Quantum criticality of the two-channel pseudogap Anderson model: Universal scaling in linear and non-linear conductance

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by  Tsan-Pei Wu

Quantum criticality of the two-channel pseudogap Anderson model: Universal scaling
in linear and non-linear conductance
Tsan-Pei Wu1 and Chung-Hou Chung1,2
1Electrophysics Department, National Chiao-Tung University,
HsinChu, Taiwan, 300, R.O.C.
2National Center for Theoretical Sciences,
HsinChu, Taiwan, 300, R.O.C.
(Dated: January 20, 2015)
5
Thequantumcriticalityofthetwo-leadtwo-channelpseudogapAndersonmodelisstudied. Based
1
0 on the non-crossing approximation, we calculate both the linear and nonlinear conductance of the
2 model at finite temperatures with a voltage bias and a power-law vanishing conduction electron
density of states, ρc(ω) ω µF r (0< r < 1) near the Fermi energy µF. Equilibrium and non-
n ∝| − |
equilibrium quantum critical properties at the two-channel Kondo (2CK) to local moment (LM)
a
phasetransitionareaddressedbyextractinguniversalscalingfunctionsinbothlinearandnon-linear
J
conductances,respectively. Cleardistinctionsarefoundonthecriticalexponentsbetweenlinearand
8
non-linear conductance. The implications of these two distinct quantum critical properties for the
1
non-equilibrium quantumcriticality in general are discussed.

] PACSnumbers:
l
e
-
r
t I. INTRODUCTION.
s
.
t
a Quantum phase transitions (QPTs)[1], the zero-
m
temperature phase transitions due to quantum fluctu-
- ations, are of fundamental importance in condensed
d
matter systems. Near the transitions, these systems
n
o show non-Fermi liquid behaviors manifested in univer-
c sal power-law scaling in all thermodynamic observables.
[ Recently, QPTs in quantum impurity problems[2], have

1 attracted much attention recently due to their relevance
v for the nano-systems, such as: quantum dots[3], realized
7 experimentally. The well-known Kondo effect[4], the an-
4
tiferromagnetic spin correlations between impurity and
3
conduction electrons, plays a crucial role in understand-
4
0 ing their low temperature behaviors. New scaling laws
. areexpectedtooccurwhentheQPTsareassociatedwith
1
the Kondo breakdownin these systems either in equilib- FIG. 1: Schematic 3D phase diagram of 2CK-LM quantum
0
phasetransition of thetwo-lead 2-channel pseudogap Ander-
5 rium (at finite temperatures) or under non-equilibrium
son modelasfunctionsof voltagebias V, temperatureT and
1 conditions (at finite voltage bias). Of particular interest
power-law exponentr of thepseudogap fermion bath.
: lies in QPTs out of equilibrium[5] where distinct univer-
v
i sal scalings are expected in contrast to the counterparts
X
in equilibrium.
r
Afascinatingplaygroundtoaddressthisissueistheex- density of states (LDOS) ρ (ω) which vanishes linearly
a c
otic two-channel Kondo (2CK)[6–11] systems with non- due to the Dirac spectrum: ρ (ω) ω r with r =1 [19–
c
∝| |
Fermiliquidgroundstateduetooverscreeningofs=1/2 21]. ThisleadstoQPTfromtheKondoscreenedphaseto
impurity spin by two independent conduction reservoirs. the unscreened local moment (LM) phase with decreas-
Much of the theoretical effort has been made for the ing Kondo correlation due to insufficient DOS of con-
2CK physics, including: via Bethe ansatz [12], confor- duction electrons. In fact, such transitions exist in the
mal field theory [13], bosonization [14] and the numeri- more generalframeworkof pseudogapKondo (or Ander-
calrenormalizationgroup[15]. Experimentally,the 2CK son) models with power-law exponent 0 < r < 1, which
ground state has been realized in semiconductor quan- have been extensively studied [22–26]. However, rela-
tumdots[16],magneticallydopednanowires,andmetal- tively less is knownonthe more exotic 2CK-LMQPT in
lic glasses [17, 18]. Recently, Kondo physics in magnet- thepseudogapAnderson(Kondo)models[27,28],inpar-
ically doped graphene has attracted much attention for ticular, when the system is subject to a non-equilibrium
the possible 2CK physics as well as the pseudogap local condition[5, 29].
2
0 < r < 1, where D = 1 is the bandwidth cutoff.
