Converging toward a practical solution of the Holstein molecular crystal model

JOURNAL OF CHEMICAL PHYSICS VOLUME 109, NUMBER 16 22 OCTOBER 1998

Converging toward a practical solution of the Holstein molecularcrystal model

Aldo H. RomeroDepartment of Physics and Department of Chemistry and Biochemistry, University of California,San Diego, La Jolla, California 92093-0354

David W. BrownInstitute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402

Katja LindenbergDepartment of Chemistry and Biochemistry, University of California, San Diego, La Jolla,California 92093-0340

~Received 5 June 1998; accepted 21 July 1998!

We present selected results for the Holstein molecular crystal model in one space dimension asdetermined by the Global–Local variational method, including complete polaron energy bands,ground state energies, and effective masses. We juxtapose our results with specific comparableresults of numerous other methodologies of current interest, including quantum Monte Carlo, clusterdiagonalization, dynamical mean field theory, density matrix renormalization group, semiclassicalanalysis, weak-coupling perturbation theory, and strong-coupling perturbation theory. Taken as awhole, these methodologies are mutually confirming and provide a comprehensive andquantitatively accurate description of polaron properties in essentially any regime. In particular, thiscomparison confirms the Global–Local variational method as being highly accurate over a widerange of the polaron parameter space, from the nonadiabatic limit to the extremes of highadiabaticity, from weak coupling through intermediate coupling to strong coupling. ©1998American Institute of Physics.@S0021-9606~98!51140-3#

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I. INTRODUCTION

Great progress has been made in the last few yearsward achieving the practical solution of the Holstein moleclar crystal model,1,2 a practical solution being one whicyields a sufficiently complete spectrum of information tosufficiently high accuracy to satisfy most purposes. A nuber of different methodologies, each important for the pticular strength it lends to the problem, are proving increingly mutually consistent if not always straightforwardconvergent in their conclusions. In this paper, we compasuite of our own recent results3–7 obtained using the Global–Local variational method with comparable results of othcontemporary approaches, including quantum Monte C~QMC!,8–14 cluster diagonalization,15–27 dynamical meanfield theory ~DMFT!,28–33 density matrix renormalizationgroup ~DMRG!,34–39 as well as both weak-coupling pertubation theory~WCPT!38,40 and strong-coupling perturbatiotheory~SCPT!.21,41–45Every such comparison we have beable to implement has shown the Global–Local results toamong the best currently available, and where a better reobtains by another method, the quantitative discrepancyvolved tends to be satisfyingly small. Thus, although it is nthe case that our methods are necessarily the most accpossible in every instance, our methods are valid overlargeenougha region of the polaron parameter space, andaccu-rate enoughover their region of validity to be sufficient towealth of practical purposes.

In several recent works@Paper I~Ref. 46!, Paper 2~Ref.47!, and Paper III~Ref. 4!# we have presented a sequence

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increasingly refined variational approaches to the problemdetermining the lowest polaron energy band and the assated energy–momentum eigenfunctions for the Holstein mlecular crystal model in one space dimension. The GlobLocal method generalizes those of Toyozawa47–53 andMerrifield46,54 by including local electron–phonon correlation channels under-represented in the former and gloelectron–phonon correlation channels under-representethe latter, both of which are of particular significance in tlocal structure of the polaron. While in the present work wlean heavily on the results of Paper III because of theirperior accuracy, this sequential approach has proven insmental in motivating our present study and presaging soof our conclusions.

This paper is organized as follows: In Sec. II, we presthe model and states upon which the present work is baand set down notation. In Sec. III, we focus on the globground state energy, displaying specific results accordinour own method and comparing our results with thoseother authors and certain approximate formulas. In Sec.we present some particular examples of complete polaenergy bands and compare Global–Local results with scific results obtained by the DMRG method and by direcluster diagonalization. In Sec. V, we turn to the polareffective mass, computing effective mass curves cuttswaths through the polaron parameter space in severagimes, and make specific comparisons of Global–Localsults with those of a variety of competing approaches. Cclusions are summarized in Sec. VI.

0 © 1998 American Institute of Physics

IP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

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6541J. Chem. Phys., Vol. 109, No. 16, 22 October 1998 Romero, Brown, and Lindenberg

II. MODEL, STATES, METHOD

As our system Hamiltonian, we choose the traditionHolstein Hamiltonian,1,2

H5Hel1Hph1Hel-ph, ~1!

Hel5E(n

an†an2J(

nan

†~an111an21!, ~2!

Hph5\v(n

bn†bn , ~3!

Hel-ph52g\v(n

an†an~bn

†1bn!, ~4!

