Brownian Scattering of a Spinon in a Luttinger Liquid

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PHYSICAL REVIEW B 90, 245434 (2014)

Brownian scattering of a spinon in a Luttinger liquid

M.-T. Rieder,1 A. Levchenko,2 and T. Micklitz3 1Dahlem Center for Complex Quantum Systems and Institut fur¨ Theoretische Physik, Freie Universitat¨ Berlin, 14195 Berlin, Germany 2Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA 3Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil (Received 8 September 2014; revised manuscript received 11 December 2014; published 29 December 2014)

We consider strongly interacting one-dimensional electron liquids where elementary excitations carry either spin or charge. At small temperatures a spinon created at the bottom of its band scatters off low-energy spin and charge excitations and follows the diffusive motion of a Brownian particle in momentum space. We calculate the mobility characterizing these processes and show that the resulting diffusion coefﬁcient of the spinon is parametrically enhanced at low temperatures compared to that of a mobile impurity in a spinless Luttinger liquid. We brieﬂy discuss that this hints at the relevance of spin in the process of equilibration of strongly interacting one-dimensional electrons, and comment on implications for transport in clean single-channel quantum wires.

DOI: 10.1103/PhysRevB.90.245434 PACS number(s): 71.10.Pm, 05.40.Jc, 72.10.−d, 73.63.Nm

I. INTRODUCTION finite coupling between spin and charge modes allows for an exchange of energy and momentum and both subsystems are The Luttinger liquid model of interacting one-dimensional then characterized by a common temperature and drift velocity. (1D) electrons advocates that spin and charge degrees of The rate of such prethermalization is relatively fast and follows freedom of electrons deconfine into elementary excitations power-law temperature dependence ∝T 3 [15]. which represent collective waves of spin and charge density In the second stage, the total momentum of excitations [1–3]. These collective bosonic modes do not interact and relaxes. The relevant processes can be viewed as the umklapp propagate with different velocities so that the charge and spin scattering of large-momentum bosonic excitations in which of an electron move apart in time [4–6]. This scenario of momentum is transferred to the zero modes of the Luttinger spin-charge separation is a remarkable example of the frac- liquid [16]. Alternatively, it can be described by backscattering tionalization of a quantum number occurring when low-energy of fermionic excitations [17], and we use both languages quasiparticles of a strongly interacting system do not evince interchangeably throughout this paper. At strong interactions much resemblance to the underlying electrons. Indeed, the the key process for equilibration is the backscattering of subsequent refermionization of the bosonized model reveals large-momentum spinons. That is, spin excitations first relax that elementary excitations are charged spinless quasiparticles- their drift velocity and then drag the fluid of charge excitations holons, and neutral spin-1/2 quasiparticles-spinons. Of course, to equilibrium. As a result, the rate of full equilibration is slow − this paradigm of spin-charge separation is an idealization and follows the activated temperature dependence ∝e σ /T , which is violated once effects of band curvature are accounted with σ being the width of the spinon band. for [7,8]. In general, spin-charge coupling leads to a plethora of Given the prominent role of spinons close to the band pronounced effects, and the properties of nonlinear Luttinger bottom in the process of equilibration it is interesting to study liquids are currently under intensive investigation [9,10]. their kinetics. Previous work [18] has studied the dynamics At small but finite spin-charge coupling, spinons are the of a mobile impurity in a spinless Luttinger liquid (which lowest energy excitations of the strongly interacting one- is also deeply related to the problem of dark soliton decay dimensional system at any given momentum. While holons inaBosesystems[19,20]). Regarding the spinon at the readily decay into a continuum of spin excitations even at zero bottom of the band as a mobile impurity, we will closely temperature [11,12], the spinon remains a stable quasiparticle. follow the approach of Ref. [18] and include spin into That is, by creating a hole with large momentum (e.g., close the picture. In a similar spirit, the recent study Ref. [21] to the spinon’s band bottom) in a strongly interacting one- considered a spin-1/2 impurity coupled antiferromagnetically dimensional electron liquid, the charge is rapidly screened, to a one-dimensional gas of electrons and showed the forma- leaving behind the neutral spin-1/2 fermionic degree of tion of an unconventional Kondo effect. Here we study the freedom. At finite but small temperatures one may then study complementary case of a ferromagnetic coupling where the the dynamics of such a large-momentum spinon. spin of impurity remains unscreened. We find that, similar Knowledge of the spinon’s dynamics is of fundamental to the antiferromagnetic case, the mobility of the spinon is interest, as it is the backscattering of large-momentum spinons parametrically suppressed at low temperatures as compared to which eventually equilibrates one-dimensional electronic liq- the impurity diffusion in a spinless Luttinger liquid. However, uids. Indeed, the relaxation of strongly interacting electrons unlike the antiferromagnetic case, mobility ceases to be occurs as a result of a multistage process [13,14]. In the universal in the low-temperature limit and strongly depends on first stage, excitations scatter of each other and relax to a the integrability breaking perturbations of the Luttinger liquid common equilibrium. The generic equilibrium of the liquid model. Suppression of the mobility hints at the relevance of the of excitations is characterized by a temperature, reflecting spin degree of freedom for equilibration of strongly interacting energy conservation, and a drift velocity, accounting for a one-dimensional electrons, and we briefly comment on its finite motion of a system with momentum conservation. A implications for the conductance in quantum wires. Similar

