Nonequilibrium occupation number and charge susceptibility of a resonance level close to a dissipative quantum phase transition
Based on the recent paper (Phys. Rev. Lett. 102, 216803, (2009)), we study the nonequilibrium occupation number and charge susceptibility of a resonance level close to dissipative quantum phase transition of the Kosterlitz-Thouless (KT) type between a de-localized phase for weak dissipation and a localized phase for strong dissipation. The resonance level is coupled to two spinless fermionic baths with a finite bias voltage and an Ohmic bosonic bath representing the dissipative environment. The system is equivalent to an effective anisotropic Kondo model out of equilibrium. Within the nonequilibrium Renormalization Group (RG) approach, we calculate nonequilibrium magnetization and spin susceptibility in the effective Kondo model, corresponding to and of a resonance level, respectively. We demonstrate the smearing of the KT transition in the nonequilibrium magnetization as a function of the effective anisotropic Kondo couplings, in contrast to a perfect jump in at the transition in equilibrium. In the limit of large bias voltages, we find and at the KT transition and in the localized phase show deviations from the equilibrium Curie-law behavior. As the system gets deeper in the localized phase, both and decrease more rapidly to zero with increasing bias voltages.
Quantum phase transitions (QPTs)sachdevQPT (); Steve ()
due to competing quantum ground states
in strongly correlated systems
have been extensively investigated over the past decades.
Near the transitions, exotic quantum critical properties are
realized. In recent years, there has been
a growing interest in QPTs in nanosystemslehur1 (); lehur2 (); zarand (); Markus (); matveev (); Zarand2 ().
Very recently, QPTs have been extended to nonequilibrium
nanosystems where little is known regarding
nonequilibrium transport near the transitions. A generic
examplechung () is the transport through a dissipative
resonance-level (spinless quantum dot)
at a finite bias voltage where dissipative bosonic
bath (noise) coming from the environment in the leads gives rise
to quantum phase transition in transport
between a conducting (de-localized) phase where resonant tunneling
dominates and an insulating (localized) phase where
the dissipation prevails.
In fact, dissipative quantum phase transitions have been investigated
in various systemsJosephson (); McKenzie ().
Nevertheless, much of the attention has been
focused on equilibrium properties; while very little is known
on the nonequilibrium properties.
The bias voltage plays a very different
role as the temperature in equilibrium systems as
the voltage-induced decoherence behaves very differently
from the decoherence at finite temperature,
leading to exotic transport properties near the quantum phase
transition compared to that in equilibrium at finite temperatures.
Based on the recent work in Ref. chung ()
on nonequilibrium transport of a dissipative
resonance-level at the Kosterlitz-Thouless (KT) type
de-localized-to-localized quantum transition,
we study in this paper
the nonequilibrium occupation number and charge susceptibility
of a resonance-level quantum dot subjected to a noisy environment near
the phase transition. In equilibrium,
it has been shown that the occupation number
of a dissipative resonance-level shows a jump at
the Fermi energy at the KT transition and in the localized phase
where is an infinitesmall shift
in the energy of the resonance-levellehur1 (); matveev ().
At finite temperatures
and in equilibrium, a crossover in
replaces the jump and in the high temperature limit it
is determined by the thermal magnetization of a free spin;
hence the Curie law behavior is expected. On the other hand,
when a large bias voltage is
applied on the system at , however, very little is known
about the nonequilibrium effects on the occupation number
and charge susceptibility. By first mapping our system onto an
effective Kondo model and applying the recently developed
frequency-dependent Renormalization Group (RG) approachRosch ()
to the nonequilibrium Kondo effect of a quantum dot, we
calculate the occupation number and charge susceptibility
of a resonance-level near the transition. Near the transition,
we find distinct nonequilibrium
behaviors of these quantities from those in equilibrium.
