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Quantum criticality of the two-channel pseudogap Anderson model: universal scaling in linear
and non-linear conductance
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2016 J. Phys.: Condens. Matter 28 175003
(http://iopscience.iop.org/0953-8984/28/17/175003)
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1. Introduction
Quantum phase transitions (QPTs) [1], the zero-temperature
phase transitions due to quantum fluctuations, are of funda-
mental importance in condensed matter systems. Near the
transitions, these systems show non-Fermi liquid behaviors
manifested in universal power-law scaling in all thermody-
namic observables. Recently, QPTs in quantum impurity
problems [2], have attracted much attention recently due
to their relevance for the nano-systems, such as: quantum
dots [3], realized experimentally. The well-known Kondo
effect [4], the antiferromagnetic spin correlations between
impurity and conduction electrons, plays a crucial role in
understanding their low temperature behaviors. New scaling
laws are expected to occur when the QPTs are associated with
the Kondo breakdown in these systems either in equilibrium
(at finite temperatures) or under non-equilibrium conditions
(at finite voltage bias). Particular interest lies in QPTs out of
equilibrium [5] where distinct universal scalings are expected
in contrast to the counterparts in equilibrium.
A fascinating playground to address this issue is the exotic
two-channel Kondo (2CK) [6–11] systems with non-Fermi
liquid ground state due to overscreening of s = 1/2 impu-
rity spin by two independent conduction reservoirs. Much
Journal of Physics: Condensed Matter
Quantum criticality of the two-channel
pseudogap Anderson model: universal
scaling in linear and non-linear
conductance
Tsan-Pei Wu1, Xiao-Qun Wang2, Guang-Yu Guo2,3, Frithjof Anders4 and
Chung-Hou Chung1,3
1
Electrophysics Department, National Chiao-Tung University, HsinChu, Taiwan, 300, Republic of China
2
Department of Physics, National Taiwan University, Taipei, 106, Republic of China
3
Physics Division, National Center for Theoretical Sciences, HsinChu, Taiwan, 300, Republic of China
4
Theoretische Physik II, Technische Universitaet Dortmund 44221 Dortmund, Germany
E-mail: chung@mail.nctu.edu.tw
Received 17 January 2016, revised 8 March 2016
Accepted for publication 9 March 2016
Published 5 April 2016
Abstract
The quantum criticality of the two-lead two-channel pseudogap Anderson impurity model
is studied. Based on the non-crossing approximation (NCA) and numerical renormalization
group (NRG) approaches, we calculate both the linear and nonlinear conductance of the model
at finite temperatures with a voltage bias and a power-law vanishing conduction electron
density of states, r
c F( )ρ ω ω µ∝ | − | (0 < r < 1) near the Fermi energy Fµ . At a fixed lead-
impurity hybridization, a quantum phase transition from the two-channel Kondo (2CK) to the
local moment (LM) phase is observed with increasing r from r = 0 to r r 1c= < . Surprisingly,
in the 2CK phase, different power-law scalings from the well-known T or V form is
found. Moreover, novel power-law scalings in conductances at the 2CK-LM quantum critical
point are identified. Clear distinctions are found on the critical exponents between linear and
non-linear conductance at criticality. The implications of these two distinct quantum critical
properties for the non-equilibrium quantum criticality in general are discussed.
Keywords: quantum criticality, two-channel Kondo physics, quantum phase transitions,
non-equilibrium quantum transport
(Some figures may appear in colour only in the online journal)
0953-8984/16/175003+10$33.00
doi:10.1088/0953-8984/28/17/175003J. Phys.: Condens. Matter 28 (2016) 175003 (10pp)