findings have been reported in a recent elegant work where a Consequences of the Fokker-Planck equation on 1D elec- particular realization of the Luttinger liquid model—a Wigner tronic transport are well studied in the literature [23–26]. crystal at low electron density—was considered [22]. Here, our main interest is on the scaling W(q) ∝ qα with the typical momentum exchange q in a collision, as this sets the scaling of the diffusion constant with temperature according II. FOKKER-PLANCK EQUATION AND DIFFUSION to D(T ) = q2W(q) ∝ T 3+α. For spin-polarized electrons, CONSTANT q scattering from low-energy charge excitations, the Brownian The lowest energy excitation for a system with a concave particle follows the scaling W(q) ∝ q2, which implies for the spectrum at a given momentum p is a hole. For weak diffusion constant in this case D(T ) ∝ T 5 [23,27,28]. The interactions the hole carries spin and charge quantum numbers latter leads to the mobility μ ∝ 1/T 4, in agreement with earlier of the removed electron. In a strongly interacting electron fluid, results [18,19]. To gain intuition as to how this result may on the other hand, the charge is screened and a neutral spin-1/2 change upon adding a spin degree of freedom to the problem, particle, the spinon, remains with a twofold degeneracy of we first discuss the limit of weak interactions. the energy levels protected by spin-rotation symmetry. In the following we assume that a spinon in the vicinity of the band III. WEAKLY INTERACTING ELECTRONS bottom has a quadratic dispersion We consider weakly interacting electrons described by the p2 εσ = − , (1) Hamiltonian p σ 2m∗ † 1 ∗ H = ε c c + V − . (4) with m being the effective mass. At finite temperature√ T the p p,ς p,ς 2L q q,ς q,ς typical momentum of a spinon is of the order p ∼ m∗T and pς qς should be compared to the momentum exchanged in collisions Here we assume a quadratic spectrum ε = p2/2m, where ∼ p with low-energy charge or spin excitations δp T/vρ(σ ), = † m is the electron effective mass, q,ς p cp+q,ςcp,ς is the where vρ(σ ) are their respective velocities. If temperatures ∗ charge density of electrons with spin quantum number ς, V are sufficiently low, T m v , relative changes are small q ρ(σ ) denotes the Fourier component of the interaction potential, and δp/p 1 and the spinon may be viewed as a heavy Brownian L is the system size. particle propagating in a gas of light particles, viz. the low- For weak interactions the physical mechanism of relaxation energy excitations. Collisions with the light particles render the in the one-dimensional system was attributed to three-particle motion of the spinon to be diffusive in momentum space, and collisions [17,23]. Kinematic considerations suggest that the the kinetic equation describing the spinon distribution F (p) required momentum transfer of 2p to backscatter from the is approximated by the Fokker-Planck form (throughout the F left to the right Fermi point cannot be accommodated within paper we set = kB = 1): a single three-particle collision. Rather, the momentum 2pF D p is transferred within a sequence of three-particle scattering ∂ F (p) = ∂ − + ∂ F (p). (2) t 2 p m∗T p events accommodating small momentum transfer δp pF . In the course of such multistage scattering processes a hole Here we employed that by spin rotation symmetry F (p)is passes through the bottom of the band between the two Fermi independent of the spin orientation, and for simplicity we points at ±pF and multiple particle-hole pairs are created at will concentrate on homogeneous liquids. The microscopics the Fermi level. of this Brownian motion in momentum space is governed by The kinetic equation for the distribution F (p) of a hole in the diffusion constant the vicinity of the bottom of the band again takes the form of a Fokker-Planck equation (2). The diffusion constant (3)inthis 2 D(T ) = q W(q), case is expressed in terms of the probability for scattering of a q hole (3) ςς W(q) = (q; kk )nl(k)[1 + nl (k )]. ςς ll W(q) = f (p )[1 − f (p )]f (p )[1 − f (p )], pp 2 2 3 3 ll ςς ,kk Q2Q3 Q2Q3 Here W(q) is the probability for a collision in which the spinon (5) ςς changes its momentum by q, and we already anticipated that at where pp is the quantum mechanical rate for a three- the low momenta of interest it depends only on the transferred electron scattering process from initial states I ={Q1,Q2,Q3} ςς ={ } momentum. Scattering rates (q; kk ) describe processes into final states F Q1,Q2,Q3 , characterized by quantum ll ={ } in which a spinon with momentum p and spin-projection numbers Qi pi ,ςi and correspondingly for Qi .Thesum value ς is scattered into a state with p + q,ς by absorbing runs over all intermediate states involving the scattering of ={ } ={ + } a spin (charge) excitation l = σ(ρ) with momentum k and hole-state Q1 p,ς into Q1 p q,ς . Notice that = − upon linearizing the dispersion near the Fermi points εp = emitting an l excitation with momentum k k q.The ± ± ± = vF p/T + bosonic occupation numbers for charge (spin) excitations at vF p, Fermi distribution functions f ( p) (e −1 momentum k are n (k), and we have set the occupation of a 1) and one can perform momentum summations exactly, ρ(σ ) + − = L − − − missing large-momentum spinon 1 − F (p + q) 1. Finally, p f (p k)[1 f (p)] 2π kn(k) and p f ( p k)[1 − = L + = vF k/T − 1 the mobility of the spinon is related to the diffusion constant f (p)] 2π k[1 n(k)], with n(k) (e 1) the Bose by the usual kinetic formula μ(T ) = T/D(T ). distribution function. Equation (5) then becomes structurally