10
8 r=0.09 Graphene and highT -cuprate superconductors are pos-
r=0.11 c
r=0.115 sible realizations of the pseudogap leads with r = 1,
r=0.13 while semiconductors with soft gaps are candidates with
r=0.15
6 r=0.19 0<r <1. The Hamiltonian reads [8, 32]
A A r=0.23 1
ρ (ω) 4 ρ(ω) H = τ,Xα,σ,k(ǫkσ −µα)cαkσ†τcαkστ +Xσ,τ ǫσd†σ,τdσ,τ
1
02 A-A0.A4AAAAA-A0AA.A2AAAAAAAAAAAA AAAAAωAAAAAAAAAAAAAAAAAAA0AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAA/AAAAA ADAA AAA AAA0AA.A2AAAAAA0A.4AA 0-1.1×1A0A-6AAA-A5A×AA1A0AA-A7AAAAAAAAAAAA AAAAAAωAAAAAAAAAAAAAAAAAAAAA0AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAA/AAAA AADAAAAAAA5AA×AA1AA0A-7AAAA1A×1A0-6 ++ τ2,XαU,σσ,k,σX(′V,τkα,στc′nαkσ†στ,τdnσσ+′,τh′(.c1.)−δσ,σ′δτ,τ′) (1)
FIG. 2: The equilibrium impurity spectral function ρ(ω) (in where µα=L/R = V/2 is the chemical potential of lead
±
arbitraryunit)versusenergyω(inunitsofthebandwidthD) α = L/R. The operators cα† (cα ) create (destroy)
with V = 0, Γ = 0.28, ǫd = 0.2, T = 5 10−7. (a). The an electron in the leads withkmστomeknσtτum k. Spin flavors
− ×
spectral function ρ(ω) in the full energy range. For r < rc are represented by σ,σ′ = 1, N and τ,τ′ = 1, M
(rc 0.115), the Kondo peaks are located at ω = 0 (Fermi ··· ···
∼ correspondsto M independent electronreservoirs. Here,
energy),whileforr>rc,dipsnearω=0aredeveloped. (b).
Same plots as in (a) with energies close to thepeaks/dips. N =M =2. The U term describes the on-site Coulomb
energyontheimpurity,andVα representsthehybridiza-
kσ
tionstrengthbetweenthegrapheneelectronsandtheim-
purity.
TheHamiltonianEq.(1)canbesolvedviaNCAinthe
Motivatedbythesedevelopments,weaddressthe2CK-
large-U limit U with the following Hamiltonian in
LM quantum phase transition in the two-channel pseu- →∞
the pseudofermion slave-bosonrepresentation:
dogap Anderson model both in and out of equilibrium
in a Kondo quantum dot subject to a voltage bias and
H = (ǫ µ )cα† cα +ǫ f†f
finite temperature. By studying experimentally accessi- k− α kστ kστ d σ σ
X X
kστα σ
ble steady-state transport, we search for universal scal-
ings inboth linear andnon-linearconductancenear crit- + (Vkασ(fσ†bτcαkστ)+h.c)
X
icality via a large-N method based on Non-Crossing kστα
Approximation (NCA) [8, 31–33], a reliable approach (2)
for multi-channel Kondo systems with non-Fermi liq-
where the local (impurity) electron operator d† is de-
uid ground states. A fundamentally important but less σ,τ
composedinthepseudofermionrepresentationasaprod-
addressed issue–non-equilibrium quantum criticality– is
uct of pseudofermion f† and a slave-boson b : d† =
emphasized here by identifying and comparing differ- σ τ σ,τ
entuniversalscalingbehaviorsbetweenequilibrium(zero fσ†bτ subject to the local constraint Q = σfσ†fσ +
bias)andnon-equilibrium(finitebias)conductancesnear τ b†τbτ = 1 via the Lagrange multiplier λPto ensure
the 2CK-LM transition[34]. Psingle occupancy on impurity.
Here, we employ the NCA to address the equilib-
rium and non-equilibrium transport at criticality based
on Eq. (1). This approach has been known to cor-
II. THE MODEL HAMILTONIAN.
rectly capture the non-Fermi liquid properties of the
two-channel Anderson model[4, 28, 29]. Recently, it has
The Hamiltonian of the two-lead two-channel single- been generalized to address the 2CK-LM crossover in
impurity pseudogap Anderson model is formulated non-equilibrium transport in a voltage-biased 2-channel
within the Non-Crossing Approximation (NCA) [8, 31– pseudogap Anderson model with r = 1, relevant for
33], a large-N approach based on the SU(N) SU(M) graphene[30]. Wegeneralizethisapproachherefurtherto
×
generalizationoftheSU(2)modelwithN ,M the voltage-biased2-channelpseudogapAndersonmodel
→∞ →∞
being the number of degenerate flavors of spins σ = with 0<r <1.
↑
, , N and the number of Kondo screening channels WithinNCA,theGreen’sfunctionsfortheconduction
↓ ···
K = 1,2,···M. In the physical SU(2) two-channel electronsGσ,iλ(t),pseudo-fermionsGfσ(t),andtheslave
Kondo system, N = M = 2. The two leads are de- bosons D(t) are given by[8, 32]:
scribedbyapower-lawvanishingdensityofstates(DOS)
at Fermi energy defined as ρ (ω) ω rΘ(D ω ) with Gr (t) = iθ(t)< c (t),c†(0) >
c ∼ | | −| | σ,iλ − { σ σ } iλ
3
d=0.038 0.15
0.4
1 ρ(ω) r=0.07
r=0.09
0.3 r=0.11
ω) T) 0.1 r=0.13
ρ( -0.004 0 ω 0.004 G(0, rrr===000...111579
0.5
r=0.21
A Ar=0.23
Kondo Peak 0.05
0 0 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4
ω /
D 10-7 10-6 10-5 10-4 10-3 10-2 10-1
T/D
FIG.3: Thenon-equilibriumimpurityspectral function ρ(ω)
(in arbitrary unit) with Γ = 0.3D, ǫd = 0.3D, r = 0.05, FIG.4: Theequilibrium(linear)conductanceG(0,T)(inunit
V = 0.038D, T = 5 10−7D. The Kond−o peak splits into of 2e/~) versus temperature T (in unit of D=1) for various
×
twopeakswherethewidthofthetwosplitedpeaksequalsto r with Γ=0.3D, ǫd = 0.3D.