T-P Wu et al
2
of the theoretical effort has been made for the 2CK physics,
including: via Bethe ansatz [12], conformal field theory [13],
bosonization [14] and the numerical renormalization group
(NRG) [15]. Experimentally, the 2CK ground state has been
realized in semiconductor quantum dots [16], magnetically
doped nanowires, and metallic glasses [17, 18]. Recently,
Kondo physics in magnetically doped graphene has attracted
much attention for the possible 2CK physics as well as the
pseudogap local density of states (LDOS) c( )ρ ω which van-
ishes linearly due to the Dirac spectrum: r
c( )ρ ω ω∝ | | with
r = 1 [19–21, 23]. This leads to QPT from the Kondo
screened phase to the unscreened local moment (LM) phase
with decreasing Kondo correlation due to insufficient DOS of
conduction electrons. In fact, such transitions exist in the more
general framework of pseudogap Kondo (orAnderson) models
with power-law exponent 0 < r < 1, which have been exten-
sively studied [24–28]. However, relatively less is known on
the more exotic 2CK-LM QPT in the pseudogap two-channel
Anderson (Kondo) models [29, 30], especially in transport
properties in the two-lead setup. In particular, three funda-
mentally important issues are yet to be addressed: (1). Does
the equilibrium conductance in the 2CK phase still exhibit the
T scaling as in the case of r = 0? (2). What are the scaling
properties near 2CK-LM QPT in conductances both in equi-
librium and out of equilibrium? (3). Does the scaling in non-
equilibrium conductance at criticality show distinct behaviors
from its equilibrium counterpart [5, 23, 31]?
Motivated by these developments, we address the above
issues on the 2CK-LM quantum phase transition in the two-
lead two-channel pseudogap Anderson model both in and out
of equilibrium in a Kondo quantum dot subject to a voltage
bias across the impurity and finite temperature. By studying
experimentally accessible steady-state transport, we search
for universal scalings in both linear and non-linear conduct-
ance in the 2CK regime and near criticality. Two different
theoretical approaches are applied here: (1) a large-N method
based on the non-crossing approximation (NCA) [8, 32–34],
a reliable approach for multi-channel Kondo systems with non-
Fermi liquid ground states, and (2) the NRG approach [35],
a well-established method for Anderson/Kondo impurity sys-
tems. A fundamentally important but less addressed issue—
non-equilibrium quantum criticality—is emphasized here by
identifying and comparing different universal scaling behav-
iors between equilibrium (zero bias) and non-equilibrium
(finite bias) conductances near the 2CK-LM transition [36].
2. The NCA approach
2.1. The model Hamiltonian
We start from the original Hamiltonian [22, 37]
( )
[ ]
†
∑ ∑ ∑
∑
µ= + + −
+ +
σ
σ σ σ
τ
τ τ τ
σατ
α στ
α
στ
α
σατ
στ
α
σ τ στ
α
εH E X E X c c
V X c h.c.
k
k k k
k
k
, ,
,
(1)
where the Hubbard operator 〉〈σ α= | |σ αX , induces a trans
ition from the impurity state ⟩τ| containing N electrons and
with energy Eτ to an impurity state ⟩σ| containing N + 1 elec-
trons with energy Eσ. The impurity part of the Hamiltonian
comprises only fluctuations between the ground states of the
two valence configuration; all other local states are omitted
due to the local Coulomb interaction. For derivation of
such models we refer to [22, 38] and the review on multi-
orbital Kondo models [17]. Introducing the energy difference
E Ed = −σ τε for energy degenerate multiplets ⟩τ| and ⟩σ|
and the slave-boson representation of the transition operator
X f b, → †
σ α σ τ allows us to map equation (1) to (2) (see below).
The Hamiltonian of the two-lead two-channel single-impu-
rity pseudogap Anderson model defined above can be form
ulated within the NCA [8, 32–34, 38], a large-N approach
based on the SU N SU M( ) ( )× generalization of the SU(2)
model with N M,→ →∞ ∞ being the number of degen-
erate flavors of spins N, ,σ =↑ ↓ and the number of Kondo
screening channels K M1, 2,= . In the physical SU(2)
two-channel Kondo system, N = M = 2. The two leads are
described by a power-law vanishing DOS at Fermi energy
defined as Dr
c( ) ( )ρ ω ω ω∼ | | Θ −| | with 0 r 1, where
D = 1 is the bandwidth cutoff. Graphene and high Tc-cuprate
superconductors are possible realizations of the pseudogap
leads with r = 1, while semiconductors with soft gaps are can-
didates with 0 r 1.
The Hamiltonian of the model via NCA in the large-U limit
U → ∞ has the following form in the pseudofermion slave-
boson representation [8, 33]:
( )
( ( ) )
† †
†
∑ ∑
∑
µ= − +
+ +
στα
α στ
α
στ
α
σ
σ σ
στα
σ
α
σ τ στ
α
ε εH c c f f
V f b c h.c
k
k k k d
k
k k

(2)
where µ = ±α= V /2L/R is the chemical potential of lead
L R/α = . The operators ck
†
στ
α
(ckστ
α
) create (destroy) an elec-
tron in the leads with momentum k. Spin flavors are repre-
sented by N, 1,σ σ =′ and M, 1,τ τ =′ corresponds to
M independent electron reservoirs. Here, N = M = 2. The
Vkσ
α
term represents the hybridization strength between the
conduction electrons and the impurity. Here, the local (impu-
rity) electron operator d ,
†
σ τ is decomposed in the pseudofer-
mion representation as a product of pseudofermion f†
σ and
a slave-boson bτ: d f b,
† †
=σ τ σ τ subject to the local constraint
Q f f b b 1† †
= ∑ + ∑ =σ σ σ τ τ τ via the Lagrange multiplier λ to
ensure single occupancy on impurity.
We employ the NCA to address the equilibrium and non-
equilibrium transport at criticality based on equation (2). This
approach has been known to correctly capture the non-Fermi
liquid properties of the two-channel Anderson model [4, 17,
30, 31, 39]. Recently, it has been generalized to address the
2CK-LM crossover in non-equilibrium transport in a voltage-
biased two-channel pseudogap Anderson model with r = 1,
relevant for graphene [23]. We generalize this approach here
further to the voltage-biased two-channel pseudogapAnderson
model with 0 r 1. Note that though the NCA formalism
in our case is the same as that in [23] except for the value of
r, we find surprising new scaling behaviors in conductances
J. Phys.: Condens. Matter 28 (2016) 175003

T-P Wu et al
3
both in and out of equilibrium both in the 2CK phase and near
2CK-LM QCP.
To facilitate our discussions, let us briefly summarize the
NCA formalism for the 2CK pseudogap Anderson model in
U → ∞ limit. Within NCA, Green’s functions for the conduc-
tion electrons G t,i ( )σ λ , pseudo-fermions G tf ( )σ , and the slave-
bosons D(t) are given by [8, 33]:
G t t c t c
t D t G t D t G t
i , 0
i .
r
f f
,i i( ) ( ) { ( ) ( )}
( )[ ( ) ( ) ( ) ( )]
†
θ
θ
= −
= − − − −
σ λ σ σ λ
σ σ

(3)
G f t f G f f ti 0 , i 0 ,f fi i( ) ( ) ( ) ( )† †
≡ − ≡ σ σ σ λ σ σ σ λ

(4)
and
D b t b D b b ti 0 , i 0 ,i i( ) ( ) ( ) ( )† †
≡ − ≡ τ τ λ τ τ λ

(5)
where the notation and represents lesser and greater
Green functions. The lesser (greater) Green functions can be
written as [8, 33]:
ω ω ω ω
ω ω
= Π
= Σσ σ σ σ

D D D
G G G
,
,
r a
f f
r
f f
a
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( )
(6)
where Da r
( )( )
ω and G f
a r
( )( )
ωσ are advanced (retarded) Green
functions of the boson and fermion, respectively, and the sub-
script iλ and the slave-bosons impurity via Lagrange mul-
tiplier λ when evaluating these correlation functions. The
( )( )
ωΠ
and f ( )( )
ωΣ σ

are the self-energies of the slave-boson
and pseudofermion greater (lesser) Green functions, respec-
tively. The NCA expressions for the lesser self-energy of the
pseudofermion, G Gf f
r 2
( ) ( ) ( )ω ω ω= Σ | |σ σ

