The Effect of Randomness on the Mott State

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by  Thomas Nattermann

The Effect of Randomness on the Mott State
Thomas Nattermann, Aleksandra Petkovi´c, Zoran Ristivojevic, and Friedmar Schu¨tze
Institut fu¨r Theoretische Physik, Universit¨at zu K¨oln, Zu¨lpicher Str. 77, 50937 K¨oln, Germany
(Dated: February 6, 2008)
We reinvestigate the competition between the Mott and the Anderson insulator state in a one-
dimensional disordered fermionic system by a combination of instanton and renormalization group
methods. Tracingbackboththecompressibility andtheac-conductivitytoavanishingkinkenergy
7
oftheelectronicdisplacementfieldwedonotfindanyindicationfortheexistenceofanintermediate
0
(Mott glass) phase.
0
2
PACSnumbers: 71.10Hf,71.30.+h

n
a
J Introduction. Two non-trivial routes to insulating be- an instanton calculation of the conductivity [13, 14] we

5 havior in solids are connected with the names of Ander- show that an incompressible phase has always a gap in

son and Mott. Non-interacting electrons in disordered the low frequency conductivity, excluding the possibil-
]
l solids are always localized in one and two space dimen- ity of a MG phase. A number of further consideration
e
sions resulting from a combination of classical localiza- support this finding.
-
tr tion and quantum-mechanical interference [1, 2]. This The model. Following Haldane [15] we relate the mass
.s Andersoninsulator (AI) is characterizedby both a finite density ρ(x) of the electrons to their displacement field
at ac conductivity at low frequenciesand a finite compress- ϕ(x): ρ(x) = ρ0 + π1∂xϕ + ρ0cos(2ϕ + 2kFx) + ....
m ibility [3]. Interaction between electrons reduces the ef- Throughout the paper we will use e = ~ = 1. The
- fectofdisorderandmayleadto metallic behaviorintwo ground state energy of the system follows from E0 =
d dimensions [4]. In Mott insulators, on the other hand, lim T ln Dϕe−S where the action is [6, 11]
T→0
n insulating behavior results from the blocking of sites by −
R
o
repulsive interactionbetween electrons [5] and hence are
c LΛλTΛ
[ dominated by correlation effects. The Mott-insulating 1
1 (MI) phase is incompressible and has a finite gap in the S = 2πK Z Z dxdτn(∂τϕ)2+(∂xϕ+µ(x))2− (1)
conductivity. A natural question is then: what happens 0 0
v
in systems when both disorder and interactions are non- 2πκ
4 Fϕ (ζ(x)e−2ϕi+h.c.) wcos2ϕ +S .
1 negligible? Is there, depending on the strength of in- − Λ2 − − o diss
1 teraction or disorder, a single transition between these
1 two phases or is the scenario more complex? A particu- Here we introduced dimensionless space and imaginary
0
larly interesting case to answer these questions is that of time coordinates by the transformation Λx x and
7 →
electrons in one space dimension: interactions are very Λvτ τ where v is the plasmon velocity and Λ a
0
→
/ strong, destroying Fermi liquid behavior, but interac- large momentum cut-off. λT = v/T and L and de-
t
a tions can be partially incorporated into the harmonic note the thermal de Broglie wavelength of the plas-
m (bosonized) theory [6]. Additional motivation to study mons and the system size, respectively. w denotes the
- this case comes from its physical realization in quantum strengthoftheUmklappscatteringterm(orthestrength
d
wires [7] and ultra-cold gases [8]. of a commensurate potential) and F the external driv-
n
o One-dimensionalsystemsofthistypehaveindeedbeen ing force. µ(x) and ζ(x) result from the coupling of
c considered in a number of publications. Early work by the random impurity potential to the long wave length
: Ma [9] using real-space renormalization group (RG) for and the periodic part of the density, respectively, with
v
i thedisordered1DHubbardmodelsuggestsadirecttran- ζ(x)ζ∗(x′) =u2δ(x x′), µ(x)µ(x′) =σδ(x x′), all
X h i − h i −
sition from an AI to an MI phase. This result contrasts othercorrelatorsvanish. Finallyweaddadissipativeac-
ar with more recent work (see e.g. [10, 11] and references tionSdiss describing anOhmic resistancewhich may e.g.
therein)whereanewtypeoforder,differentfromtheMI result from the interaction with an external gate [16].
