arXiv:cond-mat/0401149v3 [cond-mat.str-el] 4 Aug 2004 eairo h rniini xetdt emean-field be to expected both is for transition like the of behavior ftepsil nepeain fti nml ssponta- is anomaly this one 1D of attention; interpretations a possible considerable of the attracted of conductance has the wire of electron density-dependence the in hr saqatmciia on.O h experimental the On point. “0 so-called thus critical the side, found quantum second-order, a also is was is transition there it ferromagnetic which by the in dictated recently that [5], not established work are only numerical was (that through theorem) models ferromagnetic Lieb-Mattis 1D of the existence in ferromag- The states of ground models. possibility these the in out netism rules [4], certain thus spin-independent Mattis for with interactions, and singlet models a Lieb one-dimensional is of of state theorem classes ground a the to that recently. states due which part until in systems, is in This electron transition ferromagnetic (1D) of one-dimensional possibility the to devoted [3]. study theoretical extensive transition systems under the electron currently itinerant of is three-dimensional nature be- and The critical two- the [3]. in change transition the and of Millis, gradients havior and its Hertz power-counting or of the invalidate parameter, analysis may order singularities func- the such energy of [3]; 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gapless out odrv h ffciebsncter ntrso h fer- the of terms one parameter in allows order theory powerful, bosonic romagnetic effective extremely the and derive systems to 1D spe- is to which cific method, bosonization The the Abelian study behavior. using to critical group transition renormalization use the and describes bosonization, that bosonic theory effective the field derive We 1D. in transition magnetic work. that in studied ferromag- not the was of behavior transition developed critical netic is the liquid although [8], electron Ref. 1D in the a of in- of description subband phase a developments, ferromagnetic the these actually by Inspired is dex. index electrons, pseudospin is the the interpretation where of corresponding magnetization the polariza- pseudospin case spin spontaneous this opposite in field with [7]; magnetic tion subbands of of presence the crossings in near behavior similar observa- of the tion from support further receives interpretation ttetasto a relvl,tu h pe rtcldi- critical upper is the thus mension tree-level), (at transition the at hl eomlzto ru obndwith momentum combined using group behavior fixed critical renormalization interacting the an shell study by We controlled is point. transition behav- the critical the of thus ior 1D, in dimension critical upper its iet lta-iebhvo er0 near give behavior would plateau-like which to electrons, the rise of magnetization neous near rn n1D: in trons elec- interacting describing Hamiltonian Hubbard-like ing l pndpnet ute egbrrplin depend- repulsion; neighbor further spin-dependent) bly > U where xaso.Cmaioswl emd ihthe with made be will Comparisons expansion. H nti ae esuyteciia eairo ferro- of behavior critical the study we paper this In oe n Bosonization and Model niini ihrdimensions. higher in ansition = sa nierplin and repulsion, onsite an is 0 d utn hoyi hw ohv dynamical have to shown is theory sultant c t X ij ijσ eairo h rniini controlled is transition the of behavior ,a eoa ela o temperature. low as well as zero at 2, = d oel ecie ermgei transi- ferromagnetic describes roperly c r h igeeeto opn arxelements, matrix hopping electron single the are .Tu n ieso sbelow is dimension one Thus 2. = t d ij c c 4 = iσ † c e,Foia32306 Florida see, jσ − z + .A eutteter is theory the result a As 2. = U X i crnSystems ectron n without i ↑ n osdrtefollow- the Consider — i ↓ V + ij σσ . ijσσ aigt integrate to having 5(2 X ′ ersns(possi- represents e ′ 2 V ) ij /h σσ ǫ ′ n expansion 6.This [6]. iσ n z jσ below 2 = ′ not (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 −−). On the other hand energy cost of charge and spin current fluctuations. the sign of coefficient b depends on the details of single We emphasize that HLL is an approximation of the electron dispersion; we assume b to be positive here [13], original electron Hamiltonian Eq. (1) with generic sin- so that the transition is second order which corresponds gle electron dispersion relation and two-body interaction. to the situation in Ref. [5]. Other possible terms that are For example, a nonlinear term of the form dx cos(√8φs) allowed by symmetry can be shown to be irrelevant at that describes back scattering of electronsR with opposite the ferromagnetic critical point to be studied below [14]. ′ spins is neglected here. As pointed by Haldane [11], non- Thus combining Hs with H , switching from Hamiltonian linearity in single-electron dispersion gives rise to terms to action which is more convenient for RG analysis, and beyond quadratic order in Π and φ, which represent in- after rescaling the spin field (φ = √2aφs), we arrive at teractions among the bosons. Also the non-locality of the following effective action:

2 2 1 2 1 2 2 r 2 4 To order O(ǫ ) the field φ receives a non-zero anoma- S = dτdx (∂τ φ) + (∂ φ) + (∂xφ) + u(∂xφ) , Z 2 2 x 2  lous dimension η, which also leads to a correction to the (8) dynamical exponent z =2 η/2 [15]. RG and Critical Behavior− at Finite T — The where τ is imaginary time, r = vNs/(4πa), and u = phase transition occurs at T = 0 only. However the b/(4a2). At mean field level, the ferromagnetic transi- quantum critical point has very significant influence on tion occurs at r = 0. We note that the last three terms the thermodynamic properties at finite T , if the system in Eq. (8) (or the static parts) take the usual form of is sufficiently close to the critical point. More specifi- Landau theory, upon identifying ∂xφ as the local mag- cally, as we will see below, finite temperature introduces −1/z netization m(x): r m2 + 1 (∂ m)2 + um4. The first term a thermal length scale ξT T , and depending on its 2 2 x ∼ in Eq. (8) controls the fluctuations along the imaginary interplay with the correlation length ξ of the system at time direction, and is responsible for the quantum na- T = 0, one can divide the temperature-coupling space ture of the theory. The effective action (8) is the basis into three regions, which are separated from each other ∗ zν of our study of the critical behavior of the ferromagnetic by two crossover lines of the form T r r [16]: ∗ ∼ | − ∗ | transition, which we now turn to. (i) Quantum Disordered: r > r (where r is the crit- Renormalization Group and Critical Behavior ical coupling) and ξT > ξ, in which the system behaves at T = 0 — We perform a renormalization group (RG) as an ordinary Luttinger liquid; for example the specific analysis of the action (8), and as is standard in such heat C T and susceptibility χ constant. ∼ ∼ ∗ analysis, we treat the spatial dimension d as a contin- (ii) Renormalized Classical: r < r and ξT > ξ, the uous variable, even though the action (8) is derived in system behaves like a ferromagnetic Luttinger liquid [8] 1D. We perform the following space-time transforma- whose magnetic order is suppressed by thermal fluctu- tion with scale factor s > 1: x′ = x/s,t′ = t/sz, and ations; here depending on the symmetry there can be φ′(x′) = φ(x)s∆, that leaves the first two terms in (8) two types of behavior: (a) if the system possesses full invariant. This leads to z =2and∆= d/2 1, which in Heisenberg symmetry, the gapless transverse spin fluctu- − ation with spectrum ω k2 gives rise to specific heat turn lead to the (tree-level) scaling relation for r and u: ∼ ′ 2 ′ 2−d relevant C √T , while the susceptibility χ 1/T 2 [17]; (b) if r = rs and u = us . We thus find u is be- ∼ ∼ low the upper critical dimension dc = 2, and the critical the system possesses Ising symmetry only, then the trans- property of the transition is controlled by an interacting verse spin fluctuation is gapped, the longitudinal fluctu- fixed point for d < 2. Obviously this fixed point is very ation with linear spectrum (the usual Luttinger liquid different from the Luttinger liquid fixed point (which is behavior) gives rise to specific heat C T , while the susceptibility χ exp(J/T ), which is the∼ usual behavior non-interacting and has dynamical exponent z = 1), even ∼ though we are in 1D. In the following we study the crit- of an Ising ferromagnet (J is an energy scale of order the ical property using momentum shell RG combined with domain wall energy of the Ising ferromagnet). ǫ = 2 d expansion, and hope it gives a reasonable de- (iii) Quantum Critical: ξT < ξ, in which the thermo- scription− at d = 1. dynamic property is controlled by the quantum critical We assume there is a momentum cutoff Λ, while no fixed point studied above. In the following we focus on such cutoff exists in frequency. Integrating out modes the quantum critical region. To study finite T proper- with Λ/s

3 critical region, to order O(ǫ). To determine the behavior only difference being the field is complex in their work, of specific heat, we need the singular contribution to the which is crucial for the Heisenberg (O(3)) symmetry of free energy from critical fluctuations, which obeys the their model. If the symmetry were reduced to Z(2) U(1), hyper-scaling law for theories below their upper critical the corresponding theory would involve a real field,× then dimensions: the theory becomes identical to Eq. (8) [15]. It was con- jectured [15,18] that this action properly describes ferro- F T 1+d/zΦ( r r∗ zν /T ), (13) ∼ | − | magnetic transition in 1D itinerant electron systems. It where Φ(x) is a universal scaling function. From Eq. is remarkable that two very different approaches lead to (13) we immediately obtain C T 1−ǫ/2 in the quantum the same effective theory for the transition. critical region. ∼ Incidentally, a 2D version of the action (8) was used Summary and Discussion — In this work we de- to study transitions between different valence bond solid veloped a bosonic field theory that describes ferromag- states in 2D recently [19]. The physics of these transitions netic transition in 1D