Capacitance of a resonant level coupled to Luttinger liquids

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by  Moshe Goldstein

Capacitance of a resonant level coupled to Luttinger liquids
Moshe Goldstein and Richard Berkovits
The Minerva Center, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
In this paper we study the differential capacitance of a single-level quantum dot attached to
quantumwiresintheLuttingerliquidphase,ortofractionalquantumHalledges,bybothtunneling
andinteractions. Weshowthatlogarithmic orevenpower-law divergenceofthecapacitance atlow
temperatures may result in a substantial region of the parameter space (including the ν = 1/3
filling). ThisbehaviorisamanifestationofgeneralizedtwochannelKondophysics,andcanbeused
toextract Luttinger liquid parameters from thermodynamic measurements.
PACSnumbers: 71.10.Pm,73.43.Jn,73.21.La,73.21.Hb
1
1 I. INTRODUCTION universal15, so it cannot be used to extract LL behavior.
0
2
The study of low-dimensional strongly-correlated sys- In this work we thus proceed to investigate intrinsi-
n tems has been at the focus of numerous experimental cally multi-channel scenarios, such as the side-coupled
a
and theoretical efforts in the last decades. Two impor- and embedded geometries, depicted in Figs. 1(b) and
J
tant categories in this field are quantum impurities, sys- 1(c),respectively. Anequivalentsystemisaquantumdot
9
tems of a few interacting degrees of freedom coupled to tunnel-coupled to two FQHE edges [Fig. 1(d)]. A gate
1
noninteracting environments (e.g., the spin-boson, An- electrodecanbeusedbothtocontrolthelevelenergyand
] derson, and Kondo models1); as well as gapless one- to measure the capacitance through AC conductance18;
l
l dimensional systems, whose low energy physics is gov- the capacitance could also be probed using a quantum
a
erned by the Luttinger liquid (LL) theory2. With recent point contact charge sensor. We find that in these con-
h
- advance in fabrication techniques, quantum impurities figurationsthereareregimesofpower-lawbehaviorofthe
s andLLs arebecoming the basic building-blocks ofnano- differential capacitance at low energies. Hence, we have
e
electronic circuits: both can be realized in semiconduc- here the rare opportunity of experimentally demonstrat-
m
tor heterostructures,metallic nanograinsand nanowires, ingLLbehaviorandextractingLLparametersfromther-
.
t or carbon-based materials. LL physics also applies to modynamic measurements. Furthermore, we show how
a
fractionalquantumHalleffect(FQHE)edges3. Research these phenomena expose generalizedtwo-channelKondo
m
of circuits composed of components from both families (2CK) physics1, and may thus enable its experimental
d- thusnotonlyallowsustostudyquantumimpuritiescou- realization. We introduce our model in Sec. II, present a
n pled to nontrivial environments in order to shed light preliminary analysis in Sec. III and a general treatment
o on both systems, but also is of much experimental rel- in Sec. IV, and summarize our findings in Sec. V.
c evance. There is thus no wonder that such problems
[
have attracted significant attention recently. However,
1 most of it was devoted to transport properties2–7, while
v other (e.g., thermodynamic) phenomena received much
3 less treatment8–15. This situation is now rapidly chang-
2
ing, with a surge of interest in the capacitance of low-
7
dimensional systems16–20.
3
. Previousstudies8,12,15 haveaddressedthe staticdiffer-
1
ential capacitance of a level coupled to a single FQHE
0
1 edge, or, equivalently21, to the end of a single quan-
1 tum wire, cf. Fig. 1(a). This problem was found to be
: equivalenttoananisotropicsingle-channelKondomodel,
v
where the occupied and unoccupied states of the level
i
X correspondtothe twospinstatesofthe Kondoimpurity.
r The system can thus be in one of two phases: (i) An
a
antiferromagnetic-Kondo type phase, in which the level
is strongly-coupled to its environment at low energies,
leadingtoaconstantlowtemperaturestaticcapacitance;
(ii) A ferromagnetic-Kondo type phase, where the level
is decoupled at low energies, leading to a 1/T diver-
gence of the capacitance when the level en∼ergy crosses FIG.1: DifferentconfigurationsofalevelcoupledtoLLwires:
the the Fermi energy. Moreover, the dependence of the (a) end-coupled; (b) side-coupled; (c) embedded; (d) level
coupled to FQHEedges (with FQHE bars hatched).
