Finite-temperature Wigner solid and other phases of ripplonic polarons on a helium film
aa r X i v : . [ c ond - m a t . o t h e r] J un Finite-temperature Wigner solid and other phases of ripplonic polaronson a helium film
S. N. Klimin , J. Tempere , , V. R. Misko , , and M. Wouters TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium Lyman Laboratory of Physics, Harvard University, USA Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium (Dated: September 16, 2018)Electrons on liquid helium can form different phases depending on density, and temperature. Alsothe electron-ripplon coupling strength influences the phase diagram, through the formation of so-called “ripplonic polarons”, that change how electrons are localized, and that shifts the transitionbetween the Wigner solid and the liquid phase. We use an all-coupling, finite-temperature variationalmethod to study the formation of a ripplopolaron Wigner solid on a liquid helium film for differentregimes of the electron-ripplon coupling strength. In addition to the three known phases of theripplopolaron system (electron Wigner solid, polaron Wigner solid, and electron fluid), we defineand identify a fourth distinct phase, the ripplopolaron liquid. We analyse the transitions betweenthese four phases and calculate the corresponding phase diagrams. This reveals a reentrant meltingof the electron solid as a function of temperature. The calculated regions of existence of the Wignersolid are in agreement with recent experimental data.
I. INTRODUCTION
The two-dimensional (2D) electron system formed onthe surface of liquid helium has been widely investigated,especially with regard to the formation and melting of aWigner crystal, or Wigner solid (WS) . In the WS phase,the electrons are self-trapped in a commensurate surfacedeformation of liquid He called the dimple lattice .The self-trapping effect of the surface electrons is similarto the formation of polaron states where electrons aredressed by self-induced lattice deformations, or virtualphonons .Being driven by a force parallel to the surface of liquid He, the WS moves as a whole keeping the hexagonal or-der. The electron motion on liquid helium is associatedwith surface excitations, or ripplons (see, e.g., ). Whentraveling faster than the ripplon phase velocity, as in thecase of the Cherenkov radiation, an electron radiates sur-face waves and the ripplons emitted by different electronsinterfere constructively if the wave number of the ripplonsequals the reciprocal lattice vector of the Wigner solid(the Bragg condition). This resonant Bragg-Cherenkovemission of ripplons gives rise to the limitation of theelectron velocity . Another intriguing nonlinear phe-nomenon is a sharp rise in mobility at a much highervelocity which was attributed to the decoupling ofthe WS from the dimple lattice. This decoupling can beexplained within a hydrodynamic model assuming thatthe dimple lattice deepens due to the Bragg-Cherenkovscattering which bridges the two above-mentioned phe-nomena.One of the most actively developing research directionsin the field, which became possible due to the recent ad-vances in the microfabrication technology, is the studyof the Wigner solid in confined geometries using devicessuch as microchannel arrays , single-electron traps ,field-effect transistors (FET) and charge-coupled de-vices . One of the advances in this direction was ac- cessing the “quantum wire” regime, when the effectivewidth of a conductive channel is less than the thermalwavelength of the electrons. This was achieved in therecent experiments where the transport propertiesof electrons were measured in a microchannel with theconfinement potential controlled on the scale of the inter-electron separation ( ≈ . µ m). Note that in these exper-imental studies, electrons are confined in channels withconstrictions. In our theoretical model developed for aninfinite system we only used typical experimental valuesof the electron density, n , and the thickness of the he-lium film (helium depth), h .The motion of electrons, or in general charged parti-cles, in quasi-one-dimensional (Q1D) channels has beenanalyzed, using numerical simulations, in early works.For example, the structural, dynamic properties andmelting of a Q1D system of charged particles, interact-ing through a screened Coulomb potential were stud-ied using Monte Carlo simulations. However, theexperiments revealed new interesting behavior, suchas oscillations in the single-electron conductance in shortand long constrictions, which required understanding andtherefore stimulated new theoretical and numerical stud-ies. Thus, step-like electric conduction of a classical 2Delectron system in a microchannel with a narrow constric-tion has been analyzed . Related numerical studies ,using molecular-dynamics simulations of Langevin equa-tions of motion of interacting electrons on surface ofliquid He, revealed a significant difference in the elec-tron dynamics for long and short constrictions. Thepronounced current oscillations found for a short con-striction were shown to be suppressed for longer con-strictions , in agreement with the experimental ob-servations. Also, an asymmetric FET-like structure hasbeen proposed that allows an easy control of rela-tively large electron flows and can be used for rectifica-tion of an ac-driven electron flow. Furthermore, the au-thors addressed the important issue of the so-called“non-sequential ordering of transitions (non-SOT)” char-acterized by inversions in the subsequent number of elec-tron rows in a Q1D channel, e.g., “1-2-4-3” (see, e.g., ).In particular, they found the sequence of transitions “1-2-4-3-6-4-5” with two striking inversions “2-4-3” and “3-6-4” and demonstrated that some amount of fluctuations(i.e., in the number of particles) restores the usual se-quential order, i.e., “1-2-3-4-5”. The role of the potentialprofile and the form of the interparticle interaction (e.g.,the screening length for electrons) in the appearance ofthe non-SOT has been recently further analyzed in de-tail .Despite the above technological, experimental and the-oretical advances in the study of the Wigner solid, someof the fundamental properties of this system still re-main not well-understood. Moreover, recent experimen-tal studies revealed a number of related issues to be ad-dressed. For example, the experiment revealed a verygradual increase in the electron effective mass as the tem-perature drops below the WS transition temperature,while the theory predicted a full formation of dimplesat the transition temperature and a very weak temper-ature dependence. Also, the mechanisms of the decou-pling of the Wigner solid from the dimple lattice towardsan electron Wigner solid are not yet understood in de-tail. Also, despite great efforts to observe a bound single-polaron state experimentally, this is still an open prob-lem. The polaronic Wigner crystal is a well-establishedphenomenon, but the situation for single polarons (whichhas a particular interest in view of the polaron liquid dis-cussed below) is not yet clear. The work on the observa-tion of a single polaron was strongly debated.Here, we analyze in detail various phases of theelectron-ripplon system, i.e., when the WS is coupled tothe dimple lattice, when the WS still exists but is decou-pled from the dimple lattice, and when the WS melts, de-pending on such parameters of the system as the strengthof the electron-ripplon interaction, temperature and theelectron concentration. Also a polaron liquid phase ispredicted in the present work at sufficiently high tem-peratures combined with high coupling strengths. Tothe best of our knowledge, this phase was not yet consid-ered in the literature. The treatment is performed withinthe variational scheme similar to that used in Ref. formultielectron bubbles in liquid helium. II. ELECTRON-RIPPLON INTERACTION
The Hamiltonian of a single electron on a flat heliumsurface is given byˆ H = ˆp m + X q ω q (cid:18) ˆ a + q ˆ a q + 12 (cid:19) + 1 √ S X q V q (cid:0) ˆ a q + ˆ a + − q (cid:1) e i q . r , (1) where ˆp is the electron momentum operator, m is theelectron mass S is the surface area, ω q is given by , ω q = s(cid:18) g ′ q + σρ q (cid:19) tanh ( qh ) , (2)where σ ≈ . × − J m − is the surface tension of he-lium, ρ = 145 kg m − is the mass density of helium, and g ′ = g (cid:0) c/ρgh (cid:1) is the acceleration of the liquid dueto its van der Waals coupling to the substrate (where g is the acceleration due to gravity and c is the van derWaals coupling of the helium to the substrate). In theHamiltonian (1), we restrict ourselves to 2D position andmomentum operators, assuming that the part of the wavefunction of the electrons relating to the direction perpen-dicular to the surface can be factored out exactly. Thesecond-quantization operators ˆ a + q , ˆ a q create/annihilate aripplon with planar wave number q . The electron-ripploncoupling amplitude is given by V q = s ~ q ρω q tanh ( qh ) eE, (3)where E is the electric field perpendicular to the surface(the so-called ‘pressing field’), e is the electron charge,and h is the thickness of the helium film. The pressingfield pushes the electrons with a force eE towards thehelium surface. A 1 eV barrier prevents electrons frompenetrating the helium surface. The total electric field isa sum of an external (manually applied) field E ext and theelectric field induced by the image charge in the substratewith the dielectric constant ε : E = e h ε − ε + 1 + E ext . (4)It should be noted that the areas of parameters fordifferent phases of a ripplonic polaron system must bedetermined with a special care on the stability of thesystem itself. For example, at very high densities, therecan exist an instability of the polaron when the pressingfield becomes too large . Also there is a maximum sur-face density of electrons when a uniform distribution isstable on a flat surface .The self-induced trapping potential of the electron onthe helium surface is manifested by the appearance ofa dimple in the helium surface underneath the electron,much like the deformation of a rubber sheet when a per-son is pulled down on it by a gravitational force. Theresulting quasiparticle consists of the electron togetherwith its dimple and can be called a ripplonic polaron orripplopolaron .The Hamiltonian (1) for the ripplopolarons is very sim-ilar to the Fr¨ohlich Hamiltonian describing polarons ;the role of the phonons is now played by the rip-plons. Methods suitable for the study of single polaronshave been used to analyze the single polaron on a flatsurface . The path integral treatment for a Wignersolid of polarons has been developed in Refs. . InRef. , we adapt their method so that it becomes suit-able for the treatment of a lattice of ripplopolarons inmultielectron bubbles. III. HAMILTONIAN FOR A RIPPLOPOLARONIN A WIGNER SOLID
