Rate of equilibration of a one-dimensional Wigner crystal
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Rate of equilibration of a one-dimensional Wigner crystal
K. A. Matveev a , ∗ , A. V. Andreev b , M. Pustilnik c a Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA b Department of Physics, University of Washington, Seattle, Washington 98195, USA c School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
Abstract
We consider a system of one-dimensional spinless particles interacting via long-range repulsion. In the limit of stronginteractions the system is a Wigner crystal, with excitations analogous to phonons in solids. In a harmonic crystalthe phonons do not interact, and the system never reaches thermal equilibrium. We account for the anharmonism ofthe Wigner crystal and find the rate at which it approaches equilibrium. The full equilibration of the system requiresumklapp scattering of phonons, resulting in exponential suppression of the equilibration rate at low temperatures.
Key words: equilibration, one-dimensional systems, Wigner crystal
PACS:
The low-temperature physics of interacting electronsystems is usually described in the framework of the so-called Luttinger liquid theory [1]. The phenomenologi-cal nature of this approach enables one to study the sys-tems with any interaction strength, provided that thephysics is controlled by the low-energy excitations. Onthe other hand, even at low temperature T some phe-nomena involve excitations with energies much higherthan T . In such cases microscopic approaches are usu-ally more effective.An example of such a phenomenon is the equilibra-tion of a one-dimensional system. The latter is under-stood most easily in the case of weak interactions be-tween electrons, when the usual picture of quasiparticleexcitations is applicable. Because of the conservationof momentum and energy in electron-electron scatter-ing, the two-particle collisions do not affect the distri-bution function. Thus at weak interactions the equi-libration is accomplished via the three-particle colli-sions, Fig. 1(a). To equilibrate the chemical potentialsof the right- and left-moving electrons, the collisionsmust change the numbers of electrons on each branch,i.e., the backscattering events, such as the one shown in ∗ Corresponding author. E-mail: [email protected] µ q (b)(a) ǫ ( q ) k k F k F E k Fig. 1. (a) Three particle scattering process leading to equi-libration of weakly-interacting one-dimensional electrons. (b)Spectrum of the hole excitation. Backscattering of the holeoccurs when its wave vector crosses the point q = k F . Fig. 1(a), are needed. The most effective such processincludes backscattering of a particle at the very bot-tom of the band [2]. The small probability of findingan available final state deep below the Fermi level E F results in exponential suppression of the equilibrationrate τ − ∝ e − E F /T [3].The key event in the process of equilibration is thebackscattering of a hole at the bottom of the band. Itis therefore helpful to focus on the motion of such ahole in momentum space. For electrons with quadraticspectrum E k = ~ k / m the energy of the hole exci-tation is ǫ ( q ) = E k F − E k F − q = ~ v F q (1 − q/ k F ), as-suming that the hole is created by moving an electron Preprint submitted to Physica B 6 September 2018 rom state k F − q to the right Fermi point k F . Here k F is the Fermi wave vector and v F = ~ k F /m is the Fermivelocity. Backscattering occurs when the wave vectorof the hole q crosses k F , Fig. 1(b).Because the equilibration of the system involves ex-citations with energies of the order of the bandwidth,this phenomenon is not captured by the conventionalLuttinger liquid theory [1]. This makes the generaliza-tion of the above picture