Ground state of the impurity Anderson model revisited: A projector operator solution

Roura‐Bas, P.; Hamad, I. J.; Anda, E. V.

doi:10.1002/pssb.201451520

Cited by 3 publications

(23 citation statements)

References 38 publications

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“…The inclusion of the Zeeman interaction within the SCH procedure is straightforward and it is described in Ref. (). The impurity magnetization can be obtained by deriving of the ground state energy with respect to the magnetic field and removing from this the conduction band contribution,

M_{imp} false(B false) = - \frac{\partial}{\partial B} false(E false(B false) - ε_{T} false(B false) false) .

…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The effective Hamiltonian that operates on the

S_{1}

subspace obeys the equation ()

\tilde{H} false| φ_{1} 〉 = (H_{11} + H_{12} \frac{1}{E - H_{22}} H_{21}) false| φ_{1} 〉 = E false| φ_{1} 〉,

where

H_{ij} = P_{i} {HP}_{j}

. The term

H_{21}

comes from the hybridization term and so connects the subspace

S_{1}

with

S_{2}

.…”

Section: Model and Formalismmentioning

confidence: 99%

“…As we stated in the previous work (), in order to get an explicit expression for the eigenvalues in , it is enough to calculate the matrix elements of the operator

{false(E - H_{22} false)}^{- 1}

between the states

false| φ_{kd} 〉

. These states are created by a single application of the

H_{12}

term of the Hamiltonian to

false| φ_{1} 〉

in the

S_{1}

subspace.…”

Section: Model and Formalismmentioning

confidence: 99%

“…Although it is in principle possible to obtain all the matrix elements

g_{ij}

of the resolvent operator

G = {false(E - H_{22} false)}^{- 1}

, it can be noticed that to calculate the ground state energy, given in , it is only necessary to calculate the diagonal

g_{kk}

and nondiagonal matrix elements

g_{{kk}^{'}}

corresponding to the states below the Fermi level. Following closely the treatment done in (), and working in the thermodynamic limit, we begin by examining only the diagonal contribution

g_{kk}

, which is expressed as a continuous fraction that ultimately, in the thermodynamic limit, can be written in a closed expression as follows:

g_{kk} false(E false) = \frac{1}{E + ε_{k} - E_{d} - F_{0} false(E + ε_{k} false) - F_{2} false(E + ε_{k} false)},

where the functions

F_{0}

and

F_{2}

satisfy a set of three self‐consistent equations, exact up to terms proportional to

V^{2}

\begin{matrix} right F_{0} (E) & center = & left \frac{V^{2}}{N} \sum_{K} \frac{1}{E - ε_{K} - F_{1} (E - ε_{K})}, \\ right F_{1} (E) & center = & left 2 \frac{V^{2}}{N} \sum_{k} \frac{1}{E + ε_{k} - E_{d} - F_{0} (E + ε_{k}) - F_{2} (E + ε_{k})}, \\ right F_{2} \end{matrix}

…”

Section: Model and Formalismmentioning

confidence: 99%

“…Recently, an improvement of these calculations incorporating higher order processes when calculating the occupancy and magnetization of the impurity was done by three of us (). The agreement with exact results was remarkable.…”

Section: Introductionmentioning

confidence: 99%

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Self‐consistent hybridization expansions for static properties of the Anderson impurity model

Hamad

Roura‐Bas

Aligia

et al. 2015

Physica Status Solidi (b)

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Building an effective Hamiltonian utilizing a projector‐operator procedure, we derive an approximation based on a self‐consistent hybridization expansion to study the ground state properties of the Anderson impurity model. We applied the approximation to the general case of finite Coulomb repulsion U, extending previous work with the same formalism in the infinite‐U case. The ground state energy and their related zero temperature properties are accurately obtained in the case in which U is large enough, but still finite, as compared with the rest of energy scales involved in the model. The results for the valence of the impurity are compared with exact results that we obtain from equations derived using the Bethe ansatz and with a perturbative approach. The magnetization and magnetic susceptibility is also compared with Bethe ansatz results. In order to do this comparison, we also show how to regularize the Bethe ansatz integral equations necessary to calculate the impurity valence, for arbitrary values of the parameters.

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M_{imp} false(B false) = - \frac{\partial}{\partial B} false(E false(B false) - ε_{T} false(B false) false) .

