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Singularities and Pseudogaps in the Density of States of Peierls Chains Lorenz Bartosch and Peter Kopietz Institut fu¨r Theoretische Physik, Universität Göttingen, Bunsenstrasse 9, D-37073 Göttingen, Germany (October 23, 1998) isindeedexact[6,7],thereexistsanothernon-triviallimit 9 We develop a non-perturbative method to calculate the where the exact ρ(ω) is known: if in Eq.(1) we let 9 densityofstates(DOS)ρ(ω)ofthefluctuatinggapmodelde- h i ξ 0, K , with K ξ D = const, the right- 9 scribing the low-energy physics of electrons on a disordered ha→ndsideo0f→Eq.∞(1)reduces0to→2Dδ(x x′). As shownby 1 Peierls chain. For a real order parameter field we calculate − OvchinnikovandErikhman(OE)[8],inthelimitL n ρ(0) (i.e. the DOS at the Fermi energy) exactly as a func- →∞ (where L is the length of the chain) the exact ρ(ω) a tionalofthedisorderforachainoffinitelengthL. Averaging h i can then be obtained from the stationary solution of a J ρ(0)withrespecttoaGaussianprobabilitydistributionofthe Fokker-Planckequation [9]. For small ω and ∆ =0 one 5 fluctuatingPeierlsorderparameter,weshowthatforL→∞ 0 theaveragehρ(0)idivergesforanyfinitevalueofthecorrela- finds [8] hρ(ω)i ∝ |ωln3|ω||−1. Singularities of this type ] tion length above thePeierls transition. Pseudogap behavior at the band center of a random Hamiltonian have been l e emerges only if the Peierls order parameter is finite and suf- discovered by Dyson [10], and have recently also been - ficiently large. found in one-dimensional spin-gap systems [11,12]. It is r t important to note that in the FGM the singularity is a s PACS numbers: 71.23.-k, 02.50.Ey, 71.10.Pm consequence of the charge conjugation symmetry of the . at underlying Dirac Hamiltonian, andisnot relatedto con- m creteprobability propertiesof∆(x) [9,13]. Inparticular, Atlowtemperaturesmanyquasione-dimensionalcon- the singularity is not an artefact of the exactly solvable - d ductors become unstable and developlong-rangecharge- limit ξ 0 considered by OE [8]. It is therefore reason- n density-wave order, i.e. undergo a Peierls-transition [1]. ableto→expectthatforanyξ < theaverageDOSofthe o Within a mean field picture a finite value ∆0 of the FGM exhibits a singularity at ω∞=0. This general argu- [c Peierls order parameter leads for frequencies |ω| < |∆0| mentisindisagreementwithRef.[4],whereforlargebut to a gap in the electronic density of states (DOS) ρ(ω) finite ξ a pseudogap(and hence no singularity)has been 3 [2]. To achieve a better understanding of the effect of obtained. Inthisworkweshallresolvethiscontradiction v fluctuations on the Peierls transition, Lee, Rice, and by calculatingthe averageDOSatthe Fermi energy(i.e. 2 Anderson [3] introduced the so-called fluctuating gap ρ(ω =0) ) exactly for arbitrary ξ. 6 model(FGM).Inthismodelthefluctuatingpart∆˜(x)= h The locialDOS ρ(x,ω) of the FGM for a givenrealiza- 3 0 ∆(x) ∆0 of the order parameter is approximated by a tion of the disorder can be written as − 1 Gaussian stochastic process with covariance 8 ρ(x,ω)= π−1ImTr[σ3 (x,x,ω+i0)], (2) 9 ∆˜(x)∆˜(x′) K(x,x′)=K e−|x−x′|/ξ . (1) − G / h i≡ 0 where the 2 2 matrix Green’s function satisfies t × G a Here ... denotes averaging over the probability distri- [i∂ +ωσ i∆(x)σ ] (x,x′,ω)=δ(x x′)σ . (3) m butionh ofi∆(x), K is a positive constant, and ξ is the x 3− 2 G − 0 0 - order parameter correlationlength. We assume that the Here σi are the usual Pauli matrices and σ0 is the 2 2 d × field∆(x)isreal,correspondingtoacharge-densitywave unit matrix. Note that in Eq.