−
bias V (see Inset).
Here, Γ (ω) Γ ρ (ω) with Γ = π Vα 2 with
= iθ(t)[D>( t)G< (t) D<( t)G> (t)]. ρ (ω) =α 1I≡mG α(ωc),α= r+1 ω rθα(D ω| kσµ| ) and
− − fσ − − fσ c,α −π c 2Dr+1| | −| − α|
(3) D =1 being the bandwidth of the conduction bath, and
f(ω)= 1 is the Fermi function.
1+eβω
G> i<f (t)f†(0)> ,G< i<f†(0)f (t)> , The relation between greater Green function and re-
fσ ≡− σ σ iλ fσ ≡ σ σ iλ(4) tardedGreenfunctionare,D>(w)=2iImDr(w), G>fσ =
and 2iImGr , where the self-energies, Π>(w) = 2iImΠr(w)
fσ
and Σ> (w) = 2iImΣr (w). The NCA expressions for
D> i<b (t)b†(0)> ,D< i<b†(0)b (t)> , fσ fσ
≡− τ τ iλ ≡ τ τ iλ theselfenergiesofretardedGreensfunctionsforpseudo-
(5)
fermionGr (ω)=[ω ǫ Σr(ω)]−1 andforslave-boson
wherethenotation<and>representslesserandgreater fσ − d−
Dr(ω)=[ω Πr(ω)]−1 are given by[8, 30, 32]:
Greenfunction. Thelesser(greater)Greenfunctionscan −
be written as [8, 32]: 2
Σr(ω) = dǫΓ (ω ǫ µ )f(ǫ ω µ )Dr(ǫ),
D>(<)(ω) = Dr(ω)Π>(<)(ω)Da(ω), π Xα Z α − − α − − α
G>(<) = Gr (ω)Σ<(>)(ω)Ga , (6) (9)
fσ fσ fσ fσ
where Da(r)(ω) and Ga(r)(ω) are advanced (retarded) Πr(ω) = 2 dǫΓ (ǫ ω µ )f(ǫ ω µ )Gr (ǫ).
Green function of bosonfσand fermion, respectively, and π XZ α − − α − − α fσ
α
thesubscriptiλreferstotheenforcementofthelocalcon- (10)
straint on the impurity via Lagrange multiplier λ when
Thephysicalimpurityspectralfunction, ρ (ω,V),isob-
evaluatingthesecorrelationfunctions. TheΠ>(<)(ω)and σ
tained via the convolution of pseudo-fermion and slave-
Σ<(>)(ω) are the self-energies of slave-boson and pseud-
fσ bosonGreensfunctionbasedonEqs.(7),(8),(9),and(10)
ofermion greater (lesser) Green functions, respectively.
as[30]:
The NCA expressions for the lesser self-energy of the
pseudofermion, G<fσ(ω) = Σ<(ω)|Grfσ(ω)|2, and slave- ρ(ω,V)= i dǫ[ImDr(ǫ)G<(ω+ǫ)
boson, D<(ω)=Π<(ω)Dr(ω)2, are[8, 30, 32]: 2π2Z Z −
| | D<(ǫ)ImGr(ω+ǫ)]. (11)
Σ<fσ(ω)= π2 XZ dǫΓα(ω−ǫ−µα)f(ω−ǫ−µα)D<(ǫ), TGh<e(ωn)o]rmenafloizracetisonthfeacctoonrsZtra=in2tiπ, R<dQω[M>=×D1<w(iωth)−MN×=
α
N = 2 here. The current going through the impurity
(7)
therefore reads[30]:
2
Π<(ω)= dǫΓ (ǫ ω µ )f(ω ǫ+µ )G< (ǫ). 2e 2ΓL(ω)ΓR(ω)
π XZ α − − α − α fσ I(V,T) = ~ Z dωΓ (ω)+Γ (ω)ρ(ω,V,T)
α L R
(8) [f(ω+eV/2) f(ω eV/2)]. (12)
× − −
4
1 1×10-1 0.25
σ Τ 0.2
T))/ Br σΣ*))Tr - 0.01.510 0.02 0.04 0.06 0.08
(G(0,0)-G(0, 0.1 rrrrr=====00000....00003579 G(0,T/T*)/((T/T 1×10-2 T*11100|0 --r063 - rc| Fit to | r - rc |4 rrr===000...000357
r=0.11 10-9 r=0.09
0.5 r=0.11
fit to T 10-2 10-1 0.23
| r - r | fit to T
c
0.01
10-1 100 101 102 1×10-4 1×10-2 1×100 1×102 1×104 1×106
0 T/T*
T/T
2CK
FIG. 5: The √T scaling of equilibrium (linear) conductance
t(vhGaer(i0ot,wu0so)-rc−hwaGni(tn0he,lTΓK))o=/nBd0ro.2(t8ienDm,upneǫirdtat=oufre2−ef0/o.~r2)Drv=ewrsi0tu.hsHTTe20/rCeTK,20CBKbreaifnorger F(oifInGteu.mn6ipt:eorUfa2ntueiv/ree~rTsaan/lTds∗ncaonlreimnagra2liiCnzeLdli-nLteoMaΣrqrcu(oTann/dtTuu∗cm)tσapTnhc)aeasseGat(r0fau,nTnsc/ittTiioo∗nn)
non-universal functions of r. The dashed line is a fit to √T for variousvalues of r. Parameters are thesame as in Fig. 4.