in which an† creates an electron in the rigid-lattice Wann

state at siten, andbn† creates a quantum of vibrational e

ergy in the Einstein oscillator at siten. We presume periodicboundary conditions on a one-dimensional lattice ofN sites.The electron transfer integral between nearest-neighboris denoted byJ, v is the Einstein frequency, andg is thedimensionless local coupling strength.~Except where dis-played for formulaic clarity, the reference energyE is set tozero throughout.! The lattice constant does not appear expitly in this formulation provided wave vectors are measurrelative to it, as will be our convention. Two dimensionlecontrol parameters can be constructed from the three prpal Hamiltonian parameters in different ways. One such ndimensionalizing scheme involves selecting the phonquantum\v as the unit of energy; in these terms, the natudimensionless parameters are the remaining coefficientthe electronic hopping term (J/\v) and electron–phononinteraction term (g). This scheme is particularly appropriawhen considering dependencies onJ and/or g at fixed v,such as we shall be concerned with in most of this papThis scheme is not so convenient near the adiabatic lihowever, where bothJ/\v andg diverge in a certain fixedrelationship. In the adiabatic regime, it is convenient to ndimensionalize by selecting the electron half-bandwidthJas the unit of energy; in these terms, the natural dimensless parameters are the remaining coefficients of the phoenergy (g5\v/2J) and the electron–phonon interactioterm, expressed in the formAl/g, where l5g2\v/2J.~Here, we follow the convention of including the site coodination numberz in the definition of l, such that l5g2\v/zJ.) The adiabatic limit is reached by allowingg tovanish at arbitrarily fixedl. We distinguish these two options as nonadiabatic and adiabatic scaling conventionsspectively. Except where explicitly noted, we conform to tnonadiabatic scaling convention.

Our central interest in this paper is in the polaron eneband, computed as

E~k!5^C~k!uHuC~k!&, ~5!

whereinH is the total system Hamiltonian andk is the totaljoint crystal momentum label of the electron–phonon stem.

All of our calculations are performed using normalizBloch states


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uC~k!&5uk&/^kuk&1/2. ~6!

These states are eigenfunctions of the appropriate totalmentum operator and orthogonal for distinctk, makingvariations for distinctk independent.55 Since selectedk val-ues such as the global polaron ground state (k50) are gen-erally insufficient to convey a very complete picture of plaron structure, we have found it important to examicompletevariational solutions both for quality assurance ato extract the best description of polaron structure. By ‘‘coplete’’ variational solutions, we mean a set ofN variationalenergiesE(k) andN polaron Bloch statesuC~k!&, the latterbeing described by a distinct set of variational parameterseachk. The set ofE(k) so produced constitute an estima~upper bound! for the polaron energy band.48,55

The Global–Local method4 represents polaron structurthrough three classes of variational parameters$an

k ,bqk ,gq

k%,

uk&5(nna

eiknana2nk ana

†

3expH 2N21/2(q

@~bqke2 iqn2gq

ke2 iqna!bq†

2h.c.#J u0&. ~7!

We note here for later emphasis that a solution ofGlobal–Local method for any particulark is contained in the3N complex quantities$an

k ,bqk ,gq

k%, and that acompletesolution is contained in 3N2 complex quantities, reducible to

O$ 32N

2% independent real quantities utilizing Hamiltoniasymmetries. We typically computed complete energy bastructures on 32-site lattices for every parameter(J/\v,g) we considered; thus, the solution for any particuk is encoded in 96 independent real numbers~48 for theground state!, and the complete solution~all k’s! in 1536independent real numbers~see, e.g., Fig. 3 in Sec. IV!.

Since our solutions are encoded in a relatively smnumber of variational parameters~compared to some othecalculations we shall consider!, even ‘‘complete’’ bandstructure solutions can be stored compactly, allowing a ‘brary’’ of polaron band structures to be archived and revited at leisure.

We solve the variational equations by relaxatitechniques3,56,57 through which an initializing state is iteratively refined toward the self-consistent target state. Unlmany other methods, nearby solutions can be used to iniize new calculations, accelerating convergence and reduthe need to obtain new solutions from scratch. Such initizing solutions might be obtained from nearby points in prameter space, for example, or might be lower-precisionlutions obtained from prior calculations at the samparameters andk. A library of polaron structures samplinthe polaron parameter space thus facilitates the acquisitionew solutions.

The computational time required for our calculation vaies according to the region of the parameter space examithe error tolerance required, and the ‘‘scope’’ of the calcution. In the intermediate coupling regime, complete ba


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6542 J. Chem. Phys., Vol. 109, No. 16, 22 October 1998 Romero, Brown, and Lindenberg

structures computed to tolerances adequate for this ptake 1–2 h on a single-processor Sun Microsystems USparc I workstation; optimized to compute only the effectmass, only a few minutes per effective mass value arequired.

More intensive calculation is required as one movesfrom the intermediate coupling region, but in no reasonageneral case for which we have achieved converged rehas a complete set ofN band energies andN Bloch statesrequired more than 2–4 h for one (J/\v,g) point. Somespecial cases presented greater difficulty, e.g., the limiweak coupling forJ/\v;1/4 proved difficult due to deteriorating numerical precision, and convergence in the vicity of the self-trapping transition grew more challenging wincreasing adiabaticity.

The self-consistency equations that follow from applyithe variational principle to this class of trial states, tmethod of solving those equations, and sample results hbeen detailed in Paper III.4 In the following sections, wecompare polaron energy band characteristics as determby the Global–Local method with a variety of alternatidescriptions, but apart from the one complete solution illtrated in Fig. 3, we defer to a subsequent work7 any detaileddiscussion of the particulars of internal polaron structurereflected in the variational parameters themselves.

III. GROUND STATE ENERGY

Without question, the single most important state in aquantum system is the ground state, and, by associationsingle most important energy is the ground state energy. Tbeing said, we emphasize quickly that a particular numervalue of the ground state energyin itself contains very littleinformation about the system that is of practical use. Fmost purposes, we require relationshipsbetweenenergies,e.g., effective masses, bandwidths, densities of states.such purposes we require multiple energies, all but onewhich areexcited statesin the global sense. Variational approaches such as ours divide the total space of the probinto subspaces of the total crystal momentum within whthe ground state of eachk sector is determined independently. In this sense, theglobal ground state energyE(0) isonly one of N independentk-sector ground state energieE(k) computed on an equal footing. There is no guaranthat the numerical value ofE(0) constitutes any more~orless! accurate an estimate of its particular target value tany otherE(k) so determined.

