PHYSICAL REVIEW B 84, 165123 (2011) Validity of the Tomonaga Luttinger liquid relations for the one-dimensional Holstein model Ka-Ming Tam,1,* S.-W. Tsai,2 and D. K. Campbell1 1Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA 2Department of Physics and Astronomy, University of California, Riverside, California 92521, USA (Received 21 June 2011; published 24 October 2011) For the one-dimensional Holstein model, we show that the relations among the scaling exponents of various correlation functions of the Tomonaga Luttinger liquid (LL), while valid in the thermodynamic limit, are significantly modified by finite-size corrections. We obtain analytical expressions for these corrections and find that they decrease very slowly with increasing system size. The interpretation of numerical data on finite-size lattices in terms of LL theory must therefore take these corrections into account. As an important example, we re-examine the proposed metallic phase of the zero-temperature, half-filled one-dimensional Holstein model without employing the LL relations. In particular, using quantum Monte Carlo calculations, we study the competition between the singlet pairing and charge ordering. Our results do not support the existence of a dominant singlet pairing state. DOI: 10.1103/PhysRevB.84.165123 PACS number(s): 71.10.Fd, 71.30.+h, 71.45.Lr I. INTRODUCTION A recent quantum Monte Carlo study25 further suggested The study of quantum fluctuations arising from dynamical that the dominant power-law ordering is the singlet BCS phonons and their effects on the charge ordering instability superconducting pairing (SS) correlation. This conclusion was of one-dimensional systems has a long history.1 One of the based on data from finite-size simulations which showed simplest models to describe a one-dimensional solid under that the Luttinger charge exponent (Kρ ) is larger than 1 (Ref. 26). [It is important to realize at the outset that this is not the influence of the quantum fluctuations is the molecular- 27 2,3 an off-diagonal long-range ordering, as breaking the U(1) crystal or Holstein model, which is given by the following 28,29 Hamiltonian: gauge invariance to generate a mass gap is forbidden in one dimension.30–32 SS should thus be understood as a charge H = Hel + Hph + Hel−ph, (1) gapless, dominant off-diagonal power-law decaying ordering.] In the (possibly singular) limit of infinite phonon frequency, the where Holstein model becomes equivalent to the attractive Hubbard model.4 For the attractive Hubbard model at half-filling, the =− † + Hel t (ci,sci+1,s H.c.), (2) SS and the charge density wave (CDW) order parameters are i,s degenerate because of the particle-hole symmetry. Moreover, p2 K both order parameters possess continuous symmetry with H = i + q2 , (3) 4 ph 2M 2 i respect to the spin rotation. Therefore, long-range ordering i of either sort is forbidden in this case. However, from the Hel−ph = gep qi ni,s. (4) numerical simulations for finite-size lattices, Kρ is always i,s larger than 1 and converges to 1 very slowly with increasing 33 system size. The similar behaviors of Kρ for both the attrac- In this model, the electrons move in a tight-binding band tive Hubbard model and the Holstein model at finite phonon interacting with the lattice via the coupling between the local frequency have led to the suggestion that the Holstein model electron density and the local lattice displacement. In the even at finite phonon frequency may have a charge gapless † Hamiltonian, ci,s (ci,s) creates (annihilates) an electron at site metallic phase with degenerate SS and CDW correlations, 33 i with spin s, and ni,s is the electron number operator. The ions even though Kρ > 1 from finite-size calculations. We note 34 are described by momentum pi and displacement qi for site that this coexistence was proposed years ago. i. The mass of the ions is M, and K is the harmonic√ lattice In the present article we present analytical and numerical coupling constant, so that the phonon frequency ω0 = K/M. calculations of the Luttinger liquid (LL) relations in the one- Following convention, we shall call these dispersionless dimensional (1D) Holstein model with finite-size corrections. optical excitations “Einstein phonons.” The hopping energy Our results extend to Einstein phonons previous studies involv- 35,36 of the electrons between nearest neighbors is t, and gep is ing acoustic phonons and complement prior multiscale the electron-phonon coupling energy. In what follows we functional renormalization group studies of retardation effects will use units such that t = M = 1, except where otherwise due to Einstein phonons.37 In essence, we seek to answer specified. For the