itinerant electron systems, based are very different from the one discussed here however. on Abelian bosonization. This approach is quite different This work was initiated while the author was visiting from that of the Hertz-Millis theory, because in principle the Max Planck Institute for Physics of Complex Systems the bosonization procedure keeps all the degrees of free- in Dresden, during her Workshop on Quantum Phase dom of the original fermionic system, and allows us to Transitions. He benefited from stimulating discussions arrive at a bosonic theory without integrating out gap- with Peter Kopietz. He is particularly grateful to Subir less fermionic degrees of freedom, as was done in the Sachdev and T. Senthil for explaining to him the connec- Hertz-Millis theory. We thus believe the theory devel- tion between Ref. [18] and the present work, and bringing oped here is free of the possible singularities associated Ref. [19] to his attention. This work was supported by with integrating out gapless fermions. NSF grant No. DMR-0225698. The bosonic theory developed here, when generalized to arbitrary dimensions, was found to have upper criti- cal dimension dc = 2. Thus for d = 1, which is where the theory applies, the system is below its upper critical dimension, and the universal critical behavior of the tran- sition is controlled by an interacting fixed point, which [1] J. A. Hertz, Phys. Rev. B 14, 1165 (1976). we have studied in some detail using ǫ expansion. This is [2] A. J. Millis, Phys. Rev. B 48, 7183 (1993). again quite different from the Hertz-Millis theory, which [3] See, e.g., D. Belitz and T. R. Kirkpatrick, Phys. Rev. is above its upper critical dimension dc = 1 for d = 2 Lett. 89, 247202 (2002); T. R. Kirkpatrick and D. Belitz, and d = 3 where it applies; there the critical behavior Phys. Rev. B 67, 024419 (2003); A. V. Chubukov, C. is controlled by a Gaussian fixed point, and some of the Pepin, and J. Rech, cond-mat/0311420 (2003). critical properties are non-universal due to the presence [4] E. H. Lieb and D. Mattis, Phys. Rev. 125, 164 (1962). of dangerously irrelevant operators. [5] S. Daul and R. M. Noack, Phys. Rev. B 58, 2635 (1998). The Abelian bosonization procedure can be applied to [6] K. J. Thomas et al., Phys. Rev. Lett. 77, 135 (1996); D. systems either with full Heisenberg symmetry or Ising J. Reilly et al., Phys. Rev. Lett. 89, 246801 (2002). 91 symmetry only. However since it does not exhibit the [7] A. C. Graham et al., Phys. Rev. Lett. , 136404 (2003). 67 Heisenberg symmetry explicitly, this symmetry is most [8] L. Bartosch, M. Kollar, and P. Kopietz, Phys. Rev. B , 092403 (2003). likely lost due to the approximate nature of the deriva- [9] H. Pu et al., Phys. Rev. Lett. 87, 140405 (2001). tion and treatment of the bosonic theory. Thus the crit- [10] For a recent review, see, e.g., J. Voit, Rep. Prog. Phys. ical behavior discussed above probably applies to sys- 58, 977 (1995). tems with Ising symmetry only. In solid state systems [11] F. D. M. Haldane, J. Phys. C 14 2585 (1981). the Heisenberg symmetry of electron spins are often re- [12] The importance of some of these “irrelevant terms” for duced to Ising due to the ubiquitous spin-orbit coupling; the stability of the ferromagnetic phase has been pointed in systems with pseudospin transitions the Heisenberg in Ref. [8]. A very similar situation occurs in the field symmetry is absent in the first place. Thus the results theory description of the reconstructed phase as well as presented here are highly relevant. Nevertheless it would the reconstruction transition at the edge of a quantum be highly desirable to maintain the Heisenberg symmetry Hall liquid, which is an example of chiral Luttinger liquid: 91 when present, using for example non-Abelian bosoniza- K. Yang, Phys. Rev. Lett. , 036802 (2003). [13] If b is negative then higher order terms in ∂xφs(x) must tion, and study if and how the extra symmetry affects be retained, which will lead to a first-order ferromagnetic critical behavior of the transition. We leave this for fu- transition. Only even powers of ∂xφs(x) are present as ture investigation. guaranteed by spin symmetry. In an earlier work, Sachdev and Senthil [18] studied [14] In fact there are also non-linear terms of the form 2 ferromagnetic transitions in lattice rotor models, and ar- dx(∂xφc(x))(∂xφs(x)) etc that actually couple spin rived at an effective action very similar to Eq. (8); the R

4 and charge sectors; since the transition happens in the spin sector, the charge sector can be integrated out which ′ generates terms that are already present in Hs and H . No singularities are generated here. [15] S. Sachdev, private communication. [16] S. Chakravarty, B. I. Haperin and D. R. Nelson, Phys. Rev. B 39, 2344 (1989). [17] M. E. Fisher, Am. J. Phys. 32, 343 (1964). [18] S. Sachdev and T. Senthil, Ann. Phys. 251, 76 (1996). [19] A. Vishwanath, L. Balents, and T. Senthil, cond- mat/0311085; E. Fradkin, David A. Huse, R. Moessner, V. Oganesyan, and S. L. Sondhi, cond-mat/0311353.

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