behavior. Here, T∗, σT, and Σr are defined in the text. The magenta
dashed line is a fit to T0.23. Top Inset: The power-law expo-
nentσT inlinearconductanceGQCP(T)closetocriticalityas
a function of r rc. Solid line is a fit to a linear relation:
| − |
where Γα(ω) = Γ(ω−µα) with α = L,R. The equilib- σtoTm=InβseTt:−CαrTos|rso−verrct|ewmiptheraβtTur≈eT−∗0v.1e4rs5u,sαTr ≈rc1..2T5.heBroetd-
rium (linear) conductance is directly obtained via dashed line is a fit to r rc 4. | − |
| − |
2e2 2Γ (ω)Γ (ω) ∂f(ω)
L R
G(0,T) = dω
~ Z ΓL(ω)+ΓR(ω)(cid:16)− ∂ω (cid:17)
ρ(ω,V =0). (13)
×
And the nonlinear conductance G(V) is given by
dI(V). Eqs.(6)-(10) form a self-consistent set of Dyson’s
dV
equations within NCA. We solve these equations self-
consistently and evaluate Eqs.(11)-(13) based on these
solutions.
2CK pseudogap Anderson model considered here shows
particle-hole (ph) asymmetry, giving rise to an overall
III. RESULTS. asymmetric impurity spectral function with respect to
the Fermi energy (see Fig. 2). For r <r , ρ (ω) shows a
c σ
A. Quantum critical point at 2CK-LM phase non-LorentzianKondo peak, a characteristic of the non-
transition: impurity spectral function Fermi liquid 2CK state[4, 8]. In fact, ρ (ω) exhibits a
σ
power-lawsingularitynearω =0withanexponentbeing
Theexistenceofaquantumcriticalpoint(QCP)sepa- r: ρ (ω) ω−r[28]. With increasing r, the Kondo peak
σ
∼
rating 2CK for r < r from the LM for r > r phases gets narrower with reduced spectral weight. For r r ,
c c c
≤
exists in the PSG Anderson model has been studied however,the Kondopeaksplitsintotwo,andtheground
extensively[28, 35]. The generic phase diagram is shown state is in the LM phase. The critical value r 0.115
c
≈
as in Fig. 1. We focus here on the transport properties for Γ 0.3D, ǫ 0.2D (see Fig. 2). The spectral
α d
∼ ∼ −
for the two-lead setup near criticality both in and out weight of the Kondo peaks gets further suppressed with
ofequilibrium,especiallyonthedistinctnon-equilibrium increasing r until it completely disappears. At a finite
quantum critical properties (see Fig. 1). In equilibrium bias voltage for r < r , the impurity local DOS shows
c
(µ = 0), the 2CK-LM phase transition is studied here splitedKondopeaksatω = V/2(seeFig.3). Notethat
α
±
bytuningrofthepseudogappower-lawDOSoftheleads the non-zero local DOS of ρ (ω =0) for r >r is due to
σ c
with fixed hybridization parameter Γ and the impurity thelimitationofthelowesttemperatureT 5 10−7D
α 0
∼ ×
energy ǫ . The value of r is extracted from the local we can reach numerically. As T 0 and for r r , we
d c c
→ ≥
impurity spectral function ρ (ω,V = 0) via solving the expect a complete dip developed in local DOS such that
σ
self-consistent Dyson’s equations. Since ǫ = U/2, the ρ (ω =0) 0.
d σ
6 − ∼
5
0.2
r=0.03
r=0.05
r=0.07 1×102
0.15 rrr===000...011913 0.5T)01×102
G(V,T)0 0.1 0,T)-G(V,T)/(001×101 1×101
G(
0.05
r=0.03 r=0.03
r=0.07 r=0.07
fit to (eV/kT)2 fit to (eV/KT)0.5
b 0 b 0
1×10-8 1×10-6 1×10-4 1×10-2 1×104 2 1×108 1×100 1×101 0.5 1×102
V / D (eV/k T ) (eV/k T )
B 0 B 0
FIG. 7: The non-equilibrium (non-linear) conductance FIG. 8: The 2CK scaling of thenon-equilibrium (non-linear)
G(V,T0) (in unit of 2e/~) versus bias voltage V (in unit of conductance(G(0,T0) G(V,T0))/T00.5 (inunitof2e/~) ver-
D=1) for various r with Γ=0.3D, ǫd =−0.3D. sus (a) (eV/kBT)2 an−d (b) (eV/kBT)0.5 for various r with
Γ=0.3D, ǫd = 0.3D.
