Recursive Method for the Density of States in One Dimension

1.5 MB·English

by  Imke Schneider

A recursive method for the density of states in one dimension
Imke Schneider and Sebastian Eggert
Department of Physics and Research Center OPTIMAS,
University of Kaiserslautern, D-67663 Kaiserslautern, Germany
(Dated: January 19, 2010)
We derive a powerful yet simple method for analyzing the local density of states in gapless
one dimensional fermionic systems, including extensions such as momentum dependent interaction
parameters and hard-wall boundaries. We study the crossover of the local DOS from individual
densitywavestothewell-knownasymptoticpowerlawsandidentifycharacteristicsignsofspincharge
0 separation in possible STM experiments. For semi-infinite systems a closed analytic expression is
1 found in terms of hypergeometric functions.
0
2 PACSnumbers: 71.10.Pm,73.21.Hb,73.63.-b

n
a
The density of states (DOS) is a central quantity in Accordingly, the low energy theory is altered substan-
J
the study of electronic condensed matter systems. The tially and the generically predicted powerlaw behavior
9
corresponding expression for the local DOS with energy, momentum, and position is changed into a
1
more complicated behavior of the DOS.
] ρ(ω,x) = |hωm|Ψ†(x)|0i|2δ(ω−ωm) In this paper, we address the question of how to gen-
el Xm erally calculate the local DOS including complications
r- = 1 ∞ eiωthΨ(x,t)Ψ†(x,0)idt (1) whichmakethecentralinteractionparameterseffectively
t 2π Z energy and momentum dependent. The calculation is
s −∞
. basedon the Fourier transformin Eq. (1) for vertex cor-
t describes the probability of inserting an electron at a
a relation functions in finite systems, which yields an ex-
m given energy ω and can be probed in tunneling exper-
pansionindelta-functionsforthediscreteDOSwithcoef-
iments or, averagedovera range x, in photoemission ex-
- ficients that follow a recursion relation for arbitrary mo-
d periments. In two or more dimensions the DOS is typ-
mentum dependent interaction parameters. It is shown
n ically peaked at renormalized single particle excitation
o that boundaries cause a natural scale dependence in the
energies. In one dimensional systems, however, interac-
c descriptionofthelocalDOS,whichleadstothecrossover
tioneffects areenhanced, sothat the DOSis determined
[ fromboundaryto bulk behavior. Using a continuousde-
2 by collective many-body states instead [1]. Accordingly, scription, we obtain a closed analytic expression of the
therehasbeenconsiderabletheoreticalinterestinanalyz-
v local DOS as a function of energy and position. In or-
ing the one dimensional DOS for several decades [1–7].
3 der to identify spin- and charge separation by scanning
0 Typical features that have been predicted are separate
tunneling microscopy (STM) it is useful to analyze the
0 spin and charge density excitations and a characteristic
spatial Fourier transform of the local DOS.
0 depletion of the DOS at low energies and near bound-
. The starting point is the general expression of chiral
0 aries, which have been seen experimentally in some spe-
fermionicfieldsintermsofvertexoperatorsinthenormal
1 cial cases, such as cleaved edge overgrowthwires [8], su-
ordered form
9 perstructuresonsurfaces[9], andcarbonnanotubes [10].
:0 Recent theoretical activities in the field have produced O†(x,t):=c eiP∞ℓ=1√1ℓeiℓ∆ωtA†ℓ(x)eiP∞ℓ=1 √1ℓe−iℓ∆ωtAℓ(x),(2)
v notable advances in the areas of non-linear corrections
Xi [11], numerical simulations [12, 13], and applications of whereAℓ arelinearcombinationsofbosonicannihilation
r exact methods [14]. operators,e.g.oftheformAℓ(x)=αℓeikℓxbRℓ +βℓe−ikℓxbLℓ
a On the other hand, there remain a number of open for a periodic system with length L, energy spacing
questions, especially in regards to the applicability of ∆ω = 2Lπv, and kℓ = 2Lπℓ [15]. Such operators O† are
typicaleffectivelowenergytheoriessuchastheLuttinger usedto representleft- or rightmoving fermionoperators
liquidformalismtomorerealisticsystemsandmodels. In ψ† . In the case of several fermion channels (e.g. spin
L/R
fact a number of energy and length scales can affect the and charge) the chiral fermion field will be the product
behavior, including perturbations from other degrees of of operators O†(x,t) for each channel separately, all of
freedom, longer range electron-electron interactions, im- whichhavetheforminEq.(2),butwithdifferentenergy
purities, andfinite systemsizes,so that realistic systems spacings ∆ω. The magnitude of the prefactor c is typi-
are never truly scale invariant. The renormalizationdue callyunknownunlessacomparisonwithexactresultscan
to higher order perturbing operators leads to an energy be made. Zeromode terms havebeen omitted in Eq.(2)
and momentum dependence of the interaction parame- since they only shift the spectrum of the DOS.
