase diagram of the original spin-phonon model (3.13) and the effective spin model (3.18) with spin-phonon couplings of types I (a) and II (b).

way. A second order theory is not capable to describe the observed critical line. The upturn of αc (g) at small g was reproduced quite recently by linked series expansion techniques [40]. By contrast αc (g) is a monotonous decreasing function of g for the coupling of type II. It appears that one would get the same shape also for a second order theory. However, to enlarge the application area of our approximation taking into account higher order contributions is still appropriate. For α ≡ 0, di?erently from the coupling case I, gc tends to zero in the antiadiabatic limit ω0 → ∞. While the q = 0 and the q = π phonon mode compete in the case of coupling I, allowing for a stable gapless phase up to a critical II g, there is no interaction with the q = 0 mode in Hsp . Therefore the q = π mode induces long ranged exchange more e?ciently, leading to a vanishing gc for ω0 → ∞. 3.3 RPA approach at T = 0: Soft-mode vs. central-peak behaviour

The absence of a soft phonon mode at the displacive SP in CuGeO3 has been puzzling for a long time, since the behaviour was believed to be inconsistent with the standard Cross-Fisher RPA approach to SP transitions [41]. Even worse, the relevant phonon modes harden by about 5% with decreasing temperature [42], pointing to a central peak scenario. A way to reconcile these results within the framework of the Cross-Fisher theory has been proposed by Gros and Werner [43], with the result that a complete phonon softening occurs only for bare phonon frequencies less than a critical value (ω0 < ω0,c = 2.2Tc , where Tc is the SP transition temperature. For higher ω0 a central peak develops reaching Tc from above. For the magnetostrictive Heisenberg spin chain model, however,

1/ω0

0.3

gapless

0.3

12

Fehske et al.

it was not possible to analyze the complete pole structure of phonon spectral function. To test this scenario we restrict ourselves to the simpler XY model ? XY Hs =

i y y x x (Si Si+1 + Si Si+1 ) ,

? Hp =

i

p2 K i + (ui ? ui+1 )2 2M 2

(3.21)

with the spin-phonon coupling term λ ? Hs?p = 2

+ ? ? + (ui ? ui+1 )(Si Si+1 + Si Si+1 ) . i

(3.22)

The above magnetostrictive XY model has been proposed as the minimal model to describe the SP transition [44]. Performing a Jordan-Wigner transformation [45] it can be mapped onto a model of spinless fermions only interacting ? with the phonon system Hp = q ωq b? bq : q ? XY Hs ? Hs?p where η(k, q) = ?i ω0 N ωq

1 2

? JW → He = ?

k

cos(k)c? ck , k η(k, q)(bq + b? )c? ck?q , ?q k

k,q

(3.23) (3.24)

? JW → He?p = gω0

[sin(k ? q) ? sin(k)] .

(3.25)

with ωq = 2ω0 sin(q/2), i.e ωπ = 2ω0 . Determining the equation of motion for the Matsubara phonon Green’s func? tion D(q, iωn ) within the scheme worked out by Bennett and Pytte for the magnetostrictive Heisenberg model [46], the main advantage is that the higherorder spin-spin correlation functions can be calculated without further approx? imations for the XY-case. The retarded Green’s function Dret (q, ω) is obtained ? iωn ) by analytical continuation iωn → ω. In the uniform phase above from D(q, Tc , the RPA propagator of the q = π-phonon, being responsible for the phase transition, then results as ? Dret (π, ω) = 4ω0 , 2 ? ω 2 ? 4ω0 ? 2ω0 P (π, ω)

π 0 β dk sin2 (k) tanh[ 2 cos(k)] . π ω + 2 cos(k)

(3.26)

? P (π, ω) = ?(2gω0)2

(3.27)

Lattice dynamical e?ects on the Peierls transition

soft?mode regime ω0=0.25 κ=0.4

T/Tc=20 T/Tc=3.94 T/Tc=1.97 T/Tc=1.15 T=Tc

13

central?peak regime ω0=1.05 κ=0.4

C(π,ω)

C(π,ω)

T/Tc=3.94 T/Tc=1.31 T=Tc

(a)

(b)

0.8

0.0 1.0 ω 2.0 3.0

0.0

0.2

0.4

ω

0.6

Figure 5 Dynamical structure factor for the magnetostrictive XY model.