and AI phase, was found. In particular in [11] the exis- Rigidities. We begin with a discussion of the gener-
tence of a new Mott glass (MG) phase was postulated alized rigidities, which are related to the compressibility
which is supposed to be incompressible but has no gap andtheconductivityofthesystem. Wefirstconsiderthe
in the optical conductivity. application of a fixed strain ϑ by imposing the bound-
Inthis paper wereinvestigatethis problembyrelating ary conditions ϕ(0,τ) = 0 and ϕ(L,τ) = πϑLΛ. The
both the compressibility and the low frequency conduc- boundary condition in the τ-direction is assumed to be
tivity to the energy of kinks in the displacement pattern periodic. For ϑ 1 and L the corresponding in-
≪ → ∞
ofthebosonizedelectrons,resortingtoideasknownfrom crease∆E (ϑ,0)=E (ϑ,0) E (0,0)ofthegroundstate
0 0 0
−
the discussion of the roughening transition [12]. Using energyE (ϑ,0)is clearlyanevenbut notnecessarilyan-
0
2
kink anti-kink
j w states (see Fig.1). The latter are the classical ground
MI
states in the absence of the driving force which follow
2 p
from each other by a shift of a multiple of π. This ap-
w
1 c AI proximate treatment is reliable for not too large K and
L smallF. Theconductivitylnσ(F,0)= S (L ,L )is
x I x,s τ,s
−
x u then relatedto the actionS ofa criticaldroplet(the in-
I
stanton) of the new metastable state on the background
FIG. 1: Left: Kink and anti-kink in the displacement profile
of the old one [13, 14]. Here L ,L are the saddle
x,s τ,s
ϕ(x). The thin lines represent the minima of the potential
point values of the extension of the instanton which fol-
energy in the absence of the driving force, the bold line one
low from
metastable state and the dashed line the instanton configu-
ration, respectively. Applying a fixed strain to the system
2Σ (L ) L 1
only kinks (or anti-kinks) are enforced in the system. Right: S x x L +L (2Σ +ηln τ) FL L . (6)
I τ x τ x τ
schematic phase diagram with wc ∼σ2. ≈ Λv ξ −πv
Forsimplicityweassumedthattheinstantonisrectangu-
lar, a choice which is motivated by the fact that the dis-
alytic function of ϑ. Thus
orderiscorrelatedintime[13]. Σ /(Λv)and Σ playthe
x τ
∆E (ϑ,0) Υ ϑ2/2 if Σ =0, (2) roleofthesurfacetensionoftheinstanton. Thelastterm
0 x x
L (cid:12)(cid:12)L→∞ ≈ (cid:26) Σx|ϑ| if Υ−x1 =0. (3) itsertmherebsuullktscfornotmribthuetiodnissfirpoamtiotnhewehxitcehrnaalllowfiesldre.laTxhateioηn-
(cid:12)
Ther.h.s. ofthisrelationhastobeunderstoodasfollows: in the next metastable state [13].
if Σ = 0 then the stiffness Υ = Λ2/κ describes the Thelowfrequencyac-conductivity σ(0,ω)resultsfrom
x x
response to the twisted boundary conditions, Υ−1 is the the spontaneoustunneling processesbetweenmetastable
x
isothermal compressibility [13]. In this case the change statesandtheir instantonconfigurations(1and2inFig.
of ϕ is spread over the whole sample. If however Υ 1). Tunneling leads to a level splitting of the order
x
diverges, i.e. the system becomes incompressible, then
tohfeϕkfirnokmen0ertgoyπΣoxc/cΛurissinnona-nzearror.owInktihnikscraegseiotnheofchwaidntghe δE ≈q4Σ2x(Lx)Λ−2+C(vΛξK)2e−4LxΣτ, (7)
ξ much smaller than L. Creating a kink corresponds to whichhastomatchtheenergyωoftheexternalfield[14].
adding(orremoving)anelectronatthe kinkposition. A 1/Σ Kξ plays the role of the tunneling length, C is
τ
non-zero kink energy resembles the step free energy of a a num∼erical factor. This mechanism was first considered
surface below the roughening transition [12] [27]. for non-interacting electrons by Mott [3] and extended
Inasimilarmannerwecanapplynon-trivialboundary to the interacting case via instantons in [14]. Thus, to
conditionsintheτ-directionbychoosingϕ(x,0)=0and get a non-zero σ(0,ω) for arbitrary low frequency ω,
ϕ(x,Lτ) = πjLτ/v. This corresponds to imposing an 2Σx(Lx) < ωΛ 0 which requires 2Σx(Lx) 0 for
external current j = ∂τϕ /π at x=0 and x=L: a finite density o→f kink positions, which implies→a finite
h i
compressibility. The distance L between the sites in-
∆E (0,j) Υ j2/2 if Σ =0, (4) x
0 τ τ volved in the tunneling is then L (ω) ξKln(1/(κξω)).