In their treatment of the electron Wigner solid embed-ded in a polarizable medium such as a semiconductorsor an ionic solid, Fratini and Qu´emerais described theeffect of the electrons on a particular electron througha mean-field lattice potential. The (classical) lattice po-tential V lat is obtained by approximating all the elec-trons acting on one particular electron by a homogenouscharge density in which a hole is punched out; this holeis centered in the lattice point of the particular electronunder investigation and has a radius given by the latticedistance d . Thus, in their approach, anisotropy effects,e. g., related to the lattice orientation, are neglected. Asecond assumption implicit in this approach is that theeffects of exchange are neglected. This can be justifiedby noting that for the electrons to form a Wigner solid itis required that their wave function is localized to withina fraction of the lattice parameter as follows from theLindemann criterion .Within this particular mean-field approximation, thelattice potential can be calculated from classical electro-statics and we find that for a 2D electron gas it can beexpressed in terms of the elliptic functions of first andsecond kind, E ( x ) and K ( x ), V lat ( r ) = − e πd ( | d − r | E " − rd ( d − r ) + ( d + r ) sgn ( d − r ) K " − rd ( d − r ) . (5)Here, r is the position vector measured from the latticeposition. We can expand this potential around the originto find the small-amplitude oscillation frequency of theelectron lattice:lim r ≪ d V lat ( r ) = − e d + 12 mω lat r + O (cid:0) r (cid:1) , (6)with the confinement frequency ω lat = r e md . (7)The ‘phonon’ frequency ω lat of the electron Wignersolid corresponds closely to the longitudinal plasmon fre-quency that can be derived using an entirely differentapproach based on a more rigorous study of the modesof oscillations of both the helium surface and the chargedistribution on the surface. From this, and from thesuccessful application of this mean-field approach to po-laron crystals in solids, we conclude that the approach based on that of Fratini and Qu´emerais describes the in-fluence of the other electrons well in the framework ofsmall amplitude oscillations of the electrons around theirlattice point. The phenomenological Lindemann melt-ing criterion suggests that the Wigner solid will meltwhen the electrons are on average displaced more than acertain value δ < H = ˆ p m + V lat ( ˆr ) + X q ~ ω q ˆ a + q ˆ a q + X q V q e − i q . r (cid:0) ˆ a q + ˆ a + − q (cid:1) . (8) IV. THE RIPPLOPOLARON WIGNER SOLIDAT FINITE TEMPERATURE
The simple but intuitive approach of the previous sec-tion describes the system in the limit of zero temperature.To study the ripplopolaron Wigner solid at finite temper-ature (and for any value of the electron-ripplon coupling),we use the variational path-integral approach . Thisvariational principle distinguishes itself from Rayleigh-Ritz variation in that it uses a trial action functional S trial instead of a trial wave function.The action functional of the system described byHamiltonian (8), becomes, after elimination of the rip-plon degrees of freedom, S = − ~ ~ β Z dτ n m r ( τ ) + V lat [ r ( τ )] o + X q | V q | ~ β Z dτ ~ β Z dσG ω ( q ) ( τ − σ ) e i q · [ r ( τ ) − r ( σ )] , (9)with G ν ( τ − σ ) = cosh[ ν ( | τ − σ | − ~ β/ β ~ ν/ . (10)In preparation of its customary use in the Jensen-Feynman inequality, the action functional (9) is writtenin imaginary time t = iτ with β = 1 / ( k B T ) where T isthe temperature. Following an approach analogous for alattice of polarons in an ionic crystal , and to that ofDevreese et al. for N polarons in a quantum dot , weintroduce a quadratic trial action of the form S trial = − ~ ~ β Z dτ (cid:20) m r ( τ ) + m Ω r ( τ ) (cid:21) − M w ~ ~ β Z dτ ~ β Z dσG w ( τ − σ ) r ( τ ) · r ( σ ) . (11)where M, w, and Ω are the variationally adjustable pa-rameters. This trial action corresponds to the Lagrangian L = m r + M R − κ r − K r − R ) , (12)from which the degrees of freedom associated with R have been integrated out. This Lagrangian can be inter-preted as describing an electron with mass m at position r , coupled through a spring with spring constant κ to itslattice site, and to which a fictitious mass M at position R has been attached with another spring, with springconstant K . The relation between the spring constantsin (12) and the variational parameters w, Ω is given by w = p K/m, (13)Ω = p ( κ + K ) /m. (14)Based on the trial action S trial , the Jensen-Feynmanvariational method allows one to obtain an upper boundfor the free energy F of the system (at temperature T )described by the action functional S by minimizing thefollowing function: F var = F − β h S − S trial i S trial , (15)with respect to the variational parameters of the trial ac-tion. In this expression, F is the free energy of the trialsystem characterized by the action S trial , β = 1 / ( k b T )is the inverse temperature, and the expectation value h S − S trial i S trial is to be taken with respect to the groundstate of this trial system.The evaluation of expression (15) is straightforwardthough lengthy. We find F var = 2 β ln (cid:20) (cid:18) β ~ Ω (cid:19)(cid:21) + 2 β ln (cid:20) (cid:18) β ~ Ω (cid:19)(cid:21) − β ln (cid:20) (cid:18) β ~ w (cid:19)(cid:21) − ~ X i =1 a i Ω i coth (cid:18) β ~ Ω i (cid:19) − √ πe √ D e − d / (2 D ) (cid:20) I (cid:18) d D (cid:19) + I (cid:18) d D (cid:19)(cid:21) − π ~ Z k c dq q | V q | Z ~ β/ dτ cosh[ ω q ( τ − ~ β/ β ~ ω q / × exp (cid:20) − q D − D τ ) (cid:21) . (16)In this expression, I and I are Bessel functions of imag-inary argument, k c = ( ρg ′ /σ ) / is the capillary wavenumber . The capillary wave number serves as a cutoffin the integral over q for the polaron free energy. Thefunction D τ is given by: D τ = ~ m X j =1 a j Ω j cosh[ ~ Ω j ( τ − β/ ~ Ω j β/ , (17) with the coefficients a = s Ω − w Ω − Ω ; a = s w − Ω Ω − Ω . (18)The frequencies Ω i ( i = 1 ,