? A structural instability occurs if Dret (q, ω) exhibits a pole at a certain q-value for ω = 0. In that case an excitation with arbitrary low energy is possible and the lattice becomes unstable with respect to a (static) deformation with wave number q. From this condition the equation for the inverse transition temperature βc = 1/Tc easily follows as

π

1=κ

0

dk

sin2 (k) tanh[ βc cos(k)] 2 , cos(k)

(3.28)

where κ = g 2 ω0 /π. A more detailed study of the complete pole structure of ? Dret (q, ω) in both the undimerized and dimerized phases is presented in Ref. [47], where also the ultrasound properties of the system were discussed. Here we focus on the temperature dependence of the dynamical structure factor above Tc , C(q, ω) = ? ? 1 Im Dret (q, ω + iδ) , lim π δ→0 1 ? e?βω (3.29)

which is displayed in Fig. 5 for two characteristic phonon frequencies ω0 = 0.25 and ω0 = 1.05, corresponding to the soft-mode (a) and central-peak (b) regimes, respectively. For small bare phonon frequencies the maximum in C(π, ω) is located around ω = ωπ at high temperatures and moves with decreasing temperatures to lower frequencies until a real divergence of C(π, ω) appears for T = Tc at ω = 0. For large bare phonon frequencies a completely di?erent behaviour is found. Now the high-temperature phonon peak stays around ω = ωπ and even hardens in some degree as T → Tc . At the same time a peak structure evolves at q = 0 which becomes divergent at T = Tc . This so-called central peak corresponds to a new collective magneto-elastic excitation of the coupled spinphonon system, which occurs at the SP transition and moves to higher energies as the temperature is lowered further [47].

14

Fehske et al.

4

Conclusions

In this report the lattice dynamical e?ects on the Peierls transition in coupled electron/spin-phonon systems were discussed. Our approach was based on several generic model Hamiltonians obtained from the one-dimensional Holstein/SSH Hubbard model in important limiting cases. Applying both basically exact numerical and approximative analytical methods we are able to calculate both ground-state and spectral properties of these simpli?ed models for the complete range of model parameters. The focus, however, was on non-adiabatic e?ects due to the phonon dynamics. The presented results con?rm previous ?ndings that quantum phonon ?uctuations destroy the Peierls distorted state at su?ciently weak electron/spin-phonon interactions. For the spinless fermion model this means that for weak electron-phonon couplings the system resides in a metallic (gapless) phase described by two nonuniversal Luttinger liquid parameters. The renormalized charge velocity and the correlation exponent are obtained from ?nite-size scaling relations, ful?lled with great accuracy. The Luttinger liquid phase splits in an attractive and repulsive regime at low and high phonon frequencies, respectively. Here the polaronic metal, realized for repulsive interactions, is characterized by a strongly reduced mobility of the charge carriers. Increasing the electron-phonon coupling, the crossover between Luttinger liquid and charge-density-wave behaviour is found in excellent agreement with very recent DMRG results. The transition to the CDW state is accompanied by signi?cant changes in the optical response of the system. Most notably seems to be the substantial spectral weight transfer from the Drude to the regular (incoherent) part of the optical conductivity, indicating the increasing importance of inelastic scattering processes in the CDW (Peierls distorted) regime. As a simple model for a spin-Peierls system we considered a (frustrated) Heisenberg spin chain model supplemented by a coupling to optical phonons with frequencies comparable to the magnetic exchange coupling, which is, e.g.. the relevant limit for the spin-Peierls compound CuGeO3 . The magnetic excitations inherently include a local lattice distortion requiring a multi-phonon-mode treatment of the lattice degrees of freedom. When compared to the static model of an alternating, dimerized spin chain the magnetic properties are strongly renormalized due to the coupled spin and lattice dynamics. For the quantum spin-phonon model the dimerization dependence of the spin triplet excitation gap is found to be in qualitative agreement with experiment. In the anti-adiabatic phonon frequency range we used the concept of unitary transformations in order to integrate out phonon degrees of freedom, where the spin-phonon interaction results in an additional frustration in the e?ective spin-only model. For two types of spinphonon couplings, the critical lines separating gapless from gapped ground states are found to agree for the original spin-phonon and e?ective spin models. Finally

Lattice dynamical e?ects on the Peierls transition

15

the reliability of the RPA approach to SP transitions was assessed in terms of an XY model with additional magneto-elastic interaction. With increasing phonon frequency the dynamical displacement structure factor shows a crossover from a soft-mode to central-peak behaviour, normally linked to displacive and orderdisorder types of transition respectively. The central peak predicted to appear at Tc for large bare phonon frequency corresponds not to a real phononic but rather to a new magneto-elastic excitation.

Acknowledgements

The authors would like to thank B. B¨ chner, H. B¨ ttner, R. J. Bursill, A. P. u u Kampf, A. W. Sandvik, J. Schliemann, G. S. Uhrig, G. Wellein and R. Werner for valuable discussions. The ED calculations were performed at the LRZ M¨ nchen, u NIC J¨ lich, and HLR Stuttgart. u

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