, and slave-boson,
D Dr 2
( ) ( ) ( )ω ω ω= Π | |
, are [8, 23, 33]:
f D
2
d ,f ( ) ( ) ( ) ( )∫∑ω
π
ω µ ω µΣ = Γ − − − −σ
α
α α α

ε ε ε ε
(7)
f G
2
d .f( ) ( ) ( ) ( )∫∑ω
π
ω µ ω µΠ = Γ − − − +
α
α α α σ

ε ε ε ε
(8)
Here, we define an effective hybridization with a power-
law spectral density: Vk k k
r2
( ) ( )ω π δ ω ωΓ ≡ ∑ | | − ∝ | |α σ
α
σ
α
ε .
The spectral density of the conduction bath is given by
ρ ω ω ω θ ω µ= − = = | | −| − |α π
ω
α
Γ
Γ
+α
α
+G DImc
r
D
r
,
1
c
1
2 r 1( ) ( ) ( )
( )
with
d( )∫ ω ωΓ = Γα α and D = 1 being the bandwidth of the con-
duction bath. Here, f
1
1 e
( )ω = + βω is the Fermi function.
The relation between the greater Green function and
the retarded Green function are, D w D w2iIm r
( ) ( )=
,
G G2iImf f
r
=σ σ

, where the self-energies, w w2iIm r
( ) ( )Π = Π
and w w2iImf f
r
( ) ( )Σ = Σσ σ

. The NCA expressions for the
self energies of the retarded Green functions for pseudo-
fermion G f
r
d
r 1
( ) [ ( )]ω ω ω= − − Σσ
−
ε and for slave-boson
Dr r 1
( ) [ ( )]ω ω ω= − Π −
are given by [8, 23, 33]:
∫∑ω
π
ω µ ω µΣ = Γ − − − −
α
α α αε ε ε εf D
2
d ,r r
( ) ( ) ( ) ( )

(9)
( ) ( ) ( ) ( )∫∑ω
π
ω µ ω µΠ = Γ − − − −
α
α α α σε ε ε εf G
2
d .r
f
r
(10)
The physical impurity spectral function, V,( )ρ ωσ , is
obtained via the convolution of the pseudo-fermion and slave-
boson Green function based on equations (7)–(9) and (10) as
[23, 38]:
∫ρ ω
π
ω
ω
= +
− +
σ σ
σ

ε ε ε
ε ε
V
Z
D G
D G
,
i
2
d Im
Im .
r
f
f
r
2
( ) [ ( ) ( )
( ) ( )]

(11)
The normalization factor [ ( )∫ ω ω= × −π

Z M Dd
i
2
( )]ω×
N G enforces the constraint,Q =1 with M = N = 2
here.The current going through the impurity therefore reads [23]:
( )
( ) ( )
( ) ( )
( )
[ ( ) ( )]
∫∑ ω
ω ω
ω ω
ρ ω
ω ω
=
Γ Γ
Γ + Γ
× + − −
σ
σI V T
e
V T
f eV f eV
,
2
d
2
, ,
/2 /2 ,
L R
L R
(12)
where ( ) ( )ω ω µΓ = Γ −α α with L R,α = . The equilibrium
(linear) conductance is directly obtained via
∫∑ ω
ω ω
ω ω
ω
ω
ρ ω
=
Γ Γ
Γ + Γ
−
∂
∂
× =
σ
σ
⎛
⎝
⎜
⎞
⎠
⎟G T
e f
V
0,
2
d
2
, 0 .
2
L R
L R
( )
( ) ( )
( ) ( )
( )
( )

(13)
And the nonlinear conductance G(V) is given by
I V
V
d
d
( )
.
Equations (6)–(10) form a self-consistent set of Dyson’s equa-
tions within NCA. We solve these equations self-consistently
and evaluate equations (11)–(13) based on these solutions.
Note that via equations (11) and (12) and the power-law
spectrum of the effective hybridization ( )ωΓα , we expect that
the conductance shows power-law behaviors as a function of T
or V. It is therefore interesting to clarify the dependence of the
power-law exponents in conductance on the pseudogap bath
exponent r both in the 2CK and in the 2CK-LM quantum crit-
ical regime. To this end, we access the critical point by tuning
r (instead of conventional ways by tuning hybridization Γα
or dε ).
2.2. Results
2.2.1. Quantum critical point at 2CK-LM phase transition:
impurity spectral function. The existence of a quantum criti-
cal point (QCP) separating 2CK for r rc from the LM for
r rc phases exists in the PSG Anderson model has been
studied extensively [29, 30]. The generic phase diagram is
shown as in figure 1. We focus here on the transport prop-
erties for the two-lead setup near criticality both in and out
of equilibrium, especially on the distinct non-equilibrium
quantum critical properties (see figure 1). In equilibrium
( 0µ =α ), the 2CK-LM phase transition is studied here by
tuning r of the pseudogap power-law DOS of the leads with
fixed hybridization parameter Γα and the impurity energy dε .
The value of rc is extracted from the local impurity spectral
function V, 0( )ρ ω =σ via solving the self-consistent Dyson

T-P Wu et al
4
equations. Since , 0dµ ≠ε , the 2CK pseudogap Anderson
model considered here shows particle-hole (ph) asymmetry,
giving rise to an overall asymmetric impurity spectral function
with respect to the Fermi energy (see figure 2). For r = 0, the
NCA approach to Anderson models is known to give rise to a
non-Lorenzian power-law cusp in impurity spectral function,
1( )ρ ω ω∼ −σ [17, 40]. For r r0 c , ( )ρ ωσ in our two-
channel pseudogap Anderson model exhibits a power-law
singularity near 0ω = : r
( )ρ ω ω∼σ
−
[30]. With increasing r,
the Kondo peak gets narrower with reduced spectral weight.
For r rc⩾ , however, the Kondo peak is strongly suppressed,
and the ground state is in the LM phase. The critical value
r 0.115c ≈ for D0.3Γ ∼α , D0.2d ∼ −ε (see figure 2). The spec-
tral weight of the Kondo peaks gets further suppressed with
increasing r until it completely disappears. At a finite bias
voltage for r rc , the impurity local DOS shows split Kondo
peaks at ω = ±V /2 (see figure 3). Note that the non-zero
local DOS of 0( )ρ ω =σ for r rc is due to the limitation of
the lowest temperature T D5 100
7
∼ × −
we can reach numer
ically. As T 0→ and for r rc⩾ , we expect a complete dip devel-
oped in local DOS such that 0 0( )ρ ω = ∼σ .
2.2.2. Universal scaling in linear (equilibrium) conductance
in 2CK regime and near criticality. Clearer signatures of the
2CK-LM quantum phase transition can be obtained via lin-
ear and non-linear conductances. First, we analyze the lin-
ear (equilibrium) conductance at finite temperatures but zero
bias voltage G(V = 0, T ). Figure 4 shows G(0, T ) for differ-
ent r with D0.28Γ = , 00µ = , D0.2d = −ε , and D = 1. For
r = 0 it is well known that G(0, T ) follows the 2CK scaling
function [8]: G T G B T T0, 0, 0 r 2CK
0
( ) ( ) /− = for T T2CK
0