.2 Model Hamiltonian
The starting point is a spin-polarized quantum dot coupled to two Fermi-liquid leads subjected to noisy Ohmic environment, which coupled capacitively to the quantum dot. The noisy environment here consists of a collection of harmonic oscillators with the Ohmic correlation: with being the circuit resistance and being the quantum resistance. For a dissipative resonant level (spinless quantum dot) model, the quantum phase transition separating the conducting and insulating phase for the level is solely driven by dissipation. Our Hamiltonian is given by:
where is the hopping amplitude between the
lead and the quantum dot, and are electron operators for the
Fermi-liquid leads and the quantum dot, respectively,
is the chemical potential (bias voltage)
applied on the lead , while is the energy level of the dot.
that the electron spins have been polarized by a
strong magnetic field. Here, are the boson operators of the
dissipative bath with an ohmic spectral density
lehur2 (): with being the
strength of the dissipative boson bath.
First, through similar bosonization and refermionization procedures as in equilibrium lehur1 (); lehur2 (); Markus (); matveev (), we map our model to an equivalent anisotropic Kondo model in an effective magnetic field with the effective left and right Fermi-liquid leadschung (). The effective Kondo model takes the form:
where is the electron operator of the effective lead , with spin . Here, the spin operators are related to the electron operators on the dot by: , , and where describes the charge occupancy of the level. The spin operators for electrons in the effective leads are , the transverse and longitudinal Kondo couplings are given by and respectively, and the effective bias voltage is , where . Note that near the transition ( or ) where the above mapping is exact. The spin operator of the quantum dot in the effective Kondo model can also be expressed in terms of spinful pseudofermion operator : . In the Kondo limit where only the singly occupied fermion states are physically relevant, a projection onto the singly occupied states is necessary in the pseudofermion representation, which can be achieved by introducing the Lagrange multiplier so that . An observable is defined asRosch ():
In equilibrium, the above anisotropic Kondo model
exhibits the Kosterlitz-Thouless transition from a de-localized
phase with a finite conductance
for to a localized phase for
with vanishing conductance. The nonequilibrium
transport near the KT transition exhibits distinct profile from that
in equilibrium and it has been addressed in Ref. chung ().
We will focus here on the nonequilibrium occupation number and
charge susceptibility near the KT transition.
At the KT transition () and in the localized phase,
we expect in equilibrium
a perfect jump in (or ): for
and for lehur1 (). At a finite bias voltage, however,
instead of a jump
we expect shows a smooth crossover as a function of .
.3 Nonequilibrium RG formalism
The non-equilibrium perturbative renormalization group (RG) equations for the effective Kondo model in a magnetic field are obtained by considering the generalized frequency dependent Kondo couplings in the Keldysh formulation followed Ref. Rosch ():
where , are dimensionless frequency-dependent Kondo couplings with being density of states per spin of the conduction electrons (we assume symmetric hopping ), , is the running cutoff, and is the decoherence (dephasing) rate at finite bias which cuts off the RG flow Rosch (), given by
where is the Fermi function given by
. Note that the Kondo couplings
exhibit the following symmetries: , . We have solved the RG equations
subject to Eq. 5 self-consistently. The solutions for
and at the transition
are shown in Fig. 1. Similar behaviors for
are obtained in the localized phase.
The decoherence rate is plotted
in Fig. 3.
Note that, unlike the equilibrium RG at finite temperatures
where RG flows are cutoff by temperature ,
here in nonequilibrium the RG flows will be cutoff by the
decoherence rate , a much lower energy scale than ,
. This explains the dips (peaks) structure
in in Fig. 1 and Fig. 2. In contrast, the
equilibrium RG will lead to approximately frequency independent
couplings, (or “flat” functions
In the absence of field , show
dips (peaks) at .
In the presence of both bias and field ,
shows dips at ; while
show peaks at
()Rosch (). At , two dips of
at and merge into
a large dip at .
We use the solutions
of the frequency-dependent Kondo couplings
to compute the occupation number and charge susceptibility
of the resonance-level near the transition.
.4 Occupation number and magnetization
From the mapping, the occupation number of the resonance level is related to the magnetization of the pseudofermion in the effective Kondo model by where . Since the occupation number in the dissipative resonance-level is related to the pseudospin magnetization in the effective Kondo model by a simple linear relation, in the following we will use the properties of the magnetization to represent those of the occupation number. The nonequilibrium occupation number of the pseudofermion in the effective model can be determined by solving the Keldysh component of the Dyson equation for the pseudofermion self-energyRosch (), given by
Here, the nonequilibrium pseudofermion self-energies are obtained via renormalized perturbation theory up to second order in :
with being the component of the Pauli matrices. Similarly, are obtained by interchanging and in . The nonequilibrium occupation number is given by:
The nonequilibrium magnetization is therefore given by:
We can further simplify as:
At , magnetization takes the following simple form:
Note that occupation number can also be determined by the rate equationRosch (): or where is the spin-flip rate of pseudofermion from spin-down to spin-up state.
