Nonequilibrium occupation number and charge susceptibility of a resonance level close to a dissipative quantum phase transition

Nonequilibrium occupation number and charge susceptibility of a resonance level close to a dissipative quantum phase transition

Chung-Hou Chung, K.V.P. Latha Electrophysics Department, National Chiao-Tung University, HsinChu, Taiwan R.O.C. 300
Departments of Physics and Applied Physics, Yale University, New Haven, CT, 06511 USA
Institute of applied physics, Academia Sinica, NanKang, Taipei, Taiwan R.O.C. 11529
July 9, 2019
Abstract

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.

pacs:
72.15.Qm,7.23.-b,03.65.Yz

.1 Introduction

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:

(1)

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. We assume 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.

Figure 1: (Color online) versus at the KT transition. (a). . (b). . Arrows indicate spin of the corresponding curve. The bare couplings are ; bias voltage is fixed at ; the effective magnetic fields are fixed at . Here, for all the figures.

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:

Figure 2: (Color online) versus in the localized phase. (a). . (b). . Arrows indicate spin of the corresponding curve. The bare couplings are , ; bias voltage is fixed at ; the effective magnetic fields are fixed at . Here, for all the figures.
(2)

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 ():

(3)

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

Figure 3: (Color online) versus for different bare Kondo couplings. Here, the bare Kondo couplings are in units of , and with .

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 ():

(4)

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

(5)

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.

Figure 4: (Color online) Nonequilibrium magnetization in the effective Kondo model at fixed and fixed bias voltage (small bias with for lower pannel and large bias with for upper panel) versus the initial (bare) Kondo couplings across the KT transition between the delocalized phase () and the localized phase (). Here, the bare Kondo couplings are in units of with .

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

Figure 5: (Color online) Nonequilibrium magnetization in the effective Kondo model at the KT transition and in the localized phase. Here, the bare Kondo couplings are in units of , and with .

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

(6)

Here, the nonequilibrium pseudofermion self-energies are obtained via renormalized perturbation theory up to second order in :

(7)

where

(8)

with being the component of the Pauli matrices. Similarly, are obtained by interchanging and in . The nonequilibrium occupation number is given by:

(9)
Figure 6: (Color online) Nonequilibrium magnetization in the effective Kondo model at the KT transition and in the localized phase. The dot-dash (dot) lines are results via Eq. 14 (15). Here, the bare Kondo couplings are in units of , and with .

The nonequilibrium magnetization is therefore given by:

(10)

We can further simplify as:

At , magnetization takes the following simple form:

(12)

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 suppression.
To gain more understanding of the numerical results, we obtain an analytic approximated form for for . For , in the following approximated form:

(13)

while in the large bias limit, , we find has the following approximated form:

(14)

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).

Figure 7: (Color online) versus at the KT transition and in the localized phase. Here, the bare Kondo couplings are in units of , and with .

In the extremely large bias limit, , is well approximated bychung ()

(15)

In this limit, the explicit voltage dependence of at the KT transition are given bychung ():

(16)

Similarly, in the localized phase take the following forms:

where , , with . Here, we have neglected the subleading terms in Eq. LABEL:gperloc1 and Eq. LABEL:gperloc2 which depend logarithmically on .
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:

(19)

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.

.5 Susceptibility

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):

(20)
Figure 8: (Color online) versus at the KT transition and in the localized phase. Here, the bare Kondo couplings are in units of , and with .

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):

(21)

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 .

Figure 9: (Color online) versus at the KT transition and in the localized phase. Here, the bare Kondo couplings are in units of , and with .

Conclusions

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 and susceptibility 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. For small 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.).

References

  • (1) S. Sachdev, Quantum Phase Transitions, Cambridge University Press (2000).
  • (2) S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Rev. Mod. Phys. 69, 315 (1987).
  • (3) K. Le Hur, Phys. Rev. Lett. 92, 196804 (2004); M.-R. Li, K. Le Hur, and W. Hofstetter, Phys. Rev. Lett. 95, 086406 (2005).
  • (4) K. Le Hur and M.-R. Li, Phys. Rev. B 72, 073305 (2005).
  • (5) P. Cedraschi and M. Büttiker, Annals of Physics (NY) 289, 1 (2001).
  • (6) A. Furusaki and K. A. Matveev, Phys. Rev. Lett. 88, 226404 (2002).
  • (7) L. Borda, G. Zarand, and D. Goldhaber-Gordon, cond-mat/0602019.
  • (8) G. Zarand et al., Phys. Rev. Lett. 97, 166802 (2006).
  • (9) G. Refael, E. Demler, Y. Oreg, and D. S. Fisher, Phys. Rev. B 75, 014522 (2007).
  • (10) J. Gilmore and R. McKenzie, J. Phys. C. 11, 2965 (1999).
  • (11) C.H. Chung, K. Le Hur, M. Vojta and P. Wölfle, Phys. Rev. Lett 102, 216803 (2009).
  • (12) A. Rosch et al., Phys. Rev. Lett. 90, 076804 (2003); A. Rosch, J. Paaske, J. Kroha, P. Wöffle, J. Phys. Soc. Jpn. 74, 118 (2005).
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
54938
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description