245434-2 BROWNIAN SCATTERING OF A SPINON IN A . . . PHYSICAL REVIEW B 90, 245434 (2014) identical to Eq. (3), as should be expected since the com- l = which diagonalize the respective Hamiltonians H0 binations of Fermi distributions f (1 − f )in(5) describe l l† l ω bq b ,l= ρ,σ. In general, the dispersion of bosonic particle-hole excitations which correspond to the bosonic q q q excitations is nonlinear and has an acoustic form ωl = v q modes that are emitted/absorbed in a scattering process (3). q l only at low momenta q → 0. For repulsive interactions v > The central step in determining the diffusion constant is the ρ v , and LL parameters obey K = v /v . ςς = σ l F l calculation of transition rates via Fermi’s golden rule, pp To account for interactions between excitations, we intro- |Aςς |2 − 2π pp δ(EI EF ), where EI/F labels energies of the duce couplings in the density-density and spin-spin channels, = σ + ρ initial and final states. Then following previous works [27,29] H1 H1 H1 , whose structures are dictated by locality and and carefully taking into account exchange contributions, we spin-rotational symmetry find from second-order perturbation theory [30], ρ = σ = H λρ dx (x) d (x),H1 λσ dx S(x)s(x). (9) − 1 ςς ς1ς2ς3 VpF (VpF V2pF ) pF A ∝ ς ς ς δ . (6) pp 1 2 3 2 PI PF † † εF L q = σ = Here s ςς ςς dς dς and d ς dς dς are the spinon’s spin and particle density, while λρ(σ ) are respective coupling Here δPI PF ensures momentum conservation with PI/F being momenta of the initial and final states. The spin structure constants. The corresponding densities for bosonic excitations ς ς ς are conveniently expressed in terms of the fields φ : 1 2 3 = l of the scattering rate is governed by the matrix ς1ς2ς3 √ √ δς ς δς ς δς ς − δς ς δς ς δς ς . Crucially, we notice that this 2 2 1 2 2 3 3 1 1 3 2 1 3 2 (x) =− ∂ φ (x),Sz(x) =− ∂ φ (x), (10) particular spin structure forces all spin-polarized contributions π x ρ π x σ to compensate each other. Amplitudes involving different spin x 1 y 1 orientations and spin flips, however, remain singular in the S (x) = cos[φσ (x)],S(x) = sin[φσ (x)], (11) transferred momentum. That is, for spin-polarized electrons πa πa ∼ −1 the leading contribution from the maximally exchanged terms where a kF is the short distance cutoff. The above cancels and the three-particle transition rate is dominated Hamiltonian is introduced based on the recently developed |Aςς |2 ∝ 2 | | by subleading contributions in q, ςς pp ln (pF / q ) phenomenology [11,12]. Coupling constants λl can in prin- [27]. On the other hand, taking into account the spin degree of ciple be fixed microscopically employing Galilean invariance |Aςς |2 ∝ 2 and SU(2) symmetry, but are treated as mere parameters of the freedom gives ςς pp 1/q . This suppression of the scattering amplitude in the spin-polarized case can be traced model in the following. back to Pauli’s exclusion principle in three-particle collisions [31,32]. A. Transition rates Building on the previous discussion, and noting that p We have prepared the stage for a calculation of the spinon summations in Eq. (5)givetwoextrapowersinq, one diffusion coefficient (3) beyond the weak interaction limit. 3 5 finds D(T ) ∝ T , in contrast to D(T ) ∝ T in the spinless ςς Generalizing the preceding analysis, we study the rates case. Correspondingly, μ(T ) ∝ 1/T 2 and μ(T ) ∝ 1/T 4 in the ll of transition from an initial into a final spinon state accom- two cases, so that the spin degree of freedom parametrically panied by absorption and emission of bosonic spin or charge suppresses the mobility of a hole at the band bottom. A more † l† | = | |= | l excitations, i.e., I dp,ς bk 0 and F 0 dp+q,ς bk , with detailed calculation gives = l,l σ,ρ (see Fig. 1). 2 2 3 V (Vp − V2p ) T As in the weakly interacting limit, kinematic constraints pF F F 2 D(T ) pF εF (7) enforce vanishing of scattering rates calculated in first-order v4 ε F F perturbation theory. Leading-order contributions thus arise up to a numerical factor of order 1 [30]. from the next, second order–Raman scattering processes. To separate then contributions where spinons scatter off spin IV. SPINON IN A LUTTINGER LIQUID