2) are the eigenfrequencies ofthe trial system, and w is the third (auxiliary) frequencywhich is also the variational parameter. The parameter d is the inter-electron distance on the helium surface, whichis related to the concentration as: d = 2 (cid:0) √ n (cid:1) / . The two first lines in the expression (16) for the vari-ational free energy describe the free energy of the modelsystem and the averaged influence phase of the modelsystem. The third line in (16) is the averaged energy ofthe Coulomb interaction of the electron with the self-consistent field induced by other electrons. In otherwords, this is the averaged potential energy of the elec-tron in the Wigner solid. Finally, the last line is thepolaron contribution to the free energy.Optimal values of the variational parameters are de-termined by numerical minimization of the variationalfunctional F as given by expression (16). The result ofthe variational path-integral method allows us to intro-duce different measurable quantities, e. g., temperatureto examine the melting of the Wigner solid, and the effec-tive mass of a polaron. The latter one can be estimatedas ( m + M ) where M is the mass of the fictitious particle.A signature of the polaron phase can be a drastic changeof the mobility when varying the coupling strength, thatcan allow one to distinguish between polaron and elec-tron WS experimentally. V. RESULTS AND DISCUSSION
In this section, we calculate different parameters for aripplopolaron Wigner solid on the liquid helium surface.This is performed using the variational approach for thepolaron free energy as described above. Optimal valuesof the variational parameters are determined by the nu-merical minimization of the variational functional F var given by expression (16). We can consider, as a start-ing point for the treatment, the experimental conditionsas obtained from Ref. , where the thickness of the he-lium film was h ≈ µ m to h ≈ . µ m, the temperaturewas T ≈ T ≈ . n ≈ . × cm − and n ≈ . × cm − . We however vary temperaturesand concentrations in a rather wide range around thosevalues.The electron-ripplon coupling is measured through thedimensionless coupling constant α determined as : α = ( eE ) πσ m ~ k c , (19)where E is the electric field applied perpendicular to thesurface. It includes both the image field induced by a po-lar substrate and an external field which can be controlledartificially. Fig. 1 shows the correspondence between E and α for the aforesaid set of material parameters. Notethat for h ≈ µ m, the contribution to α from the imagefield is negligibly small: even with a metallic substrate, α . − , so that the electron-ripplon coupling can becompletely controlled by an external field. E ( V / c m ) FIG. 1: Dependence of the electric field E measured in V/cmon the dimensionless electron-ripplon coupling constant α . For the numeric calculation, the dimensionless unitsare used with ~ = 1, the electron mass m = 1 and theunit for the energy is ~ k c m = 1, where k c ≈ × cm − isthe capillary wave number from Ref. . Also the effectiveacceleration g ′ = 10 g is taken from Ref. . Note that,despite a substantial dependence of g ′ on the helium filmthickness, this dependence can only slightly change thephase diagrams calculated below, because at given α , itinfluences the results only through ripplon frequencies(which are very small for any reasonable g ′ ).The polaronic aspects in the formation of the elec-tron Wigner solid on the liquid helium surface are al-ready thoroughly studied both experimentally and theo-retically, see, e. g., the recent review and referencestherein. However, some questions remain unexplored.The transition between two types of the Wigner solid(the electron and polaron Wigner solid) at different tem-peratures is of a particular interest, because this problemrequires an arbitrary-coupling finite-temperature polarontheory. We successfully applied this polaron theory to in-vestigate polaron Wigner solids in multielectron bubbles.Here, the same approach is used for the calculation of thephase diagrams on the flat helium surface.The phenomenological Lindemann criterion is fre-quently used in the literature for the determination ofa melting point in a Wigner solid. This criterion statesin general that a crystal lattice of objects (be it atoms,molecules, electrons, or polarons) will melt when the av-erage motion of the objects p h r i around their lattice site is larger than a critical fraction δ of the lattice pa-rameter d . It would be very hard to calculate from firstprinciples the exact value of the critical fraction δ , butfor the particular case of electrons on a helium surface, wecan make use of an experimental determination. Grimesand Adams found that the Wigner solid melts whenΓ = 137 ±
15, where Γ is the ratio of potential energy tothe kinetic energy per electron. In their experiment, theelectron density varied from 10 cm − to 3 × cm − while the melting temperature T c varied from 0.23 K to0.66 K. As estimated in Ref. using the experimentaldata by Grimes and Adams , the critical fraction equals δ ≈ .
