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Kondo temperature equation for the Fermi liquid and the non-Fermi liquid regime for a two-channel anderson model
Journal of Magnetism and Magnetic Materials 499 (2020) 166259
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Kondo temperature equation for the Fermi liquid and the non-Fermi liquid regime for a two-channel anderson model C.A. Philippsa, P.G.B. Shiotaa, F.M. Zimmerb, J.V.B. Ferreirab, a b
T
⁎
Programa de Pós-Graduaçāo em Ciências dos Materiais, Instituto de Física, Universidade Federal de Mato Grosso do Sul, Campo Grande-MS, Brazil Instituto de Física, Universidade Federal de Mato Grosso do Sul, Campo Grande-MS, Brazil
A R T I C LE I N FO
A B S T R A C T
Keywords: Kondo temperature Anderson model Kondo effect Fermi liquid Non-Fermi liquid
Multichannel Kondo resonance causes non-Fermi liquid behavior in many systems, as nanostructured metallic quantum dots and metallic alloys. The Kondo effect occurs at low temperatures and its indicating parameter is the Kondo temperature TK . This parameter can be used to renormalize thermodynamic and transport properties. The Kondo temperature is an important quantity to analyze the antiferromagnetic interaction between a localized spin and itinerant ones in a fermionic bath. There are many different definitions of the Kondo temperature. One of them is that it universalizes magnetic impurity susceptibility. This version was created for the onechannel Anderson model, but it is also valid in two-channel Anderson models, where a single magnetic impurity interacts with the conduction band electrons by two different channels. If the two coupling parameters are different, the model exhibits the well known Fermi liquid regime. When the coupling parameters are equal, the model displays the non-Fermi liquid regime. Each one shows different universal magnetic susceptibility behavior with appropriate TK value. In this article, Numerical Renormalization Group method is applied to study the Kondo temperature systematically. Different values of the coupling parameters in a two-channel Anderson model are examined. Finally, a single analytic expression for the Kondo temperature, valid in the Fermi and non-Fermi liquid regime, is presented.
1. Introduction The Kondo temperature TK indicates the energy scale where the Kondo effect becomes important in the system of a magnetic impurity coupled to an electronic reservoir [1,2]. For example, electronic conductance at low temperatures can be affected by Kondo correlations in different systems: (i) quantum dot system [3,4] (ii) metallic alloys with diluted 4f 2 orbitals [5] or Kondo entanglement [6]. The parameter TK can be used to universalize thermal properties, for instance, the magnetic susceptibility of the atomic impurity, which has been observed both in simple metals, and in metallic alloys [5]. In general, simple metals can be described by Landau Fermi liquid theory (FL) [7]. However, with FL theory many phenomena of metallic alloys can not be covered properly, since they exhibit the so called non-Fermi liquid behavior (NFL) [8]. Therefore, it is important to know how to determine the Kondo temperature for different single impurity magnetic models for the FL and the NFL regime. Usually, a magnetic atom in a non-magnetic solid is represented by single impurity Anderson models (SIAMs). These models show the Kondo effect due to the antiferromagnetic interactions
⁎
between the localized spin (atom) and the spins of the conduction band electrons. At first, SIAMs could only yield the FL behavior. These models stipulated that the localized spin was completely quenched by the itinerant ones at sufficiently low temperatures (T ⩽ TK ). The two wellknown models were the Kondo and the Anderson models [9,10]. In these models, the procedures to obtain TK derive from perturbation theory [11] or numerical methods [12]. In numerical methods, the Kondo temperature is obtained from universalizing the magnetic susceptibility temperature curve. It was realized that these models were oversimplified and were not able to describe metallic alloys that exhibited the NFL behavior. In order to make the Kondo model more realistic, P. Nozières and A. Blandin introduced a concept called channel [13]. It is a new quantum number considered in the SIAM Hamiltonian representing the universality and symmetry of the physical system, and intends to catch intricate effects occurring in metals, as the crystal field splitting [14], rare earth impurity and spin orbit coupling [15]. There are three different cases that can be achieved by changing the relation between the number of channels n and the impurity spin S
Corresponding author. E-mail address: [email protected] (J.V.B. Ferreira).
https://doi.org/10.1016/j.jmmm.2019.166259 Received 14 July 2019; Received in revised form 27 November 2019; Accepted 1 December 2019 Available online 16 December 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 499 (2020) 166259
C.A. Philipps, et al.