x
L (cid:12)(cid:12)L→∞ ≈ (cid:26) Στ|j| if Υ−τ1 =0. (5) Fixed points and phases. We come n≈ow to the discus-
(cid:12) sion of the possible phases of model (1) by attributing
Υ is related (at T = 0) to the charge stiffness D =
τ them to their RG fixed points (denoted by superscript
1/Υ = κv2. D determines the Drude peak of the con-
τ ∗). Bare values will get a subscript . For small u and w
0
ductivity σ(ω) = Dδ(ω) + σ (ω) [6, 17]. In Lorentz
reg the lowest order RG-equations read [19]:
invariant systems (like the MI) Υ = (Λv)2Υ and
x τ
Σx =ΛvΣτ, provided they are finite. dK = K(au2+bw2), dσ =σ(1 cw2) (8)
So far we assumed that Υ , Σ are self-averaging
x/τ x/τ dl − dl −
if L . In the same way we may introduce local dw 2
→ ∞ = w(2 K σ) (9)
rigidities by applying twisted boundary conditions over
dl − − π
a large but finite interval [x,x+LxΛ], Λ−1 Lx L. du2 1
TheresultforΥ andΣ willthendepen≪dinge≪neral = u2(3 2K)+ σw2 (10)
x/τ x/τ dl − π
on the size L of the interval , i.e. Σ Σ (L ) .
Conductivitxy. In cases where D vaxn/τish→es, xth/τe fiexld or dκ = κ(z 1+ cw2) (11)
dl − − 2
frequency dependent conductivity σ(F,ω) may still be
non-zero, provided F or ω are finite. The non-linear dc- wherelisthelogarithmofthelengthscaleanda,b,care
conductivity σ(F,0) follows from an instanton approach positive non-universal constants. z denotes the dynam-
in which the current is calculated from the tunneling ical critical exponent which has to be determined from
probability of electrons between consecutive metastable the fixed point condition.
3
Luttinger liquid (LL). The LL phase is characterized one in a sample of length L is of the order ξ/L and
x x
by u∗ =w∗ =0 and hence Σ =Σ =0. z =1, K∗ >0 hence Σ Λ/(L κ ), i.e. the kink energy vanishes for
L L x τ L x ≈ x 0
and κ∗ =K∗/(πv∗)> 0. The fixed point is reached for L [13] and hence the system is compressible.
L L L x →∞
sufficiently large values of K and σ. The long time and Twisted boundary conditions in the y-direction give
large scale behavior of the system is that of a clean LL Σ (K ξ )−1 [13]. The non-linear conductivity is
y 0 A
∼
characterized both by a finite compressibility κ∗ and a given by σ(F,ω = 0) exp F /(FK2) where F
L A A
finite charge stiffness DL = κ∗LvL∗2. The dynamical con- 1/(κξA2) [13]. This res∼ult can−pbe understood as electro∼n
ductivity is given by σreg = iDL/(πω). The presence of tunneling between sites at which Σx 1/Lx. As ex-
∼
the forward scattering term ∼ µ(x)∂xϕ does not change plained already a vanishing Σx is also crucial for the
theseresultssinceitcanbealwaysremovedbythetrans- existence of the low frequency conductivity σ(0,ω)
formation ϕ=ϕ˜ xdx′µ(x′). ξ K(ωκ Kξ ln(κ ξ ω))2 ashasbeendiscussedinde≈-
− 0 M 0 M 0 M
Mott insulator (MRI). Here K∗ =κ∗ =u∗ =σ∗ =0 tail in [14]. This result can be understood in terms of
M M M M
but w∗ 1. Clearly the fixed point w∗ is outside the tunneling processes between rare positions at which the
M ≫ M
applicability range of the (8)-(11) but nevertheless some kink energies Σ (x) are much smaller than 1/(κ L ).
x 0 x
general properties of this phase can be concluded. The
system is in the universality class of the 2-dimensional
TABLE I: Properties of phases.