capacitance on the various interactions in the system is
2
II. MODEL nels of spinful electrons1. Its anisotropic version is:
We consider a single level quantum dot embedded be-
H =H +H +h S +S [J σ (0)+J σ (0)]+
tween the ends of two LL wires [cf. Fig. 1(c)] or between 2CK 1 2 z z z 1z 1z 2z 2z
two FQHE edges [Fig. 1(d)], and use the terminology S+[J1xyσ1 (0)+J2xyσ2 (0)]/2+H.c., (1)
− −
“leads” (denoted by ℓ = L,R) to refer to both possi-
bilities. Assuming spin-polarized electrons (the natural
where h is a local magnetic field, the J’s are exchange
situation in the FQHE case), both systems are then de- z
couplings, S are the components of the impurity spin
scribed by the Hamiltonian H =H +H +H +H . α
D W T U (α=x,y,z),andσ (x)arethecomponentsofspinden-
H = ε d d is the level Hamiltonian, with d the level iα
D 0 † sity at x of channel i = 1,2, governed by the nonin-
Fermioperator,andε itsenergy. The bosonizedHamil-
0 teracting Hamiltonian H . The latter can be bosonized,
tonian of the leads is HW =v/(4π) ℓ ∞ [∂xφℓ(x)]2dx and expressed as (usingiagain a chiral representation)
wchhirearel Bvoissetfiheeldvseloobcietyyinogf tehxecitcaotmPiomnsuR,t−aa∞tniodnφrℓe(lxa)tioanres Hi =v/(4π) ∞ [∂xφiρ(x)]2+[∂xφiσ(x)]2 dx, featur-
[φ (x),φ (y)] = iπδ sgn(x y)2. This representation ing charge-spRi−n∞(ρn-σ) separation. Only thoe spin fields
ℓ ℓ′ ℓℓ′
is natural in the FQHE cas−e3, but also describes the couple to the impurity, with σ (0) = ∂ φ (0)/(π√2)
iz x iσ
quantum wires system, after unfolding the decoupled and σ = [D/(2πv)]exp[ i√2φ (0)]. The 2CK prob-
i iσ
(Bogolubov-transformed) right and left moving fields21. lem is±equivalent to our o∓riginal system, provided that
Finally, the level and the leads are connected by tunnel- g = 1/2, with φ (φ ) being identified with φ (φ ),
1σ 2σ L R
ing terms HT = ℓtℓd†ψℓ(0)+H.c. (characterized by and S+ (S ) being identified with d† (d), so Sz
a1m/2p)litudUes∂tℓφ),(a0n)P/d(2lπoc)a(lwiintthersatcrteinogntshsHUU )=. T√hge(dF†edrm−i d†d − 1/222−. Under this mapping J1z = √2gUL, an→d
ℓ ℓ x ℓ ℓ J1xy = 2 2πv/DtL (and similarly for the other chan-
operaPtors at the end of lead ℓ is related to the bosonic nel/lead).pIt is well known that the 2CK model exhibits
fields through ψ (0) = D/2πvχ eiφℓ(0)/√g, where χ
ℓ ℓ ℓ a logarithmicallydivergingimpurity susceptibility in the
are Majorana fermions,pand D is the bandwidth. In channel-symmetric case1, which immediately implies a
quantum wires g is the LL interaction parameter (g <1
similar behavior for the capacitance of our system at
for repulsion, g > 1 for attraction)2; for edges of FQHE
g = 1/2. Furthermore, for g = 1/2 our system would
at simple filling fraction ν we have g =ν, assuming only be equivalent to a 2CK model6 with LL channels, with
electron tunneling, i.e., a dot outside the quantum Hall spin LL parameter g = 2g2. Experimental realization
σ
bar, cf. Fig. 1(d)3.
of our system will thus enable the investigation of this
Inarecentwork7 wehavedemonstratedthatthe side- generalized 2CK problem. For g < 1/2 (g < 1) we
σ
coupledgeometryofFig.1(b)isrelatedtotheembedded might then suspect that a behavior more singular than
configuration[Fig.1(c)]byadualitytransformationg logarithmic may arise. This expectation is indeed borne
↔
1/g, under which transmission maps onto reflection, but out by our subsequent calculations.
thecapacitanceremainsinvariant. Wewillthuscontinue
to analyze the embedded system, and later translate our
results to the side-coupled geometry.