We proceed to study the fate of the above result beyond (a) (b) the weak interaction limit by including situations in which electrons fractionalize into spin and charge modes. To this end, we start out from a free model for the relevant excitations = d + σ + ρ described by the Hamiltonian H0 H0 H0 H0 , where d = σ † H0 ς εp dp,ς dp,ς is the mobile free spinon in the vicinity ρ(σ ) of the band bottom Eq. (1). H0 are the standard Luttinger liquid Hamiltonians for bosonic charge and spin excitations described by the displacement ﬁelds φρ(σ ). The latter are conveniently expressed in terms of bosonic creation and annihilation operators FIG. 1. Schematics of two of the most relevant scattering processes contributing to the spinon’s mobility. Panel (a) A high-energy spinon is backscattered absorbing and emitting a spin excitation of πKl iqx−η|q| l† l φ (x) = i e b− + b , (8) the LL. Panel (b) A mixed backscattering process with the absorption l 2|q| q q q of a spin and emission of a charge excitation.

245434-3 M.-T. RIEDER, A. LEVCHENKO, AND T. MICKLITZ PHYSICAL REVIEW B 90, 245434 (2014)

ςς projection value Mςς are subleading in q/k 1. A careful and/or charge excitations, it is convenient to write = ρσ F ll ςς 2 calculation of all the relevant contributions [30] results in 2π|M | δ(E − E ), where ll F I the diffusion coefficient (again up to a numerical factor of ll ll H + H 1 order 1) ςς = | 2 2 | ll = l l Mll F I ,H2 H1 H1 1 + δll EI − H0 √ (12) λ K 4 T 3 λ2 v3 D(T ) σ σ 1 + ρ σ ε p2 , (14) 2 3 σ F and to discuss individual contributions separately. To this end, vσ εσ 8λσ vρ we bring the effective Hamiltonian into the form ςς h l l † l j l j ; lj lj describing the scattering of a Brownian spinon from spin H = d X X d , 2 p+k+k ,ς k E − H − εσ k p,ς = ςς,jj pkk I 0 p+k and charge excitations of a Luttinger liquid, where εσ ∗ 2 (13) m vσ /2. Equations (7) and (14) are the main results of this paper. where Xσj = Sj , with j = x,y,z the components in the spin As mentioned above, in the case of antiferromagnetic channel and Xρ = the charge channel. We next discuss the coupling the diffusion coefficient also scales with T 3,how- main results for the different scattering channels. ever, the physics reason for this behavior is different and can be traced back to the two-channel Kondo problem B. Scattering off charge excitations [21]. For the strongly repulsive interactions of the Wigner crystal limit, D also scales with T 3 [22]. We conclude The scattering of a spinon by absorption and emission of that independent of the interaction strength spin degrees of ςς ∝ 2 a low-energy charge excitation is described by hρρ λρδςς freedom suppress the mobility of large-momentum excita- ςς in Eq. (13). The corresponding transmission amplitude Mρρ tions in one-dimensional quantum liquids in a parameter ∼ 2 can then be calculated straightforwardly. We leave aside cal- (T/εF ) 1. This includes situations where spin and charge culational details (see Ref. [30]) and note here only that upon decouple. inserting typical momenta dictated by kinematic constraints, ∗ ςς ∝ 2 2 one finds that Mρρ (λρ q/m vρ )δςς . This, of course, just leads to the result D(T ) ∝ T 5, such as for scattering of an impurity in a spinless