(2) we have factored out n o that is commensurate with the lattice. a Pauli matrix σ3, so that the differential operator i∂x c TwentyyearsagoSadovskii[4]foundanapparentlyex- in Eq.(3) is proportional to the unit matrix. To solve : actalgorithmto calculate the averageDOS ofthe FGM. Eq.(3), we try the ansatz (suppressing for simplicity the v His calculations showedthat for temperatures above the frequency label) i X Peierls transition,in a regime whereξ is large but finite, (x,x′)=U(x) (x,x′)U−1(x′), (4) r theaverageDOSexhibitsasubstantialsuppressioninthe G G1 a vicinity of the Fermi energy, a so-called pseudogap. The whereU(x)isaninvertible2 2matrix. Eq.(4)resembles algorithmconstructedbySadovskiihasalsobeenapplied thetransformationlawofthe×comparatorinnon-Abelian in a different context to explain the weak pseudogapbe- gauge theories [14]. In fact, Eq.(4) can be viewed as haviorintheunderdopedcuprates[5]. However,recently a gauge transformation which generalizes the Schwinger it has been pointed out [6,7] that Sadovskii’s algorithm ansatz [15] to the non-Abelian case. It is easy to show contains a subtle flaw and hence does not produce the that the solution of Eq.(3) can indeed be written in the exactDOSoftheFGM.Itisthereforeimportanttocom- form (4) provided and U satisfy 1 pare this algorithmwith limiting cases where ρ(ω) can G be calculated without any approximation. h i [i∂x+ωσ3] 1(x,x′)=δ(x x′)σ0 , (5) G − Besides the limit ξ where Sadovskii’s algorithm i∂ U(x)=ω[U(x)σ σ U(x)]+i∆(x)σ U(x). (6) → ∞ x 3− 3 2 1 Eq.(5) defines the Green’s function of free fermions, and ∂ ψ~ = H(x)ψ~ , H(x)=2iωJ +2∆(x)J , (11) x 3 1 − can be solved trivially via Fourier transformation. The difficult partof the calculationis the solution of the ma- where Ji are spin J =1 operators in the representation trix equation (6). We parameterize U(x) as follows, 1 0 0 0 1 0 1 U(x)=eiΦ+(x)σ−eiΦ−(x)σ+eiΦ3(x)σ3 , (7) J3 =0 0 0  , J1 =  1 0 1  . (12) √2 0 0 1 0 1 0 where σ = 1[σ iσ ], and the three functions Φ (x),  −    ± 2 1± 2 ± Φ3(x) have to be chosen such that U(x) satisfies Eq.(6). Eq.(11) is a linear multiplicative stochastic differential A parameterization similar to Eq.(7) has recently been equation[18]. Formallythisequationlooksliketheimag- usedbySchopohl[16]tostudytheEilenbergerequations inary time Schrödinger equation for a J = 1 quantum of superconductivity. We find that the ansatz (7) solves spin in a random magnetic field, with x playing the role Eq.(6) if Φ±(x) and Φ3(x) satisfy ofimaginarytime. AlthoughtheoperatorH inEq.