with T D3 102CK
0 5
∼ × −
being the 2CK Kondo temperature
at r = 0 and Br being a non-universal constant. Here, we set
G G T D0, 0 0, 5 100
7
( ) ( )= × −
. For r r0 c , however,
the 2CK T scaling in G(0, 0) − G(0, T ) ceases to exist. As
shown in figure 4, the T behavior in G(0, 0) − G(0, T ) is
Figure 1. Schematic 3D phase diagram of 2CK-LM quantum
phase transition of the two-lead two-channel pseudogap Anderson
model as functions of voltage bias V, temperature T and power-law
exponent r of the pseudogap fermion bath.
Figure 2. The equilibrium impurity spectral function ( )ρ ω
(in arbitrary units) versus energy ω (in units of the bandwidth D)
via NCA with V = 0, Γ = 0.28, = −ε 0.2d , = × −
T 5 10 7
.
(a) The spectral function ( )ρ ω in the full energy range. For r rc
( ∼r 0.115c ), the Kondo peaks are located at ω = 0 (Fermi energy),
while for r rc, dips near ω = 0 are developed. (b) Same plots as in
(a) with energies close to the peaks/dips.
A
A
A
A
A
A
A
A
A
A AAAAA
A
A
A
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A A A A A A A A A A
-0.4 -0.2 0 0.2 0.4
ω /
0
2
4
6
8
ρ(ω)
r=0.09
r=0.11
r=0.115
r=0.13
r=0.15
r=0.19
r=0.23A A
A A A A A A A A AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA A A A A A A A A
-1×10
-6
-5×10
-7 0 5×10
-7
1×10
-6
ω /
0.1
1
10
ρ(ω)
D D
Figure 3. The non-equilibrium impurity spectral function ( )ρ ω (in
arbitrary units) via NCA with Γ = D0.3 , = −ε D0.3d , r = 0.05,
V = 0.038 D, = × −
T D5 10 7
. The Kondo peak splits into two
peaks where the width of the two split peaks is equal to bias V
(see inset).
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4
ω /
0
0.5
1
ρ(ω)
-0.004 0 0.004
ω
0.3
0.4
ρ(ω)
Kondo Peak
d=0.038
D
Figure 4. The equilibrium (linear) conductance G(0, T ) (in units of
/e2 ) versus temperature T (in units of D = 1) via NCA for various
r with Γ = D0.3 , = −ε D0.3d .
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
T/D
0
0.05
0.1
0.15
G(0,T)
r=0.07
r=0.09
r=0.11
r=0.13
r=0.15
r=0.17
r=0.19
r=0.21
r=0.23A A

T-P Wu et al
5
clear for r = 0, but it deviates from T more with increasing
r rc . We propose that this deviation from the conventional
2CK behavior for T T2CK for r r0 c can be due to the
following two scenarios: (i) the emergence of distinct universal
power-law scaling behaviors when the system approaches the
2CK-LM quantum critical regime for T T ∗
with T∗
being the
crossover energy scale above which quantum critical behaviors
are observed, and (ii) the existence of a distinct 2CK scaling
form other than T in conductance in low temperature regime
T T2CK for r r0 c with T2CK being the two-channel Kondo
temperature for r r0 c . As indicated in figure 5, the generic
2CK behaviors for r r0 c can not be manifested in G(0,
0) − G(0, T ) as a universal power-law in T as it does for the
r = 0 case. Therefore, instead of analyzing G(0, 0) − G(0, T ),
we address the above two scenarios below by trying the pos-
sible scaling behaviors of G(0, T) itself.
First, as r rc→ , the existence of a quantum critical regime
for T T ∗
requires the divergence of the correlation length ξ
in a power-law fashion: r rc →ξ ∝ | − | ∞ν−
[1, 2]. As a result,
all thermaldynamical observables, including conductances,
are expected to exhibit universal scaling properties. As shown
in figure 6, we indeed find numerically that the linear con-
ductance G T0,QCP( ) shows a universal power-law in T near
criticality within a temperature range of (approximately)
D T D5 10 5 107 4
× ×− −
as:
G T T T0, ,r r
QCP
T T T c
( ) ∝ =σ β α− | − |
(14)
where the exponent Tσ exhibits a linear relation to r rc| − |
with 0.145Tβ ≈ − and 1.25Tα ≈ being non-universal con-
stant pre-factors dependent on Γ and dε , and T TT T
=σ β
as r r 0.115c= ∼ . Based on the above analysis, we divide
G(0, T) by the power-law function T Tσ
, gives:
( ) ( )
( )
( ) ( )
= =
=
∼
∼
σ
β α
σ
− | − |
G T G T T
G T
T
G T G T T
0, 0, /
0,
,
0, 0, /
r r
QCP QCP
T
T T
T
c