−
B. Universal scaling in linear (equilibrium)
conductance near criticality First, as r r , the existence of a quantum critical
c
→
regime for T > T∗ requires the divergence of the corre-
lation length ξ in a power-law fashion: ξ r r −ν
More clear signatures of the 2CK-LM quantum phase c
∝| − | →
transitioncanbe obtainedvia linearand non-linearcon- [1, 2]. As a result, all thermaldynamical observables,
∞
including conductances, are expected to exhibit univer-
ductances. First, we analyze the linear (equilibrium)
sal scaling properties. As shown in Fig. 6, we indeed
conductance at finite temperatures but zero bias volt-
findnumericallythatthelinearconductanceG (0,T)
age G(V = 0,T). Fig. 4 shows G(0,T) for different r QCP
showsanuniversalpower-lawinT nearcriticalitywithin
with Γ = 0.28D, µ = 0, ǫ = 0.2D, and D = 1.
0 d
− a temperature range of (approximately) 5 10−7D <
For r = 0 it is well known that G(0,T) follows the 2CK
×
scaling function[8]: G(0,T) G(0,0)=B T/T0 for T <5 10−4D as:
− r 2CK ×
T <T0 withT0 3 10−5Dbeingthpe2CKKondo
2CK 2CK ∼ × G (0,T) TσT =TβT−αT|r−rc|, (14)
temperature at r =0 and Br being a non-universal con- QCP ∝
stant. Here, we set G(0,0) G(0,T = 5 10−7D).
For 0 < r < rc, however,≃the 2CK0 √T ×scaling in wwihtehreβtheexpo0n.1e4n5tσaTndexαhibitsa1.l2in5ebareirneglantioonn-utoni|vre−rsrac|l
G(0,0) G(0,T) ceases to exist. As shown in Fig. 4, T ≈ − T ≈
the √T−behavior in G(0,0) G(0,T) is clear for r = 0, constant pre-factors dependent on Γ and ǫd, and TσT =
but it deviates from √T m−ore with increasing r < rc. TdiβvTidaesGr(0=,Trc)∼by0t.1h1e5p.oBwaesre-dlawonfuthnectaiobnovTeσaTn,agliyvseiss:, we
We propose that this deviation from the conventional
2CK behavior for T < T for 0 < r < r can be due
2CK c G(0,T)
to the following two scenarios: (i). the emergence of dis- G(0,T) = G(0,T)/TσT = ,
TβT−αT|r−rc|
tinctuniversalpower-lawscalingbehaviorswhenthesys-
G e(0,T) = G (0,T)/TσT (15)
temapproachesthe2CK-LMquantumcriticalregimefor QCP QCP
T > T∗ with T∗ being the crossover energy scale above e
where G (0,T) in the quantum critical region be-
which quantum critical behaviors are observed, and (ii). QCP
the existence of a distinct 2CK scaling form other than comes aeconstant: GQCP(0,T) ∼ G0QCP. The univer-
√T inconductanceinlowtemperatureregimeT <T2CK salscalingfunction Ge(0,TT∗)inlinearconductance is ob-
for 0 < r < rc with T2CK being the two-channel Kondo tained by rescaling G(0,T) by a non-universal factor Σr
temperature for 0 < r < rc. As indicated in Fig. 5, the and T by the crossoveerenergy scale T∗ (see Fig. 6):
generic 2CK behaviors for 0 < r < r can not be mani-
c
fested in G(0,0) G(0,T) as an universal power-law in T G˜(0, T )/Σ
− G(0, ) T∗ r , (16)
T as it does for r =0 case. Therefore, instead of analyz- T∗ ≡ (T/T∗)βT−αT|r−rc|
ingG(0,0) G(0,T),wethereforeaddresstheabovetwo
−
scenarios below via trying the possible scaling behaviors where T∗ is proportional to the inverse of correlation
of G(T) itself. length 1/ξT r rc νT, vanishing in a power-law of
∝ | − |
6
10-1 B. Universal scaling in non-linear (non-equilibrium)
100 conductance near criticality
fit to | r - r |
c
σV10-1
- We now add the bias voltage in the leads and study
)Br 10-2 the scalings in non-equilibrium conductance near 2CK-
V 10-2 10-1 LMquantum criticalpoint. It is generallyexpected that
σT)/(V 10-2 | r - rc | tichaelistcyaalirnegdbiesthianvcitofrrsoomfntohna-tliinneaeqruciolnibdruiucmta[n5c].enTehaerncorint--
G(V, 10-4 fit to | r - rc |0.5 rr==00..0035 linear conductance G(V,T0) is obtained at a fixed low
V*10-5 rr==00..0079 temperature T0 = 5 10−7, symmetrical hybrizidation
×
10-6 10-2 10-1 rf=it 0t.o1 1V0.23 tΓhLe=noΓnR-li=nea0r.3cDo,ndanudctaǫdnc=e −G0(V.3,DT0.)AfosrsheaocwhnvianluFeigo.f7r,
| r - rc | saturates as V 0, while a small “kink”-like minima
10-3 →
