Singularities and Pseudogaps in the Density of States of Peierls Chains

5 Pages·0.16 MB·English

by  Lorenz Bartosch

Singularities and Pseudogaps in the Density of States of Peierls Chains
Lorenz Bartosch and Peter Kopietz
Institut fu¨r Theoretische Physik, Universit¨at G¨ottingen, Bunsenstrasse 9, D-37073 G¨ottingen, 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 Schr¨odinger 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 condi-
tions [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)iioisanqduaMli´etlaintiv[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. M´elin, 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
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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.
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cond-mat/9804129.
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(cid:21)
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c
[7] E. Z. Kuchinskii and M. V. Sadovskii, cond- (cid:23) (cid:23)c((cid:21))
mat/9808321. ), 0.5
(cid:21)
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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