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arXiv:0908.4058 · Physical Review B

Nonequilibrium dynamics of a two-channel Kondo system due to a quantum quench

Zurab Ratiani, Aditi Mitra

Published Mar 9, 2010• 8 citations
arXivHTMLPDF

Abstract

Recent experiments by Potok et al. have demonstrated a remarkable tunability between a single-channel Fermi-liquid fixed point and a two-channel non-Fermi-liquid fixed point. Motivated by this we study the nonequilibrium dynamics due to a sudden quench of the parameters of a Hamiltonian from a single-channel to a two-channel anisotropic Kondo system. We find a distinct difference between the long-time behavior of local quantities related to the impurity spin as compared to that of bulk quantities related to the total (conduction electrons plus impurity) spin of the system. In particular, the local impurity spin and the local spin susceptibility are found to equilibrate but in a very slow power-law fashion which is peculiar to the non-Fermi-liquid properties of the Hamiltonian. In contrast, we find a lack of equilibration in the two-particle expectation values related to the total spin of the system.

DOI
10.1103/physrevb.81.125110

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Recent experiments by Potok et al. have demonstrated a remarkable tunability between a single-channel Fermi-liquid fixed point and a two-channel non-Fermi-liquid fixed point. Motivated by this we study the nonequilibrium dynamics due to a sudden quench of the parameters of a Hamiltonian from a single-channel to a two-channel anisotropic Kondo system. We find a distinct difference between the long-time behavior of local quantities related to the impurity spin as compared to that of bulk quantities related to the total (conduction electrons plus impurity) spin of the system. In particular, the local impurity spin and the local spin susceptibility are found to equilibrate but in a very slow power-law fashion which is peculiar to the non-Fermi-liquid properties of the Hamiltonian. In contrast, we find a lack of equilibration in the two-particle expectation values related to the total spin of the system.
Over recent decades, a growing number of systems, many of them quantum critical, have been shown to exhibit non-Fermi-liquid behavior, but a full analytic understanding of such systems out of equilibrium is still lacking. In this paper, we provide a distinct example with broad applications in correlated mesoscopic systems to address this issue---a two-channel Kondo-Luttinger model where a Kondo impurity couples to two voltage-biased interacting electron leads, experimentally realizable in a dissipative quantum dot. Therein, an exotic quantum phase transition has been known to exist since the 1990s from the one-channel to two-channel Kondo ground states by enhancing electron interactions in the leads, but a controlled analytic approach to this quantum critical point has not yet been established due to the breakdown of weak-coupling perturbation theory near this strong-coupling critical point. We present a controlled method to this long-standing problem by mapping the system in the strong-coupling regime to an effective spin-boson-fermion Hamiltonian. Another type of non-Fermi-liquid quantum critical point is discovered with a distinct logarithmic-in-temperature and -voltage dependence in transport. We further obtain an analytical form for the universal differential conductance out of equilibrium near the transition. Our approach can be further generalized to study nonequilibrium physics of other strong-coupling low-dimensional non-Fermi-liquid fixed points. The relevance of our results for recent experiments is discussed.
The effects of interchannel scattering of conduction electrons by the impurity and repulsion of conduction electrons at the impurity site on the two-channel Kondo model are simultaneously considered in this paper. It is shown that these two perturbations will substantially modify the usual local non-Fermi liquid behavior of the two-channel Kondo model. With bosonization and unitary transformations we find that the system can be transformed into a single channel Kondo model with anisotropy between longitudinal and transverse exchange couplings. Whatever for originally antiferromagnetic or ferromagnetic isotropic coupling, the system always flows to strong-coupling limit, which exhibits local Fermi liquid behavior at low temperatures.
We investigate the equilibrium and real-time properties of the spin-correlation function $\ensuremath{\langle}{\stackrel{P\vec}{S}}_{1}{\stackrel{P\vec}{S}}_{2}\ensuremath{\rangle}$ in the two-impurity Kondo model for different distances RRR between the two-impurity spins. It is shown that the competition between the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and the Kondo effect governs the amplitude of $\ensuremath{\langle}{\stackrel{P\vec}{S}}_{1}{\stackrel{P\vec}{S}}_{2}\ensuremath{\rangle}$. For distances RRR exceeding the Kondo length scale, the Kondo effect also has a profound effect on the sign of the correlation function. For ferromagnetic Heisenberg couplings JJJ between the impurities and the conduction band, the Kondo effect is absent and the correlation function only decays for distances beyond a certain length scale introduced by finite temperature. The real-time dynamics after a sudden quench of the system reveals that correlations propagate through the conduction band with Fermi velocity. We identify two distinct timescales for the long-time behavior, which reflects that for small JJJ the system is driven by the RKKY interaction while for large JJJ the Kondo effect dominates. Interestingly, we find that at certain distances a one-dimensional dispersion obeying $\ensuremath{\epsilon}(k)=\ensuremath{\epsilon}(\ensuremath{-}k)$ may lead to a local parity conservation of the impurities such that $\ensuremath{\langle}{\stackrel{P\vec}{S}}_{1}{\stackrel{P\vec}{S}}_{2}\ensuremath{\rangle}$ becomes a conserved quantity for long times and does not decay to its equilibrium value.