10-4 10-2 100 102 104 106 in conductance is developed at eV k T. For r = 0
V/V* ≈ B
andT <T0 ,it followsthe well-known2CK scaling[8]:
2CK
G(0,T0)−G(V,T0) = ΣrT12H(keBVT0), where the univer-
FIG. 9: Universal scaling in non-linear conductance salfunctionH( eV )behavesas( eV )2 foreV k T ,
Gtu(rVe,VT/0V)/∗(VnVeσaBrr2)C(Lin-LuMniqtuoafn2teu/m~)pahsasaefturannctsiiotinonoffotremvaprieoruas- and (kBeVT0)12 forkBeVT0≫kBT0. HowkeBvTe0r,for 0<r≤<rBc a0s
values of r. Parameters are the same as in Fig. 4. Here, V∗, thesystemgetsclosertocriticality,while( eV )2 behav-
kBT0
σT, and Br are defined in the text. Black dashed line is a ior remains in G(0,T0) G(V,T0) for eV kBT0, more
power-law fit to the conductance in the 2CK regime. Top deviations from the 2CK−( eV )12 behavior∼are observed
Inset: Thepower-lawexponentσV innon-linearconductance for k T < eV < k T .kBFTu0rthermore, similar to the
B 0 B 2CK
GQCP(V,T0)closetocriticalityasafunctionof|r−rc|. Solid caseinouranalysisforequilibriumconductance,asindi-
line is a fit to a linear relation: σV = βV αV r rc with
− | − | ∗ catedinFig.8,the universal2CKbehaviorinnon-linear
βV 0, αV 2.5. Bottom Inset: Crossover temperature V
vers≈us r r≈c . The red dashed line is a fit to r rc 0.5. conductance for 0 < r < rc in general are not mani-
| − | | − |
fested in G(0,T ) G(V,T ) in the power-law fashion
0 0
−
as it is for r = 0 case. We therefore analyze the univer-
salscalingpropertiesbasedonG(V,T )itselfratherthan
0
G(0,T ) G(V,T ). Weperformthesimilaranalysishere
0 0
|erx−pornc|enwti:ththeexponentνT beingthecorrelationlength taopthowaterin-−laewqudileibpreinudme.ncAesorn→Vrci,nwtehfienqduGan(Vtu,mT0)crsihtoicwasl
regime approximately 10−4 < V/D < 10−6, defined as
1 G (V,T ) with power-law exponent linear in r r :
T∗ =D r r νT (17) QCP 0 | − c|
c
∝ ξ | − |
T
G (V,T ) VσV =VβV−αV|r−rc| (19)
QCP 0
∝
with ν being the correlation length exponent corre-
T withβ ,α beingnon-universalprefactorsdependenton
V V
sponding to the power-law exponent of crossover scale
Γ and ǫ . At criticality r = r , σ 0, yielding a con-
T∗ versus r r . We find ν 4 here. (see Inset of d c V ∼
Fig. 6). A|s r− c|r from beloTw,∼we find a perfect data stant non-linear conductance: GQCP(V,T0)|rc ∼ const..
collapse in G(0→, T c) const. for 1 < T/T∗ < 103, indi- We further perform a re-scaling on G(V,T0) as:
T∗ ≈
catingtherangeofquantumcriticalregion. Surprisingly, G(V,T ) G(V,T )
at low temperatures T < T2CK ≈ 0.1T∗ where the sys- G˜(V,T0) ≡ VσV0 = VβV−αV|r0−rc| (20)
tem is governed by the 2CK phase, we find the above
function G(0,TT∗) exhibits as a distinct universal func- whereG˜(V,T0)becomesaconstantinthecriticalregime.
tion power-law scaling function (see Fig. 6): We may further define the universal scaling function
G¯( V ,T ) for non-linear conductance via the following
V∗ 0
G(0,TT∗)2CK ∝ (TT )σ2TCK (18) rFei-gs.ca9l)i:ngs: V → V/V∗, G˜(V,T0) → G˜(V,T0)/Σr (see
2CK
V G( V ,T )/Σ
walietnhtlσy2TCGK( ∼T 0).23 f(orTthe)σp2TaCrKa+mσeTt)e.rsWweemseaty(othreerqeufoivre- G¯(V∗,T0) = (VV∗)Vβ∗V−α0V|r−rrc| (21)
regard thisT2lCowK te∝mpTe2rCaKture behavior in G(0,T) as the where V∗ is the crossover energy scale, and Σ a non-
r
distinct 2CK scaling in equilibrium (linear) conductance universalconstantpre-factor. Here,wefindtheexponent
for the pseudogap 2-channel Anderson model. σ depends linearly on r r with β 0, α 2.5
V c V V
| − | ≈ ≈
7
by the best fit of the data (see Inset of Fig. 9), and to the well-known√T,√V scalings for G(0,0) G(0,T)
−
the crossover scale V∗ is inversely proportional to the andG(V,0) G(V,T ), respectively. Our resultsprovide
0
−
correlation length ξ with a power-law dependence on further insights on two-channel Kondo physics and on
V
the distance to criticality: V∗ ∝ ξ1V ∝ |r −rc|νV with thenon-equilibriumquantumcriticalityinnano-systems.