IV. GENERAL ANALYSIS
III. PRELIMINARY CONSIDERATIONS
We first apply the transformation H˜ = H with
†
U U
Before going into an elaborate analysis we will try to U = ei√g(d†d−12)[ULφL(0)+URφR(0)]/(2πv), to eliminate
the interaction term from the Hamiltonian, at the
use two limiting cases in order to obtain some insights
cost of making the tunneling term more complicated:
into the problem. At g = 1 (and U = 0) we have
just a noninteracting resonant level, wℓhich has a con- H˜T = ξ−1d† ℓyℓχℓeiPℓ′Kℓℓ′φℓ′(0) + H.c., where yℓ =
stant low-temperature capacitance, proportional to the tℓ ξ/(2πv),KPℓℓ′ =[δℓℓ′ gUℓ′/(2πv)]/√g,andξ D−1
− ∼
inverse level width. As tunneling into a LL is enhanced ispashort-timecutoffscale. ToproceedwithRGanalysis,
forg >12,4,thisnonsingularbehaviorshouldapplythere we need to include in H˜ another term (invariant under
), H = 2ξ 1y d d 1 χ χ ei[φL(0) φR(0)]/√g +
foranyinteractionstrength. Moreover,fermionicpertur- U LR − LR † − 2 L R −
bative (in the electron-electroninteraction)renormaliza- H.c., describing inter(cid:0)-lead cot(cid:1)unneling with dimension-
tion group (RG) calculations5 indicate that this simple less amplitude yLR, generated by RG via virtual pro-
cesses.
behavior is not changed qualitatively for g smaller but
close to unity. Thus, interesting behavior is expected Usingthescalingdimensionsandoperatorproductex-
only for sufficiently small g. pansions of the various terms in the Hamiltonian, we
A deeper insight can be obtained from the 2CK prob- can derive RG equations for the various terms (Here
lem,describingaspin1/2impuritycoupledto twochan- δ gU /(2πv), δ (δ δ )/2, and ℓ˜= R for ℓ = L
ℓ ℓ L R
≡ ± ≡ ±
3
and vice-versa)2,23:
TABLE I: Definition of the different phases of the system,
ddlynℓξ =(cid:20)1− (1−2gδℓ)2 − (δ2ℓ˜g)2(cid:21)yℓ+yℓ˜yLR, (2) czoerroretsepmonpdeirnagtutroeothne-repshoansaencdeia(gorffa-mre,soFniga.nc2e.) lGinoena(orffc)oinsdtuhce-
tance. (T/TK)2g−1/TK becomes ln(T/TK)/TK at g = 1/2.
dyLR 1 In the side-coupled system G is replaced by e2/h−G, and g
= 1 y +y y , (3)
dlnξ (cid:18) − g(cid:19) LR L R by1/g7. Seethetext for furtherdetails.
dδ
dln+ξ =(1−2δ+)(cid:0)yL2 +yR2(cid:1), (4) SPCh-OasFeF eG2/onh eG2/offh Ca∼pa1c/itTaKnce
ddlδn−ξ =(cid:0)yL2 −yR2(cid:1)−2(cid:0)yL2 +yR2(cid:1)δ−. (5) SSCC--OONN21 ee22//hh 00 ∼(T/∼TK1/)2TgK−1/TK
These equations show that left-right asymmetry is rele- WC 0 0 ∼1/T
vant in general. Thus, in L-R asymmetric case the sys-
tem will end up (below an energy scale T , behaving as
a
(tL tR)2g/[2g−(1−δ+)2−δ+2] for UL = UR) in a fixed so that S assumes one of its possible values ( 1/2),
∼ − x
point where the dot is strongly-coupled to one of the ±
and φ (0) is confined to one of the resulting minima of
a
leads. The capacitance is thus the same as that of a dot
this term. One can then expand the partition function
coupledtoonelead[Fig.1(a)],whichdoesnotexhibitany
in the amplitude for tunneling events of φ (0) between
a
LL-type power-law behavior, as discussed above. As for
these minima (instantons), and find that it is relevant
transport, we effectively have a simple tunnel-junction
for g < 1/4 and irrelevant for g > 1/4. Off resonance
between two leads, corresponding to the dot and the
only the cotunneling process is important at low ener-
weakly-coupled lead in the original system4.
