Interchain conductivity of coupled Luttinger liquids and organic conductors

4 Pages·0.14 MB·English

by  Antoine Georges

Interchain conductivity of coupled Luttinger liquids and organic conductors
Antoine Georges1,2, Thierry Giamarchi2,1, Nancy Sandler3
1CNRS UMR 8549- Laboratoire de Physique Th´eorique, Ecole Normale Sup´erieure, 24 Rue Lhomond 75005 Paris FRANCE
2 Laboratoire de Physique des Solides, CNRS-UMR 8502, Batiment 510, Universit´e de Paris-Sud, 91405 Orsay, FRANCE
3 Laboratoire de Physique de la Matiere Condens´ee, Ecole Normale Sup´erieure, 24 Rue Lhomond 75005 Paris FRANCE
(February 1, 2008)
We reconsider the theory of dc and ac interchain conductivity in quasi-one dimensional systems.
0
0 Ourresultsareingoodagreementwiththemeasuredc-axisopticalconductivityof(TMTSF)2ClO4
0 and suggest that the c-axis dc-conductivity of (TMTSF)2PF6 in the 150K < T < 300K range
2 is dominated by precursor effects of Mott localization. The crossover from a Luttinger liquid at
highenergytoaFermiliquidatlowenergyisalsoaddressed,withinadynamicalmean-fieldtheory.
n
Implicationsfortheinter-chainresistivityandDrudeweightintheFermiliquidregimearediscussed.
a
J
Inter-chain(orinter-plane)coherenceis akeyissuefor temperature scale than the early estimates based on the

6 the physicsofanisotropicstronglycorrelatedsystems. It NMR.Furthermore,itisclearfromthe opticaldatathat

isofspecialimportanceforthequasi-onedimensionalor- another important energy scale is present in these com-
]
l ganic conductors TMTSF X and TMTTF X which are pounds,associatedwithaMottgapsignaledbytheonset
e 2 2
- systems of weakly coupled chains [1]. Since the one di- of optical absorption at 100cm−1 150K. Transverse
r mensional (1D) interacting electron gas has non Fermi hoppingisactuallyrespo∼nsibleforth≃e metalliccharacter
t
s liquid (Luttinger liquid) properties [2], one expects in of the (TMTSF) X compounds, which otherwise would
. 2
t these compounds a crossover from Luttinger liquid be- be Mott insulators [11,15]. These findings raise a num-
a
m havior to a coherent behavior as the temperature or fre- ber of important and largely unanswered questions such
quency is lowered. The latter can be either a metallic as the precise nature of the dimensional crossover, the
-
d three-dimensional conductor (presumably Fermi-liquid origin of the anomalies observed for 10K < T < 100K
n
like), or a phase with long-range order. (calling in particular for a reinterpretation of the NMR
o
Although there is general agreement on this qualita- in this range), and the actual range of the LL behavior.
c
[ tive scenario, there is considerable debate on how pre- Inthisletter,wefocusoninter-chaintransportandop-

cisely this crossover takes place [3–7]. As was pointed ticalresponse,whichareidealprobesofthese issues. We
1
outearlyon[8],theinteractionsrenormalizetheeffective first reconsider the theoretical analysis of the ω and T-
v
3 interchain hopping of weakly-coupled Luttinger chains. dependence of the inter-chain conductivity for the sim-
6 This leads to a crossover scale E∗ t(t⊥/t)1/(1−α) = plest case of weakly coupled metallic Luttinger liquids.
∝
0 t (t /t)α/(1−α) for the onset of coherence along one of We findapowerlawofthetemperatureorthe frequency
⊥ ⊥
1 the transverse directions with interchain hopping t . In σ (T,ω) (T,ω)2α−1. Theexponentisatvariancewith
⊥ ⊥
0 ∝
this expression, t is the intra-chain hopping and α previous theoretical predictions [6,16,14], but is in rea-
0 ≡
0 41(Kρ +1/Kρ)− 21 is the Fermi surface exponent. The sonable agreement with recent measurements of c-axis
t/ scale E∗ can be much smaller than the bare interchain optical conductivity of the (TMTSF)2ClO4 compound
a hopping t . With this idea in mind, analysis of NMR [17]. We then show that understanding the observed
⊥
m data [9,10] led to believe that the 1D Luttinger liquid T-dependence of c-axis transport in (TMTSF) PF re-
2 6
- (LL) to Fermi liquid (FL) crossover took place in the quires to take into account the physics associated with
d
(TMTSF) Xfamilyatalowtemperature,oftheorderof theMottgapapparentinopticalmeasurements. Finally,
n 2
10 15K. This scale seemed consistent with the above we consider a dynamical mean-field theory (DMFT) for
o
−
c estimate of E∗ since the interchain hopping along the b- the description of the dimensional crossover, and draw
: axis is t 300K, an order of magnitude smaller than someconsequencesfortheinter-chainconductivityinthe
v b
∼
i the hopping ta along the chains. Recent analysis of the low-temperature FL regime.