Nonetheless, it is the global ground state energyE(0)that is the most common denominator of diverse variatioapproaches, including those based on localized stateother states that may not be well adapted to the propeexpected of energy–momentum eigenfunctions. Thus,have compiled in Tables I and II a number of ground stenergies from our own calculations together with a sampof others for two particular sets of parameters of the HolstHamiltonian.

We have chosen the parameter sets (J/\v,g)5(1,1)and ~1,&! largely because these points are commotransected by the ground state curves of different approain the literature and thus permit the widest possible comp


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sons to be made. We have also chosen these parameteues because they aremoderate, falling in a regime where noparticular limiting theory is favored. Indeed, several resuincluded in our tabulations~indicated by dagger symbols! arewell-known asymptotic results not strictly valid in these itermediate scenarios.

In making the selection of data to display in Tables I aII, we have not attempted to be exhaustive, but to presesampling of results from differing methodologies, highlighing more recent examples. With the exception of the DMvalues, all tabulated data have been computed directly byobtained from cited analytic formulas, or obtained direcfrom the original authors by private communication. The vues shown for dynamical mean field theory have been msured from published figures, and are believed by us to fa

TABLE I. Ground state energies as computed by various methods

(J/\v,g)5(1,1), (g,l)5(12,

12). The table is broken at the most likel

location of the true ground state energy. Values indicated by dagger symare obtained from limiting formulas that at these parameter valuesclearly beyond their limits of validity.

Value @\v# Type of method Reference

21.735 75 † Nonadiabatic small polaron 2, Eq.~9!21.97 QMCN532, b51, m510 8, 1022.000 00 † Adiabatic strong coupling 43, Eq.~10!22.086 14 Semiclassical variation 58, 5922.35 QMCN532, b55, m532 8, 1022.40 Dynamical mean field 3122.447 21 Second-order WCPT Equation~8!22.456 11 Merrifield variation 4622.46 QMCN532, b520, m5256 1022.468 69 Toyozawa variation 4722.469 31 Global–Local variation Fig. 1, this paper22.469 68 DMRGN532 37, 38

22.471 Cluster diagonalization,N56 17, 19, 6023.089 59 † Second-order SCPT 21, 44, Eq.~11!

TABLE II. Ground state energies as computed by various methods

(J/\v,g)5(1,&), (g,l)5(12,1). The table is broken at the most likel

location of the true ground state energy. Values indicated by dagger symare obtained from limiting formulas that at these parameter valuesclearly beyond their limits of validity.

Value @\v# Type of method Reference

22.270 67 † Nonadiabatic small polaron 2, Eq.~9!22.500 00 † Adiabatic strong coupling 43, Eq.~10!22.518 15 Semiclassical variation 58, 5922.73 QMCN532, b51, m510 8, 1022.86 QMCN532, b55, m532 8, 1022.89 Dynamical mean field 3122.894 42 Second-order WCPT Equation~8!, this paper22.933 01 Merrifield variation This paper22.991 72 Toyozawa variation This paper22.998 02 Global–Local variation Figure 1, this paper22.998 83 DMRGN532 37, 38

23.000 Cluster diagonalization,N56 17, 19, 6023.00 QMCN532, b520, m5256 1023.052 79 † Second-order SCPT 21, 44, Eq.~11!


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represent the published data to the number of significantits displayed.

Since not all of the approaches sampled in Tables III are variational calculations, we should note that a compson of a variational energyEvar with a lower nonvariationalenergyEnon (Evar.Enon) in itself yields no conclusion regarding which value is ‘‘better’’ unless there exists indepedent proof that the nonvariational energy liesabove theground state. In the opposite circumstance (Enon.Evar),however, the variational principle alone assures thatvariational energy is the better estimate.

The most probable value of the true ground state enecan be sorted out from the data in Tables I and II as folloIn view of the fact that the uncertainties in the best QMvalues tabulated are of the order of 1% or greater,10 the QMCvalues cannot be viewed as any more definitive thanGlobal–Local, DMRG, and cluster values that are all detmined to higher precision and fall within the QMC uncetainties. Further, since the density matrix renormalizatgroup method has been shown to have a variatiorealization,34–39 it would appear that the true ground staenergy should lie below the DMRG values. On the othhand, as is discussed in greater detail in Sec. IV, cluenergies appear to convergeupward with increasing clustersize, suggesting that the true bulk ground state energyabove the N56 cluster values. Thus, each of the tabshown has been broken at the point where the best appupper bound to the ground state energy meets the best aent lower bound.

Turning our attention to the broader landscape ofsystem parameter space, Fig. 1 shows the dependence oglobal ground state energyE(0) on the coupling strength foassorted values ofJ/\v from 1/4 to 5. The overall behavioof the ground state energy is to trend between tasymptoticg2 dependencies with differing coefficients anoffsets. The crossover between these asymptotic trends

FIG. 1. The global ground state energyE(0) vs g2 for J/\v51/4, 1/2, 1,2, 3, 4, and 5 in order from top to bottom@note that limg→0 E(0)522J#.The straight dotted lines asymptoting each Global–Local curve at smag2

are given by the second-order WCPT formula~8!. The arched dashed lineasymptoting each Global–Local curve at largeg2 are given by the secondorder SCPT formula~10!.