(15)
where G T0,QCP( )
∼
in the quantum critical region becomes a
constant: G T G0,QCP QCP
0
( ) ∼
∼
. The universal scaling func-
tion G 0,
T
T
( )∗ in linear conductance is obtained by rescaling
G T0,( )
∼
by a non-universal factor rΣ and T by the crossover
energy scale T∗
(see figure 6 ):
G
T
T
G
T T
0,
0,
,
T
T r
r rT T c
( )
˜( )/
( / )
≡
Σ
β α∗ ∗ − | − |
∗
(16)
where T *
is proportional to the inverse of correlation length
r r1 T c
T
/ξ ∝ | − |ν
, vanishing in a power-law of r rc| − | with the
exponent Tν being the correlation length exponent:
Figure 5. The T scaling of equilibrium (linear) conductance
(G(0, 0) − G(0, T ))/Br (in units of /e2 ) versus /T T2CK
0
via NCA for
various r with Γ = D0.28 , = −ε D0.2d with T2CK
0
being the two-
channel Kondo temperature for r = 0. Here, Br are non-universal
functions of r. The dashed line is a fit to T behavior.
10
-1
10
0
10
1
10
2
T/T
0
2CK
0.01
0.1
1
(G(0,0)-G(0,T))/Br
r=0
r=0.03
r=0.05
r=0.07
r=0.09
r=0.11
fit to T
0.5
Figure 6. Universal scaling in linear conductance ( / )∗
G T T0,
(in units of /e2 and normalized to ( / )Σ σ∗
T Tr
T) as a function of
temperature / ∗
T T via NCA near 2CL-LM quantum phase transition
for various values of r. Parameters are the same as in figure 4.
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 exponent σT in linear
conductance ( )G TQCP close to criticality as a function of| − |r rc .
Solid line is a fit to a linear relation: σ β α= − | − |r rT T T c with
β ≈ −0.145T , α ≈ 1.25T . Bottom inset: crossover temperature ∗
T
versus| − |r rc . The red dashed line is a fit to| − |r rc
4
.
1×10
-4
1×10
-2
1×10
0
1×10
2
1×10
4
1×10
6
T/T*
1×10
-2
1×10
-1
G(0,T/T*)/((T/T*)
σT
Σr
)
r=0.03
r=0.05
r=0.07
r=0.09
r=0.11
fit to T
0.23
0 0.02 0.04 0.06 0.08
| r - rc
|
0.1
0.15
0.2
0.25
-σΤ
10
-2
10
-1
| r - rc
|
10
-9
10
-6
10
-3
10
0
T*
Fit to | r - rc
|
4
Figure 7. The non-equilibrium (non-linear) conductance G(V, T0)
(in units of /e2 ) versus bias voltage V (in units of D = 1) via NCA
for various r with Γ = D0.3 , = −ε D0.3d .
1×10
-8
1×10
-6
1×10
-4
1×10
-2
V / D
0.05
0.1
0.15
0.2
G(V,T0
)
r=0.03
r=0.05
r=0.07
r=0.09
r=0.11
r=0.13

T-P Wu et al
6
T D r r
1
T
c
T
ξ
∝ = | − |ν∗
(17)
with Tν being the correlation length exponent corresponding
to the power-law exponent of crossover scale T *
versus
r rc| − |. We find 4Tν ∼ here. (see inset of figure 6). As r rc→
from below, we find a perfect data collapse inG 0, const.
T
T
( ) ≈∗
for T T1 103
/ ∗
, indicating the range of quantum critical
region. Surprisingly, at low temperatures T T T0.12CK ≈ ∗
where the system is governed by the 2CK phase, we find the
above function G 0,
T
T
( )∗ exhibits as a distinct universal power-
law scaling function (see figure 6):
G
T
T
T
T
0, 2CK
2CK
T
2CK( ) ( )∝ σ
∗(18)
with 0.23T
2CKσ ∼ for the parameters we set (or equivalently
G
T
T
T
T
T
T
2CK 2CK
2CK( ) ( )∝ σ σ+
). We may therefore regard this low
temperature power-law behavior in G(0, T) as the distinct 2CK
scaling in equilibrium (linear) conductance for the pseudogap
two-channel Anderson model. Note that the constants T
2CKσ ,
Tα , and Tβ in the exponent of equation (18) are in general func-
tions of various parameters in our model Hamiltonian, and
shall be addressed elsewhere. Nevertheless, we would like to
emphasize that the power-law in the temperature dependence
of G(0, T ) is found to be a generic feature of 2CK phase in the
pseudogap Anderson model.
2.2.3. Universal scaling in non-linear (non-equilibrium) con-
ductance in the 2CK regime and near criticality. We now add
the bias voltage in the leads and study the scalings in non-
equilibrium conductance near the 2CK-LM quantum critical
point. It is generally expected that the scaling behaviors of
non-linear conductance near criticality are distinct from that
in equilibrium [5]. The non-linear conductance G(V, T0) is
obtained at a fixed low temperature T D5 100
7
= × −
, sym-
metrical hybrizidation D0.3L RΓ = Γ = , and D0.3d = −ε .
As shown in figure 7, the non-linear conductance G(V, T0)
for each value of r saturates as V 0→ , while a small ‘kink’-
like minima in conductance is developed at eV k TB≈ . For
r = 0 and T T2CK
0
, it follows the well-known 2CK scal-
ing [8]: G T G V T T H0, , r
eV
k T0 0
1
2
B 0
( ) ( ) ( )− = Σ , where the uni-
versal function H
eV
k TB 0
( ) behaves as
eV
k T
2
B 0
( ) for eV k TB 0⩽ , and
eV
k T
1
2
B 0
( ) for eV k TB 0. However, for r r0 c as the system
gets closer to criticality, while
eV
k T
2
B 0
( ) behavior remains in
G T G V T0, ,0 0( ) ( )− for eV k TB 0∼ , more deviations from the
2CK
eV
k T
1
2
B 0
( ) behavior are observed for k T eV k TB 0 B 2CK .
Furthermore, similar to the case in our analysis for equilib-
rium conductance, as indicated in figure 8, the universal 2CK
behavior in non-linear conductance for r r0 c in general
are not manifested in G T G V T0, ,0 0( ) ( )− in the power-law
fashion as it is for the r = 0 case. We therefore analyze the
universal scaling properties based on G(V, T0) itself rather
than G T G V T0, ,0 0( ) ( )− . We perform the similar analysis
here to that in equilibrium. As r rc→ , we find G(V, T0) shows
a power-law dependence on V in the quantum critical regime
approximately V D10 104 6
/ − −
, defined as G V T,QCP 0( )
with power-law exponent linear in r rc| − |:
G V T V V, r r
QCP 0
V V V c
( ) ∝ =σ β α− | − |
(19)
with Vβ , Vα being non-universal prefactors dependent on Γ
and dε . At criticality r rc= , 0Vσ ∼ , yielding a constant non-
linear conductance: G V T, const.rQCP 0 c
( )| ∼ . We further per-
form a re-scaling on G(V, T0) as:
G V T
G V T
V
G V T
V
,
, ,
r r0
0 0
V V V c
˜( )
( ) ( )
≡ =σ β α− | − |(20)
Figure 8. The 2CK scaling of the non-equilibrium (non-linear)
conductance ( ( ) ( ))/−G T G V T T0, ,0 0 0
0.5
(in units of /e2 ) via
NCA versus (a) ( / )eV k TB
2
and (b) ( / )eV k TB
0.5
for various r with
Γ = D0.3 , = −ε D0.3d .
1×10
4
1×10
8
(eV/kB
T0
)
2
1×10
1
1×10
2
G(0,T0
)-G(V,T0
)/(T0
0.5
)
r=0.03
r=0.07
fit to (eV/kb
T0
)
2
1×10
0
1×10
1
1×10
2
(eV/kB
T0
)
0.5
1×10
1
1×10
2
r=0.03
r=0.07
fit to (eV/Kb
T0
)
0.5
Figure 9. Universal scaling in non-linear conductance
( )/( )σ
G V T V B, V r0 (in units of /e2 ) as a function of temperature
/ ∗
V V via NCA near 2CL-LM quantum phase transition for various
values of r. Parameters are the same as in figure 4. Here, ∗
V , σT,
and Br are defined in the text. Black dashed line is a power-law
fit to the conductance in the 2CK regime. Top inset: the power-
law exponent σV in non-linear conductance ( )G V T,QCP 0 close to
criticality as a function of| − |r rc . Solid line is a fit to a linear
relation: σ β α= − | − |r rV V V c with β ≈ 0V , α ≈ 2.5V . Bottom inset:
crossover temperature ∗
V versus| − |r rc . The red dashed line is a fit
to| − |r rc
0.5
.
10
-4
10
-2
10
0
10
2
10
4
10
6
V/V*
10
-3
10
-2
10
-1
G(V,T)/(V
σV
Br
)
r=0.03
r=0.05
r=0.07
r=0.09
r=0.11
fit to V
0.23
10
-2
10
-1
| r - rc
|
10
-2
10
-1
10
0
-σV
fit to | r - rc
|
10
-2
10
-1
| r - rc
|
10
-6
10
-5
10
-4
V*
fit to | r - rc
|
0.5