ters, so that a description in terms of a single constant The formof the vertex operatorsin Eq.(2) is believed
Luttinger liquid parameter is in general not adequate. to apply to gapless interacting fermion systems in one
2
dimension. However, it is important to note that the statesareingeneralnotexactlydegenerateattheenergy
expression (2) already implies that the possible energy m∆ω. Therefore, the weights ρ are spread in energy
m
levels are assumed to be evenly spaced at ω =ℓ∆ω rela- [16], which is also the case in all examples that follow.
tivetotheFermienergyǫ ,whichisgenerallynotexactly However, the total value of ρ is expected to be rather
F m
justifiedinrealisticsystemsandisalsoexplicitlyviolated accurate [13].
in anyfinite lattice model. The reasonwhy bosonization Next, we consider fermions in a finite system with
remains useful is that the central information about the hard-wallboundaryconditionsΨ(0)=Ψ(L)=0. Inthis
electron-electron interactions is encoded in the Bogoli- case, the operators in Eq. (2) are of the form A (x) =
ℓ
ubovrotations,i.e.theexactformofthe linearcombina- (α eikℓx +β e−ikℓx)bR with k = πℓ [17]. The Green’s
ℓ ℓ ℓ ℓ L
tions of bosons A (x) in Eq. (2). These do not crucially functionsplitsintoauniformanda2k xoscillatingpart
ℓ F
dependontheassumptionofanequallyspacedspectrum, hΨ(x,t)Ψ†(x,0)i = (G (x,x,t)−ei2kFxG (x,−x,t))+
R R
since they are derived in momentum space. Therefore, c.c.,whereG (x,y,t)=hψ (x,t)ψ†(y,0)i[5–7,17]. For
R R R
the typical bosonization theory does not describe the ex- spinfulfermionsthosecorrelationfunctionshavetheform
act energy locations of the levels very accurately, but it ofEq.(3)forthespinfactor(K , v )andthechargefac-
s s
appears to predict the spectral weights surprisingly well, tor (K , v ) separately
c c
as can also be seen by numerical studies [13].
∞
Thegeneraltimecorrelationfunctioncanbecalculated 1
G (x,x,t)=|c |2 exp( e−iℓ∆ωνtγuni(x)) (7)
from Eq. (2) R x ℓ ν,ℓ
νY=c,s Xℓ=1
∞
1
hO(x,t)O′†(x,0)i=|c|2exp( e−iℓ∆ωtγ (x)), (3)
ℓ ℓ ∞ 1
Xℓ=1 G (x,−x,t)=|c |2 exp( e−iℓ∆ωνtγosc(x)),
R x ℓ ν,ℓ
whereγ (x)=[A (x),A′†(x)]. Inaccordancewiththepe- νY=c,s Xℓ=1
ℓ ℓ ℓ
riodicity in time, the Fourier transform in Eq. (1) yields where the commutators are dependent on position x
an expansion in delta-functions for the DOS and mode ℓ in this case γuni(x) = 1 + Kν +
ν,ℓ 4Kν 4
( 1 − Kν)cos(2k x) and γosc(x) = ( 1 − Kν) +
1 ∞ 4Kν 4 ℓ ν,ℓ 4Kν 4
2π Z−∞dteiωthO(t)O′†(0)i=Xm ρmδ(ω−m∆ω). (4) (cid:16)4K1ν + K4ν(cid:17)cos(2kℓx) + 2i sin(2kℓx) with ∆ων = vνLπ
for ν = s,c. Also the prefactor is position dependent
By partial integration a recursion formula for the indi- |cx|2 ∝ ν(sinπLx)4K1ν−K4ν from normal ordering. Us-
vidual spectral weights ρm is obtained ing Eq. (Q5) for the uniform/oscillating and spin/charge
parts separately ρuni/osc = 1 m ρuni/oscγuni/osc, it is
1 ν,m m ℓ=1 ν,m−ℓ ν,ℓ
ρm = (ρm−1γ1+ρm−2γ2+...+ρ0γm) (5) possible to solve the Fourier tPransform (1) even for this
m
more complicated case. The local DOS at the energy
withρ =|c|2. Thissimple butcentralresultisthemain ω =m ∆ω +m ∆ω is then simply the folded product
0 c c s s
instrument for analyzing the DOS in what follows. Be- of spin and charge, i.e.