beyond the weakly-interactinglimit rather challenging. On the other hand, it is possi-ble to develop a microscopic theory of this phenomenonin the case of strong Coulomb interactions [4]. Here werevisit this approach and obtain the full expression forthe equilibration rate for arbitrary strong long-rangerepulsion. The key idea is that as repulsion of electronsbecomes stronger, the system minimizes its energy byforming a periodic structure known as the Wigner crys-tal. Although the long-range order in such a systemis destroyed by quantum fluctuations [5], the presenceof the strong short-range order enables us to treat thesystem as anharmonic chain described by the Hamil-tonian H = X l p l m + 12 X l,l ′ V ( x l − x l ′ ) . (1)Here p l and x l are the momentum and coordinate ofthe l -th particle and V ( x ) is the interaction potential.The excitations of the system are essentially thephonons in the electronic crystal. They are conve-niently described in terms of the displacements u l = x l − la of electrons from their equilibrium positions,where a is the mean interparticle distance. Strongrepulsion means small displacements, | u l − u l ′ | ≪| l − l ′ | a , and the Hamiltonian can be approximated bythat of a harmonic chain H = X l p l m + 14 X l,l ′ V (2) l − l ′ ( u l − u l ′ ) , (2)where the r -th derivative of V ( x ) is denoted as V ( r ) l = d r V ( x ) dx r (cid:12)(cid:12)(cid:12)(cid:12) x = la . (3)The phonon modes of this Hamiltonian are easilyfound, ω q = 2 m ∞ X l =1 V (2) l [1 − cos( ql )] . (4)At small wave vector qa → ω q = s | q | , where s = ( P l V (2) l l /m ) / is the“sound velocity” in the Wigner crystal measured inunits of lattice spacings per unit time. The spectrumis periodic in q , with the Brillouin zone − π < q < π ,Fig. 2 q q ′ q ′ q q ′ − π ω q − π π q π Fig. 2. Excitation spectrum of a one-dimensional Wigner crys-tal. Equilibration processes involve a phonon q being scat-tered to a state q ′ outside the Brillouin zone as a result ofa collision with a thermal phonon q . Conservation of energyand momentum requires the latter to be backscattered to q ′ .This umklapp process is analogous to a hole crossing the point q = k F in the weakly interacting Fermi gas, Fig. 1(b). (Notethat if the distances are measured in units of a , the Fermi wavevector k F = π .) Any eigenstate of the harmonic chain (2) can be de-scribed by the set of occupation numbers N q of thephonons. In the absence of interaction of phonons, thelifetime of any such state is infinite, and the systemnever reaches thermal equilibrium. On the other hand,the harmonic Hamiltonian (2) is merely the leadingterm of the expansion of Eq. (1) in the small param-eter K = π ~ /ma s . The next term is proportional tothe third power of the displacement u l and generatesscattering processes involving three phonons. At lowtemperature T ≪ ~ ω π the typical quasimomenta ofphonons are small, q ≪ T / ~ s , umklapp scattering issuppressed, and the phonons remain in the first Bril-louin zone upon scattering. This means that apart fromenergy, collisions conserve the total quasimomentum ofthe phonons. This yields the equilibrium phonon dis-tribution N q = 1 e ~ ( ω q − uq ) /T − T and the velocity u of the phonon gas with respect tothe crystal.On the other hand, even at low temperatures thereare rare umklapp collisions of phonons, such as the oneshown in Fig. 2, which do not conserve their quasimo-mentum. As a result, one expects the velocity u to re-lax gradually as ˙ u = − u/τ with a small relaxation rate τ − . To find it we notice that the most efficient umk-lapp processes involve a phonon q near the boundary q = π of the Brillouin zone colliding with a thermalphonon with q ∼ T / ~ s , Fig. 2. At low temperature theresulting change of quasimomentum | q ′ − q | ∼ T / ~ s issmall compared to the typical scale q T ∼ ( T / ~ | ω ′′ π | ) / of the distribution (5) near the edge of the Brillouinzone. (Here ω ′′ q = ∂ q ω q .) Thus the high-energy phononperforms a slow diffusive motion in the momentum2pace, and its distribution function N q ( t ) obeys theFokker-Planck equation ∂ t N q = ∂ q (cid:20) B ( q )2 (cid:18) ~ ω ′ q T + ∂ q (cid:19)(cid:21) N q . (6)Here B ( q ) = X δq ( δq ) W q,q + δq (7)has the meaning of the diffusion constant in momentumspace and W q,q + δq is the rate at which a phonon q changes its wave vector by δq as a result of collisionswith other phonons.The Fokker-Planck equation should be solved withthe boundary conditions N q = e − ~ ω q /T e ± π ~ u/T , q T ≪ ∓ ( q − π ) ≪ π (8)obtained by extending the distribution (5) beyond thefirst Brillouin zone. Such solution [4] gives the relax-ation law ˙ u = − u/τ with the rate τ − = 3 B (cid:18) ~ sT (cid:19) (cid:18) ~ | ω ′′ π | πT (cid:19) / e − ~ ω π /T , (9)where B = B ( π ). The temperature dependence of therelaxation rate is dominated by the exponentially smallprobability of the occupation of phonon states nearthe edge q = π of the Brillouin zone. Expression (9)is analogous to the result τ − ∝ e − E F /T for weakly-interacting electrons. The strong interactions betweenelectrons renormalize the activation temperature from E F to ~ ω π in Eq. (9).The temperature dependence of the prefactor inEq. (9) is determined by that of the diffusion constant B and by T − / explicitly present in (9). The formercan be deduced phenomenologically [6] by treating thephonon near q = π as a mobile impurity in a Luttingerliquid, for which B is known [7] to scale as B = χT , T → . (10)We therefore conclude that the equilibration rate scaleswith temperature as τ − ∝ T / e − ~ ω π /T .The constant χ in Eq. (10) has to be determined bymicroscopic evaluation of the scattering rate W q,q + δq in Eq. (7). The dominant scattering process, illustratedin Fig. 2, involves two phonons in both the initial andfinal states. Such scattering can be accomplished ei-ther in the first order in four-phonon scattering ampli-tude or in the second order in three-phonon scatteringamplitude. The resulting expression for the scatteringrate has the form [4] W q ,q ′ = 2 π ~ m N X q ,q ′ Λ N q ( N q ′ + 1) ω q ω q ω q ′ ω q ′ δ q + q ,q ′ + q ′ × δ ( ω q + ω q − ω q ′ − ω q ′ ) . (11) Here N is the total number of particles in the systemandΛ = − f ( q , q ) f ( q ′ , q ′ ) ω q + q − ( ω q + ω q ) + f ( q , − q ′ ) f ( q , − q ′ ) ω q − q ′ − ( ω q − ω q ′ ) + f ( q , − q ′ ) f ( q , − q ′ ) ω q − q ′ − ( ω q − ω q ′ ) + m f ( q , q , − q ′ ) , (12)with the functions f and f defined as f ( q , q ) = ∞ X l =1 V (3) l { sin[( q + q ) l ] − sin( q l ) − sin( q l ) } ,f ( q , q , q ) = ∞ X l =1 V (4) l { − cos( q l ) − cos( q l ) − cos( q l ) − cos[( q + q + q ) l ]+ cos[( q + q ) l ] + cos[( q + q ) l ]+ cos[( q + q ) l ] } . The expressions (11) and (12) are valid for any q and q ′ . At low temperature their magnitudes are small, | q | , | q ′ | . T / ~ s . Thus to find χ in Eq. (10) we needto expand Λ in powers of q and q ′ . Carrying out suchexpansion and taking into account conservation of mo-mentum and energy we findΛ = ( δq ) m n ω q v Υ q , (13)Υ q = ( ∂ n ω q ) ∂ n ( v − v q ) − ( v − v q ) ∂ n ω q + ω ′′ q ( ∂ n ω q ) n − , (14)where δq = q − q ′ is the small momentum change asa result of scattering, n = 1 /a is the particle density, v = s/n and v q = ω ′ q /n are the physical velocities ofthe phonons with wave vectors 0 and q , respectively.To leading order in δq we have replaced q , q ′ → q .Equations (11), (13), and (14) determine the scat-tering rate W q,q + δq for any q . It is easy to see that W q,q + δq ∝ ( δq ) , and