T-P Wu et al
7
where G V T, 0
˜( ) becomes a constant in the critical regime. We
may further define the universal scaling function G T,
V
V 0
¯( )∗
for non-linear conductance via the following re-scalings:
V V V→ / ∗
, G V T G V T, , r0 0
˜( ) → ˜( )/Σ (see figure 9):
G
V
V
T
G T
,
,
V
V r
V
V
r r
0
0
V V c
¯( )
( )/
( )
=
Σ
β α∗ − | − |
∗
∗
(21)
where V∗
is the crossover energy scale, and rΣ a non-
universal constant pre-factor. Here, we find the exponent
Vσ depends linearly on r rc| − | with 0Vβ ≈ , 2.5Vα ≈ by the
best fit of the data (see inset of figure 9), and the crossover
scale V∗
is inversely proportional to the correlation length Vξ
with a power-law dependence on the distance to criticality:
V r r
1
c
V
V
∝ ∝ | − |ξ
ν∗
with 0.5Vν ∼ (see inset of figure 9). Note
that these critical exponents out of equilibrium are distinct
from those in equilibrium, and can be considered as char-
acteristics of non-equilibrium quantum criticality. The dis-
tinction between equilibrium and nonequilibrium quantum
critical properties is expected due to the different role played
by the temperature and voltage bias near criticality, leading
to different behaviors in decoherence rate s
Γ (the broadening
of impurity DOS) in equilibrium T
s
Γ versus out of equilibrium
V
s
Γ [5, 41].
For T V V0 ∗
, the system approaches 2CK state at a
characteristic energy scale V V V0.12CK∼ ≈ ∗
below which a
universal power-law scaling is observed (see figure 9):
G
V
V
T
V
V
, 0 2CK
V
2CK¯( ) ( )∝
∗
σ
∗(22)
where the exponent 0.23V
2CKσ ∼ . In the low bias limit
V T0, conductance reaches a constant equilibrium value
G(0, T0) and therefore deviates from the 2CK scaling (see
figure 9).
Note that we find equation (22) for non-equilibrium con-
ductance to exhibit the same form as that in equilibrium in
equation (18) with the same exponent ( 0.23T V
2CK 2CKσ σ= ∼ ).
Further studies are required to clarify if this relation holds
in general. Though the exact value (0.23) of the above expo-
nents depend in general on physical parameters, such as: dε
and Γ, equations (18) and (22) correctly reproduce the well-
known V and T behaviors for r = 0 in the 2CK regime in
G T G V T0, ,0 0( ) ( )− and G(0, T0) − G(0, T ), respectively.
3. The NRG approach
In order to confirm the generic and robust nature of the NCA
results obtained above, we perform the calculation via a dif-
ferent approach—via the numerical renormalization group
(NRG) [4, 35, 42], a widely used approach to the Kondo
Figure 10. The equilibrium zero temperature impurity spectral
function versus energy ( )ρ ω via NRG in units of the band width
D, with hybridization strength Γ = D0.3 , and the impurity level
= −ε D0.3d . For r rc the Kondo peaks form. At ≈r 0.12c , ( )ρ ω
becomes flat near ω = 0, whereas a dip at the Fermi level develops
for r rc, signature of the local moment (LM) phase.
-5 0 5
ω /D
0
0.2
0.4
0.6
0.8
ρ(ω)
r = 0.09
r = 0.11
r = 0.12
r = 0.13
r = 0.15
r = 0.19
r = 0.23
-4×10
-8
-2×10
-8 0 2×10
-8
4×10
-8
ω /D
10
-3
10
-2
10
-1
10
0
Figure 11. The equilibrium conductance G(0, T) via NRG zero
temperature impurity spectral function versus energy ( )ρ ω in units
of the band width D, with hybridization strength Γ = D0.3 , and the
impurity level = −ε D0.3d .
A
A
A
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
T/D
0
0.05
0.1
0.15
0.2
G(0,T)
r = 0.07
r = 0.09
r = 0.11
r = 0.13
r = 0.15
r = 0.17
r = 0.19
r = 0.21
r = 0.23A A
Figure 12. The T scaling of equilibrium (linear) conductance
(G(0, 0) − G(0, T))/Br via NRG (in units of /e2 ) versus /T T2CK
0
for various r with Γ = D0.3 , = −ε D0.3d with ≈ −
T D102CK
0 5
being
the two-channel Kondo temperature for r = 0. Here, Br are non-
universal functions of r. The dashed line is a fit to T behavior.
10
-1
10
0
10
1
10
2
T/T
0
2CK
0.01
0.1
(G(0,0)-G(0,T))/Br
r = 0.00
r = 0.03
r = 0.05
r = 0.07
r = 0.09
r = 0.11
fit to T
0.5

