Ferromagnetic Transition in One-Dimensional Itinerant Electron Systems Kun Yang Department of Physics, Florida State University, Tallahassee, Florida 32306 (November 13, 2018) We use bosonization to derive the effective field theory that properly describes ferromagnetic transi- tion in one-dimensional itinerant electron systems. The resultant theory is shown to have dynamical exponent z = 2 at tree level and upper critical dimension dc = 2. Thus one dimension is below the upper critical dimension of the theory, and the critical behavior of the transition is controlled by an interacting fixed point, which we study via ǫ expansion. Comparisons will be made with the Hertz-Millis theory, which describes the ferromagnetic transition in higher dimensions. Packs numbers: 71.10.Hf,71.10.Pm Ferromagnetic transitions in itinerant electron systems neous magnetization of the electrons, which would give are among the very first examples of quantum phase rise to plateau-like behavior near 0.5(2e2)/h [6]. This transitions studied theoretically [1]. In the approach interpretation receives further support from the observa- pioneered by Hertz, one decouples the electron-electron tion of similar behavior in the presence of magnetic field interaction using Hubbard-Stratonovish transformation, near crossings of subbands with opposite spin polariza- integrates out the fermionic degrees of freedom, and ar- tion [7]; in this case the corresponding interpretation is rives at a Ginsburg-Landau-Wilson like free energy func- spontaneous pseudospin magnetization of the electrons, tional that involves the ferromagnetic order parameter where the pseudospin index is actually the subband in- only, which are bosonic degrees of freedom. The quan- dex. Inspired by these developments, a description of the tum nature of the theory lies in the (imaginary) time de- ferromagnetic phase of a 1D electron liquid is developed pendence of the order parameter. This effective bosonic in Ref. [8], although the critical behavior of the ferromag- theory, known as the Hertz-Millis theory, was argued to netic transition was not studied in that work. have upper critical dimension dc = 1, thus the critical In this paper we study the critical behavior of ferro- behavior of the transition is expected to be mean-field magnetic transition in 1D. We derive the effective bosonic like for both d = 2 and d = 3 [1,2]. It has been pointed field theory that describes the transition using Abelian out recently, however, that the procedure of integrating bosonization, and use renormalization group to study the out gapless fermions may lead to subtle singularities in critical behavior. The bosonization method, which is spe- the expansion of the resultant bosonic free energy func- cific to 1D systems and extremely powerful, allows one tional in terms of the order parameter, or its gradients to derive the effective bosonic theory in terms of the fer- [3]; such singularities may invalidate the power-counting romagnetic order parameter without having to integrate analysis of Hertz and Millis, and change the critical be- out gapless fermions; thus the resultant theory does not havior of the transition [3]. The nature of the transition suffer from singularities that the Hertz-Millis theory may in two- and three-dimensional itinerant electron systems encounter. A straightforward power-counting analysis of is currently under extensive theoretical study [3]. the theory indicates that the dynamical exponent z = 2 Comparatively speaking, much less attention has been at the transition (at tree-level), thus the upper critical di- devoted to the possibility of ferromagnetic transition in mension is dc =4 z = 2. As a result the theory is below one-dimensional (1D) electron systems, until recently. its upper critical dimension− in 1D, thus the critical behav- This is in part due to a theorem of Lieb and Mattis [4], ior of the transition is controlled by an interacting fixed which states that the ground state is a singlet for certain point. We study the critical behavior using momentum classes of one-dimensional models with spin-independent shell renormalization group combined with ǫ expansion interactions, thus rules out the possibility of ferromag- near dc = 2, at zero as well as low temperature. netism in these models. The existence of ferromagnetic Model and Bosonization — Consider the follow- arXiv:cond-mat/0401149v3 [cond-mat.str-el] 4 Aug 2004 ground states in 1D models (that are not dictated by ing Hubbard-like Hamiltonian describing interacting elec- the Lieb-Mattis theorem) was only established recently trons in 1D: through numerical work [5], in which it was also found † ′ σσ ′ that the ferromagnetic transition is second-order, thus H = tij ciσcjσ + U ni↑ni↓ + Vij niσnjσ , ′ there is a quantum critical point. On the