13. Recently, a modified Lindemann criterion hasbeen derived in Ref. , which is based on the calculationof a two-site correlation function for the Wigner solid,describing the correlation of displacements for the near-est neighbors. When combined with the Monte Carlocalculation , this leads to the modified value δ ≈ . we used twoparallel Lindemann criteria following to the scheme de-veloped in Ref. . According to this scheme, areas ofstasbility for different phases of a ripplopolaron systemare determined, at least qualitatively, by the parameters δ c ≡ p h R c i /d and δ ρ ≡ p h ρ i /d , where R c and ρ are,respectively, the center-of-mass and the relative coordi-nate for the model polaron system. The averaged squaredradii are explicitly determined using the variational pa-rameters for the ripplopolaron system , (cid:10) R c (cid:11) = w (Ω − Ω ) [Ω Ω − w (Ω + Ω )] × " Ω (cid:0) Ω − w (cid:1) Ω coth (cid:18) β Ω (cid:19) , + Ω (cid:0) w − Ω (cid:1) Ω coth (cid:18) β Ω (cid:19) , (20) (cid:10) ρ (cid:11) = 1Ω − Ω (cid:20) Ω Ω − w coth (cid:18) β Ω (cid:19) + Ω w − Ω coth (cid:18) β Ω (cid:19)(cid:21) . (21)In principle, four combinations are possible:(1) The case when both δ c < δ and δ ρ < δ corre-sponds to the polaron Wigner solid. In this case, elec-trons are strongly localized in the Wigner solid togetherwith the dimple lattice on the helium surface. Thisregime is called in Ref. the polaron anchoring of theWigner crystal.(2) When δ c < δ and δ ρ > δ , the electron Wignersolid exists without anchoring to dimples (electrons leavedimples but the Wigner solid still exists). However,the parameters of this system are still influenced by theelectron-phonon interaction through scattering of rip-plons on the electrons. Therefore, this regime can beconsidered as the electron Wigner solid. In other words,this is the Wigner solid of weak-coupling polarons.(3) When δ c > δ and δ ρ < δ , the ripplonic polaronsare chaotically moving but electrons are in dimples. Thiscase can be interpreted as a polaron liquid. It can berealized when both the coupling strength and the tem-perature are sufficiently high.(4) When both δ c > δ and δ ρ > δ , this is the casewhen the Wigner solid melts to the electron liquid (withthe polaron effect). c T = 0.01K R e l a ti v e s qu a r e d r a d ii a b T = 0.1K R e l a ti v e s qu a r e d r a d ii a T = 1K h = 1 m m n = 2.58×10 cm - (< R >) / d = d c (< r >) / d = d r Melting criterion d R e l a ti v e s qu a r e d r a d ii FIG. 2: Parameters δ c (solid curves) and δ ρ (dashed curves)for a ripplopolaron Wigner lattice as a function of the couplingconstant α for the electron system on the surface of the heliumfilm of the width h = 1 µ m. The concentration of electronsis n = 2 . × cm − , the temperatures are T = 1 K ( a ), T = 0 . b ), and T = 0 .
01 K ( c ). The dot-dashed line showsthe critical value δ = 0 . In Fig. 2 ( a ), we plot parameters δ c (the solid curve)and δ ρ (the dashed curve) for a ripplopolaron Wignersolid as a function of the coupling constant α for theelectron system on the surface of the helium film. The dot-dashed curve shows the critical value for the modifiedLindemann melting criterion δ = 0 . h = 1 µ m. The concentration of electrons is n = 2 . × cm − , the temperature is T = 1 K. Underthese conditions, the relative averaged squared oscillationamplitude δ ρ decreases when strengthening the electron-ripplon coupling, passing the critical value δ c at α = α c ≈ .
53. The center-of-mass relative averaged squaredoscillation amplitude, δ c , varies extremely slightly, beingsmaller than the critical value at all coupling strengths.In terms of the aforesaid four regimes, this means that theWigner solid exists at these conditions for all α , changingat α = α c from the electron Wigner solid at α < α c tothe polaron Wigner solid at α > α c .In the experiments , where there is no additionalexternal field to enhance α , electrons are attracted to thesurface of liquid helium by a rather small image charge.The coupling constant in this regime is small. For exam-ple, in the conditions of Ref. , α ∼ . × − . Accordingto Fig. 2 ( a ), the ripplopolaron Wigner lattice for small α is stable, thus the result in Fig. 2 ( a ) is in line withthese experiments.The graphs 2 ( a ) and ( b ) show the analogous depen-dence of the parameters ( δ c , δ ρ ) for lower temperatures: T = 0 . T = 0 .
01 K, respectively. For a suffi-ciently low temperature T = 0 .
01 K, we can see a sharptransition between two regimes at certain α ≡ α c , whichcan be qualitatively attributed to a weak and strong-coupling polaron regimes. It was found in Ref. thatat T = 0, there is a crossover between weak-couplingand strong-coupling polaron regimes when varying α .This transition is not discontinuous at non-zero temper-atures, although at low temperatures it can be sharp. At T = 0 .