(11)is not hermitian, we may perform an analytic continuation ∂ Φ = 2iωΦ +∆(x)[1 Φ2], (8a) x + − + − + to imaginary frequencies (ω =iE) to obtain a hermitian ∂xΦ− =2iωΦ− ∆(x)[1 2Φ+Φ−], (8b) spin Hamiltonian. We thus arrive at the remarkable re- − − ∂ Φ = i∆(x)Φ . (8c) sult that the average DOS of the FGM can be obtained x 3 + − fromtheaverage state-vectorofaJ =1spininarandom Non-linear differential equations of the type (8a) are magneticfield. Ournon-lineartransformation(10)iswell called Riccati equations. The set of equations obtained knowninthequantumtheoryofmagnetism: withthefor- by Schopohl [16] has a similar structure but is not iden- mal identification R J , √2Z J , √2Φ b†, 3 ± ± − → → ∓ → tical with Eqs.(8a-8c). Note that Eq.(8a) involves only and √2Φ b, Eq.(10) is precisely the Dyson-Maleev + → Φ+. IfwemanagetoobtainthesolutionΦ+,Eqs.(8b,8c) transformation [19], which expresses the spin operators become simple linear equations which can be solved ex- in terms of boson operators b,b†. actly. From Eqs.(2,4) and (7) it is easy to see that the The solution of Eq.(11) with initial condition ψ~(0) = local DOS can be written as ψ~ is ψ~(x)=S(x)ψ~ , where the S-matrix is 0 0 ρ(x,ω)=π−1ReR(x,ω+i0) , R=1 2Φ+Φ− . (9) x − S(x)= exp dx′H(x′) . (13) Thus, to calculate the average DOS we have to average T (cid:20)−Z (cid:21) 0 the product Φ Φ over the probability distribution of + − Here is the usual time ordering operator. The proper the field ∆(x). In the limit where the right-hand-side T of Eq.(1) reduces to 2Dδ(x x′) we can use the fact choice of boundary conditions requires some care. For − simplicity, let us assume that ∆(x) is non-zero only in that Φ and Φ satisfy first order differential equations + − a finite interval 0 x L. Outside this regime we to express the average Φ Φ in terms of the solution + − ≤ ≤ h i find that Z (x) = exp[ 2i(ω + i0)(x x )]Z (x ) is oftwo coupledone-dimensionalFokker-Planckequations ± 0 ± 0 ∓ − the solution of Eq.(11). Physically it is clear that expo- [17]. Thus, our method leads to an algorithm for ob- nentially growingsolutions areforbidden, which requires tainingtheexact ρ(ω) withoutusingthenodecounting h i Z (0) = Z (L) = 0 and implies R = 1 for x 0 and theorem [9]. However, the Fokker-Planck equation ob- − + ≤ x L. We conclude that at the boundaries tained by OE [8] within the phase formalism [9] is easier ≥ to solve than our system of two coupled Fokker-Planck Z (0) 0 equations. Hence, for δ-function correlated disorder our + approach does not have any practical advantage. ψ~0 = 1  , ψ~(L)= 1  , (14) 0 Z (L) On the other hand, if the disorder is not δ-function    −  correlated, probability distributions of physical quanti- where Z (0) and Z (L) are determined by ψ~(L) = ties do in general not satisfy Fokker-Planck equations, + − anditisnotsoeasytoperformcontrolledcalculationsor S(L)ψ~0. This implies Z+(0) = S12(L)/S11(L). Be- − evenobtainexactresults[18]. Wenowshowthatforreal cause the matrix elements of S(L) depend on the disor- ∆(x) the local DOS at the Fermi energy can be calcu- der,the initialvectorψ~ is stochastic. Note thatintext- 0 lated exactly. To derive this result, let us introduce the book discussions of multiplicative stochastic differential complex vector equations one often assumes deterministic initial conditions [18]. After simple algebra we obtain for the second √2(1 Φ+Φ−)Φ+ Z+ component of ψ~(x)=S(x)ψ~ in the interval 0 x L ψ~ = − 1 −2Φ+Φ−   R  . (10) 0 ≤ ≤ − ≡  √2Φ−  Z−  R(x)=S22(x)−S21(x)S12(L)/S11(L). (15) NotethatbyconstructionR2 2Z+Z− =1forallx,and In general we have to rely on approximations to calcu- − that according to Eq.