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curs at the self-trapping transition (g'gST), which throughanalyses presented elsewhere5–7 has been found to be accurately located by the relationgST511AJ/\v.

Using weak-coupling perturbation theory, one can shthat the leading dependence of the ground state energy ocoupling constant is given for anyJ by

E~0!'22J2g2\v

A114J/\v. ~8!

This is, in fact, what we find to within numerical precisiowithin the weak coupling regime of the Global–Locmethod~see Fig. 1!.

The weak-coupling estimate~8! is superior to the nonadiabatic small polaron estimate2,41,42

E~0!'2g2\v22Je2g2~9!

for most J provided the coupling strength is not too larg(g,gST), though the two approximations agree when boJ/\v andg are small. When extrapolated into the adiabaregime, however, Eq.~9! is not even qualitatively sensible aweak coupling forJ/\v.1/2, and does not even approacthe weak-coupling estimate untilg'gST. At strong coupling(g@gST), Eq. ~9! approaches2g2\v for anyJ, which gen-erally differs significantly from the true value of the grounstate energy. Thus, it appears that there isnowherein theadiabatic regime where the nonadiabatic small polaronproximation provides a meaningful estimate of the groustate energy.

On the other hand, the adiabatic strong-coupling perbation result~in one dimension!43

E~0!'22JlS 111

4l2D52g2\v2J2

g2\v~10!

correctly describes the asymptotic behavior at strong cpling for g@gST. Both this adiabatic result and the nonadibatic result ~9! can be recovered from the second-ordstrong-coupling perturbation result21,44

E~0!'2g2\v22Je2g2

22

\vJ2e22g2

@ f ~2g2!1 f ~g2!#, ~11!

f ~x!5Ei~x!2g2 ln~x!, ~12!

in which Ei(x) is the exponential integral andg is the Eulerconstant. Examples of this strong-coupling result are plotin Fig. 1 together with comparable Global–Local and weacoupling results. The second-order SCPT result is goodall g providedJ/\v,1/4, and for allJ providedg@gST,but breaks down rather dramatically otherwise.

Thus, while both weak- and strong-coupling perturbatitheories are clearly limited in scope, both are in quantitatagreement with results of the Global–Local method inappropriate regimes, suggesting that a quite complete picis available in the results of the Global–Local method suported by the leading orders of perturbation theory. Indethis blended approach proves to be broadly useful, aborne out in results presented elsewhere.5–7


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IV. ENERGY BANDS

The archetypical result of polaron theory in the strocoupling limit is the band form2,21,41,42,44

E~k!5~E22J!2EB2 12B@12cos~k!#. ~13!

The detailed dependences of the binding energyEB andbandwidthB on J, \v, and g depend on regime, and it ithrough these dependences that changes in polaron struare manifested in this limiting result. The binding energy abandwidth are related quite simply in the nonadiabaticgime (EB;g2\v, B;4Je2g2

), and less simply in the adiabatic regime. When this band form is valid, the polaronheavily dressed by phonons and the polaron bandwidthB issmall relative toboth the bare electron bandwidth 4J and thebare phonon energy\v; i.e.,B!min$4J,\v%. This narrowingof the polaron band and the related increase in the effecmass are commonly characterized as aspects of polarondistortion.

More generally, however, away from the strong couplilimit, polaron energy bands arenot simple narrowed andshifted replicas of the bare band, but are more stronglytorted shapes whose nonsinusoidal dependence onk is a cru-cial reflection of polaron structure. For such bands, thelaron binding energy, effective mass, and polaron bandwno longer stand in any simple relationship to each other.

In the limit g→01 of the adiabatic regime~J/\v.1/4!,one finds that the polaron energy band assumes aclippedform,

E~k!5~E22J!12J@12cos~k!#, uku,kc ~14!

5~E22J!1\v, uku.kc , ~15!

reflecting the difference in the character of polaron staabove and below the wave vectorkc ~given by 2J@12cos(kc)#5\v) at which the bare electron energy bacrosses into the one-phonon continuum. Although tcrisply clipped band form is strictly valid only in the limig→01, its essential characteristics persist to nontrivial v

FIG. 2. Comparison of Global–Local~GL! energy band results with comparable results of the density matrix renormalization group~DMRG! methodfor J/\v51, g51. kc5p/3. DMRG data kindly provided by E. Jeckelmann~Ref. 38!.


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ues of the electron–phonon coupling strength. Figures 24, for example, demonstrate the persistence of the clipband form and ofkc as the relevant scale parameter forg oforder unity.

Figure 2 includes a comparison of a complete GlobaLocal energy band with a comparable energy band as cputed independently by the DMRG method37,38 for the caseJ/\v51, g51. As was the case in the ground state enecomparisons of Tables I and II, the DMRG energies aslightly lower than the GL energies, the difference rangifrom about 0.015% atk50 to about 0.46% atk5p; nofitting or numerical adjustment of any kind was madenormalize these two results.

The several surfaces completely detailing the electrophonon structure underlying this energy band are shownFig. 3.