X
transport and optical properties [11] have led to a re- We consider the hamiltonian of coupled chains:
r examination of this commonly accepted view. Measure-
a ment of the optical conductivity along the chains agrees H = H1(mD)− t⊥ (c+imσcim′σ +h.c) (1)
withaLLpictureathighenoughfrequencyandprovides Xm hmX,m′i Xiσ
a determination of the LL exponent K [12,13](of order
ρ
K 0.23). This value is consistent with the temper- in which H(m) stands for each individual chain m, and
ρ 1D
≃
ature dependence of ρa(T) in the 100K 300K range the inter-chain hopping t⊥ will be first considered, for
−
(suitably corrected for thermal expansion [1]). However, simplicity, to be identical for all transverse directions.
these studies andmostnotably the recentmeasurements UsingtheDMFT approachdetailedbelow,ageneralfor-
of the interchain dc conductivity [14] revealed that the mula(Eq.6)forthetransverseconductivitycanbeestab-
1D regime only extends down to 100K, a much higher lished. When the inter-chain hopping t⊥ can be treated
∼
in perturbation theory to lowest order, this formula re-
1
ducestothatestablishedinRef.[14]withinthetunneling relevant. This suggests that if the transverse hopping
approximation [18]: was entirely suppressed in these compounds, they would
actually be one-dimensional (quarter filled) Mott insu-
Reσ (ω,T)pert t2 dx dω′A (x,ω+ω′) lators. Fortunately, optical experiments extend to high
⊥ ∝ ⊥Z Z 1D × frequencies compared to the Mott gap. Thus the com-
A (x,ω′)f(ω′)−f(ω′+ω) (2) plications associated with the proximity to the Mott in-
1D
× ω sulating state can be treated perturbatively, making the
comparison to Eq.(3) relevant. Recently, optical stud-
where A (x,ω) is the one-electron spectral function of
1D
iesof(TMTSF) ClO havebeenperformed[17],andthe
asinglechain,andf the Fermifunction. Letusconsider 2 4
high-frequencyopticalconductivityσ (ω)alongthe least
firstthe simplest casein whicheachdecoupledchainis a c
conducting c-axis was found to be a broad, very slowly
LuttingerliquidcharacterizedbyagivenvalueofK ,and
ρ increasing function of frequency in the range 50cm−1 <
all intra-chain umklapp processes can be neglected. For
ω <105cm−1. Thisslowlyincreasingcontinuumisessen-
thevaluesofK ofinteresthere,thetransversehoppingis
ρ
tially independent of temperature for 10K< T < 300K,
thenarelevantperturbationintherenormalizationgroup
andis ingoodagreementwithourresultσ (ω) ω2α−1.