(9) the second component of ψ~ is late the time-ordered exponential in Eq.(13). However, related to the local DOS. Using Eqs.(8a,8b) we find there are two special cases where S(x) can be calculated 2 exactly. The first is obvious: if ∆(x) = ∆ is indepen- λ = ξ/L, and ν = ∆ /(2K ξ). Numerical results for 0 0 0 dentofx,ourspinHamiltonianH(x)isconstant,sothat ρ(x,0) are shown in Fig.1. Due to symmetry with re- h i the time-ordering operator is not necessary. Eq.(15) can spect to x = L/2 the local DOS assumes an extremum thenbe evaluatedexactlyforarbitraryL[17]. Ifwetake atx=L/2,which in the limit L approacheseither →∞ the limit L holding x/L fixed, we recover the well zero or infinity. Using the fact that at x=L/2 the cosh → ∞ known square root singularity at the band edges, in the numerator of Eq.(19) is unity, we obtain lim ρ(x,ω)= Θ(ω2−∆20)|ω| , 0<x/L<1. (16) ρ(L/2,0) = eL2˜f(λ) ∞ dsexp −s2/(2σ2) , (21) L→∞ π ω2−∆20 h i π√2πσ2 Z−∞ co(cid:2)sh[s+νL˜] (cid:3) p There exists another, more interesting limit where S(x) where σ2 =L˜[1 λ(1 e−1/λ)] and can be calculatedexactly. Obviously,at ω =0 the direc- − − tion ofthe magneticfieldinourspinHamiltonian(11)is f(λ)=1 λ[3 4e−1/(2λ)+e−1/λ]. (22) constant. Althoughinthis caseH(x) isx-dependent, we − − may omit the time-ordering operator in Eq.(13). After As shown in Fig.2, f(λ) is positive and monotonically straightforwardalgebra we obtain from Eq.(15) decreasing. Let us first consider the case ν = 0. This correspondsto the modeldiscussedbySadovskii[4]with R(x)=cosh[A(x) B(x)]/cosh[A(x)+B(x)], (17) − real ∆(x). For L˜ 1 the s-integration in Eq.(21) is ≫ where A(x) = 0xdx′∆(x′) and B(x) = xLdx′∆(x′). In Keaeseilpyindgoinne,mainndd twheatfifnodr ahρn(yL2fi,n0i)tie∝ξ tLh˜e−1p/a2reaxmpe[Lt2˜efr(λλ)=]. Eq.(17)itisunRderstoodthatR(x)standsRforR(x,i0),so that ρ(x,0)=π−1R(x). We have thus succeeded to cal- ξ/L vanishes for L , it is obvious that in this limit → ∞ ρ(L/2,0) is infinite. From Eq.(19) it is easy to show culatethelocalDOSρ(x,ω =0)oftheFGMattheFermi h i energyforagiven realization ofthedisorder. Thespecial numerically that this is also true for limL→∞ ρ(x,0) in h i the open interval 0 < x/L < 1. We have thus proven symmetries of random Dirac fermions at ω = 0 have re- that for L and arbitrary ξ < the average DOS centlybeenusedbySheltonandTsvelik[20]tocalculate → ∞ ∞ of the FGM is infinite at the Fermi energy, in agreement the statistics of the corresponding wave-functions. with general symmetry arguments [9,13]. To calculate the disorder averageof Eq.(17), we intro- Forfiniteν acarefulanalysis[17]ofEq.(21)showsthat duce P(x;a,b)= δ(a A(x))δ(b B(x)) and write h − − i there exists a critical value ν (λ) such that for ν > ν c c | | ∞ ∞ cosh(a b) the local DOS ρ(L/2,0) scales to zero in the thermo- R(x) = da dbP(x;a,b) − . (18) h i dynamic limit, and a pseudogap emerges. We obtain h i Z Z cosh(a+b) −∞ −∞ ν (λ)=[1 λ(1 e−1/λ)]1/2[f(λ)]1/2 , (23) Assuming that the probability distribution of ∆(x) is c − − Gaussian with average ∆ and covariance K(x,x′), the 0 seeFig.2. Forλ=0Eq.