We use two different methods, the Hirsch-Fye quantum Monte Carlo simulation and the slave-boson mean field analysis to estimate the effect of spin-orbit coupling on the Kondo temperature in a quantum wire. In quantum Monte Carlo simulation, we calculate the product of spin susceptibility and temperature for different spin-orbit couplings and impurity energies. The variation of the Kondo temperature is estimated via the low temperature universal curves. In the mean field analysis, the Kondo temperature is estimated as the condensation temperature of slave-boson. Both the two methods demonstrate that the Kondo temperature is almost a linear function of the spin-orbit energy, and that the Kondo temperature is suppressed by the spin-orbit coupling. Our results are dramatically different from those given by the perturbative renormalization group analysis.
We use two different methods, the Hirsch-Fye quantum Monte Carlo simulation and the slave-boson mean field analysis to estimate the effect of spin-orbit coupling on the Kondo temperature in a quantum wire. In quantum Monte Carlo simulation, we calculate the product of spin susceptibility and temperature for different spin-orbit couplings and impurity energies. The variation of the Kondo temperature is estimated via the low temperature universal curves. In the mean field analysis, the Kondo temperature is estimated as the condensation temperature of slave-boson. Both the two methods demonstrate that the Kondo temperature is almost a linear function of the spin-orbit energy, and that the Kondo temperature is suppressed by the spin-orbit coupling. Our results are dramatically different from those given by the perturbative renormalization group analysis.
Abstract The analysis of the impact of spin–orbit coupling (SOC) on the Kondo state has generated considerable controversy, mainly regarding the dependence of the Kondo temperature T K on SOC strength. Here, we study the one-dimensional (1D) single impurity Anderson model (SIAM) subjected to Rashba ( α ) and Dresselhaus ( β ) SOC. It is shown that, due to time-reversal symmetry, the hybridization function between impurity and quantum wire is diagonal and spin independent (as it is the case for the zero-SOC SIAM), thus the finite-SOC SIAM has a Kondo ground state similar to that for the zero-SOC SIAM. This similarity allows the use of the Haldane expression for T K , with parameters renormalized by SOC, which are calculated through a physically motivated change of basis. Analytic results for the parameters of the SOC-renormalized Haldane expression are obtained, facilitating the analysis of the SOC effect over T K . It is found that SOC acting in the quantum wire exponentially decreases T K while SOC at the impurity exponentially increases it. These analytical results are fully supported by calculations using the numerical renormalization group (NRG), applied to the wide-band regime, and the projector operator approach, applied to the infinite- U regime. Literature results, using quantum Monte Carlo, for a system with Fermi energy near the bottom of the band, are qualitatively reproduced, using NRG. In addition, it is shown that the 1D SOC SIAM for arbitrary α and β displays a persistent spin helix SU(2) symmetry similar to the one for a 2D Fermi sea with the restriction α = β .
Studies of the effects of the Hund's rule coupling JH in multiple orbit impurities or quantum dots using different models have led to quite different predictions for the Kondo temperature TK as a function of JH. We show that the differences depend on whether or not the models conserve orbital angular momentum about the impurity site. Using numerical renormalization-group calculations, we deduce the renormalized parameters for the Fermi liquid regime and show that, despite the differences between the models, the low-energy fixed point in the strong-correlation regime is universal, with a single energy scale TK and just two renormalized interaction parameters, a renormalized single orbital term, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}=4{T}_{K}$, and a renormalized Hund's rule term, ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}}_{H}=8{T}_{K}/3$.
In this work, using the finite-U slave boson mean-field approximation to solve the single-impurity Anderson model, the authors apply two different strategies to study the Kondo cloud, by analyzing quantities that are dependent on the distance to the magnetic impurity and then finding a universal distance scale ${\ensuremath{\xi}}_{K}$ through the collapse of the results into an universal function. The first method is based on the analysis of the local density of states of the conduction electrons (denoted as ${\ensuremath{\xi}}_{K}^{L})$, while the second relies on the analysis of spin correlations $({\ensuremath{\xi}}_{K}^{\mathrm{\ensuremath{\Sigma}}})$. Our calculations show that there is exact quantitative agreement, in the way ${\ensuremath{\xi}}_{K}$ depends on $U/\mathrm{\ensuremath{\Gamma}}$, between ${\ensuremath{\xi}}_{K}^{\mathrm{\ensuremath{\Sigma}}}$ and the results obtained through the heuristic expression ${\ensuremath{\xi}}_{K}\ensuremath{\propto}{v}_{F}/{T}_{K}$, while there is very close quantitative agreement between ${\ensuremath{\xi}}_{K}^{\mathrm{\ensuremath{\Sigma}}}$ and ${\ensuremath{\xi}}_{K}^{L}$. The use of the slave boson technique to calculate the spin correlations, which eliminates finite size effects, allowed us to study very large Kondo clouds, something that is very difficult using other techniques, like the density matrix renormalization Group method, for example. In addition, the very smooth curves obtained for the spin correlations allowed us to qualitatively identify a region in the Kondo cloud, adjacent to the impurity, that had been connected to the free orbital fixed point in previous numerical renormalization group calculations [Phys. Rev. B 84, 115120 (2011)].