ν 0.5 (see Inset of Fig.9). Note that these critical Furtheranalyticandnumericalinvestigationsareneeded
V
∼
exponents out of equilibrium are distinct from those in to address issues on the mechanism behind the different
equilibrium, and can be considered as characteristics of scalingsbetweenequilibriumandnonequilibriumconduc-
non-equilibrium quantum criticality. The distinction be- tances.
tween equilibrium and nonequilibrium quantum critical We thank S. Kirchner and K. Ingersent for use-
properties is expected due to the different role playedby ful discussions. C.H. C. acknowledges the NSC
thetemperatureandvoltagebiasnearcriticality,leading grant No.98-2112-M-009-010-MY3, No.101-2628-M-009-
to different behaviorsin decoherence rate Γs (the broad- 001-MY3,theNCTS,theMOE-ATUprogramofTaiwan
ening of impurity DOS) in equilibrium Γs versus out of R.O.C.
T
equilibrium Γs [5, 36].
V
For T < V < V∗, the system approaches 2CK state
0
at a characteristic energy scale V V 0.1V∗ be-
2CK
∼ ≈
lowwhichanuniversalpower-lawscalingisobserved(see
[1] S.Sachdev,Quantumphasetransitions,CambridgeUni-
Fig. 9):
versity press (2000); S. L. Sondhi, S. M. Girvin, J. P.
V V Carini, and D.Shahar, Rev.Mod. Phys.69, 315 (1987).
G¯(V∗,T0)2CK ∝ (V )σ2VCK (22) [2] M. Vojta, Phil. Mag. 86, 1807 (2006).
∗ [3] L. Kouwenhovenand L.Glazman, Physics World 14, 33
where the exponent σV 0.23. In the low bias limit (2001); D. Goldhaber-Gordon et al., Nature 391, 156
2CK ∼ (1998); W. G. van der Wiel et al., Science 289, 2105
V < T , conductance reaches a constant equilibrium
0
(2000); L.I. Glazman and M.E. Raikh,Sov.Phys. JETP
value G(0,T )andthereforedeviates fromthe 2CKscal-
0 Lett.47,452(1988);T.K.Ng,P.A.Lee,Phys.Rev.Lett.
ing (see Fig. 9).
61, 1768 (1988).
NotethatwefindEq.(22)fornon-equilibriumconduc- [4] A.C. Hewson, The Kondo Problem to Heavy Fermions
tance to exhibit the same form as that in equilibrium in (Cambridge University Press, Cambridge, UK, 1997).
Eq.(18)withthesameexponent(σT =σV 0.23). [5] Chung-HouChung,KarynLeHur,MatthiasVojta,Peter
2CK 2CK ∼
Further studies are required to clarify if this relation Wlfle, Phys. Rev. Lett. 102, 216803 (2009); Chung-Hou
Chung,KarynLeHur,GlebFinkelstein,MatthiasVojta,
holds in general. Though the exact value (0.23) of the
Peter Woelfle, Phys. Rev. B 87, 245310 (2013); Chung-
above exponents depend in general on physical param-
Hou Chung, Kenneth Yi-Jie Zhang, Phys. Rev. B 85,
eters, such as: ǫ and Γ, Eqs. (18) and (22) correctly
d 195106 (2012);S. Kirchner and Q. Si, Phys. Rev. Lett.
reproduce the well-known √V and √T behaviors for 103, 206401 (2009).
r = 0 in the 2CK regime in G(0,T0) G(V,T0) and [6] P.Nozieres, A.Blandin, J.Phys.(Paris) 41, 193(1980).
−
G(0,T ) G(0,T), respectively. [7] I. Affleck, and A.W.W. Ludwig, Nucl. Phys. B 360, 641
0
−
(1990); I.Affleck and A.W.W. Ludwig, Phys. Rev.Lett.
67, 161 (1991).
V. CONCLUSIONS. [8] M. H. Hettler, J. Kroha, and S. Hershfield, Phys. Rev.
Lett. 73, 1968 (1994); M.H. Hettler, J. Kroha, and S.
Hershfield, Phys.Rev. B 58, 5649 (1998).
In conclusions, we have studied quantum phase tran-
[9] P. Fendley, F. Lesage, and H. Saleur, J. Stat. Phys. 79,
sitions in and out of equilibrium between two-channel Nos. 5/6, 799 (1995).