sense, so that it is legitimate to treat it perturbatively c ∝
The exponent 2α 1 = (K + 1/K )/2 2 depends
only at sufficiently high energy or temperature. Eq. (2) − ρ ρ −
quite sensitively on the value of K (2α 1 = 0.12 for
thus applies only when kT E∗ or h¯ω E∗, where ρ −
≫ ≫ K = 0.25 and 2α 1 = 0.6 for K = 0.2). The latter
E∗ = t(t⊥/t)1/(1−α) is the single-particle crossover scale ρ − ρ
value 2α 1 0.6 provides a reasonable fit to the ex-
mentionedabove[8,3,19]. Inthishigh-temperature/high- − ≃
perimental data of [17]. Note that values of K in the
energy regime, the system can be considered to be quasi ρ
0.2-0.25range are quite consistent with the other exper-
one-dimensional. Using knownproperties of the spectral
imental findings discussed above. In contrast, the ω2α
functions in a Luttinger liquid, we then obtain from (2):
law advocated in [6,16,14] would require K 0.35 to
ρ
≃
t 2 T 2α−1 ω explain these data, a value quite inconsistent with that
Reσ⊥(ω,T)pert ⊥ Fα (3) obtained from the longitudinal optical conductivity. Fi-
∝(cid:18) t (cid:19) (cid:18)t (cid:19) (cid:16)T(cid:17)
nally, an important issue in view of the success of this
In this expression, F is a scaling function such that simple theory is whether it can also account for the be-
α
F (0) = const and F (x 1) x2α−1. Thus, we find havioroftheopticalconductivityalongtheb-axis. Since
α α
≫ ∼
Reσ (T ω) T2α−1 for kT E∗, while at high- t /t 0.1,anincoherentregimeshouldalsoapplythere
⊥ b a
≫ ∝ ≫ ≃
frequency (h¯ω E∗), we obtain σ (ω T) ω2α−1. atthosehighfrequencies. Preliminaryanalysis[20]ofthe
⊥
≫ ≫ ∝
This behavior of σ (T) differs from the conclusions σ data of[17]seems in agreementwith sucha behavior,
⊥ b
ofpreviousauthors. In[6], the authorsuseda somewhat but more detailed comparison with the data is needed.
differentapproachthanthedirectKuboformulaandcon- We now discuss the measurements of the c-axis trans-
cluded that σ (T) T2α. In [14], it was noted that this portin(TMTSF) PF [14,21]. Becauseofthelargether-
⊥ 2 6
∝
temperature dependence also results from (2) provided malexpansion,thesewereperformedatseveralpressures
that the spatial integral is cutoff at a scale of the order so that a volume correction could be made. This yields
of the lattice spacing. According to these authors, this data for the c-axis resistivity ρ (T) which can be inter-
c
cutoff is justified by the incoherent nature of the propa- preted as corresponding to a compound with constant
gationof a physicalelectron within each chain (in which latticeparameters. Thisvolume-correctedρ (T)displays
c
the true quasiparticles are spinons and holons). In our a marked increase by more than a factor of 3 upon cool-
opinion however, this information is already encoded in ing from T = 350K down to T = 100K (followed by a
the spectral function of the physical electron A (x,ω) rapid downturn at lower temperature which presumably
1D
and no extra cutoff has to be introduced in (2). This marks the onset of inter-chain coherence). Such a large
conclusion would of course be changed in the presence variationofρ (T)cannot be explainedbythedependence
c
ofspecificformsofintra-chaindisordersuchasarandom ρ 1/T2α−1 found above with acceptable values of K
c ρ
∝
forwardscattering,whichwouldspoilmomentumconser- (which correspond to rather small values of 2α 1). In
−
vation and bring in a natural cutoff of the order of the [14],itwasfittedtoa1/T1.4powerlaw,whichhappensto
mean-free path. Note however that more realistic forms be in reasonable agreement with the 1/T2α dependence
of disorder would be signaled by localisation effects. advocated by these authors. However, the temperature
We now turn to a comparison with experiments on rangeinwhichthesedataareobtainedissuchthatkT is
(TMTSF) X. Itisincreasinglyrecognizedthattheprop- never significantly larger than the “would-be” 1D Mott
2
erties ofthese compounds resultfromthe commensurate gap which can be extracted from the optical measure-
quarter-fillingoftheband[11,13,14]. Transportandopti- ments (100 200cm−1) leading to transport gaps (half
−
calmeasurementsleadto consistentvalues ofthe LL pa- the optical gap) in the range ∆ 75 150K. As a re-
≃ −