classical sine Gordon model which describes inter alia
the MI to LL transition and the roughening transition Phase κ∗ K∗ D Σx Σy σ(ω≪Σx)
2
of a 2-dimensional classical crystalline surface [12]. In LL κL >0 KL >0 κLvL 0 0 iDL/(πω)
sthtaeteMiIspghivaesnebΥyxϕa(nxd) =Υynπdivweirtghe.nTinhteegcelra.ssTichael gsyrostuenmd MAII κA0>0 00 00 ∼ξ0M−1 ∼∼ξξMA−−11 ∼ω20ln2ω
is characterizedby a finite kink energyΣ Λ(κ ξ )−1
x 0 M
∼
whereξM denotes the correlationlengthofthe MI-phase Mott-glass (MG). This new hypothetical phase was
[12]. From (6) we get σ(F,0) exp( FM/(KF)) where proposedin [11] to be characterizedby a vanishing com-
∼ −
FM ≈ 4πΣxΣτ/Λ ∼ 1/(κ0ξM2 ) is a characteristic depin- pressibility, κ∗G = 0, but a non-zero optical conductivity
ning field[22]. Accordingto (7)the acconductivity van- at low frequencies [11]. Since the phase is considered to
ishAesndfoerrsωon.i2nΣsuxl/aΛto.r (AI). Here w∗ = K∗ = 0 but tboe tghlaessAy,I,btohthe figrxoeudnpdoisntattvealcuaens wbeG∗,fouu∗Gn≫d b1y. Smiminiilmarilzy-
A A
u∗A ≫ 1. κ∗L ≈ κ0 is finite which is the result of the ing first the two backward scattering terms followed by
so-called statistical tilt symmetry [23] corresponding to minimization of the elastic energy. Although the ground
z = 1. The fixed point Hamiltonian is in the univer- state solution is now more involved than for the MI and
sality class of the 1+1 dimensional sine Gordon model AI case, for F = 0 it is clearly periodic with period π.
with a random phase correlated in the τ-direction. The As before kinks (or anti-kinks) with δϕ = π allow the
transition to the LL phase occurs at K =K∗(u) 3/2. accommodation of twisted boundary condit±ions and the
A ≥
NextwelookatΣ (L )finite lengthscaleξ L formationofinstantons. Avanishingcompressibilitycor-
x x A x
≪ ≪
L such that the parameters are close to their fixed point responds to a finite kink energy Σ > 0 which,
x
≥ C
values. ξ isthe correlationlengthwhichdivergesatthe according to (7), leads to a gap in the ac-conductivity.
A
Beresinskii-Kosterlitz-Thouless (BKT) transition to the Here we make the reasonable assumption that it is the
LL phase. To find the classical ground state of the sys- instantonmechanismwhichdominatesthelowfrequency
tem under periodic boundary conditions we have first to response [20]. Thus, in a system with a non-zero σ(ω)
choose 2ϕ + α = 2πn with n integer, and secondly for small ω also the compressibility has to be non-zero,
i i i i
the elastic term has to be minimized with respect to contrary to the claims in [11].
the n . The subscript i refers to the sites of the lat- Phase diagram. We come now to the discussion of the
i
tice with spacing ξ and ζ = ζ eiαi. The solution phase diagram of our model (1). From (10) follows that
A i i
| |
is n = [(α α )/2π] [13]. [x] denotes the therandombackwardscatteringtermisgeneratedbyfor-
i j≤i j − j−1 G G
closest inPteger to x. The ground state is uniquely deter- wardscatteringandthecommensuratepinningpotential.
mined by the α apart from the pairs of sites (of mea- Since σ(l) = σ el the two Eigenvalues λ = 3 2K and
j 0 1
−
sure zero) at which α α = π. At such pairs the λ =4 2K 4σ/π describingtheRG-flowofu2 andw2
j j−1 2
− ± − −
ground state bifurcates since two solutions are possible. around the the LL fixed point u∗ = w∗ = 0 have oppo-
In the case of non-interacting electrons (K = 1) bifur- sitesign: u(l)increaseswhereasw(l)decreases. Thusthe
cation sites are states at the Fermi energy. For pairs hypothetical MG phase, if it existed, cannot reach up to
at which α α = π +ǫ, ǫ 1 we can go over the point u = w = 0, in contrast to the findings in [11].
j j−1
− ± | | ≪
to an excited state by creating a kink which costs at Fromthisweconcludethatfornottoolargevaluesofw
0
most the energy Σ ǫΛ/(κξ ) [13]. Those ’almost’ the AI phase is stable, as shown in Fig. 1. To find the
x A
≈
bifurcating sites correspond to states close to the Fermi phaseboundarytotheMIphaseweconsiderthestability
energy. The smallest ǫ found with probability of order of the MI phase with respect to the formation of a kink
4
by the disorder. To lowest order in the disorder we get ity. Thus we exclude the possibility of the existence of a
for the kink energy in the MI phase Mott-glass phase.
WeacknowledgehelpfulremarksbySergeyArtemenko
Σx ∼ √κw0 1− 21(σ02/w0)14 −u0/(π2w03/4) (12) andAchimRoschandthe SFB608forfinancialsupport.
0 (cid:2) (cid:3)
which gives for the phase boundary between the MI and
theAIphaseasdepictedinFig.1. Asimilarresultfollows
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finite unless L .
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→∞
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ordered Mott insulator (1), tracing back both the com- [27] If we apply instead of the fixed boundary conditions an
pressibility and the ac-conductivity to the kink energy external stress to the system, then Σx is of the order of
Σ of the electronic displacement field that an incom- the critical stress to generate thefirst kink.
x
pressible system has also a vanishing optical conductiv-