experimental Xijσ Xi ijσσX side, the so-called “0.7(2e2)/h” anomaly (or 0.7 anomaly) (1) in the density-dependence of the conductance of a 1D where tij are the single electron hopping matrix elements, electron wire has attracted considerable attention; one ′ σσ of the possible interpretations of this anomaly is sponta- U > 0 is an onsite repulsion, and Vij represents (possi- bly spin-dependent) further neighbor repulsion; depend- 1 ing on the spin dependence of V the system may possess electron-electron interaction (due to V for example) leads either full Heisenberg (or O(3)) symmetry, or just Ising to non-trivial wave-vector dependence in Fourier space, (or Z(2) U(1), the latter is responsible for conservation which gives rise to terms that are quadratic in φ but tot × of Sz ) symmetry. It has been shown [5] that in the pres- involve higher gradients. These terms, however, are ir- ence of further neighbor hopping (t with i j > 1), relevant in the renormalization group sense, at the Lut- ij | − | the Lieb-Mattis theorem no longer applies and a ferro- tinger liquid fixed point described by HLL (Eq. (2)); they magnetic ground state is stabilized for large enough U, scale to zero in the long-wave length and low-energy limit; when V = 0. One can also stabilize the ferromagnetic their physical effect is to renormalize the parameters of phase by having V terms that are spin-dependent and HLL. Thus the long-wave length, low-energy properties ferromagnetic; this possibility has been considered in the of the system are well described by HLL, albeit with context of atoms trapped in 1D optical lattices [9]. renormalized parameters. This is the essence of the Lut- One of the most powerful methods of tackling such 1D tinger liquid theory [11,10]. What we are going to see models is Abelian bosonization [10]. In this scheme, one below however, is that some of the terms that are irrel- takes advantage of the fact that in 1D, all particle-hole evant and neglected at the Luttinger liquid fixed point excitations can be generated by electron density and cur- (which describes the paramagnetic phase only) are cru- rent operators which satisfy bosonic commutation rela- cial for the stability of the ferromagnetic phase, as well tions, and expresses both kinetic energy and interaction as the ferromagnetic critical point; they must be retained terms of H in terms of the electron density and current for a proper description of the ferromagnetic phase as well operators. If one keeps terms that are quadratic and with as the transition. This should not be surprising, as oper- least number of gradients in the density and current op- ators irrelevant at one fixed may well be relevant at other erators, one arrives at the familiar Luttinger liquid (LL) fixed points [12]. Hamiltonian, which describe decoupled spin and charge We now consider approaching the second order phase excitations of the paramagnetic phase [10]: boundary from the paramagnetic side, by changing pa- rameters in the Hamiltonian. As we approach the criti- HLL = Hc + Hs; (2) cal point, the spin susceptibility χ diverges; thus v Ns ∝ 1 2 2 2 1/χ 0! As we move further into the ferromagnetic Hc = dx[π vJcΠc(x)+ vNc(∂xφc(x)) ]; (3) → 2π Z phase, one expects vNs to become negative; when this 1 2 2 2 happens it becomes energetically favorable to have a non- Hs = dx[π vJsΠ (x)+ vNs(∂xφs(x)) ]. (4) 2π Z s zero expectation value of ∂xφs(x), which is the sponta- neous magnetization. Physically this occurs because the Here φc(x) and φs(x) are the charge and spin fields re- gain of exchange energy from the magnetization over- lated to the charge and spin densities of the system: comes the loss of kinetic energy, as is standard in ferro- magnetism in itinerant electron systems. Clearly with a 1 1 2 ρ(x)= ∂ φ (x), S (x)= ∂ φ (x); (5) negative coefficient for (∂xφs(x)) , Hs is not stable, and π x c z 2π x s higher order terms in gradients of φs(x) and powers of while Πα are their conjugate fields satisfying ∂xφs(x) must be retained to maintain stability: ′ ′ ′ ′ ′ 2 2 4 [φα(x), Πα (x )] = iδαα δ(x x ), (6) H = dx[a(∂ φs(x)) + b(∂xφs(x)) + ], (7) − Z x ··· with α being c or s. Physically Πα(x) represents local where in the charge or spin current. Clearly the velocity parameters 2 2 +− ++ weak coupling limit a (1/8π ) j j (Vn,n+j Vn,n+j ) vNc and vNs parametrize the energy cost of charge and 3 ≈ − 1 d ǫ(k) P spin density fluctuations respectively, and are thus pro- and b 3 F (ǫ(k) is the single electron dis- ≈ 24π dk |k=k portional to the inverse charge and spin susceptibilities, persion). It is clear that the coefficient a is positive while vJc and vJs are proportional to the charge and spin for generic repulsive interactions when there is Ising stiffness of the system respectively, as they measure the anisotropy (V +− > V ++ = V −−).