Luttinger liquid [18,19,21]. V. DISCUSSION AND SUMMARY We have studied the diffusion coefficient of a spinon in C. Scattering off spin and charge excitations a Luttinger liquid and shown that the resulting mobility of the spinon is parametrically suppressed at low temperatures Scattering of a spinon accompanied by absorption and emis- ςς compared to that of a mobile excitation in the spinless case. The ∝ 2 + sion of spin excitations is described by hσj,σj λσ (δjj δςς motion of an impurity in a quantum liquid is one of the central σ i iijj ςς )inEq.(13), where ij k is the Levi-Civita tensor. concepts of LL theory, with applications, e.g., to hole dynamics ςς in semiconducting nanowires or impurities in ultracold quan- The first contribution in hσj,σj preserves the projection ςς tum gases. The diffusion coefficient discussed in this work sets value of spin and is structurally identically to hρρ .The −1 ∝ ςς the equilibration rate in generic Luttinger liquids as τ − /T second contribution in hσj,σj is structurally different from D(T )e σ . Relatedly, it also defines transport properties x y x the previous ones. As the products of operators S , S and S , of clean quantum wires. Specifically, the interaction-induced z S involve odd numbers of creation/annihilation operators, backscattering process results in corrections to the quantized these combinations do not contribute to the scattering rate of conductance G =√(e2/π)(1 − δg), displaying activation be- − interest. This leaves us with scattering processes involving ∝ ∗ 3 σ /T y z havior δg (LD/ m T )e [33]. An activated behavior the product S , S , describing spin-flip events. By further of δg has also been observed in the recent experiments making use of normal ordering of the spin operators we may Refs. [34–36]. Verification of the pre-exponential temperature y = − linearize S φσ /(πa). This allows us to directly calculate dependence δg ∝ T 3/2e σ /T predicted here would provide the corresponding transition amplitude [30], which turns an important test for our understanding of equilibration effects out to be the leading amplitude of all allowed scattering in clean quantum wires. ςς ∝ channels. Invoking kinematic constraints we find Mσσ 2 ∗ 2 σ y (λσ kF /m vσ ) ςς , which implies that the weak interaction result D(T ) ∝ T 3 holds at arbitrary interaction strength. ςς ACKNOWLEDGMENTS Technically, the difference between contributions hρρ and ςς hσσ results from the noncommutativity of spin operators, We would like to thank K. A. Matveev, A. V. Andreev, and which prevents a cancellation of the leading terms in the A. D. Klironomos for numerous discussions and for sharing amplitude. results of their work [22] prior to their publication. T.M. Similarly, mixed processes involving spinon scattering off acknowledges support by the Brazilian agencies CNPq and ςς ∝ charge and spin excitations give to leading order Mρσ FAPERJ. Work by M.-T.R. was supported by the Alexander ∗ σ y (λρ λσ kF /m vρ vσ ) ςς . Notice that these again involve spin- von Humboldt Foundation. The work at MSU (A.L.) was flip processes, while those processes conserving the spin- supported by NSF Grant No. DMR-1401908.

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