half-filled band, early work suggested that three important questions raised by previous studies: (i) Are the Holstein model displays dimerization and that long-range the LL scaling relations valid in the 1D Holstein model? charge ordering sets in at any finite phonon frequency.4–9 (ii) What are the dominant correlations in this model? (iii) These conclusions have been challenged by various studies What can be said about the possibility of a charge gapless which find that for small enough electron-phonon coupling metallic phase at half-filling for weak electron-phonon cou- the system has zero charge gap and thus no long-range pling? As we shall see, we are able to provide clear answers order.10–24 to the first two questions but can only comment on the third. 1098-0121/2011/84(16)/165123(6)165123-1 ©2011 American Physical Society KA-MING TAM, S.-W. TSAI, AND D. K. CAMPBELL PHYSICAL REVIEW B 84, 165123 (2011) In Sec. II of the paper, we begin our study by describing the are coupled with the electrons. The Hamiltonian for the forward scatterings of the Holstein model in terms of bosonic phonons with momentum k = 0is fields. In Sec. III, we solve analytically the forward scattering 1 part of the Holstein model to show that electron-phonon H ≈ dx ζ −12 + ζω2d2 , (6) ph d0 0 0 coupling modifies the relations between scaling exponents, 2 and we calculate these modifications for finite-size systems. where ζ is the mass density, d0 is the phonon field with = This enables us to answer question (i). In Sec. IV,weemploy momentum k 0, and d0 is the conjugate momentum. The quantum Monte Carlo calculations for the full Hamiltonian Hamiltonian for the electron-phonon coupling is [Eq. (1)] to resolve question (ii) and to gain some insight into ≈ † question (iii). In Sec. V, we discuss the reasons why all the Hel−ph gep dx(ψs ψs d0). (7) perturbative renormalization group studies obtain long-range s CDW phase and under what circumstances the metallic phase By keeping only the term which is linear in the bosonic field can be obtained. Section VI contains a brief conclusion. † 35,36,39 to express the density operator ψs ψs , we obtain 2 H − ≈ g dx (∂ ϕ )d . (8) II. FORWARD SCATTERINGS IN THE el ph ep π x ρ 0 1D HOLSTEIN MODEL The remaining terms will generate the back scatterings and To answer the question about the validity of the LL relations Umklapp scatterings. They also lead to spin-charge coupling for the Holstein model in the gapless regime, including due to the retardation.39 the gapless half-filled case if present, it is sufficient to When only forward scatterings are considered, the electron- diagonalize exactly the forward scatterings (scatterings with phonon coupling affects only the charge part of the Hamil- zero momentum transfer, often denoted as g2 and g4 in the tonian, and we will thus focus on the charge part only in “g-ology” terminology)38 of the Hamiltonian. The neglect the following. The spin part is unchanged from the standard of anything beyond forward scatterings is unquestionably an Luttinger model.35,36,39 The charge part of the Hamiltonian assumption which cannot be apriorijustified in the full [Eq. (1)] arising solely from forward scatterings is model, but this neglect is the fundamental conjecture in u 1 the description of the gapless one-dimensional system: any H ≈ dx u K 2 + ρ (∂ ϕ )2 + dx ζ −12 forward ρ ρ ρ x ρ d0 Kρ 2 contributions beyond forward scatterings can be incorporated into the renormalization of the forward scatterings. Therefore, 2 + ζω2d2 + g dx (∂ ϕ )d . (9) although we are considering forward scatterings only, our 0 0 ep π x ρ 0 approach is completely general for the description of any gapless regime regardless of the filling. In what follows, the Since the phonon fields appear at quadratic order, we can 35,36 right-hand side of the symbol, ≈, is understood to include follow the work by Martin and Loss, who considered the only those terms in which the forward scatterings exclusively case of acoustic phonons (as opposed to the Einstein phonon are retained. case we are considering here) to integrate out the phonon fields By formulating the one-dimensional problem in the con- d0, which results in the collective inverse propagator for the tinuum limit and retaining only the low-energy degrees of charge field ϕρ , freedom, Hel can be expressed in terms of bosonic fields ϕν 1 b(kω u )2 and their corresponding momenta with explicit spin(σ )- D (k,ω) = ω2 + u2 k2 − 0 ρ , (10) ν ρ ρ 2 + 2 charge(ρ) separation39,40 Kρ uρ ω ω0 = 2 2 where b 2Kρ gep/πuρ ω0. The first part of the inverse propagator is the contribution from the bosonized electrons, Hel = Hρ + Hσ , (5) and the second part comes from the electron-phonon coupling.