It is well known that the free electron Kondo problem can be described by a one-dimensional (1D) model because only the s-wave part of the electronic wavefunction is affected by the Kondo coupling. Moreover, since only the spin degrees of freedom are involved in the Kondo interaction, and due to spin–charge separation in 1D, the universal low energy long distance physics of the Kondo model also arises when a magnetic impurity is coupled to the end of a gapless antiferromagnetic J 1 –J 2 spin- chain, where J 1 ( J 2 ) is the (next) nearest neighbor coupling. Experimental realizations of such spin chain models are possible and, using various analytical and numerical techniques, we present a detailed and quantitative comparison between the usual free electron Kondo model and such spin chain versions of the Kondo problem. For the gapless J 1 –J 2 spin chain two cases are studied, with zero next nearest neighbor coupling, J 2 = 0, and with a critical second-neighbor coupling, J 2 = J 2 c . We first focus on the spin chain impurity model with a critical second-neighbor antiferromagnetic exchange where a bulk marginal coupling present in the spin chain model for J 2 < J 2 c vanishes . There, the usual Kondo physics is recovered in the spin chain model in the low energy regime (up to negligible corrections, dropping as powers of inverse length or energy). At J 2 c the spin chain model is not exactly solvable and we demonstrate the equivalence to the Kondo problem by comparing density matrix renormalization group calculations on the frustrated spin chain model with exact Bethe ansatz calculations of the electronic Kondo problem. We then analyze the nearest neighbor model ( J 2 = 0) where a new kind of Kondo effect occurs due to the presence of the bulk marginal coupling. This marginal coupling slightly alters the β function for the Kondo coupling, leading to a slower variation of the Kondo temperature, T K , with the bare Kondo coupling. In their exact Bethe ansatz solution of this spin chain impurity model ( J 2 = 0), Frahm and Zvyagin noted this relation as well as the connection to the Kondo problem. Here, by numerically solving the Bethe ansatz equations we provide further evidence for the connection to Kondo physics and in addition we present low temperature quantum Monte Carlo results for the impurity susceptibility that further support this connection.
Recent interest in the fundamental physics of the Kondo effect has been driven by the observation of Kondo physics in mesoscopic systems such as quantum dots [1, 2] and carbon nanotubes [3], which can act analogously to magnetic impurities in a bulk metal. Highly tunable mesoscopic systems such as these present the possibility of observing the controversial Kondo length scale εK associated with the cloud of conduction electrons that screen the spin of the impurity [4]. This plays a similar role in scaling theories as the Kondo temperature Tk. One proposal [5] for detecting this length scale is to measure the finite-size dependence of persistent currents in an isolated conducting ring coupled to a quantum dot. The screening cloud should be 'trapped' in the closed system, and will not form if the size of the ring L is much smaller than εK. In particular, the current in the Kondo regime should be a universal scaling function j = L^-1 f(Φ, L/εK, T/Tk) (here Φ is the applied flux) [6, 7]. Considerable disagreement has arisen in the theoretical estimates of these persistent currents as different analytical treatments yield contradictory predictions [6, 8, 9, 10]. This thesis presents a new Quantum Monte Carlo (QMC) technique for measuring persistent currents in such systems, based on the Hirsch-Fye Impurity QMC algorithm [11] which is ideally suited to treating systems with a single impurity such as the quantum dot. The algorithm provides exact numerical results at finite temperatures. The complexity of the algorithm does not scale directly with the size of the system, making it particularly attractive for investigating a wide range of system sizes.
In this thesis we study the physical properties of quantum magnetic impurities in con- densed matter systems. More precisely, we are interested in the quantum many-body phenomena emerging in metallic quantum wires with spin-orbit coupling (SOC) in the presence of magnetic impurities. In particular, we investigate Kondo effect in topological superconducting quantum wires sustaining Majorana zero modes (MZMs) bound to their edges. The first part of this work is dedicated to a general digression on the formation of localized magnetic moments in solid state materials and the emergence of the Kondo effect in single-impurity systems. We further study a system composed by a single level quantum dot coupled to a metallic reservoir and to two MZMs with opposite polarizations. The system is described by a version of single impurity Anderson model (SIAM) adapted to account for the MZMs. To access the Kondo physics we used the projection approach to derive an effective Kondo-like Hamiltonian from the SIAM. We show the coexistence of the many-body Kondo singlet and a local Majorana singlet which results from the robustness of the Kondo fixed points even in the presence local Majorana operators. Moreover, we show that the free MZMs of the system are responsible for the impurity residual entropy S res = N/ 2 k B ln(2), where N is the number of free MZMs. This result reveals a fractional residual entropy for an odd number of free MZMs associated with a local non-Fermi liquid behavior. Both results were confirmed by the perturbative Anderson’s Poor Man’s scaling (PMS), and the nonperturbative Wilson’s numerical renormalization group (NRG). The second part of the thesis is dedicated to the low-temperature physics of two-impurity systems. We revisit the well known Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two spatially separated magnetic impurities in a quantum wire with SOC. To compute the RKKY coupling we use the original Ruderman-Kittel approach, based on the traditional second-order perturbation theory. This allows us find also the Dzaloshinskii-Moriya and Ising interactions between the impurities, both originated from the SOC. Our results show that due to momentum-spin-lock, provided by the SOC, the RKKY coupling does not decay with the inverse of the inter-impurities distance, as in the usual case. Instead, the coupling survives even at large distances oscillating with a characteristic wave-vector 2k R, where k R is proportional to the Rashba SOC.