These surfaces show polaron structure to be compoof multiple distinguishable and characteristic features acorrelations. It is not our purpose here to analyze this interstructure in depth;4,46,47,61–64however, a few outstanding features are worth noting. First, in all components of the sotion, there exist distinguishable characters in the inner Blouin zone (uku,kc) and the outer Brillouin zone (uku.kc). These distinct characters are least evident in the etron amplitudesaq

k , and most evident in the primary phonoamplitudebq

k . In the inner zone, phonon structure consiof phonon amplitudes correlated with the electronic comnent in a manner that is structured, but largely local in chacter. In the outer zone, phonon structure continues to halocalized component, but this component is dominated bstrong, momentum-rich, largely delocalized componentflecting the strong influence of the one-phonon continuum‘‘clipped’’ polaron energy bands. Although these inner aouter zone features become muted with increasing coupstrength and become nearly indistinguishable above thetrapping transition, their qualitative character is pervasive

Figure 4 overlays the appropriate variational enerband computed by the Global–Local method upon data frseveral finite-cluster diagonalization results for the particucaseJ/\v51.25 andg5A0.5•1.25'0.79 . No fitting hasbeen performed to achieve the impressive degree of agment evident in these independent results. The variatioresults and theN520 cluster results agree to multiple significant digits across the entire Brillouin zone, the relatidifference being approximately 0.01% atk50 and 0.4% atk5p. This degree of agreement is actually evenbetterthanit appears at first glance, as may be inferred from the cvergence trends in the cluster data.

Unlike variational bounds that convergedownward to-ward the exact target energy, the trends evident in the daWellein and Fehske25,26 suggest that cluster energies covergeupwardwith increasing cluster size. Figure 5 displathe ratioEN

M510(k)/uEN532GL (k)u for severalk andN, where

ENM(k) is the band energy determined by Lanczos diagon

ization on a cluster ofN sites allowing up toM totalphonons, andEN532

GL (k) is the Global–Local variational result computed for 32 sites as throughout this paper. Thk50 cluster energy is clearly well converged to a valslightly lower than, but nearly identical to the variation


band


FIG. 3. Complete Global–Local solutions for the caseJ/\v51, g51 corresponding to the ground state energy tabulation in Table I and energypresentation in Fig. 2;~a! bq

k , ~b! gqk , ~c! Re$an

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energy.~That thek50 cluster energy is nearly independeof N suggests that theN56 cluster energies17,19 included inTables I and II are probably also well-converged.! Clusterenergies are less well-converged away from the Brillozone center, where the evident trends suggest that thelimits of the finite-k energies lie near the variational valuThus, though in principle and through sufficiently comprhensive computation on sufficiently large clusters, clusteragonalization methods should best our variational approat any k, it appears that cluster sizes and retained phonumbers must be significantly larger than those attemptedate in order to improve significantly upon the Global–Locresults for the polaron energy bandas a whole.

One conclusion that can be drawn from these compsons~considering, for example, a fixedN) is that the outerBrillouin zone (uku.kc) displays a greater sensitivity tcontributions from higher phonon numbers than does thener zone (uku,kc). This is exactly what is to be expectefrom thek-dependence of the mean phonon number, whis typically considerably smaller in the inner zone than inouter zone~see, e.g., Fig. 1 of Paper II!.

Owing to ultimate limitations of computing time anphysical limitations of data storage, cluster diagonalizatiogenerally are limited by the maximum dimension of the trucated Hilbert space that can be addressed by a particcomputer. For a one-electron problem on a lattice ofN sites,the dimension of the electronic subspace isDel5N and thedimensionDph of a truncated phonon subspace containingmostM phonons is given byDph5(N1M )!/N! M !. Calcu-lations have been reported for various combinations ofN andM from N52 to N520 andM up to 50, each such calculation striking a unique balance betweenN and M consistent


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with computational resources. Table III presents values cveying the scale of the diagonalization problem for clussizes and phonon numbers up toN5M532. Clearly, thescenarios represented by the lower right half of the tabulavalues lie beyond the scope of machine diagonalization.

Implementation of the density matrix renormalizatiogroup method also involves a truncation of the total quantHilbert space; however, the limitations imposed by sutruncations appear to be less onerous than in the cascluster diagonalization, permitting relatively large calcu

FIG. 4. Comparison of our variational energy band with results of clusdiagonalizations for cluster sizesN510, 14, 18, and 20;J/\v51.25 andg5A0.5•1.2550.79 , corresponding to (g,l)5(0.4,0.25). kc

'0.2952p. Cluster data kindly provided by G. Wellein~Ref. 26!.


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tions to be accomplished in reasonable time on desktop pforms, much as the Global Local method.

By contrast, the ‘‘size’’ of a Global–Local variationdoes not scale directly with phonon number, since the naof the Global–Local approximation is not to truncate tphonon Hilbert space but to characterize the phonon dibution for M5` through a relatively small number of parameters. For example, in all the calculations presentethis paper, only 3N independent variational parameters werequired to flexibly represent the polaron state to high presion at anyk ~96 for ukuP(0,p), 48 for uku50 andp! withno restriction on phonon numbers, vastly fewer than comrable cluster diagonalizations.

It is also the case that the Global–Local method is noits best for small clusters, but improves in quality with icreasingN ~though not without limit!. This improvement isrealized because the total phonon state is representedsuperposition ofN distinct phonon coherent states; suchsuperposition becomes more flexible as the number of coent states in the superposition increases.