gies,as discussedabove. In the spin notationit becomes
We will therefore consider the more interesting sym-
S cos[φ (0) 2/g], so now instantons are relevant for
z a
metric case, tL = tR = t and δL = δR = δ+. It ∼g < 1 and irreplevant for g > 14. Both on and off reso-
shouldbenotedthatthissituationarisesnaturallyinthe
nance,relevantinstantonsimplysuppressedtransmission
side-coupled geometry [Fig. 1(b)], where the two chan-
and vice-versa.
nels are the left- and right-movers in the same lead7.
To summarize, we have three phases regarding low-
In the embedded [Fig. 1(c)] or FQHE configurations, L-
temperature L-R conductance: (i) The weak-coupling
R asymmetry can be controlled via voltages applied to
phase (WC), with suppressed low-energy conductance
theelectrodescontrollingthetunneljunctions. Sincethe
both on and off resonance; (ii) The strong-coupling on-
channelsymmetriccasefeaturesthehigheston-resonance
resonance phase (SC-ON), with good on-resonance but
transmission4 (see also below), changing tunnel-junction
suppressed off-resonance transmission; (iii) The strong-
parameters while monitoring conductance through the
coupling off-resonance phase (SC-OFF), with good on-
system can help tune the system into L-R symmetry. and off-resonancetransport. Thesephases,togetherwith
Although we are primarily interested in the capaci-
the corresponding picture for the side-coupledgeometry,
tance here, we will start with a discussion of the L-R are summarized in Table I and Fig. 27,24. Phase SC-
conductance4, in order to make the presentation more
ON is further divided into two regions according to the
self-contained. Off-resonance (ε = 0), the flow of y is
stoppedwhenξ 1 ε . Fromt0h6ispointon,theleveℓlis behavior of the capacitance, as we now discuss.
− ∼| 0| For the level population, only the t process is impor-
lockedintooneofitstwopossiblestates[nearlyoccupied
tant(ascotunnelingdoesnotchangethedotoccupancy).
(empty),fornegative(positive)ε ]. Thesystemthusbe-
0 Interestingbehaviorcanthus ariseonlyforsmall ε . In
havesasatunneljunction,whosestrengthyLR/ξ ∼t2/ε0 the WC phase the level is effectively decoupled fr|om0|the
for large enough ε . This cotunneling process will be
0 reservoirs at low energies, thus featuring discontinuous
relevant for g > 1 and irrelevant for g < 14. On reso-
dependence ofthe populationonε (as the lattercrosses
nance t is the leading low energy process. It is relevant 0
the Fermienergy),orequivalently,1/T divergenceofthe
(when small enough) for g > g = (1 δ )2+δ2 /2,
c − + + capacitance. IntheSCphasescontinuousbehaviorisex-
irrelevant otherwise. Whenever any(cid:2)of these two te(cid:3)rms
pected. Letusnowanalyzethisbehaviorinmoredetails,
is relevant, tunneling through the level enables perfect
using methods similar to those employed in Refs. 21.
conductance for small temperature and bias voltage.
Bythe Kuboformula,the(dynamic)levelcapacitance
Forstronglevel-leadcoupling,the dot-leadinteraction
isdeterminedbytheretardedpopulation-populationcor-
rapidly converges to its fixed point value δ = 1/2 (cf.