Eq. (7) immediately gives thetemperature dependence (10). To find the coefficient χ we set q = π and obtain χ = 4 π [( ∂ n ω π ) ∂ n v − v ∂ n ω π + ω ′′ π ( ∂ n ω π ) n − ] ~ m v , (15)Remarkably, χ is fully determined by the phonon spec-trum and its dependence on the particle density n .Equations (9), (10), and (15) give the complete ex-pression for the equilibration rate of a one-dimensionalWigner crystal. In the case of pure Coulomb repulsion V ( x ) = e / | x | the velocity s diverges, and our treat-ment is inapplicable. However, in the experimental re-alizations of one-dimensional Wigner crystal, there is3sually a metal gate screening the interactions at largedistances. In this case we obtain1 τ = η ∆ ~ ln / ( d/a ) a B a (cid:18) T ∆ (cid:19) / e − ∆ /T , (16)where η = 63 π ζ (3) √ ln 2 / √ π is a numerical pref-actor, a B = ~ /me is the Bohr’s radius, d is the dis-tance to the gate, and∆ = ~ ω π = (cid:18) ζ (3) ~ e ma (cid:19) / . (17)A non-trivial test of our result (15) can be performedby considering the interaction potential V ( x ) = γ sinh cx . (18)It is well known [8] that the model (18) is integrable,i.e., it has an infinite number of integrals of motion.As a result the excitations of the system have infinitelifetimes, and one expects the diffusion constant B tovanish. Our approach applies only to the limit of strongrepulsion γ → ∞ , when the Wigner crystal approxima-tion is applicable. On the other hand, the parameter c can take any value. It is easy to obtain analytic ex-pressions for the phonon spectrum in the limiting cases c ≪ n and c ≫ n .At c ≪ n one can approximate (18) with V ( x ) = γ/ ( cx ) , the so-called Calogero-Sutherland model [8].Then from Eq. (4) one finds ω q = (cid:16) γc m (cid:17) / n ( πq − q / , (19) v q = (cid:16) γc m (cid:17) / n ( π − q ) , (20)and v is given by v q at q = 0. Substitution of Eqs. (19)and (20) into (14) gives Υ q = 0, and therefore B = 0.At c ≫ n the interactions fall off very rapidly withthe distance, V ( x ) = 4 γe − cx . In this case only theinteraction of the nearest neighbor particles in theWigner crystal needs to be taken into account (Todalattice). The spectrum takes the form ω q = 8 c (cid:16) γm (cid:17) / e − c/n sin q , (21) v q = 4 c (cid:16) γm (cid:17) / n e − c/n cos q . (22)As expected, substitution of Eqs. (21) and (22) into(14) gives Υ q = 0. Finally, we have checked numericallythat the expression (15) vanishes for potential (18) forany c .To summarize, we have obtained the equilibrationrate τ − of one-dimensional system of particles withstrong long-rage repulsion. At low temperatures therate is exponentially suppressed with the activation energy given by the Debye frequency ω π of the Wignercrystal. The prefactor can be expressed in terms ofthe phonon spectrum using Eqs. (9), (10), and (15).In the case of Coulomb repulsion the result is given byEqs. (16) and (17). Finally, we have checked that theequilibration rate vanishes for the integrable model ofparticles with interactions in the form (18).The authors are grateful to A. Levchenko for help-ful comments. This work was supported by the U.S.Department of Energy under Contracts No. DE-AC02-06CH11357 and DE-FG02-07ER46452. References [1] F. D. M. Haldane, J. Phys. C 14 (1981) 2585.[2] A. M. Lunde, K. Flensberg, and L. I. Glazman, Phys. Rev.B 75 (2007) 245418.[3] T. Micklitz, J. Rech, K. A. Matveev, Phys. Rev. B 81(2010) 115313.[4] K. A. Matveev, A. V. Andreev, and M. Pustilnik, Phys.Rev. Lett. 105, 046401 (2010).[5] H. J. Schulz, Phys. Rev. Lett. 71 (1993) 1864.[6] K. A. Matveev and A. V. Andreev, arXiv:1107.4116.[7] A. H. Castro Neto and M. P. A. Fisher, Phys. Rev. B ,9713 (1996).[8] B. Sutherland, Beautiful Models, (World Scientific,Singapore, 2004).(World Scientific,Singapore, 2004).