Unlike the Kondo model generalization, the Anderson model generalization is non-trivial [19] due to the more complex interaction between the impurity and the conduction band electrons. In the Kondo model the coupling only flips impurity spin, whereas in the Anderson model the coupling ionizes the impurity atom. Hence, the diagonalization of the TCSIAM Hamiltonian with NRG is a laborious task [20,23]. The following Hamiltonian defines the two-channel single impurity Anderson model.
[16]: the number of channels n = 2S is capable of fully compensating the localized spin leading to a spin singlet ground state and usually to the emergence of the Fermi-liquid behavior; the number of conduction electron channels n < 2S is not high enough to yield a singlet ground state with the impurity spin partially compensated; and the number of channels n > 2S is larger than required to compensate the impurity spin. These situations are known as compensated, underscreened and overscreened Kondo (Anderson) models, respectively. The last two cases are the cause of the non-Fermi-liquid properties. Nowadays, the original Anderson (Kondo) model is known as the one-channel Anderson (Kondo) model and others models, with n ⩾ 2 , as n-channel Anderson (Kondo) models [17,18]. It is important to emphasize that the one-channel Anderson (Kondo) model does not show the non-Fermi liquid regime, but the two-channel or higher channel models are capable of yielding this behavior [8]. D. L. Cox proposed a two-channel Anderson model (TCSIAM) with a single impurity to study the transition from the Fermi liquid to the nonFermi liquid behavior in thermal properties of a localized magnetic atom. Each channel is considered as a specific quantum mechanism to join the conduction band with the localized spin, using the two coupling parameters, V1 ( J1) and V2 ( J2 ), in Anderson (Kondo) notation. If V1 equals V2 , localized spins are never completely quenched, so physical properties differ from the FL paradigm. Metals with d- or f-orbital valence electrons belong to a class of materials that, at low energies, have conduction band electrons that do not behave essentially as a collection of weakly interacting quasi particles – violating basic Landau theory assumption of the FL behavior [7]. The two-channel single impurity Anderson model Hamiltonian was diagonalized using Numerical Renormalization Group (NRG) and its magnetic susceptibility versus temperature curves were calculated in the FL and the NFL regime [19,20]. Remarkably, even in the non-Fermi liquid behavior, it is possible to define the Kondo temperature as a parameter that universalizes magnetic susceptibility [21]. However, these numerical data are still lacking an analytical expression. We address exactly this point in the present work, finding a general relation between the coupling parameters V1 and V2 of the two-channel single impurity Anderson model and the Kondo temperature in the Fermi liquid and the non-Fermi liquid regime. Thus, this article shows a systematic study of the TK for the TCSIAM covering FL and NFL behavior. The TK is obtained using numerical renormalization theory and a method to universalize the magnetic susceptibility curve [21]. Finally, an analytical expression describing the dependency between the Kondo temperature TK and the physical parameters, V1 and V2 of the TCSIAM, is suggested. This expression is based on Wilson’s analytic and numerical arguments [10]. A brief presentation of the two-channel single impurity Anderson model is given. Then, it is shown how Numerical Renormalization Group can be used to solve its Hamiltonian. The relation between the TCSIAM and the two-channel Kondo model is explained via the Schrieffer-Wolff transformation. Afterwards, the numerical procedure to determine the TK (K. G. Wilson’s procedure) is explained. Finally, numerical data and K. G. Wilson theory is used to obtain a generalized expression for the Kondo temperature, being valid in the Fermi liquid and non-Fermi liquid regime.