(23)yieldsν (0)=1. Atthefirst joint distribution P(x;a,b) can be calculated exactly. c sight this seems to contradict the result of OE [8], who The integration over the difference a b in Eq.(18) can − found pseudogap behavior already for ν > 1/2. One then be performed, and we obtain for the average local | | shouldkeepinmind,however,thatwehavesetω =0be- DOS ρ(x,0) =π−1 R(x) , h i h i foretakingthelimitL ,whileinRef.[8]theselimits →∞ eα(x) ∞ are taken in the opposite order. The non-commutativity ρ(x,0) = dsexp s2/(2σ2) oftheselimitsiswellknownfromthecalculationofthelo- h i π√2πσ2 Z−∞ (cid:2)− (cid:3) calDOSoftheTomonaga-Luttingermodelwithabound- cosh[β(x)s+∆0(2x L)] ary [21]. Interestingly, for K(x,x′) = 2Dδ(x x′) there − . (19) − × cosh[s+∆ L] existsaregime1/2<ν <1whereinthethermodynamic 0 limit ρ(ω) is discontinuous at ω =0. This follows from βH(exr)e=σ2[C=1C(x1)(L)C,α2((xx))]/=σ22,[Cw1i(txh)C2(x)−C32(x)]/σ2,and tfohreωfahct t0hi,atwhacecreoardsinwge thoavOeEshhoρw(ωn)ith∝at |ωρ|(20ν)−1=→ 0. − → h i ∞ For ν = 0 and large but finite ξ we conjecture that the x x C1(x)=Z0 dx′Z0 dx′′K(x′,x′′). (20) sfrimeqiulaernctoy-tdheepebnedheanvcioeroffoulinmdLb→y∞Fhaρb(rωiz)iioisanqduaMliétlaintiv[1el1y] forrandomDiracfermionswithaspecialtypeofdisorder C (x)isdefinedbyreplacingtherangeoftheintegralsin 2 [22]. Specifically, we expect that for frequencies exceed- Eq.(20) by the interval [x,L], and C (x) is obtained by 3 ing a certain crossoverfrequency ω∗ the average DOS of choosing the interval [0,x] for the x′- and [x,L] for the the FGM shows pseudogap behavior, which is correctly x′′-integration. predicted by Sadovskii’s algorithm [4]. However, this al- We now specify K(x,x′) to be of the form (1). Then gorithm misses the Dyson singularity, which emerges for σ2, α(x), and β(x) are easily calculated. It is convenient frequencies ω < ω∗ for any finite value of ξ. to introduce the dimensionless parameters L˜ = 2K0ξL, | |∼ 3 In summary, we have developed a non-perturbative [10] F. J. Dyson,Phys. Rev. 92, 1331 (1953). method to calculate the Green’s function of the FGM. [11] M. Fabrizio and R. Mélin, Phys. Rev. Lett. 78, 3382 Ourmainresultistheproofoftheexistenceofasingular- (1997); M. Steineret al.,cond-mat/9706096; A.Gogolin ityin ρ(0) foranyfinitevalueofthecorrelationlengthξ et al.,cond-mat/9707341. as lonhg as ∆i (x) is real and ∆ is sufficiently small. For [12] R. H.McKenzie, Phys. Rev.Lett. 77, 4804 (1996). 0 ∆ = 0 we have shown tha|t w|ith open boundary con- [13] M. Mostovoy and J. Knoester, cond-mat/9805085. 0 ditions ρ(L,0) exp[K ξLf(ξ/L)], where f(0) = 1. [14] See, for example, M. E. Peskin and D.V.Schroeder, An Moreovehr, i2f weile∝t L 0 keeping x/L (0,1) fixed, IntroductiontoQuantumFieldTheory,(Addison-Wesley, → ∞ ∈ Reading, 1995), chapter15. ρ(x,0) exhibits a similar singularity [17]. Thus, disor- h i [15] J. Schwinger, Phys. Rev. 128, 2425 (1962). For other der pushes states from high energies to the band center. generalizations of the Schwinger ansatz see P. Kopietz Forfinite∆ thiseffectcompeteswiththesuppressionof 0 and G. E. Castilla, Phys. Rev. Lett. 76, 4777 (1996); theDOSduetolong-rangeorder. Intheincommensurate ibid. 78, 314 (1997); P. Kopietz, ibid. 