Our own comparisons with six-cluster ground stateergies in Tables I and II proved mutually favorable, as hour comparison with theN510– 20 cluster ground states acomputed by Wellein and Fehske.25,26However, we are moregenerally concerned with thecompleteband structure of thepolaron, which is not well captured by calculations on smclusters as can be seen from the following:

From weak coupling nearly up to the self-trapping trasition, polaron bands exhibit distinct character for wave vtors above and belowkc that must be resolved if one is tfaithfully represent polaron structure. This underscoresimportance inanyfinite-lattice calculation of maintaining thlattice sizeN in relation to the tunneling matrix elementJsuch that at least one finite-k point is sampled on both thhigh and low sides ofkc . For a givenN, this implies thatonly for JP(Jmin ,Jmax) canboth the inner and outer polaroband structure be characterized independently, where

FIG. 5. Cluster energies fork50, p/2, and p and cluster sizesN510– 20 forJ/\v51.25 andg5A0.5•1.25'0.79 . Each cluster energyis divided by the absolute value of our variational energy appropriate toparticulark, so that21 indicates identical results under the two methoCluster data kindly provided by G. Wellein~Ref. 26!.


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Jmax5\v

2@12cos~2p/N!#, ~16!

Jmin5\v

2@12cos~~N22!p/N!#. ~17!

Most crucially, forJ.Jmax, it is not possible to resolve theparabolic band bottom that is characteristic of smallk fromwhich to extract a meaningful effective mass. To illustrathe gravity of this restriction, the values ofJmin andJmax sodefined have been presented in Table III for each ofcluster sizes there considered. It is clear from these vathat cluster size imposes a significant limitation on onability to describe polaron band structure over a meaninginterval of J.

V. EFFECTIVE MASS

The ground state energy as considered in Sec. III permted a fair comparison of many different approaches in pbecause in many cases approximate methodologies atheir ‘‘best’’ when applied to the global ground state; excitstates almost universally pose greater challenges. In enband theory, the minimal excursion beyond the ground sis contained in the effective mass, since in principle thetermination of the effective mass requires knowledge of oan infinitesimal excitation to finitek. In this section we com-pare effective masses as yielded by a number of differapproaches.

Our effective mass computations were based on themula

m*

m05

2J

]2E~k!

]k2 Uk50

, ~18!

using a discrete representation of thek derivative at the Bril-louin zone center. We note that them0 used in our calcula-tions was obtained by computing its value using the sadiscrete differentiation as was applied to computem* ratherthan using the limitingN→` value. Not only does this mini-mize any dependence of the computed effective mass ron lattice size, but is technically necessary in order to prerly normalize our results, e.g., to exactly recover the limlimg→0 m* /m051. For all cases presented,J,Jmax accord-ing to Eq. ~17! for lattices of 32 sites, so that our discredifferentiation yields the physically meaningful value.

In Fig. 6, we first consider the simultaneous comparisof our Global–Local effective masses@actually, ln(m* /m0)#

at.

TABLE III. Cluster facts.

Cluster Jmin /\v Jmax/\v D tot ,M510 D tot ,M516 D tot ,M532

2 sites No value No value 132 306 11224 sites 0.5000 0.500 4.03103 1.93104 2.43105

6 sites 0.3333 1.000 4.83104 4.53105 1.73107

10 sites 0.2764 2.618 1.83106 5.33107 1.531010

16 sites 0.2599 6.569 8.53107 9.63109 3.631013

20 sites 0.2563 10.22 6.03108 1.531011 2.531015

32 sites 0.2524 26.02 4.731010 7.231013 5.931019


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with comparable DMRG results and with second-ordstrong-coupling perturbation theory as contained inrelation21,44

m*

m05

eg2

11~4J/\v!e2g2f ~g2!

, ~19!

for which f (x) has been defined in Eq.~12!.The Global–Local and DMRG results agree quite we

especially considering that these particular DMRG datanot represent actual finite-k calculations, but estimationbased on DMRG ground state data.37,38 Given the very fa-vorable energy band comparison in Fig. 2, it is reasonablexpect that a future comparison based on direct excited-sDMRG calculations would show even better agreement~e.g.,see Fig. 2!. What deviations there are between the GL aDMRG results are greatest above the self-trapping transi(g.gST); it is unclear at present whether this is a discreancy of lasting significance.

On the other hand, comparison of both GL and DMReffective masses with the results of second-order SCPTfar less favorable. Agreement is excellent forJ/\v51/4 andg.gST; however, deviations appear even at this smalJvalue forg,gST. At J/\v51, it is evident that both the GLand DMRG masses asymptote to the SCPT result, butthis convergence of results does not materialize until wabove the self-trapping transition (g@gST). At J/\v55, thedisagreement between the SCPT result and both the GLDMRG masses is so severe as to render the SCPT estimuseless. We note that although we have explicitly compaonly selectedJ/\v values, it can be safely inferred that thsecond-order SCPT mass estimation ceases to be relevapractical terms by the timeJ/\v;2.7

In Fig. 7 we make a more narrow comparison withbroader range of approaches. Here, we again compareand DMRG results, this time including as well:

~i! data from direct quantum Monte Carlo calculationsthe polaron mass,13,14

FIG. 6. Effective mass ratio. Bold curves: Global–Local results forJ/\v51/4, 1, and 5. Faint curves: Second-order strong-coupling perturbatheory @see Eq. ~19!#. Points: Density matrix renormalization grou~DMRG! results as kindly provided by E. Jeckelmann~Refs. 37 and 38!.


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~ii ! the second-order SCPT estimate as contained in~19!,

~iii ! the second-order WCPT estimate as contained in38

m0

m*512

g2~112J/\v!

~114J/\v!3/2 ~20!

and~iv! the weak-coupling Migdal estimate as containin40,45

m*

m0511g2

~112J/\v!

~114J/\v!3/2. ~21!

It is clear that all the presented results except the SCare mutually consistent at sufficiently weak coupling; hoever, divergences between results appear quickly withcreasing coupling strength, such thatonly the GL, DMRG,and WCPT masses remain mutually consistent to nontricoupling.