T-P Wu et al
8
impurity problem. This approach has been generalized to
study pseudogap Anderson and Kondo models [25, 37, 39, 43].
The NRG Hamiltonian has the same form as shown in equa-
tion (2) with M = N = 2 [40]. The discretization parameter is
set to be 1.8Λ = , and the lowest 1200 states are kept in each
NRG iteration, allowing the convergence of results within
0.1% in error.
3.1. Results
The results for the equilibrium impurity spectral function at
ground state ( )ρ ω and the equilibrium conductance G(0, T )
are summarized below.
3.1.1. Impurity spectral function. As shown in figure 10, the
impurity spectral function ( )ρ ω via NRG exhibits the same
qualitative feature as that obtained via NCA (see figure 2):
the Kondo peak occurs for r rc , while the dip develops in
the local moment phase with r rc . The value of r 0.12c ≈
via NRG is in excellent agreement with that via NCA
(r 0.115c ≈ ). Some difference in detailed profiles of ( )ρ ω
between NCA and NRG are observed though: the peaks and
dips in ( )ρ ω near the Fermi level are sharper via NRG, while
they are less prounced in the NCA results. This difference
is likely due to the frequency mesh used to solve the NCA
equations.
3.1.2. Equilibrium conductance in the 2CK regime and near
criticality. We further compute the linear (equilibrium) con-
ductance G(0, T ) in the 2CK and 2CK-LM quantum critical
regime for r r0 c⩽ . As shown in figure 11, the profile of
G(0, T ) via NRG agrees qualitatively well with that via NCA
(see figure 4) though quantitative differences are noticed and
expected. Next, we examine the T scaling in conductance.
As shown in figure 12, the NRG result for G(0, 0) − G(0, T )
in the 2CK regime (T T D102CK
0 5
≈ −
) shows T scaling for
r = 0, and starts to deviate from T scaling for r r0 c ,
the same qualitative feature as that shown in figure 5 via the
NCA approach. Following the approach in section 2.2 (ii),
we perform the scaling on G(0, T) itself. As shown in fig-
ure 13, near the 2CK-LM critical point, we obtain the same
scaling form as equation (14) with r rT T T cσ β α= − | − | and
0.10Tβ ≈ − , 1.18Tα ≈ . The crossover temperature shows
a power-law scaling near criticality as: T r rc
4
∝ | − |∗
. In
the 2CK phase, the same power-law scaling form as equa-
tion (18) is observed for T T T0.12CK ∼ ∗
with slightly dif-
ferent exponent 0.17T
2CKσ ≈ (see figure 13).
4. Discussion and conclusions
Before we conclude, we would like to make a few remarks.
First, on the origin of the power-law scalings, we found in
our model system near the 2CK-LM transition. As mentioned
in section 2.2 (i), in the 2CK phase the local density of states
in equilibrium ( )ρ ω shows a power-law singularity at 0ω = :
r
( ) ρ ω ω∼ −
. Via equations (12) and (13), this power-law sin-
gularity in the local density of states is expected to give rise to
the power-law increase in linear and non-linear conductances
G(0, T ) and G(V, T ) (for V T ) with decreasing temperature
and voltage bias, respectively. This generic power-law feature
persists both in the 2CK and in the 2CK-LM quantum critical
regime, and explains the power-law scalings that we found
above for G(0, T ) and G(V, T ) with the exponents being func-
tions of r rc| − |.
Second, we access the 2CK-LM quantum critical point
via varying the bath power-law exponent r, a non-local para
meter. This is somewhat different from the more conven-
tional approach via varying local parameters, such as: Γα, dε .
Nevertheless, we would like to point out that the introduction
of the pseudogap leads in our model is equivalent to replacing
the constant hybridization Γα , a local parameter, by an effec-
tive one with a power-law (pseudogap) spectral density while
keeping the density of states of conduction bath constant (see
equations (6) and (7)). Hence, tuning r effectively changes
the spectral density of the lead-impurity hybridization, a local
parameter.
Finally, it is a challenging task to realize a generic
pseudogap fermionic bath with 0 r 1 in solid state sys-
tems. Nevertheless, due to the advance in nano-technology,
it is possible to realize the two-lead Anderson model in
the U → ∞ limit in a quantum dot coupled to two Ohmic
dissipative nano-wires via side-gate voltages where the dissi-
pative bath gives rise to a power-law (pseudogap) suppression
in the tunneling density of states (or the hybridization Γα).
(The spinless version of the setup has been realized in [44].)
For the two-channel leads, this effective system leads to
the same set of equations as shown in equations (7)–(11),
and therefore gives the same transport properties. The
Hamiltonian of such an effective system in the simplier case
of two single-channel leads reads:
Figure 13. Universal scaling in linear conductance ( / )∗
G T T0, via
NRG (in units of /e2 and normalized to ( / )Σ σ∗
T Tr
T) as a function
of temperature / ∗
T T near 2CL-LM quantum phase transition for
various values of r. Parameters are the same as in figure 12. Here,
∗
T , σT, and Σr are defined in the text. The magenta dashed line
is a fit to T 0.23
. Top inset: the power-law exponent σT in linear
conductance ( )G TQCP close to criticality as a function of| − |r rc .
Solid line is a fit to a linear relation: σ β α= − | − |r rT T T c with
β ≈ −0.10T , α ≈ 1.18T . Bottom inset: crossover temperature ∗
T
versus| − |r rc . The red dashed line is a fit to| − |r rc
4
.
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
T/T*
4×10
-2
2×10
-1
G(0,T/T*)/(T/T*)
σT
Σr
r = 0.03
r = 0.05
r = 0.07
r = 0.09
r = 0.11
fit to T
0.17
0 0.02 0.04 0.06 0.08
| r - rc
|
0.1
0.15
0.2
0.25-σΤ
2×10
-2
3×10
-2
6×10
-2
| r - rc
|
10
-8
10
-6
10
-4
T
*
fit to | r - rc
|
4