H = Hcb + Himp + Hhyb Hcb =
(1)
† ∊k ckσα ckσα
∑
(2)
k, σ , α
Himp= E0
∑
m, σ
m, σ +
σ
Hhyb =
⎛ ⎞ ⎜E0 + Δ⎟ ⎝ ⎠
∑
⎛ ⎞ ⎜E0 + Δ⎟ ⎝ ⎠
∑
∑ k, σ , α
m − 1, α
m − 1, α +
m + 1, α
m + 1, α
α
α
⎛ ⎞ Vα ⎜f−†1, σ , α ckσα + h. c. ⎟ ⎝ ⎠
(3)
(4)
Hcb, Himp and Hhyb represent the conduction band, the magnetic impurity and the band-impurity coupling respectively. Hcb represents a fermionic bath, depicted by a half filled metallic conduction band, isotropic and energy width equal to 2D (D is an adjustable parameter, in this article D = 5 eV is used) [24]. The magnetic localized impurity is modeled by six quantum states. The ground state is a double degenerate one, that has m electrons, energy E0 and spin S = 1/2 . The z component of spin is represented by the quantum number σ = ± 1/2 , therefore, the two magnetic states are m , σ . Following Nozières precept [13], the impurity in its ground states is allowed to interact with the conduction band via two different mechanisms, called channels, described by quantum number α (α = 1 or 2) each one having its own hybridization parameter Vα . After the interaction, the impurity will have gained or lost an electron, so that its spin is now zero and the state is not magnetic anymore. This new quantum state has energy E0 + Δ (Δ > 0 ), it is four times degenerated and is represented by the quantum states m ± 1, α , where the quantum number α indicates the channel that allowed the † impurity to gain or lose an electron. The operator ckσα creates a free → conduction band electron with momentum k , spin σ = ± 1/2 , channel α = 1 or 2 and energy εk . Operator f−†1, σ , α is defined in such a way that it represents the transition between the ground and the excited states (α = −1 → α = 2 ) and vice versa (α = −2 → α = 1), as well as retaining spin and channel symmetry in quantum states. f−†1, σ , α = m , σ m − 1, −α + (2σ ) m + 1, α m , −σ
(5)
2. Theoretical concepts
The behavior of the TCSIAM is determined by the physical parameters Δ, V1 and V2 . Δ corresponds to the energy difference between the ground and the excited states. V1 and V2 are hybridization parameters between the impurity and the conduction band. It is important to note that if V1 = 0 and V2 ≠ 0 , or vice versa, the TCSIAM reproduces the results of the traditional one-channel Anderson model.
2.1. Two-channel single impurity Anderson model – TCSIAM
2.2. Schrieffer-Wolff Transformation
The original Anderson and Kondo models are defined as onechannel models since they consider magnetic impurity and conduction band electrons with only one coupling mechanism between them, indicated by parameter V or J respectively. Metal alloys and new materials require a model with more degrees of freedom [13,17] to explain the non-Fermi liquid behavior [22]. D. L. Cox proposed a two-channel model which is a generalization of the one-channel Anderson model.
The Schrieffer-Wolff Transformation relates the Hamiltonian of the one-channel Anderson model to the one of the one-channel Kondo model. The relation shows that the Anderson model is equivalent to the Kondo model in the strong coupling regime [25,26]. Similar equivalence exists between the two-channel models at low excitation energies. The relation between the coupling parameters is Jα ∝ Vα2/Δ , where Jα is the hybridization parameter in the two-channel Kondo model and Vα in 2
Journal of Magnetism and Magnetic Materials 499 (2020) 166259
C.A. Philipps, et al.
the two-channel Anderson model [21]. 2.3. Numerical Renormalization Group – NRG The Numerical Renormalization Group was developed by K. G. Wilson to diagonalize the one-channel Kondo model [10] and afterwards applied by him, H. R. Krishna-murthy and J. W. Wilkins to the one-channel Anderson model [12]. It transforms the Hamiltonian into a semi-infinite series of terms in decreasing order of energy scales. Only the first term takes the coupling with the impurity into account. The eigenstates are built recursively, allowing a truncation of states based on relevant energy scales, with minor effects on impurity properties. Thus we have a non-perturbative numerical procedure from which the energy spectrum of the model is obtained. Physical properties are calculated using the NRG with interleaving discretization [27,28] and the two-step transformation [23]. Fig. 1. The magnetic susceptibility χ for a null external magnetic field as function of the temperature T / TK . The black curve is the universal behavior of the one-channel Anderson model (FL). The other curves are identified by values of V1 = V2 = 0.01D and different Δ values showing the NFL behavior. Adapted from reference [21].