01 K, as seen from Fig. 2 ( c ), this transition isfollowed by a change of the regime for the Wigner solid:for smaller α / α c , δ ρ > δ and δ c < δ , so that theWigner solid is formed by weak-coupling polarons, andfor α ' α c , we see that both δ c and δ ρ are smaller than δ , that corresponds to the polaron Wigner solid. Athigher temperatures, the crossover between the regimesof electron and polaron Wigner solids is rather smooth.We can see a manifestation of this crossover at T = 0 . δ c as a functionof the temperature. We can also conclude from the com-parison of the behavior of the parameters δ ρ and δ c atdifferent temperatures that low temperatures are favor-able for the Wigner solid formation and for its polaronanchoring.Note that we use the same critical value δ for themelting of the Wigner solid and for the polaron dissoci-ation. Only for the former one, there are experimental and numerical estimates of the Lindemann criterion, δ , and even these do not agree. However, from Fig. 2 itis clear that a different choice of δ (keeping its range ofmagnitude) will not change the results qualitatively.Figure 3 shows the parameters δ c and δ ρ for a ripplopo-laron Wigner solid at a given coupling strength α = 1 as afunction of the temperature for the ripplopolaron systemon the surface of the helium film for different electronconcentrations. The other parameters are the same asthe previous figures. We can see from this figure thatboth δ c and δ ρ increase monotonically when the temper-ature rises. The parameters δ ρ and δ c , pass the criticalLindemann value δ at different temperatures, depend-ing on the electron concentration, so that the transitionpoints between different configurations of the ripplopo-laron system depends on the concentration n . For lower n , the transitions between different configurations oc-curs at lower temperatures. = 1.0(cid:215)10 cm , n = 2.58(cid:215)10 cm , n = 5.0(cid:215)10 cm Melting criterion R e l a ti v e s qu a r e d r a d ii T (K) = 1 c , n = 1.0(cid:215)10 cm c , n = 2.58(cid:215)10 cm c , n = 5.0(cid:215)10 cm FIG. 3: Parameters δ c and δ ρ for a ripplopolaron Wignerlattice as a function of the temperature for the electron systemon the surface of the helium film with α = 1. the otherparameters are the same as in Fig. 2. In order to obtain a more detailed picture of differ-ent regimes for ripplopolaron Wigner solid, we calcu-late phase diagrams where different regimes for the rip-plopolaron system are indicated. Figure 4 contains thephase diagram for the ripplopolaron system on the he-lium film surface in the variables (
T, α ) (using the log-arithmic scale) calculated for two concentration of elec-trons n = 2 . × cm − and n = 1 . × cm − , andfor the thickness of the liquid helium film h = 1 µ m. Inthis figure, all four regimes described above can be seen.We can conclude from Fig. 4 that the electron Wignersolid as obtained in the present calculation is expectedto be stable at the experimental conditions , althoughis rather close to the melting conditions.It should be noted that the boundary between theregimes with δ c < δ and δ c > δ corresponds to meltingof a Wigner crystal, i. e., this is a true phase transi-tion. On the contrary, the other boundary – betweenthe regimes with δ ρ < δ and δ ρ > δ indicates a transi-tion between strong-coupling and weak-coupling polaronstates. According to the Gerlach-L¨owen theorem , thereis no phase transition between those regimes for a po-laron. At sufficiently high temperatures T & . , the transition between strong-coupling andweak-coupling polaron regimes is a crossover rather than a phase transition. Correspondingly, the transition be-tween the polaron and electron Wigner solids is also acrossover (indicated by grey curves at the figures).We can see from Fig. 4 that at sufficiently low densi-ties, the melting temperature for the transition between apolaron or electron Wigner solid to a polaron or electronliquid only weakly depends on the electron-ripplon cou-pling constant α , and it becomes more sensitive to α athigher densities. This weak coupling depencence of themelting temperature can be explained by the fact thatan overlap of polaron dimples at low densities is rela-tively small, increasing when rising density. The meltingtemperature is a non-monotonic function of α , which isone of manifestations of the reentrant melting discussedbelow.The other boundary at the phase diagrams in Fig. 4,which corresponds to the polaron dissociation, behavesas follows. When the coupling strength gradually in-creases at a sufficiently low temperature, the regime ofthe electron Wigner solid turns at a certain α to the po-laron Wigner solid. This critical α rises when increasingtemperature. At higher temperatures, when increasing α , the electron liquid can change to the polaron liquidwithout forming a polaron Wigner solid. It is often as-sumed that the formation of polaron dimples always leadsto their Wigner crystallization. However, according tothe present variational calculation, there exists a regimewhere the polarons are not yet dissociated but their massis not large enough to form a Wigner crystal. As seenfrom Fig. 4, it requires a combination of large α andhigh temperatures. This transition was predicted for anelectron-phonon system in a 3D polar crystal . For aripplonic polaron system, to the best of our knowledge,this regime was not yet discussed in the literature.In Fig. 5, we show the phase diagram for the ripplopo-laron system on the helium surface in the variables ( n , α )for two temperatures T = 0 . T = 1 K, keepingother parameters the same as described above. At thelower temperature, we can see in Fig. 5 ( a ) three regimesfor the ripplonic polaron system: the polaron Wignersolid, the electron Wigner solid and the electron liquid.At low temperatures, these three regimes consequentlyfollow each other when increasing the electron concen-tration. The critical concentration for the transitionbetween the polaron and electron Wigner solids mono-tonically increases with an increasing coupling strength.The other critical concentration, which indicates melt-ing of the electron Wigner solid into the electron liquidexhibits a non-monotonic behavior as a function of α .At small α , electron-ripplon scattering favors the melt-ing of the Wigner crystal, which can be explained byelectron-ripplon scattering. On the contrary, for larger α the electron-ripplon interaction favors the Wigner crys-tallization because of increase in effective mass of thepolarons.At the higher temperature T = 1 K, close to the ex-perimental conditions of Ref. , all four phases of the rip-plopolaron system can be observed in the range of densi- Polaronliquid>< d dd d c 00 r PolaronWS<< d dd d c 00 r ElectronWS<> d dd d c 00 r Electronliquid>> d dd d c 00 r n = 2.58×10 cm - PolaronliquidPolaronWS ElectronWS Electronliquid n = 1.0×10 cm