T-P Wu et al
9
( )
( )
( )
[ ( ) ( ) ]
†
†
†
†
∑ ∑
∑
∑
∑
∑
µ
ϕ ϕ
= + + +
= −
=
= +
+ +
= + + −
α
α α α
σ
σ σ
σ
ϕ
σ σ
σ
ϕ
σ σ
=
−
=
ε
ε
H H H H H
H c c
H f f
H V f bc
V f bc
H
Q
C
q
C e L
,
,
,
e h.c.
e h.c. ,
2 2
1
2
,
d T
L R k
k k k
d d
T
k
k L
k
k R
k
N
k
k k
k
leads Env
leads
,
L
i
2
R
i
2
Env
2
1
2
2 2

(23)
with µ = V /2L/R . The Hamiltonian of the environment,
HEnv [45–47] is represented by harmonic oscillators described
by inductances and capacitances with frequencies given by
L C1k k k/ω = . Here, the charge Q and the capacitance C are
defined as: Q C Q C Q C CL L R R L R/( )= + + with C C CL R= =
being the capacitance and QL R/ the charge on the left/right
tunneling junction, respectively. These oscillators are then
bilinearly coupled to the phase operator ϕ (canonically conju-
gate to the charge operatorQ Q QD L R= − on the dot) through
the oscillator phase. The phase operator ϕ induces phase fluc-
tuations of the tunneling amplitude between the dot and the
L/R lead [44, 47, 48]. The dissipative environment has the
following action [48]:
S
r
1
2 n
n nEnv
2
( )∑β
ω ϕ ω= | |(24)
where nω represents for the Matubara frequency of the har-
monic oscillator (boson) fields, R Rr K/≡ is the dimension-
less impedence of the leds with R h eK
2
/= being the quantum
resistance. The action SEnv gives the following correlator of
the ϕ fields:
t
e e
1
.
t
r
i
2
i
0
2⟨ ⟩
( ) ( )
∼
ϕ ϕ
−
(25)
The above correlator effectively introduces an
power-law vanishing spectral density of the tunneling
(hybridization) between lead and dot when evaluating the self-
energies of the pseudo-fermion fσ and slave-boson b fields:
V Dr2
( ) ( )ω π ω ω µΓ = | | Θ −| − |α α α (see also equations (7) and
(8)). This dissipative single-channel Anderson model can be
straightforwardly generalized to the two-channel one. Hence,
by tuning the dissipation strength r0 1 , we effectively
change the power-law exponent r of the pseudogap fermi-
onic bath in the original model. Our approach to the 2CK-LM
quantum critical point via tuning r can potentially be realized
in such systems.
In conclusion, we have studied quantum phase transitions
in and out of equilibrium between two-channel Kondo and
local moment phases in the two-channel pseudogap Anderson
model where the conduction leads show a power-law van-
ishing density of states with exponent 0 r 1. Via non-
crossing approximation (NCA), we solved self-consistently
for the impurity Green’s function, and therefore determined
the linear and non-linear conductances.The 2CK-LM quantum
criticality is reached by varying the power-law exponent r of
the pseudogap conduction bath with fixed lead-dot hybridiza-
tions and chemical potentials. The impurity local density of
states in equilibrium exhibits the pronounced Kondo peak at
the Fermi level for r rc ; while the Kondo peak splits and a
dip develops for r rc in the local moment regime. The linear
G(V = 0, T ) and non-linear G(V, T0) (T D5 100
7
∼ × −
) con-
ductances show distinct universal power-law scalings near
criticality with different critical exponents. Furthermore, in
the 2CK regime T V T, 2CK , we also find different character-
istic power-law scalings in G(T ) and G(V, T0) compared to
the well-known T V, scalings for G(0, 0) − G(0, T ) and
G(V, 0) − G(V, T0), respectively. An independent confirma-
tion of our NCA results is obtained by further computing the
local density of states in equilibrium and the linear conduct-
ances via the NRG approach. Good agreement is found for the
equilibrium properties between NCA and NRG. Our results
provide further insights on two-channel Kondo physics and
on the non-equilibrium quantum criticality in nano-systems.
Further analytic and numerical investigations are needed to
address issues on the mechanism behind the different scalings
between equilibrium and nonequilibrium conductances.
Acknowledgments
We thank S Kirchner and K Ingersent for useful discussions.
CHC acknowledges the NSC grant No.98-2112-M-009-
010-MY3, No.101-2628-M-009-001-MY3, the NCTS, the
MOE-ATU program of Taiwan ROC. G-Y Guo acknowledges
support from the Ministry of Science and Technology, the
NCTS and Academia Sinica of Taiwan (ROC).
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