2.4. Computational details Computational data was obtained with the following machine. Intel (R) Core(TM) i7 CPU 860 2.80 GHz 8 processors RAM 16 GB. The operation system was Linux 2.6.26-2-amd64 x86-64 GNU/Linux. The NRG programs were implemented using C++ language and double floating point precision. Auxiliary programs to do Interleaving procedure and parallel routines were done using Script language. Interleaving procedure was done with 8 values for the discretization of the conduction band. In general, each Kondo temperature value TK took 0.75 hours of run time and used 250 MB of memory. About 3200 TK values were obtained.
On the contrary, the TCSIAM shows Fermi and non-Fermi liquid behavior, even though it is not obvious that, in the NFL case, Tχ (T ) solely depends on T / TK . However, WKM procedure still works. Fig. 1 shows Tχ (T ) scaled by T / TK . The hybridization parameters Vα change the magnetic susceptibility and therefore alter the Kondo temperature. Nevertheless, all curves have the same value at low temperatures in a rescaled graph. If both coupling parameters are equal, (V1 = V2 ), the NFL behavior occurs but TK remains defined by WKM procedure. At low temperatures, these curves also present a universal behavior, although different from the one-channel Anderson model. Both universal curves, the FL and the NFL, satisfy the condition given from WKM procedure. Until now, a mathematical expression describing the Kondo temperature for the TCSIAM has not been found yet. In the next section, we propose an expression based on NRG, WKM and analytic approximations.
2.5. Kondo temperature – TK Firstly, we will present a chronological description of the Kondo temperature concept in the one-channel Kondo model. In 1964, J. Kondo designated T0 as the temperature value at which the resistivity expansion diverges [9]. Later in 1975, K. G. Wilson denoted this parameter as Kondo temperature TK [10]. He used it in scaling formulas by combining results from NRG and analytic calculations. For example, he showed that the product of the thermal energy and the magnetic susceptibility kB Tχ (T ) depends only on T / TK in the range of 0.5 TK < T < 16 TK , whereas χ (T ) is the magnetic susceptibility of the impurity. V. Melnikov in 1982 and P. Wiegmann in 1983 [29], refined this formula, leading to
kB Tχ (T ) 0.17 ≈ T (gμB )2 1 + 1.4 T
We present numerical results obtained from the TCSIAM for several sets of parameters {Δ, V1, V2 }. The hybridization parameters are related to the Fermi liquid behavior for V1 ≠ V2 and to the non-Fermi liquid in the case of V1 = V2 . In order to obtain the Kondo temperature, we adopt the following procedure. For a given set of Δ, V1 and V2 , the TCSIAM Hamiltonian is diagonalized using NRG technique, interleaving discretization and the two-step transformation methods. The magnetic susceptibility dependence of the temperature is evaluated and the TK value is obtained from WKM procedure via Eq. (7).
−1
( ) K
3. Results and discussion
(6)
where kB is the Boltzmann constant, g is the electron gyromagnetic ratio and μB is the Bohr magneton. Eq. (6) is used until today to obtain an universalized expression for the magnetic susceptibility of lattice impurities. Following the same concepts, other researchers analyzed the onechannel Anderson model. In 1980, Krishna-murthy, Wilkins and Wilson calculated the magnetic susceptibility of the Anderson model using NRG and second order perturbation theory [12]. They fitted their results to the Curie–Weiss law using Eq. (6) and showed the universal behavior of the magnetic susceptibility. Interestingly, they obtained χ (T ) via NRG using this equation and making it possible to determine TK in the case of T = TK . The magnetic susceptibility dependence of the temperature is evaluated and TK is obtained from
kB TK χ /(gμB )2 ≈ 0.07.