…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The effective Hamiltonian that operates on the

S_{1}

subspace obeys the equation ()

\tilde{H} false| φ_{1} 〉 = (H_{11} + H_{12} \frac{1}{E - H_{22}} H_{21}) false| φ_{1} 〉 = E false| φ_{1} 〉,

where

H_{ij} = P_{i} {HP}_{j}

. The term

H_{21}

comes from the hybridization term and so connects the subspace

S_{1}

with

S_{2}

.…”

Section: Model and Formalismmentioning

confidence: 99%

“…As we stated in the previous work (), in order to get an explicit expression for the eigenvalues in , it is enough to calculate the matrix elements of the operator

{false(E - H_{22} false)}^{- 1}

between the states

false| φ_{kd} 〉

. These states are created by a single application of the

H_{12}

term of the Hamiltonian to

false| φ_{1} 〉

in the

S_{1}

subspace.…”

Section: Model and Formalismmentioning

confidence: 99%

“…Although it is in principle possible to obtain all the matrix elements

g_{ij}

of the resolvent operator

G = {false(E - H_{22} false)}^{- 1}

, it can be noticed that to calculate the ground state energy, given in , it is only necessary to calculate the diagonal

g_{kk}

and nondiagonal matrix elements

g_{{kk}^{'}}

corresponding to the states below the Fermi level. Following closely the treatment done in (), and working in the thermodynamic limit, we begin by examining only the diagonal contribution

g_{kk}

, which is expressed as a continuous fraction that ultimately, in the thermodynamic limit, can be written in a closed expression as follows:

g_{kk} false(E false) = \frac{1}{E + ε_{k} - E_{d} - F_{0} false(E + ε_{k} false) - F_{2} false(E + ε_{k} false)},

where the functions

F_{0}

and

F_{2}

satisfy a set of three self‐consistent equations, exact up to terms proportional to

V^{2}

\begin{matrix} right F_{0} (E) & center = & left \frac{V^{2}}{N} \sum_{K} \frac{1}{E - ε_{K} - F_{1} (E - ε_{K})}, \\ right F_{1} (E) & center = & left 2 \frac{V^{2}}{N} \sum_{k} \frac{1}{E + ε_{k} - E_{d} - F_{0} (E + ε_{k}) - F_{2} (E + ε_{k})}, \\ right F_{2} \end{matrix}

…”

Section: Model and Formalismmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Self‐consistent hybridization expansions for static properties of the Anderson impurity model

Hamad

Roura‐Bas

Aligia

et al. 2015

Physica Status Solidi (b)

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Kondo effect under the influence of spin–orbit coupling in a quantum wire

Lopes

Martins

Manya

et al. 2020

J. Phys.: Condens. Matter

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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 SOCrenormalized 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.

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SU(4)-SU(2) crossover and spin-filter properties of a double quantum dot nanosystem

et al. 2017

Self Cite

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The SU (4)−SU (2) crossover, driven by an external magnetic field h, is analyzed in a capacitivelycoupled double-quantum-dot device connected to independent leads. As one continuously charges the dots from empty to quarter-filled, by varying the gate potential Vg, the crossover starts when the magnitude of the spin polarization of the double quantum dot, as measured by n ↑ − n ↓ , becomes finite. Although the external magnetic field breaks the SU (4) symmetry of the Hamiltonian, the ground state preserves it in a region of Vg, where n ↑ − n ↓ = 0. Once the spin polarization becomes finite, it initially increases slowly until a sudden change occurs, in which n ↓ (polarization direction opposite to the magnetic field) reaches a maximum and then decreases to negligible values abruptly, at which point an orbital SU (2) ground state is fully established. This crossover from one Kondo state, with emergent SU (4) symmetry, where spin and orbital degrees of freedom all play a role, to another, with SU (2) symmetry, where only orbital degrees of freedom participate, is triggered by a competition between gµBh, the energy gain by the Zeeman-split polarized state and the Kondo temperature T SU (4) K , the gain provided by the SU (4) unpolarized Kondo-singlet state. At fixed magnetic field, the knob that controls the crossover is the gate potential, which changes the quantum dots occupancies. If one characterizes the occurrence of the crossover by V max g , the value of Vg where n ↓ reaches a maximum, one finds that the function f relating the Zeeman splitting, Bmax, that corresponds to V max g , i.e., Bmax = f V max g , has a similar universal behavior to that of the function relating the Kondo temperature to Vg. In addition, our numerical results show that near the SU (4) Kondo temperature and for relatively small magnetic fields the device has a ground state that restricts the electronic population at the dots to be spin polarized along the magnetic field. These two facts introduce very efficient spin-filter properties to the device, also discussed in detail in the paper. This phenomenology is studied adopting two different formalisms: the Mean Field Slave Bosons Approximation, which allows an approximate analysis of the dynamical properties of the system, and a Projection Operator Approach, which has been shown to describe very accurately the physics associated to the ground state of Kondo systems.

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Ground state of the impurity Anderson model revisited: A projector operator solution

Cited by 3 publications

References 38 publications

Self‐consistent hybridization expansions for static properties of the Anderson impurity model

Self‐consistent hybridization expansions for static properties of the Anderson impurity model

Kondo effect under the influence of spin–orbit coupling in a quantum wire

SU(4)-SU(2) crossover and spin-filter properties of a double quantum dot nanosystem

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