rameterK 0.23. NotethatK <0.25isthecondition sult,theproximitytotheincipientMottinsulatorcannot
ρ ρ
∼
upon which the quarter-filled umklapp process becomes be neglected, and we propose that this effect is the one
2
responsiblefortheincreaseofρ (T). Indeed,wehaveat- formulate a controlledDMFT of strongly-correlatedsys-
c
tempted a phenomenologicalfit by an activated behavior tems [24,25]. We consider eachone-dimensionalchain to
ρ (T) 1/T2α−1exp∆/T. and found that it provides be coupled to z nearest neighbor ones, with the
c ⊥
∝ → ∞
a fit of the data of Ref. [14,21] which is as satisfactory transverse hopping scaled as t = t /√z , so that the
⊥ ⊥ ⊥
as the power-law fit used in [14]. The value of ∆ used non-interacting density of states in the transverse direc-
e
in this fit is of the same order of magnitude than the tionD(ǫ ) δ[ǫ ǫ(k )] remains well-defined. It
⊥ ≡ k⊥ ⊥− ⊥
Mott gap seen in optical measurements. The discrep- is easily shownPthat in this limit the self-energy becomes
ancybetweenthetheoreticalresult,Eq.(3)andtheexper- independent of transverse momentum k : Σ=Σ(k,iω )
⊥ n
imentally observed behavior of ρ (T) is, in our opinion, (k is the momentum along the chains). Furthermore,
c
a strong indication that an explanation based on inco- the calculation of Σ reduces to the solution of an effec-
herent tunneling between LL chains in a purely metallic tive one-dimensional problem specified by the effective
regimeis insufficient andthat the physics ofMottlocali- action:
sation plays an important role in the temperature range
β
150K<T <300K. S = dτdτ′ c+(τ) −1(i j,τ τ′)c (τ′)
A more refined theory of the interplay between inter- eff −Z Z0 Xij,σ iσ G0 − − jσ
chain hopping and Mott localisation is clearly needed,
β
including both the transverse hopping and the quarter- + dτHint[ c ,c+ ] (4)
Z 1D { iσ iσ}
filling umklapp scattering [22]. Due to the umklapp, the 0
effective Kρ will be renormalized towards smaller values whereHint is the interactingpartofthe on-chainhamil-
1D
as temperature is decreased, hence leading to larger val-
tonian in (1), while is an effective bare propaga-
0
ues of ρ , as expected. The main issue is whether the G
c tor which is determined from a self-consistency con-
intra-chain resistivity is also strongly affected by these
dition. The latter imposes that the Green’s function
renormalizations. Indeed,nosignofanupturninρa(T)is G(i j,τ τ′) c(i,τ)c+(j,τ′) calculated from
eff
observed experimentally for the TMTSF X compounds. − − ≡ −h i
2 S coincides with the on-chain Green’s function of the
eff
Sdienccreeaρsae(Tin)K∝g12is/4aTlm16oKsρt−c3o[m11p]e,nitsaistecdonbcyeivaanbalesstohcaitatthede originalproblem,withaself-energyΣ=G0−1−G−1. This
ρ condition reads:
decrease of the effective g due to t . This however,
1/4 ⊥
D(ǫ )
remains to be verified and is probably the most chal- ⊥
G(k,iω )= dǫ (5)
n ⊥
lenging issue for the picture that we propose. Transport Z iωn+µ ǫk Σ(iωn,k) ǫ⊥
− − −
in the TMTTF X family is interesting in this respect.
2 The DMFT equations (4,5) fully determine the self-
ThesecompoundsareinsulatorswitharatherlargeMott
energy and Green’s function of the coupled chains. Fur-
gap, and indeed both ρ (T) and ρ (T) display an acti-
a c thermore, inter-chain transport properties simplify in
vated behavior at low temperature. Above some tem-
thisapproach. Becausethecurrentvertexisanoddfunc-
perature scale however, ρ (T) becomes metallic, while
a tionofthetransversemomentum,vertexcorrectionsdrop
ρ (T) still displays insulating behaviour (albeit slower
c out of the inter-chain conductivity, which reads:
than activated). Studies of the transport properties of
thesecompoundsoveramoreextendedrangeofpressure dk
Reσ (ω,T) t2 dǫ D(ǫ ) dω′A(ǫ ,k,ω′)
and temperature (and in particular a detailed compari- ⊥ ∝ ⊥Z ⊥ ⊥ Z 2π Z ⊥
son of the energy scales appearing in σa(ω), ρa(T) and f(ω′) f(ω′+ω)
ρc(T)) would certainly prove very interesting. ×A(ǫ⊥,k,ω+ω′) −ω (6)
Letusfinallyaddresstheissueofthecrossoverbetween
a Mott-Luttinger liquid at highenergy/temperatureand Here, A(ǫ ,k,ω) = 1ImG(ǫ ,k,ω) is the single-
⊥ −π ⊥
acoherent(presumablyFermi-liquid)stateatlowenergy. particle spectral function of the coupled chains system.