3.1. NRG data The Kondo temperature dependency on the hybridization parameters V1 and V2 for two different Δ values is shown in Fig. 2. The references of this article show that the Kondo Effect is only perceptible in systems with low temperatures, meaning T = 100 K and lower. For a given value of V1 (or V2 ), the TK value is smaller when V1 = V2 which corresponds to the non-Fermi liquid quantum states. When V1 ≠ V2 , namely the Fermi-liquid quantum states, the TK has higher values and they are completely symmetric in relation to the principal vertical axis defined by the V1 = V2 plane. Another important result is the decrease of the Kondo temperature with increasing energy difference Δ between the ground and excited states.
(7)
We call this the WKM procedure in this article. The product Tχ (T ) depends only on T / TK and is well defined for the one-channel Kondo and Anderson models. It is important to note that the one-channel models show only Fermi liquid behavior. 3
Journal of Magnetism and Magnetic Materials 499 (2020) 166259
C.A. Philipps, et al.
This makes the TCSIAM behave as the one-channel Anderson model and displays the FL quantum behavior. The second plane is restricted by V1 = 0.03D leading to an impurity interaction with the conduction band via two channels. In channel α = 2 , the coupling value varies from zero to V1 (0 ⩽ V2 ⩽ V1). The two curves of this case represent the TCSIAM going through a transition from the Fermi liquid to the non-Fermi liquid behavior. Furthermore, the numerical results confirm the symmetry of the band-impurity coupling properly. Changing the values of the two coupling constants V1 and V2 leads to the same Kondo temperature, as the dashed lines show in Fig. 3, meaning that the computational error of the NRG is sufficiently low. The third plane is determined by the restriction V1 = V2 . In this case the TCSIAM is entirely in the NFL quantum state. 3.2. Analytic expressions for the Kondo temperature Unfortunately, the NRG procedure gives only numerical results instead of a mathematical expression for the Kondo temperature as function of the model parameters V1, V2 and Δ . However, following K. G. Wilson’s idea, it is possible via regression analysis to fit expressions obtained from perturbation theory to the numerical results. Wilson proposed an expression for the Kondo temperature for the one-channel Kondo model (Equation IX. 55 and Equation IX. 94 from reference [10]). Here we use a similar notation,
∼ D (ρJ ) = D [c0 + c1 ρJ + c2 (ρJ )2 + O ((ρJ )3)]
(8)
∼ TK (ρJ ) ∼ D (ρJ ) exp(−1/ ∣2ρJ ∣)
(9)
where ci are constants, ρ is the electronic state density near the Fermi level simplified by ρ = 1/2D , J is the hybridization parameter and ∼ D (ρJ ) is a power series expansion in the dimensionless product of ρJ . Krishna-murthy, Wilkins and Wilson proposed a similar expression for the one-channel Anderson model [12], Fig. 2. Numerical results of the TCSIAM: The Kondo temperature surface TK in Kelvin as a function of the hybridization parameters, V1 and V2 , for two energy differences between the ground and excited states, (a) Δ/ D = 0.0001 and (b) Δ/ D = 0.001. 2D is the adjustable conduction bandwidth.