We note that the agreement between the GL, DMRand WCPT masses is actually better than it may appearin Fig. 6, the DMRG data indicated by diamond symbols athe result of an estimation procedure based on the gloDMRG ground state only. The near-perfect agreementtween the GL and DMRG energy bands as shown in Figassures that the effective masses computed directly fromcomplete energy bands are essentially identical. The shigh degree of agreement can be inferred from the compson of GL and cluster diagonalization bands in Fig. 4. Tsmall apparentdiscrepancy between theindirect DMRG es-timate and the GL and SCPT results is thus essentially elnated, showing GL, DMRG, and WCPT to be in essentiacomplete agreement up to at leastg;1.

At strong coupling, the GL, DMRG, and SCPT massare again mutually consistent, while each of the remainresults presented deviates significantly.

n

FIG. 7. Inverse effective mass ratio forJ/\v51. Solid curve: Global–Local result. Diamonds: DMRG results as kindly provided by E. Jeckmann ~Refs. 37 and 38!. Squares: QMC results as kindly provided byKornilovitch ~see note added in proof!. Dotted curve: Weak-couplingMigdal approximation@see Eq.~21!#. Dashed curve: Second-order stroncoupling perturbation theory@see Eq.~19!#. Dot-dashed curve: Second-ordeweak-coupling perturbation theory@see Eq.~20!#.


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VI. CONCLUSION

In this paper, we have presented results for the Holsmolecular crystal model in one space dimension as demined by the Global–Local variational method, includincomplete polaron energy bands, ground state energies,effective masses. We have juxtaposed our results withcific comparable results of numerous other methodologiecurrent interest, including quantum Monte Carlo, clusteragonalization, dynamical mean field theory, density marenormalization group, semiclassical analysis, wecoupling perturbation theory, and strong-coupling perturtion theory.

Through these comparisons, we have been able toclude the following.

~i! Perturbation theory, while definitive in limits, iclearly superceded by more accurate methods inintermediate-coupling regime.

~ii ! Semiclassical analysis, while consistent with pertbation theory in the strong-coupling limit, does nimprove significantly over perturbation theory awafrom this limit.

~iii ! Dynamical mean field theory does not deliver superquantitative accuracy in the regimes considered inpaper.

On the positive side, very favorable comparisons wfound between the Global–Local method, the density marenormalization group method, quantum Monte Carlo, acluster diagonalization methods. GL, QMC, and DMRwere found to be consistent with perturbation theory at bweak and strong coupling, and in the weak-coupling caseleast, consistent to higher coupling strengths than any omethods. Where these depart from perturbation-theoreticsults in the intermediate coupling regime, all remain mually consistent to an impressive degree, and where dicomparison has been possible, all are consistent with cludiagonalization.

We take these results as a whole to confirm the GlobLocal variational method as being highly accurate ovewide range of the polaron parameter space, from the nodiabatic limit to the extremes of high adiabaticity, from wecoupling through intermediate coupling to strong couplin

In succeeding works,5–7 we shall present a comprehesive analysis of polaron structure and properties as demined in the Global–Local method over essentially twhole of the polaron parameter space.

Note added in proof: At the final stage of publication, wereceived data from P. E. Kornilovitch obtained by an improved QMC technique that appears to be highly accurate~P.E. Kornilovitch, cond-mat 9808155!. The new QMC groundstate energies includable in Tables I and II fall in t‘‘break’’ between DMRG and cluster results where we epect the exact ground state energy to lie, and the errorsported for these values are comparable to the gap betwthe DMRG and cluster values. This new QMC techniqalso yields effective masses of sufficient quality that we wmoved to include the new QMC masses in our Fig. 7; thresults appear to improve significantly upon the results ornally reported in Ref. 13.


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ACKNOWLEDGMENTS

This work was supported in part by the U.S. Departmof Energy under Grant No. DE-FG03-86ER13606. Thethors gratefully acknowledge the cooperation of H. De Raand A. Lagendijk,8,10 P. E. Kornilovitch and E. R. Pike,13,14

A. S. Alexandrov, V. V. Kabanov, and D. E. Ray,17,19 G.Wellein and H. Fehske,25,26E. Jeckelmann and S. White,37,38

and G. Kalosakas, S. Aubry, and G. P. Tsironis,58,59 for pro-viding numerical values of data used in parts of this pap

1T. Holstein, Ann. Phys.~N.Y.! 8, 325 ~1959!.2T. Holstein, Ann. Phys.~N.Y.! 8, 343 ~1959!.3Y. Zhao, Ph.D. thesis, University of California, San Diego, 1994.4D. W. Brown, K. Lindenberg, and Y. Zhao, J. Chem. Phys.107, 3179~1997!.

5A. H. Romero, D. W. Brown, and K. Lindenberg, Phys. Rev. B~submit-ted!.

6A. H. Romero, D. W. Brown, and K. Lindenberg, Phys. Rev. Lett.~sub-mitted!.

7A. H. Romero, D. W. Brown, and K. Lindenberg~unpublished!.8H. D. Raedt and A. Lagendijk, Phys. Rev. B27, 6097~1983!.9H. D. Raedt and A. Lagendijk, Phys. Rev. B30, 1671~1984!.