10 2 - ab FIG. 4: Phase diagram for the ripplopolaron system on thehelium surface in the variables (
T, α ) at a concentration ofelectrons n = 2 . × cm − ( a ), n = 1 . × cm − ( b ),for the helium film width h = 1 µ m. ties 3 × cm − / n / × cm − , as seen from Fig.5 ( b ). When increasing the coupling strength, the elec-tron liquid can turn into a polaron liquid, and the elec-tron Wigner solid can transform to the polaron Wignersolid, as expected. Remarkably, at the relatively hightemperature T = 1 K, both polaron and electron liquidscrystallize, respectively, to polaron and electron Wignercrystals when the electron concentration increases , con-trary to the low-temperature case. This change of se-quence of phases between ripplopolaron systems at lowerand higher temperatures finds a transparent physical ex-planation through the interplay of the following factors.On one hand, at high temperatures the formation of a PolaronWS ElectronWS Electronliquid T = 0.1KPolaronliquid PolaronWSElectronWSElectronliquid T = 1K ab FIG. 5: Phase diagram for the ripplopolaron system on thehelium surface in the variables ( n , α ) for the temperature T = 0 . a ) and T = 1 K ( b ), with the helium film width h = 1 µ m. polaron dimple can be favored by the strengthening ofthe confinement potential, because the thermal fluctua-tions of the electron motion become gradually more re-stricted by the neighboring electrons when decreasing theinter-electron distance. On the other hand, at low tem-peratures, when thermal fluctuations are less important,melting of the electron Wigner solid can be favored byzero-temperature quantum fluctuations of the electronmotion: this is the case of quantum melting . We do seequantum melting at nonzero temperature, and it is ex-pected to persist down to T = 0. This explains the dif-ferent sequence of phases between low-temperature andhigh-temperature regimes for the Wigner solid.At very high densities, the Fermi energy of electronscan be comparable with their averaged kinetic energyand, consequently, quantum melting of the Wigner crys-tal can be strongly influenced by the Fermi statistics.Using the material parameters described above, thisrange of densities is estimated as n & cm − . InRef. , quantum melting of an electron WS to a degen-erate Fermi gas was experimentally detected at n ∼ cm − , confirming our estimations. We do not con-sider here the electron-ripplon system at very high densi-ties, when quantum melting occurs to a degenerate Fermigas. This regime will be a subject of the further study.Finally, Fig. 6 shows the phase diagram for the rip-plopolaron system on the helium surface in the variables( n , T ) plotted using two values of the electron-ripploncoupling constant α = 0 . α = 0 .
01. In analogywith the phase diagrams plotted in Fig. 5, the sequencesof different phases when varying the electron concentra-tion and temperature can be described and physicallyexplained in the following way. At small concentrationsand low temperatures, the system naturally turns intoa polaron Wigner crystal. When increasing the temper-ature while keeping the concentration constant, the po-laron Wigner crystal can either melt to a polaron liquidat low densities, or shed the dimple and change to anelectron Wigner crystal at higher densities. In the for-mer case, the breakdown of the polaron Wigner crystaloccurs through the melting of the lattice, but polarondimples survive. In the latter case, the polaron Wignersolid is changed to the electron Wigner solid through thepolaron dissociation. When temperature rises further,both the electron Wigner crystal and the polaron liq-uid can change to the electron liquid but in a differentway: the electron Wigner crystal melts, while the polaronliquid dissociates. When increasing the electron concen-tration at fixed temperature, also the electron Wignercrystal can melt.Remarkably, the electron liquid phase appears not onlyfor high temperatures but also as an “island” for low tem-peratures and high electron concentrations (see Figs. 5( a )and 6). This means that the system displays a reentrantmelting transition from the electron liquid phase to theelectron solid phase at some fixed high n when increas-ing the temperature or decreasing the coupling strength.In other words, the system displays solidification by heat-ing (for high enough electron concentrations). This sortof transition, known as “freezing by heating” transition,has been predicted for mesoscopic systems and re-cently demonstrated for colloids driven by a non-uniformforce . This counter-intuitive behavior does not vio-late principles of thermodynamics and has been observedboth in non-equilibrium and equilibrium systems, see,e. g., .In Refs. , the mechanism of the inverse meltingwas explained by the fluctuation-driven increase of theeffective size of the particles (i.e., the area effectively oc-cupied by the particle during its fluctuation-driven ran-dom motion) in the molten state such that they form a PolaronliquidPolaronWS ElectronWSElectronliquid Electronliquid a = 0.01PolaronliquidPolaronWS ElectronWSElectronliquidElectronliquid a = 0.1 ba FIG. 6: Phase diagram for the ripplopolaron system on the he-lium surface in the variables ( n , T ) for the coupling strength α = 0 . a ) and α = 0 .