Themaintheoreticaldifficultyisthatthiscrossoverisas- Eq.(6) has a much wider range of applicability than its
sociated with the breakdown of perturbative expansions small-t limit, Eq.(2),sinceitholdsallthe wayfromthe
⊥
int . Ontheotherhand,perturbativeapproachesinthe LL to the FL regime.
⊥
interactionstrengthcould be usedin the FL regime, but Solvingquantitativelythese DMFT equationsisstill a
fail in the LL regime. The necessity of treating t in a challenging problem however. In the single-site DMFT
⊥
non-perturbativemannerwasrecentlynotedbyArrigoni (corresponding to the usual d = limit [24]) the map-
∞
[23],whousedalimitofinfinitetransversedimensionality pingisontoasingle-impurityAndersonmodelwithaself-
as a guidance for resumming the perturbation series in consistent bath, and severaltechniques have been devel-
t . Here, we show that this limit provides a generalized opedtohandlethisproblem. Herehowever,themapping
⊥
dynamicalmean-fieldtheory(DMFT) frameworkforthe is onto a self-consistent 1D-chain, and the problem is an
description of the crossover, and we draw some conse- order of magnitude more difficult. Nevertheless, several
quencesforinter-chainconductivity. Thelimitofinfinite conclusionscanbe drawnfromthe aboveequations even
dimensions has been used extensively in recent years to in the absence of a full solution. We shall consider again
3
the simplest case of metallic 1D LL chains, neglecting WethankD.Jerome,J.Moser,L.Degiorgi,G.Gru¨ner,
umklappscatteringandMott physics. Inthe 1Dregime, andV.Vescoliforsharinganddiscussingtheir datawith
the self-energy behaves as Σ t((ω,k)/t)1/(1−α). From us. Discussions with M. Gabay and A. M Tremblay are
∝
(5), it is clear that the inter-chain hopping becomes rel- also acknowledged. A.G. and T.G. would like to thank
evant when t > Σ, and we recover the crossover scale the ITP, Santa Barbara for support (under NSF grant
⊥
E∗ t(t /t)1/(1−α). At energies smaller than E∗, the No. PHY94-07194) and hospitality during part of this
⊥
∝
DMFT approach leads to Fermi-liquid behavior. This is work.
clearfromviewingthe effective 1Dmodelas asetofAn-
dersonimpuritymodelscoupledtogetherbythenon-local
part of the “bare” propagator . Each of the decou-
0
G
pledAndersonmodelisaFermi-liquidatlow-energyand
the non-local hybridization is a smooth perturbation on
the FL ground state. In other words, the physics below
[1] D. Jerome, in Organic Superconductors: From
E∗ is smoothly connected from small t /t to large t /t.
⊥ ⊥ (TMTSF)2PF6 to Fullerenes (Marcel DekkerInc.,1994),
When t t and for energy scales much smaller than
⊥ Chap. 10, p. 405.
≪
the bandwidth (of order t), all single-particle quantities [2] H. J. Schulz, in Mesoscopic quantum physics, Les
shouldobeyascalingbehaviorinthevariablesω˜ =ω/E∗, Houches LXI, edited by E. Akkermans et al. (Elsevier,
x˜ = xE∗/t, T˜ = T/E∗. In particular tΣ(x,ω,T) = Amsterdam, 1995).
E∗t Σ˜(x˜,ω˜,T˜), tG(x,ω,T) = (E∗/t )G˜(x˜,ω˜,T˜) with [3] C. Bourbonnais and L. G. Caron, Int. J. Mod. Phys. B
⊥ ⊥
Σ˜ and G˜ universal scaling functions associated with the 5, 1033 (1991).
crossover. Alow-frequencyexpansionofthescalingform [4] X. G. Wen,Phys. Rev.B 42, 6623 (1990).
[5] S.BrazovskiiandV.Yakovenko,J.dePhys.(Paris)Lett.
ofΣintheFLregimeyieldsafinitequasiparticuleresidue
Z ∝ (t⊥/t)1−αα ∝ E∗/t⊥ (shown in [23] to have only 4(169,9L2)1.11 (1985); V. M. Yakovenko, JETP Lett. 56, 510
weak k-dependence along the Fermi surface). Z can be
[6] D. G. Clarke, S. P. Strong, and P. W. Anderson, Phys.
much smaller than unity, while the effective mass (spe- Rev. Lett.72, 3218 (1994).
cific heat) ratio m∗/m does not become large because
[7] H. J. Schulz, in Correlated fermions and transport in
∂Σ/∂ω and ∂Σ/∂k scale in the same manner. Using the mesoscopic systems, editedbyT.Martin etal.(Editions
FL form of Σ in (6), we obtain the T = 0 inter-chain fronti`eres, Gif sur Yvette,France, 1996).
conductivityas: σ (ω,T =0)= ωD2⊥δ(ω)+σreg(ω)with [8] C.Bourbonnaiset al.,J.dePhys.(Paris)Lett.45,L755
⊥ 8 ⊥ (1984).