Kondoandlocalmomentphasesinthetwo-channelpseu- [10] PaulFendley,MatthewP.A.Fisher,ChetanNayak,Phys.
dogapAndersonmodel where the conductionleads show Rev. B 75, 045317 (2007); Gregory A. Fiete, Waheb
a power-law vanishing density of states with exponent Bishara, Chetan Nayak, Phys. Rev. Lett. 101, 176801
(2008).
0<r <1. Via Non-Crossing Approximation (NCA), we
[11] Eran Sela, Andrew K. Mitchell, Lars Fritz, Phys. Rev.
solvedself-consistentlyfortheimpurityGreen’sfunction,
Lett. 106, 147202 (2011); Andrew K. Mitchell, Eran
and therefore determined the linear and non-linear con-
Sela, and DavidE.Logan, Phys.Rev.Lett.108, 086405
ductances. The 2CK-LM quantum criticality is reached (2012).
by varying the power-law exponent r of the pseudogap [12] N.AndreiandC.Destri,Phys.Rev.Lett.52,364(1984),
conduction bath with fixed lead-dot hybridizations and P. B. Wiegmann and A. M. Tsvelik, Z. Phys. B 54, 201
chemical potentials. The linear G(V = 0,T) and non- (1985).
linear G(V,T ) (T 5 10−7D) conductances show [13] I.AffleckandA.W.W.Ludwig,Nucl.Phys.B352,849
0 0 ∼ × (1991);ibid.360,641(1991).A.W.W.Ludwig,I.Affleck,
distinctuniversalpower-lawscalingsnearcriticalitywith
Phys.Rev.Lett.67,3160(1991),E.Sela,A.K.Mitchell,
different critical exponents. Furthermore, in the 2CK
L. Fritz, Phys. Rev.Lett. 106, 147202 (2011).
regime T,V < T2CK, we also find different characteris- [14] V.J.Emery,S.Kivelson,Phys.Rev.B46,10812(1992);
tic power-law scalings in G(T) and G(V,T0) compared M. Fabrizio, A.O. Gogolin, Phys. Rev. B 51, 17827
8
(1995). [27] I. Schneider, L. Fritz, F. B. Anders, A. Benlagra, M.
[15] A. K. Mitchell, E. Sela, D. E. Logan, Phys. Rev. Lett. Vojta, Phys.Rev. B 84, 125139 (2011).
108, 086405 (2012). [28] F.Zamani,T. Chowdhury,P. Ribeiro,K.Ingersent,and
[16] R.M. Potok, I.G. Rau, H. Shtrikman, Y. Oreg, and S. Kirchner, pssin print (2013) and arXiv:1211.4450.
Goldhaber-Gordon, Nature London 446, 167 (2007). [29] S. Kirchner, F. Zamani, E. Munoz, chapter contributed
[17] D.L.CoxandA.Zawadowski,Adv.Phys.47,599(1998). to“NewMaterialsforThermoelectricApplications: The-
[18] T. Chichorek et al., Phys.Rev.Lett. 94, 236603 (2005). oryandExperiment”,SpringerSeries: NATOSciencefor
[19] K.Senguptaand G. Baskaran, Phys. Rev.B 77, 045417 Peace and Security Series - B: Physics and Biophysics,
(2008). VeljkoZlatic(Editor),AlexHewson(Editor).ISBN:978-
[20] Matthias Vojta, Lars Fritzand RalfBulla, Europ.Phys. 9400749863 (2012), arXiv:1301.3307.
Lett.90, 27006 (2010). [30] Tsung-Han Lee, Kenneth Yi-Jie Zhang, Chung-Hou
[21] L. S. Mattos et al., (un-published); L.S. Mattos, “Cor- Chung,StefanKirchner,Phys.Rev.B88,085431(2013).
related electrons probed by scanning tunneling mi- [31] D. L. Cox and A. L. Ruckenstein, Phys. Rev. Lett. 71,
croscopy”, PhD. thesis, Stanford University (2009). 1613 (1993).
[22] David Withoff and Eduardo Fradkin, Phys. Rev. Lett. [32] Y.MeirandNedS.Wingreen,Phys.Rev.Lett.68,2512
64, 1835 (1990). (1992); Ned S.Wingreen and Y.Meir, Phys.Rev.B49,
[23] C.Gonzalez-Buxton and K.Ingersent,Phys.Rev.B 57, 11040 (1994).
14254 (1998). [33] M. Vojta, Phys. Rev.Lett. 87, 097202 (2001).
[24] K. Ingersent and Q. Si, Phys. Rev. Lett. 89, 076403 [34] Tsan-Pei Wu, Mater Thesis, Department of Electro-
(2002). physics, National Chiao-Tung University,2014.
[25] Matthew Glossop et al., Phys. Rev. Lett. 107, 076404 [35] ImkeSchneider,LarsFritz,FrithjofB.Anders,AdelBen-
(2011). lagra, Matthias Vojta, Phys. Rev.B 84, 125139 (2011).
[26] LarsFritzandMatthias Vojta, Phys.Rev.B70,214427 [36] C.H. Chunget al., in preparation.
(2004).