01 ( b ) , with the helium film width h = 1 µ m. solid state with increasing temperature. For ripplonicpolarons, the revealed sequence of the reentrant elec-tron phases when increasing temperature at high elec-tron concentrations can be explained in other way: bythe fact that the melting phase transition is differentlydriven by quantum and thermal fluctuations. In (20)and (21), thermal fluctuations contribute to the tem-perature dependence of (cid:10) R c (cid:11) and (cid:10) ρ (cid:11) through the dis-tribution functions coth ( β ~ Ω j / quantum fluctuations to the temperature dependenceof the averaged squared radii occurs through the po-laron parameters { Ω j , w } (which also depend on temper-ature). The electron-ripplon interaction can become ef-0fectively stronger with rising temperature in some rangeof temperatures, and hence the ripplon-induced poten-tial for an electron becomes deeper and narrower in thatrange. Quantum fluctuations may then favor to a non-monotonic dependence of (cid:10) R c (cid:11) as a function of temper-ature. On the contrary, thermal fluctuations always con-tribute to increase (cid:10) R c (cid:11) . Thus the transition tempera-ture can result from an interplay of quantum and thermalfluctuations.When increasing the temperature, the area of the reen-trant melting transition shifts to higher concentrations,so that is not seen in Fig. 5( b ) but appears in Fig.5( a ). Comparing the phase diagrams for two couplingstrengths, we can note that the region of stability forthe electron Wigner crystal substantially expands withdecreasing α .As mentioned above, electron-ripplon coupling in theexperimental conditions of Refs. (where a stableWigner crystal has been detected) is rather weak, α ∼ . × . Hence the obtained phase diagrams are inagreement with these experiments. We can also suggestthat the Wigner solid in the experiments can be clas-sified as an electron Wigner solid. For the experimentswhere the melting point of the electron Wigner latticeat the helium surface was determined using measure-ments of the mobility and the microwave response ,the thickness of the helium film was significantly smallerthan in the phase diagrams calculated in the presentwork. However these experiments are also related to thevery weak-coupling polaron regime, where the electronWS rather than the polaron WS can exist.It should be noted that the film thickness at high densi-ties can be strongly reduced . Consequently, at elec-tron densities of 10 cm − and higher the helium filmcan hardly have a thickness h = 1 µ m. Therefore someportions of the phase diagrams shown in Figs. 4 to 6 areexperimentally not accessible. However they represent atheoretical interest for a many-polaron problem. VI. CONCLUSIONS
In the present work, we have analyzed different phasesof a ripplonic polaron system on the surface of a liq-uid helium film, and the behavior of these phases whenvarying parameters of the system: the temperature, theelectron concentration, and the electron-ripplon couplingstrength. The electron-ripplon system is considered in awide range of electron densities leaving out very highdensities where the Fermi statistics becomes important.The treatment has been performed within the arbitrary-coupling and finite-temperature variational path-integralformalism based on the Jensen-Feynman variational prin-ciple for the free energy.We demonstrated that, by varying the electron-ripploncoupling strength α and other parameters such as theelectron concentration n on the helium surface and tem-perature T , in the vicinity of typical experimental val- ues, the system displays a rich phase behavior. We havefound that the experimental conditions corresponding toRef. are favorable for the electron Wigner solid decou-pled from the dimple lattice rather than for other phases.This conclusion is in agreement with the observation ofan electron Wigner solid in that work.For a set of typical experimental parameters h =1 µ m and n = 2 . × cm − , we revealed four differ-ent phases: (i) the electron solid (lattice), at low tem-peratures and weak couplings; (ii) the polaron Wignersolid, at low temperatures and strong couplings; (iii)the electron liquid, at high temperatures and weak cou-plings; and (iv) the polaron liquid phase, when the po-laron Wigner solid melts but the electrons do not decou-ple from the dimples, due to the strong electron-ripploncoupling. Remarkably, it should be possible to observeall the predicted phases at typical experimental temper-atures close to T = 1 K , for varying electron concen-tration and the electron-ripplon coupling strength. Forlower temperatures, some of the phases disappear, likethe polaron liquid phase.Thus in addition to three phases of an electron-ripplonsystem which were studied in the literature, the phaseof a ripplopolaron liquid is possible at a combination ofsufficiently low electron densities, strong electron-ripploncouplings and high temperatures T ∼ n andtemperature T . For weak electron-ripplon couplings α ,the electron liquid phase dominates over a broad rangeof n and T , although the other above phases can alsobe observed, like the polaron solid and liquid phases atlow electron concentrations. The polaron or electron liq-uid phase appears at rather high temperatures and cancrystallize with increasing n (turning to, respectively,the polaron or electron electron solid). For high electronconcentrations, the system exhibits quantum melting ac-companied by an unusual reentrant behavior, i.e., thetransition from a liquid to solid electron state with in-creasing temperature. This transition is known as the“freezing by heating” transition when fluctuations resultin a solidification of a molten state.The mean-field Wigner approximation used in thepresent work was successfully applied to polaron Wignerlattices in ionic crystals . The application of thismean-field approach to the WS on a liquid helium surfaceneeds however some care. There exists an enhancementof coupling constant in the Wigner solid phase due to theBragg scattering of ripplons . The Bragg scatteringof ripplons from the electron lattice gives rise to an en-hanced deformation of the dimple lattice . This effectcan be taken into account, e.g., by introducing an effec-tive Debye-Waller factor in the polaron action (9). Theeffect of the Bragg scattering can be especially important1at rather weak couplings. It can lead to a quantitativechange of the polaron energy and, consequently, to someshift of the boundaries on the phase diagrams. However,it can hardly change the physical picture of phases of therippopolaron system.Therefore, we demonstrated the existence of differentphases of electrons and polarons on surface of liquid he-lium, and we analyzed the regions of their stability andthe transitions between the revealed phases. Our findingsprovide a deeper understanding of the phase behavior ofthe Wigner matter and could be useful for the interpre-tation of the experimental observations such as the tem-perature behavior of the decoupling transition (the de-coupling of the Wigner solid from the dimples). Further-more, we expect that the revealed unusual phases (like polaron liquid) and phase transitions (like the reentrantelectron lattice melting) could stimulate further studies,including new experiments. VII. ACKNOWLEDGEMENTS
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