10H. D. Raedt~private communication!.11A. Lagendijk and H. D. Raedt, Phys. Lett.108A, 91 ~1985!.12R. H. McKenzie, C. J. Hamer, and D. W. Murray, Phys. Rev. B53, 9676

~1996!.13P. E. Kornilovitch and E. R. Pike, Phys. Rev. B55, R8634~1997!.14P. E. Kornilovitch~private communication!.15A. Kongeter and M. Wagner, J. Chem. Phys.92, 4003~1990!.16J. Ranninger and U. Thibblin, Phys. Rev. B45, 7730~1992!.17A. S. Alexandrov, V. V. Kabanov, and D. E. Ray, Phys. Rev. B49, 9915

~1994!.18A. S. Alexandrov and S. N. Mott,Polarons & Bipolarons~World Scien-

tific, London, 1995!.19V. V. Kabanov~private communication!.20F. Marsiglio, Phys. Lett. A180, 280 ~1993!.21F. Marsiglio, Physica C244, 21 ~1995!.22M. I. Salkola, A. R. Bishop, V. M. Kenkre, and S. Raghavan, Phys. R

B 52, R3824~1995!.23H. Eiermann and M. Wagner, J. Chem. Phys.105, 6713~1996!.24G. Wellein, H. Roder, and H. Fehske, Phys. Rev. B53, 9666~1996!.25G. Wellein and H. Fehske, Phys. Rev. B56, 4513~1997!.26G. Wellein ~private communication!.27H. Fehske, J. Loos, and G. Wellein, Z. Phys. B104, 619 ~1997!.28S. Ciuchi, F. de Pasquale, C. Masciovecchio, and D. Feinberg, Europ

Lett. 24, 575 ~1993!.29S. Ciuchi, F. D. Pasquale, and D. Feinberg, Physica C235-240, 2389

~1994!.30S. Ciuchi, F. D. Pasquale, and D. Feinberg, Europhys. Lett.30, 151

~1995!.31S. Ciuchi, F. de Pasquale, S. Fratini, and D. Feinberg, Phys. Rev. B56,

4494 ~1997!.32J. K. Freericks, M. Jarrell, and D. J. Scalapino, Phys. Rev. B48, 6302

~1993!.33J. K. Freericks, M. Jarrell, and D. J. Scalapino, Europhys. Lett.25, 37

~1994!.34S. R. White, Phys. Rev. Lett.69, 2863~1992!.35S. R. White, Phys. Rev. B48, 10345~1993!.36S. Ostlund and S. Rommer, Phys. Rev. Lett.75, 3537~1995!.37E. Jeckelmann and S. R. White, Phys. Rev. B57, 6376~1998!.38E. Jeckelmann~private communication!.39S. Rommer and S. O¨ stlund, Phys. Rev. B55, 2164~1997!.40A. B. Migdal, Sov. Phys. JETP34, 996 ~1958!.41S. V. Tyablikov, Zh. Eksp. Teor. Fiz.23, 381 ~1952!.42I. G. Lang and Y. A. Firsov, Sov. Phys. JETP16, 1301~1963!.43A. A. Gogolin, Phys. Status Solidi B109, 95 ~1982!.44W. Stephan, Phys. Rev. B54, 8981~1996!.45M. Capone, W. Stephan, and M. Grilli, Phys. Rev. B56, 4484 ~1997!

~Paper II!.46Y. Zhao, D. W. Brown, and K. Lindenberg, J. Chem. Phys.106, 5622

~1997!.


.

g

w-

ered


47Y. Zhao, D. W. Brown, and K. Lindenberg, J. Chem. Phys.107, 3159~1997!.

48Y. Toyozawa, Prog. Theor. Phys.26, 29 ~1961!.49Y. Toyozawa, inPolarons and Excitons, edited by C. G. Kuper and G. D

Whitfield ~Plenum, New York, 1963!, p. 211.50A. Sumi and Y. Toyozawa, J. Phys. Soc. Jpn.35, 137 ~1973!.51Y. Toyozawa, inRelaxation of Elementary Excitations, edited by R. Kubo

and E. Hanamura~Springer, Berlin, 1980!, p. 3.52D. Emin, Adv. Phys.22, 57 ~1973!.53B. Schopka, Ph.D. thesis, Technische Universita¨t Munchen, 1991.54R. E. Merrifield, J. Chem. Phys.40, 4450~1964!.55T. D. Lee, F. E. Low, and D. Pines, Phys. Rev.90, 297 ~1953!.56W. H. Press, S. A. Teukolsky, B. P. Flannery, and W. T. Vetterlin

Numerical Recipes~Cambridge University Press, New York, 1992!.


,

57N. J. Giordano,Computational Physics~Prentice–Hall, Upper SaddleRiver, NJ, 1997!.

58G. Kalosakas, S. Aubry, and G. P. Tsironis, Phys. Rev. B58, 3094~1998!.59G. Kalosakas~private communication!.60Cluster data forN.6 exist, as discussed in some detail in Sec. IV; ho

ever, theN56 calculation by Alexandrovet al. ~Refs. 17 and 19! is thelargest such calculation we could find for the parameter values considin Tables I and II.

61D. W. Brown, inNonlinear Excitations in Biomolecules~Springer, Berlin,1995!, Vol. 2, p. 279.

62D. W. Brown, Proc. SPIE2526, 40 ~1995!.63D. W. Brown and K. Lindenberg, Physica D113, 267 ~1997!.64K. Lindenberg, Y. Zhao, and D. W. Brown, inComputational Physics,

edited by P. Garrido and J. Marro~Springer, Berlin, 1997!, p. 3.


Documents

Converging toward a practical solution of the Holstein molecular crystal model