sides the obvious simplicity of calculating the DOS with
Eq. (5), it has the major advantage of being free from ρu/o = ρuc,nmi/coscρus,nmi/soscδ(ω−mc∆ωc−ms∆ωs) (8)
divergences and regularization schemes. Moreover, it is mXc,ms
possible to analyze a dependence of the γ on the prin-
ℓ and ρ(ω,x)=(ρ (ω,x)−ei2kFxρ (ω,x))+c.c..
ciple energy quantum number ℓ which will be especially u o
Equations (5) and (8) can be evaluated very quickly
useful for boundaries and long range interactions.
with a few lines of code for a large number of levels and
Let us first re-examine the simplest example of a Lut-
different parameters. The analytic expressions for each
tinger liquid, which is a short-range interacting single
levelaresuperpositionsofdensitywavesuptowavenum-
channel system with periodic boundary conditions. The
commutator γ = [A ,A†] = 1 1 +K is then given in bers corresponding to ms and mc. The level density in-
ℓ ℓ 2 K creaseswithω,accordingtothenumberofwaysofchoos-
termsoftheLuttingerparamet(cid:0)erK ind(cid:1)ependentofℓ. In
ingm andm foragivenenergyintervalinEq.(8). Itis
this case, Eq. (5) is solved analytically by s c
therefore useful to average over a small energy window,
Γ(m+γ) 1 which results in a typical local DOS as a function of x
ρ =|c|2 ≈|c|2 mγ−1. (6)
m Γ(γ)Γ(m+1) Γ(γ) and ω as shown in Fig. 1 for Kc = 0.531, Ks = 0.618
and v /v = 1.618, where the oscillating part has been
c s
This is a well known result [3], which will be useful omitted. The solid lines in Fig. 1 show the characteris-
later. However,manystatescanpotentiallycontributeto tic standing density waves with higher wavenumbers as
each spectral weight ρ , the number of which increases the energy increases, with scaling of features along hy-
m
with the partitions of m. As mentioned above, those perbolaswithω =mv /x,whilev playsalesserrole. In
c s
3
FIG.1: (Coloronline) ρu(ω,x)in unitsof|cx=L/2|2 forKc = FIG. 2: (Color online) Spatial Fourier transform of the local
0.531,K =0.618andυ /υ =1.618averagedoveranenergy
s c s DOS with parameters as in Fig. 1 in arbitrary units. Inset:
interval of 0.3 in units of ∆ω .
c Cut at fixed energy.
contrast, for non-interacting systems Kc = Ks = 1 and by the following hypergeometric function
Fig. 1 is completely flat.
sepInaraotridoenrittoisgneteccelsesaarryevtiodceonncesidoefrtthheesopsicnillaantdingchpaarrgte, ρucni(y)=y2K1c−11F2(cid:16)8K1c − K8c;4K1c,4K1c + 21,−y2(cid:17),
(10)
which is best analyzed by a Fourier transform as first with F (a;b ,b ,z) = Γ(b1)Γ(b2) ∞ Γ(a+k) zk.
considered in Ref. [7]. In Fig. 2 we have plotted the 1 2 1 2 Γ(a) k=0 Γ(b1+k)Γ(b2+k) k!