+ relation function. The latter can be obtained via an-
Eq. (4); At g = 1/2, this is the Emery-Kivelson point
alytical continuation from the imaginary time Green
of the equivalent 2CK problem6). Working in terms of
function, which, in the spin notation, is χ (τ)
the symmetric and anti-symmetric combinations of the zz ≡
bmoesrontihcenfiedldosesφsn/oat(xc)ou=pl(eφtLo(xt)h±e dφoRt(.x))O/n√2r,estohneanfocre-, −TˆτhTˆtτhSezt(iτm)Se-zo(r0d)eir=ing−oTpre[er(a|tτo|−r1a/nTd)H˜ZSzteh−e|τp|Ha˜rStizt]i/oZn,fwunitch-
H˜T is again dominant. Using the equivalent 2CK spin tion (note that S˜z = U†SzU = Sz). Near the strong
operators22, it acquires the form S cos[φ (0)/√2g] coupling fixed point H˜ S cos[φ (0)/√2g], as dis-
x a T x a
∼ ∼
4
Fourier-transforming, we find that χ (ω) is regular for
zz
small frequencies for g > 1/2, i.e., it goes to a constant
for vanishing ω. This constant will be proportional to
1/T , with the strong-coupling scale T (“Kondo tem-
K K
perature”) behaving as t2g/[2g−(1−δ+)2−δ+2] for small t
[In the noninteracting ca∼se (g = 1 and U = 0) T t2
ℓ K
∼
is simply the level width]. However, for g < 1/2, the
dynamic capacitance will have a power-law singularity
χ(ω) ω2g 1. The power-law will be smeared by fi-
−
∼
nite level energy, temperature, lead-length L, or chan-
nel asymmetry. The static capacitance will thus vary as
Λ2g 1 with Λ the largest of these scales (i.e., ε , T,
− 0
∼v/L, or T , respectively)25. At g = 1/2, the pow|er|-law
a
willbecomealogarithm,asexpectedfromthe2CKanal-
ogy discussed above. The different behaviors are again
summarized in Table I and Fig. 2.
V. CONCLUSIONS
To conclude, we have shown how the differential ca-
pacitance of a level coupled to the ends of two LL wires,
orside-coupledtoasinglewire,canexhibitpower-lawdi-
vergence at low temperatures, which enables a measure-
ment of the LL parameter g. This might help to resolve
the puzzle regarding tunneling measurements in FQHE
FIG. 2: Zero temperature phase diagram (phases defined in edges at ν = 1/3 filling, which show power-law temper-
TableI)andon-resonanceRGflow,projectedontheg-tplane: ature dependence deviating from the predicted one3. It
(a) embedded geometry; (b) side-coupled geometry. g± are shouldbenotedthatinarelatedlarge-dotsystempower-
the two roots of g +(1−δ )2/g = 47,24. See the text for
+ law behavior of the capacitance was predicted to occur
furtherdetails.
only in a subleading term, and would thus be harder to
detect9,10. In addition, the behavior found here reveals
cussed above. Hence, S H˜ S = H˜ /4. Since the generalized2CKphysics. Hence,itcreatesanalternative
z T z T
different terms in H˜ change the p−opulation of one of route for its realization in the laboratory, over existing
the leads by 1, weTalso have H˜ = H˜ , where possibilities26,27.
† T T
= eiπ(NL N±R), with N the VelectroVn nu−mber opera-
− ℓ
V
tor in lead ℓ. The other terms in the Hamiltonian com-
mute with both S and . Thus, the charge suscep- Acknowledgments
z
tibility can be written asVχ (τ) = Tˆ (τ) (0) /4.
zz τ †
−h V V i
Since N = NL + NR is constant, we get lnχzz(τ) We would like to thank Y. Gefen, I.V. Lerner, A.
∼
2π2 Tˆ N (τ)N (0) . At the strong-coupling fixed Schiller, and I.V. Yurkevich for useful discussions. Sup-
τ L L
− h i
pointthetwoleadsarewell-connected,sothelattercorre- port by the Adams Foundation of the Israel Academy
latoris simply the correlationfunction ofthe occupation of Science and by the Israel Science Foundation (Grant
of one half of an infinite wire, leading to χ (τ) τ 2g. 569/07)is gratefully acknowledged.
zz −
∼
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