We have calculated the magnetization numerically at the KT transition and in the localized phase for both small bias limit and large bias limit where we have fixed at a small value. First, we demonstrate that the nonequilibrium magnetization with a fixed for both fixed small (lower panel of Fig. 4) and large (upper panel of Fig. 4) bias voltages shows a smooth crossover as a function of across the KT transition (), in contrast to a perfect jump in at the transition in equilibriumlehur1 ().
To investigate further the crossover behavior of the magnetization ,
we calculate as a function of with being fixed at
a small value .
The result is shown in Fig. 5.
First, let us examine simple limits from the numerical results.
The spin of the quantum dot gets fully polarized only when magnetic field
exceeds the bias voltage, ; while for the
magnetization is reduced due to finite spin-flip decoherence rate.
In the extreme large bias limit, , we find gets further
To gain more understanding of the numerical results, we obtain an analytic approximated form for for . For , in the following approximated form:
while in the large bias limit, , we find has the following approximated form:
Here, we have treated within the interval
as a semi-ellipse.
From Eq. 13 and Eq. 14, it is clear that the behaviors
of the magnetization depend sensitively on the dip-peak structure in
, especially on the ratio ,
and . In general, the
analytical approximated forms for at
are rather complex. Nevertheless,
the values of at
these specific values of can be obtained numerically
(see, for example Fig. 7).
In the extremely large bias limit, , is well approximated bychung ()
In this limit, the explicit voltage dependence of at the KT transition are given bychung ():
Similarly, in the localized phase take the following forms:
, with . Here, we have neglected the subleading terms in
Eq. LABEL:gperloc1 and Eq. LABEL:gperloc2 which depend logarithmically
We first look at the behavior of at the KT transition. At a general level one might expect the nonequilibrium magnetization at in the de-localized phase behave in a similar way as the equilibrium thermal magnetization with being replaced by , leading to linear behavior in at high temperatures. In equilibrium and at finite temperatures, it has been shown that the magnetization of a closely related model–a resonance-level with Ohmic dissipation– exhibits linear behavior in at the KT transition. It is clear from Eq. 15 that the magnetization in the equilibrium form based on the ”flat approximation” () always predicts a linear behavior in . At the KT transition, we find the nonequilibrium magnetization for also shows linear behavior, . This can be understood from Eq. 13 as at the KT transition for . The Curie-law (linear) behavior in here is reminiscent of the equilibrium thermal magnetization of a free spin in the high temperature regime. However, at the large bias voltages, , we find a logarithmic correction to this linear behavior in at the KT transition due to the nonequilibrium effect:
This logarithmic suppression can be understood from Eq. 15 as in this case () becomes peak (dip) and the ratio satisfies .
We now discuss in the localized phase. First, in the limit of small bias, , as the system gets deeper in the localized phase, approaches to fully polarization more rapidly than that at the KT transition. This is expected as the system gets deeper in the localized phase, the spin is more easily polarized upon applying a magnetic field. This behavior can also be explained from Eq. 13 as in the localized phase the ratio , and it only gets smaller as the system gets deeper in the localized phase. In fact, the same qualitative behavior is seen in a closely related Bose-Fermi Kondo modellehur1 () which shows the KT transition between the Kondo and local moment ground states. In the large bias limit , however, deviates from the linear behavior due to nonequilibrium effects. The correction of to linear behavior is dominated by the ratio via Eq. 15 where shows deeper dips at , making to rapidly increase with decreasing (see Fig. 7). This gives rise to a further suppression of at large bias voltages compared to that at the KT transition (see Fig. 6).
Note that from Fig. 5 and Fig. 6, as the system goes deeper into the localized phase (or with decrease in ), we find for a fixed increases for a fixed small bias voltage (); while it decreases for a fixed large bias viltage (). This is in perfect agreement with the crossover behavior for shown in Fig. 4.