81, 2120 (1998). case (where ∆(x) is complex and in Eq. (3) we should [16] N. Schopohl, cond-mat/9804064. replaceiσ ∆ σ ∆ σ ∆∗ ) it is knownthat ρ(0) is 2 → + − − h i [17] L. Bartosch and P. Kopietz, in preparation. finiteinthewhite-noiselimit[12,23]. Weexpectthatthis [18] N. G. van Kampen, Stochastic Processes in Physics and remainstrueforarbitraryξ. Infact,thereexistsnumeri- Chemistry, (North-Holland, Amsterdam, 1981). calevidence that in the incommensurate case ρ(ω) can [19] F.J.Dyson,Phys.Rev.102, 1217(1956); S.V.Maleev, h i be accuratelycalculatedfrom Sadovskii’salgorithm[24]. Zh. Eksp. Teor. Fiz. 30, 1010 (1957) [Sov. Phys. JETP For a comparison with experiments one should keep in 6, 776 (1958)]. mindthatanyviolationoftheperfectchargeconjugation [20] D.G.SheltonandA.M.Tsvelik,Phys.Rev.B57,14242 symmetry will wash out the singularity at ω = 0. It is (1998). thereforeunlikelythatthesingularityisvisibleinrealistic [21] S. Eggert et al., Phys. Rev. Lett. 76, 1505 (1996). In materials, although an enhancement might survive. The this work it is shown that, for a given distance L from factthatthesingularbehaviorof ρ(0) intheFGMwith a boundary, the local DOS of the Tomonaga-Luttinger real∆(x)isonlydestroyedif ν =h ∆0i/(2K0ξ)exceedsa model at frequencies |ω|∼< vF/L is characterized by a | | | | finite critical value implies that in commensurate Peierls different exponentthan in thebulk. chainstrue pseudogapbehaviorshouldemergegradually [22] A. Comtet et al., Ann.Phys. (N.Y.) 239, 312 (1995). belowthePeierlstransitionwhentheorderparameter∆ [23] H. J. Fischbeck and R. Hayn, Phys. Stat. Sol. B 158, 0 is sufficiently large. Lee, Rice, and Anderson[3] came to 565 (1990); R. Hayn and J. Mertsching, Phys. Rev. B a similar conclusion within perturbation theory. 54, R5199 (1996). PSfrag replacements [24] A. Millis and H.Monien, unpublished. WethankM.V.Sadovskii,K.Scho¨nhammer,R.Hayn, R.H.McKenzie,andH.Monienfordiscussionsandcom- ments. This workwas financially supported by the DFG (Grants No. Ko 1442/3-1and Ko 1442/4-1). 0.04.0 1.0 i ) 3.0 0 i x; ( 2.0 R h 1.0 [1] G. Gru¨ner, Density Waves in Solids, (Addison-Wesley, 0.0 Reading, 1994). 0.0 0.2 0.4 0.6 0.8 1.0 [2] WemeasureenergiesrelativetotheFermienergyandset x=L theFermi velocity and ¯h equalto unity. FIG.1. Disorder average hR(x,i0)i=πhρ(x,0)i for L˜ =4 [3] P. A. Lee, T. M. Rice, and P. W. Anderson, Phys. Rev. and λ = 0 as function of x/L, see Eq.(19). From top to Lett.31, 462 (1973). bottom: ν =0,0.5,0.75,1,1.25,1.5,2. [4] M. V. Sadovskii, Zh. Eksp. Teor. Fiz. 77, 2070 (1979) [Sov.Phys. JETP 50, 989 (1979)]. [5] J. Schmalian et al., Phys. Rev. Lett. 80, 3839 (19P9S8)fr;ag replacements1.0 cond-mat/9804129. ) (cid:21) [6] O.Tchernyshydov,cond-mat/9804318. ( f((cid:21)) c [7] E. Z. Kuchinskii and M. V. Sadovskii, cond- (cid:23) (cid:23)c((cid:21)) mat/9808321. ), 0.5 (cid:21) [8] A.A.Ovchinnikovand N.S.Erikhman,Zh.Eksp.Teor. f( Fiz. 73, 650 (1977) [Sov. Phys. JETP 46, 340 (1977)]. [9] I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, In- 0.0 troduction to the Theory of Disordered Systems, (Wiley, 0.0 0.5 1.0 1.5 2.0 NewYork, 1988). (cid:21) 4 FIG. 2. Plot of f(λ) and νc(λ) defined in Eqs.(22) and (23). 5