We analyze in detail all the possible fixed points of the effective Hamiltonian of a nonmagnetic impurity hopping between two sites in a metal obtained by Moustakas and Fisher (MF). We find a line of non-Fermi liquid fixed points which continuously interpolates between the two-channel Kondo fixed point (2CK) and the one-channel, two-impurity Kondo (2IK) fixed point. There is one relevant direction with scaling dimension 12\frac{1}{2}21​ and one leading irrelevant operator with dimension 32\frac{3}{2}23​. There is also one marginal operator in the spin sector moving along this line. The marginal operator, combined with the leading irrelevant operator, will generate the relevant operator. For the general position on this line, the leading low-temperature exponents of the specific heat, the hopping susceptibility and the electron conductivity ${C}_{\mathrm{imp}},{\ensuremath{\chi}}_{\mathrm{imp}}^{h},\ensuremath{\sigma}(T)$ are the same as those of the 2CK, but the finite-size spectrum depends on the position on the line. No universal ratios can be formed from the amplitudes of the three quantities except at the 2CK point on this line where the universal ratios can be formed. At the 2IK point on this line, $\ensuremath{\sigma}(T)\ensuremath{\sim}2{\ensuremath{\sigma}}_{u}{(1+aT}^{3/2})$, no universal ratio can be formed either. The additional non-Fermi-liquid fixed point found by MF has the same symmetry as the 2IK, it has two relevant directions with scaling dimension 12\frac{1}{2}21​, and is therefore also unstable. The leading low-temperature behaviors are ${C}_{\mathrm{imp}}\ensuremath{\sim}T,{\ensuremath{\chi}}_{\mathrm{imp}}^{h}\ensuremath{\sim}\mathrm{ln}T,\ensuremath{\sigma}(T)\ensuremath{\sim}2{\ensuremath{\sigma}}_{u}{(1+aT}^{3/2})$; no universal ratios can be formed. The system is shown to flow to a line of Fermi-liquid fixed points which continuously interpolates between the noninteracting fixed point and the two-channel spin-flavor Kondo fixed point discussed by the author previously. The effect of particle-hole symmetry breaking is discussed. The effective Hamiltonian in the external magnetic field is analyzed. The scaling functions for the physical measurable quantities are derived in the different regimes; their predictions for the experiments are given. Finally the implications are given for a nonmagnetic impurity hopping around three sites with triangular symmetry discussed by MF.
We consider the single-channel Kondo problem with the Kondo coupling between a spin SSS impurity and conduction electrons with spin jjj. These problems arise as multicritical points in the parameter spaces of two-and higher-level tunneling systems and some impurity models of heavy fermion compounds. In contrast to the previous Bethe-ansatz conjectures, it turns out that the dynamics of the spin sector is the same as that of a spin-SSS impurity coupled to k(j)k(j)k(j) channels of spin-\textonehalf{} electrons with k(j)=2j(j+1)(2j+1)3k(j)=2\frac{j(j+1)(2j+1)}{3}k(j)=23j(j+1)(2j+1)​. As a result, for 2S<k(j)2S<k(j)2S<k(j), the system shows non-Fermi-liquid behavior with the same exponents for the thermodynamic quantities as those of the k(j)k(j)k(j)-channel Kondo problem. However, both the finite-size spectrum and the operator content are different due to the presence of the other sectors and can be obtained by conformal field theory techniques.
A mapping between fractional quantum Hall (FQH) junctions and the two-channel Kondo model is presented. We discuss this relation in detail for the particular case of a junction of a FQH state at $\ensuremath{\nu}=1/3$ and a normal metal. We show that in the strong coupling regime this junction has a non-Fermi-liquid fixed point. At this fixed point the electron Green's function has a branch cut and the impurity entropy is equal to S=12ln2.S=\frac{1}{2}\mathrm{ln}2.S=21​ln2. We construct the space of perturbations at the strong coupling fixed point and find that the dimension of the tunneling operator is 12.\frac{1}{2}.21​. These properties are strongly reminiscent of the non-Fermi-liquid fixed points of a number of quantum impurity models, particularly the two-channel Kondo model. However we have found that, in spite of these similarities, the Hilbert spaces of these two systems are quite different. In particular, although in a special limit the Hamiltonians of both systems are the same, their Hilbert spaces are not since they are determined by physically distinct boundary conditions. As a consequence the spectrum of operators in the two problems is different.