Notice that the linear behavior of is expected in purely asymmetric but isotropic () Kondo modelRosch (). In a symmetric Kondo model with , the nonequilibrium magnetization acquires an additional positive logarithmic corrections Rosch (). In the present case, however, the deviation from the linear behavior of in the large bias limit has a different origin. It comes from the fact that our effective Kondo model is not only asymmetric () but also highly anisotropic () at the KT transition and in the localized phase. Different corrections to the linear behavior are expected.
The nonequilibrium charge susceptibility in the dissipative resonance-level is obtained from the spin susceptibility in the effective Kondo model by the mapping mentioned above. The susceptibility of a Kondo dot in equilibrium at finite temperatures is given by the Curie’s law . However, in our highly asymmetric and anisotropic Kondo model, we find the nonequilibrium susceptibility deviates significantly from the Curie law. As shown in Fig.8, at the KT transition, as bias voltage is increased, first shows Curie-law behavior, followed by an increase and a peak around . In the large bias limit, gets a logarithmic suppression (see Eq. 15):
Note that the rapid increase in at the KT transition with decreasing for is reminiscent of the spin susceptibility of a nonequilibrium Kondo dot in a magnetic field where acquires a logarithmic increase at large bias voltagesRosch (). On the other hand, the logarithmic decrease in here at large bias is a direct consequence of the dip structure in at the KT transition (see Eq. 16 and Fig. 9).
As the system gets deeper in the localized phase, gets a more pronounced peak at . As bias is further increased, shows a similar trend as that at the KT transition–a peak around but smaller magnitudes (see Fig. 8). At large bias voltages, , gets a more severe power-law suppression compared to the slower logarithmic decrease at the KT transition (see Fig. 9 and Eq. LABEL:gperloc1 and Eq. LABEL:gperloc2):
with given by Eq. LABEL:gperloc1 and Eq. LABEL:gperloc2. This comes as a result of further decrease in Kondo coupling at in the localized phase under RG.
We may compare the behavior in in our model at large bias voltages with that in different limit of the same model or with different models. In the equilibrium limit within our model where can be considered as flat functions over , , a perfect Curie law behavior is expected for . However, for isotropic Kondo model () for a simple quantum dot in Kondo regime and at large bias voltages, shows Curie law with positive logarithmic correction, with being Kondo temperature for a single quantum dot. In our dissipative resonance-level model, the suppression in at large bias voltages at the KT transition and in the localized phase comes from the dips at .
In conclusion, we have investigated the nonequilibrium
occupation and charge susceptibility of a dissipative
resonance-level with energy . For , the system
exhibits the Kosterlitz-Thouless type quantum transition
between a de-localized phase at small dissipation strength
and a localized phase with large dissipation.
We first mapped our problem onto an effective nonequilibrium
anisotropic Kondo model in the presence of a magnetic field
. The occupation number and
charge susceptibility correspond to magnetization
of the pseudospin in the effective Kondo model, respectively.
By nonequilibrium RG approach,
we solved for the frequency-dependent effective
Kondo couplings and calculated magnetization
and at finite bias voltages.
We demonstrate the smearing of the KT transition
in the nonequilibrium
magnetization at a fixed as a function
of the effective anisotropic Kondo couplings
for both small bias and large bias voltages as it
exhibits a smooth crossover at
the KT transition, in contrast to a perfect jump
in at the transition in equilibrium.
bias and effective field and ,
we find the magnetization
at the KT transition shows
linear behavior in ; while in the localized phase
increases more rapidly with approaching to from above,
consistent with the behaviors of the equilibrium
magnetization in the localized phase at finite temperatures.
In the large bias limit , however, we find corrections
to equilibrium Curie-law behavior in due to nonequilibrium effects.
At the KT transition, the corrections are logarithmic; in ;
while in the localized phase they are power-law in . Our
results have direct relevance for the transport measurements
in nanostructures, and should stimulate further experiments.
Acknowledgements.We are grateful for the helpful discussions with P. Wöelfle. This work is supported by the NSC grant No.98-2918-I-009-06, No.98-2112-M-009-010-MY3, the MOE-ATU program, the NCTS of Taiwan, R.O.C. (C.H.C.).
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