Numerical renormalization group calculation of impurity internal energy and specific heat of quantum impurity models

Merker, Lukas; Costi, T. A.

doi:10.1103/physrevb.86.075150

Cited by 12 publications

(21 citation statements)

References 97 publications

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“…In contrast to Ref. [29], however, which computes c V (T ) by the explicit numerical derivative with respect to temperature, the latter can be fully circumvented along the lines of the mixed susceptibility χ FS discussed above by directly computing the plain thermal expectation value β Ĥ imp + 1 2Ĥ cpl Ĥ tot T = β 2 (Ĥ imp + 1 2Ĥ cpl )Ĥ tot T within the fdm-NRG framework (see Appendix C2 for details).…”

Section: Discussionmentioning

confidence: 97%

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Local susceptibility and Kondo scaling in the presence of finite bandwidth

Hanl

Weichselbaum

2014

Phys. Rev. B

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The Kondo scale T K for impurity systems is expected to guarantee universal scaling of physical quantities. However, in practice, not every definition of T K necessarily supports this notion away from the strict scaling limit. Specifically, this paper addresses the role of finite bandwidth D in the strongly correlated Kondo regime. For this, various theoretical definitions of T K are analyzed based on the inverse magnetic impurity susceptibility at zero temperature. While conventional definitions in that respect quickly fail to ensure universal Kondo scaling for a large range of D, this paper proposes an altered definition of T sc K that allows universal scaling of dynamical or thermal quantities for a given fixed Hamiltonian. If the scaling is performed with respect to an external parameter that directly enters the Hamiltonian, such as magnetic field, the corresponding T sc,B K for universal scaling differs, yet becomes equivalent to T sc K in the scaling limit. The only requirement for universal scaling in the full Kondo parameter regime with a residual error of less than 1% is a well-defined isolated Kondo feature with T K 0.01 D irrespective of specific other impurity parameter settings. By varying D over a wide range relative to the bare energies of the impurity, for example, this allows a smooth transition from the Anderson to the Kondo model.

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Section: Discussionmentioning

confidence: 97%

“…Moreover, it is also unclear a priori whether and to what extent to associate the coupling termĤ cpl with the impurity or the bath. Nevertheless, an approximate expression for the impurity contribution to the specific heat can be evaluated by computing c V (T ) d dT [29]. In contrast to Ref.…”

Section: Discussionmentioning

confidence: 99%

Local susceptibility and Kondo scaling in the presence of finite bandwidth

Hanl

Weichselbaum

2014

Phys. Rev. B

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“…(9) will differ from that in Eq. (8), and this will, in part, be due to the appearance of approximate NRG eigenvalues in the former, resulting in errors and noise in the time evolution, which we shall analyze numerically in Sec. V C. We thus also expect, and find, that the long-time limit of O(t), given by…”

Section: Limiting Cases and Exact Resultsmentioning

confidence: 99%

“…4. In this case, the short-time limit for n d (t) corresponds exactly to the thermodynamic value in the initial state, calculated within the conventional approach to thermodynamics via the NRG 6,8 . As shown in Ref.…”

Section: Time-dependencementioning

confidence: 92%

Generalization of the time-dependent numerical renormalization group method to finite temperatures and general pulses

Nghiem

Costi

2014

Phys. Rev. B

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The time-dependent numerical renormalization group (TDNRG) method [Anders et al. Phys. Rev. Lett. 95, 196801 (2005)] offers the prospect of investigating in a non-perturbative manner the time-dependence of local observables of interacting quantum impurity models at all time scales following a quantum quench. Here, we present a generalization of this method to arbitrary finite temperature by making use of the full density matrix approach [Weichselbaum et al. Phys. Rev. Lett. 99, 076402 (2007)]. We show that all terms in the projected full density matrix ρ i→ f = ρappearing in the time-evolution of a local observable may be evaluated in closed form at finite temperature, with ρ +− = ρ −+ = 0. The expression for ρ −− is shown to be finite at finite temperature, becoming negligible only in the limit of vanishing temperatures. We prove that this approach recovers the short-time limit for the expectation value of a local observable exactly at arbitrary temperatures. In contrast, the corresponding long-time limit is recovered exactly only for a continuous bath, i.e. when the logarithmic discretization parameter Λ → 1 + . Since the numerical renormalization group approach breaks down in this limit, and calculations have to be carried out at Λ > 1, the long-time behavior following an arbitrary quantum quench has a finite error, which poses an obstacle for the method, e.g., in its application to the scattering states numerical renormalization group method for describing steady state non-equilibrium transport through correlated impurities [Anders, Phys. Rev. Lett. 101, 066804, (2008)]. We suggest a way to overcome this problem by noting that the time-dependence, in general, and the long-time limit, in particular, becomes increasingly more accurate on reducing the size of the quantum quench. This suggests an improved generalized TDNRG approach in which the system is time-evolved between the initial and final states via a sequence of small quantum quenches within a finite time interval instead of by a single large and instantaneous quantum quench. The formalism for this is provided, thus generalizing the TDNRG method to multiple quantum quenches, periodic switching, and general pulses. This formalism, like our finite temperature generalization of the single-quench case, rests on no other approximation than the NRG approximation. The results are illustrated numerically by application to the Anderson impurity model.

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“…55 When the side-coupled QD in either empty or double occupied and QD d is in the Kondo regime, the transport properties are dominated by the Kondo effect on QD d. As we show below, the main features observed in the single level case for the temperature dependence of the conductance, are also observed when multiple levels are considered in the side-coupled QD. We calculate the conductance through the system and the magnetic susceptibility using the full density matrix numerical renormalization group (FDM-NRG) 56,57 . In all calculations we use a logarithmic discretization parameter Λ = 10 and the z-trick 58,59 averaging over four values of z = 0.25, 0.5, 0.75, and 1.…”

Section: Numerical Resultsmentioning

confidence: 99%

Transport through side-coupled multilevel double quantum dots in the Kondo regime

2014

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We analyze the transport properties of a double quantum dot device in the side-coupled configuration. A small quantum dot (QD), having a single relevant electronic level, is coupled to source and drain electrodes. A larger QD, whose multilevel nature is considered, is tunnel-coupled to the small QD. A Fermi liquid analysis shows that the low temperature conductance of the device is determined by the total electronic occupation of the double QD. When the small dot is in the Kondo regime, an even number of electrons in the large dot leads to a conductance that reaches the unitary limit, while for an odd number of electrons a two stage Kondo effect is observed and the conductance is strongly suppressed. The Kondo temperature of the second stage Kondo effect is strongly affected by the multilevel structure of the large QD. For increasing level spacing, a crossover from a large Kondo temperature regime to a small Kondo temperature regime is obtained when the level spacing becomes of the order of the large Kondo temperature. arXiv:1401.7963v2 [cond-mat.mes-hall]

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Numerical renormalization group calculation of impurity internal energy and specific heat of quantum impurity models

Cited by 12 publications

References 97 publications

Local susceptibility and Kondo scaling in the presence of finite bandwidth

Local susceptibility and Kondo scaling in the presence of finite bandwidth

Generalization of the time-dependent numerical renormalization group method to finite temperatures and general pulses

Transport through side-coupled multilevel double quantum dots in the Kondo regime

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