ThedependenceonpositionisnoPtyetspecified. The full
absolute value of the Fourier transform in space of the
local DOS from several channels can then be obtained
local DOS including uniform and oscillating parts. It is
by folding accordingto the continuous versionof Eq.(8)
possibletoidentifyseparatelydispersingspinandcharge
whichforK =1againresultsinahypergeometricfunc-
features starting at 2k . In addition there is a third s
F
tion
maximum at 2k for all energies. The uniform DOS at
F
k = 0 shows the typical powerlaw increase. For spin- y
ρ = f(x)vc ρuni(y′)ρuni vc(y−y′) dy′ (11)
independent interactions Ks = 1,Kc < 1 the results for u vs Z0 c s (cid:16)vs (cid:17)
Figs.1and2lookqualitativelyverysimilar(notshown),
1 1
exceptforamissingslowlydispersingmaximawhichcan vc |cx|2 ω 2Kc−2
=
barely be detected in Fig. 2 starting from k =0. rv Γ(1 + 1 )(cid:18)∆ω (cid:19)
s 2 2Kc c
Afurtheranalyticalanalysisofthe localDOSispossi-
× F 1 − Kc; 1 + 1, 1 + 3,−(ωx)2
bleinthecontinuouslimitofthelevelspacingxmν ≫L. 1 2(cid:16)8Kc 8 4Kc 4 4Kc 4 vc (cid:17)
In this case, Eq. (5) becomes an integral equation
as shown in Fig. 3. The x-dependence f(x) is fixed by
the asymptotic behavior near x = 0 which yields the
ω
ωρ(ω,x)= ρ(ω′,x)γ(ω−ω′,x)dω′, (9) expression(6) with γ = 1 +1. Thereforean exactan-
Z 2Kc 2
0 alytic formula(12)has been derivedwithout the needto
evaluatetheFouriertransformofcorrelationfunctionsin
which again holds for the oscillating/uniform and
Eq.(1),whichinvolvesacomplicatedcontourintegration
spin/chargepartsseparately,whereγ(ω,x)arethecorre-
[5] and has so far not been possible analytically. Figure
sponding continuous functions. Let us focus on the uni-
3 also shows the DOS for all m and m quantum num-
s c
form charge part close to a single edge, i.e. γ(ω,x) =
bersfromthediscreterecursionformula(5). Clearly,the
1 + Kc + 1 − Kc cos(2ωx) from above. After individual weights drop with increasing ω, but the aver-
4Kc 4 (cid:16)4Kc 4 (cid:17) vc
rewriting Eq. (9) in terms of the dimensionless variable agedbehaviorfollowstheanalyticalprediction(12). The
y =ωx/v ,differentiatingthreetimes,andusingthefact boundary and bulk exponents of the effective powerlaws
c
thatγ′′ =4( 1 +Kc−γ),itispossibletoobtainathird ρ ∝ ωα are given by α ∼ γuni/osc−1 expanded for
4Kc 4 ν ν
order differential equation, which in turn can be solved small x or averaged over laPrge x, respectively, which are
4
2.5 formulas (5) in reverse: Knowing the lowest m spectral
ρ weights ρ , it is easily possible to uniquely determine
ℓ
the first m − 1 interaction parameters γ . The analo-
ℓ
2 gous statement is also true for continuous spectra. This
is especially promising considering the fast progress in
numerical [12, 13], exact [14], and experimental [8–10]
1.5
techniques in determining the DOS.
In summary, we have obtained a straight-forwardtool
for analyzing the local DOS in one dimensional inter-
1
actingsystems. The crossoverfromindividuallevelsto a
continuousspectrumhasbeenstudiedindetailforfuture
comparisonwithpossibleSTMexperiments. Aclosedan-
0.5
alytic expression was obtained for electrons in a system
with hard-wall boundaries. Long range interactions can
also be considered. The inversion of the recursion for-
00 5 10 y=ωx/v 15 mula is usefulto calculate the interactionparametersdi-
c
rectlyfromthelowenergyspectralweights,whichinturn
can be obtained from numerical, exact, or experimental
FIG. 3: (Color online) The uniform spectral weights ρu in methods. In this way the theory can be generalized to
units of |c |2 near a boundary as a function of ωx/v for systems which are not perfect Luttinger liquids, such as
x c
Kc =0.531,Ks =1,andυc/υs=1.618. Thelargenumberof harmonically trapped ultra-cold fermions.
spinandchargelevelsforx=0.1L(bluedots)followEq.(12)
We are thankful for useful discussions with I. Affleck,
(red line) after averaging.
M. Bortz, H. Johannesson, V. Meden, D. Schuricht,
S. So¨ffing, and A. Struck. This work was supported
by the DFG and the State of Rheinland-Pfalz via the
howeveronlyroughapproximationscomparedto the an-
SFB/Transregio 49 and the MATCOR school of excel-
alytic expressions above.
lence.
So far, we have considered a constant Luttinger pa-
rameterK,wherethescaledependenceoftheDOSstems
onlyfromtheboundary. However,forlongerrangeinter-
actions,K itselfisexpectedtobedependentonℓevenin
ℓ
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