ω 2/8 Z2t2/t = t(t /t)2/(1−α). The total spectral
D⊥ ∝ ⊥ ⊥ [9] C. Bourbonnais, J. de Phys.I 3, 143 (1993).
weight dωσ (ω)=ω2 /8 is, from the f-sum rule, pro-
⊥ P⊥ [10] P. Wzietek et al.,J. dePhys. I 3, 171 (1993).
portionRal to the inter-chain kinetic energy, and hence to [11] T. Giamarchi, Physica B 230-232, 975 (1997).
t2/t, so that the relative weight carried by the Drude
⊥ [12] M. Dressel, A. Schwartz, G. Gru¨ner, and L. Degiorgi,
peak is small, of order ωD2⊥/ωP2⊥ ∝(t⊥/t)2α/(1−α) =Z2 Phys. Rev.Lett. 77, 398 (1996).
This is in reasonableagreementwith the opticaldata on [13] A. Schwartz et al.,Phys. Rev.B 58, 1261 (1998).
the b-axis reflectivity of (TMTSF) PF [26], for which a [14] J. Moser et al.,Eur. Phys. J. B 1, 39 (1998).
2 6
Drudepeakisseenatlowtemperature,withω2 /ω2 [15] V. Vescoli et al.,Science 281, 1191 (1998).
D⊥ P⊥ ≃
0.03 (while t /t 0.1). Scaling considerations also lead [16] P. W. Anderson, T. V. Ramakrishnan, S. Strong, and
b a
toaninter-chainr≃esistivityρ (T)/ρ =(t/E∗)R(T/E∗), D. G. Clarke, Phys.Rev. Lett. 77, 4241 (1996).
with R(x 1) x1−2α , ⊥R(x 01) x2 and ρ = [17] W. Henderson et al.,Eur. Phys.J. B 11, 365 (1999).
0
≫ ∼ ≪ ∼ [18] Transverse conductivities are given in the units of
hV /(e2a2) [18]. This implies that ρ /ρ =A(T/t)2 in
m ⊥ ⊥ 0 e2a2⊥/(hVm),witha⊥ theinter-chainspacingandVmthe
the FL regime, with an enhanced value of the coefficient
volume of the unit cell, and the on-chain lattice spacing
3
A (t/t⊥)1−α. Note that ρ⊥ is typically much bigger is normalized to unity.
∝
than the Mott limit ρ0, as observed experimentally. [19] D.Boies,C.Bourbonnais,andA.-M.S.Tremblay,Phys.
In conclusion, we have reconsidered in this paper the Rev. Lett.74, 968 (1995).
theory of interchain conductivity in quasi-1D systems. [20] L. Degiorgi, privatecommunication.
When applied to organic compounds, good agreement is [21] J. Moser, Ph.D. thesis, Paris XI University,1999.
found with the frequency dependence of σ (ω). while c- [22] S.Florens,A.Georges,andT.Giamarchi,inpreparation.
c
axisdctransportappearstobedominatedbytheproxim- [23] E. Arrigoni, Phys. Rev. Lett. 83, 128 (1999), ; see also
cond-mat/9910114.
ity ofthe Mott gap∆. This calls for additionaltheoreti-
[24] A.Georges,G.Kotliar,W.Krauth,andM.J.Rozenberg,
calworkinthe regimewhereE∗ t(t /t)1/(1−α) <∆<
∝ ⊥ Rev. Mod. Phys. 68, 13 (1996).
t⊥. It would also be valuable to see whether the regime [25] T.Pruschke, M.Jarrell, and J.Freericks, Adv. Phys. 42,
∆ < E∗, which is simpler to analyze theoretically, can
187 (1995).
be realized experimentally in some family of compounds
[26] V. Vescoli, phDthesis, unpublished.
(e.g by applying uniaxial stress to TMTTF X).
2
4