TK ∼ D∣2ρJ ∣1/2 exp(−1/ ∣2ρJ ∣),
(10)
V 2/Δ
is obtained by the Schrieffer-Wolff transformation where J ∝ [25]. Note that the term J 1/2 is a generalization to non-integer exponents of the polynomial ansatz in Eq. (8). 3.2.1. TK expression for the Fermi liquid behavior The TCSIAM can reproduce the one-channel Kondo and Anderson models when V1 = 0 and V2 ≠ 0 (or vice versa), where the Kondo and Anderson coupling parameters are related by Jα ∝ Vα2/Δ. To find an expression for the Kondo temperature that reproduces the data of Figs. 2 and 3 we use Eq. (9) as starting point. Thus,
TK (V1, V2, Δ) = c (2ρJFL) exp(− c /(2 ̃ ρJFL)),
(11)
)2 /Δ
is a parameter to account for the FL behavior, where JFL = (V1 − V2 and fitting constants, c and c ,̃ are introduced. Simplifying in the case of single channel interaction, V1 = 0 , Eq. (11) can be written as
a ̃Δ TK ⎛⎜0, V2, Δ⎞⎟ = a1 ⎛⎜V22/Δ⎞⎟ exp ⎛⎜− 1 2 ⎞⎟, V2 ⎠ ⎝ ⎝ ⎠ ⎝ ⎠
(12)
where a1 = 2cρ . Regression parameters are shown in Table 1, obtained by fitting this expression to the numerical data from Fig. 3. A further simulation with the value of Δ = 0.005 is included.
Fig. 3. Numerical results of the TCSIAM: The Kondo temperature TK in Kelvin as function of V2 for constant Δ/ D values of 0.0001 and 0.001. Blue and yellow curve correspond to V1 = 0.03D , green and red to V1 = 0 , purple and brown to V1 = V2 . These curves are 2-dimensional cut-outs from Fig. 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 1 Fitting parameters of Eq. (12) in units of Kelvin K divided by different powers of the conduction band parameter D for diverse Δ values.
In order to facilitate the analysis, Fig. 3 shows curves when vertical planes intersect the two axes of Fig. 2. Three types of planes are considered. One plane determined by the condition V1 = 0 . In this case, the impurity only interacts with the conduction band via channel α = 2 . 4
Δ [D]
a1 [K / D]
a1/Δ [K / D 2]
a1̃ [D]
0.0001 0.0010 0.0050
13.68 133.0 693.5
136800 133000 138700
0.2270 0.2290 0.2317
Journal of Magnetism and Magnetic Materials 499 (2020) 166259
C.A. Philipps, et al.
Table 1 shows that (a1/Δ) ∼ (1.35 × 105) K / D 2 and a1̃ ∼ 0.23D for different Δ . This aspect allows to further simplify Eq. (12). It can be written, with the new constant a, as
0.23Δ ⎞ TK ⎛⎜0, V2, Δ⎞⎟ = a 105 V22 exp ⎛⎜− ⎟. V22 ⎠ ⎝ ⎝ ⎠
Table 3 Fitting parameters of Eq. (19) for the two curves with V1 = 0.03 of Fig. 3 in units of Kelvin K divided by different powers of the conduction band parameter D for diverse Δ values. Δ[D]
a [K / D 2]
b [K / D3]
c [K / D 4]
(13)
This is the Kondo temperature expression for the TCSIAM in the case of V1 = 0 and V2 ≠ 0 corresponding to the pure Fermi liquid behavior. The value of the constant a will be investigated later on. The same expression is used when V2 = 0 and V1 ≠ 0 , simply changing V1 and V2 .
0.0001
1.39
9.41
3.10·103
0.0010
1.33
9.48
3.66·103
3.2.2. TK expression for the non-Fermi liquid behavior Although Eq. (9) was not designed for the non-Fermi liquid behavior, V1 = V2 ≠ 0 , we adopt a similar expression starting with
TK (V1, V2, Δ) = c (2ρJNLF )1.5 exp(− c /(2 ̃ ρJNLF )),
(14)
where JNLF = (V1 V2)/Δ , is a parameter to denote the NFL behavior and c and c ̃ are regression constants. Considering V1 = V2 , Eq. (14) turns into
⎛ TK ⎜V2, V2, ⎝
b ̃Δ ⎞ Δ⎟ = b1 (V22/Δ)1.5exp ⎛⎜− 1 2 ⎞⎟. ⎝ V2 ⎠ ⎠
(15)
/Δ1.5
is approxiAnalysis of the regression parameters shows that b1 mately b105K / D3 and b1̃ ∼ 0.23D for different Δ values, see Table 2. This aspect is remarkable, since it allows further simplification, leading to the following equation as the Kondo temperature expression for the TCSIAM in the case of the non-Fermi liquid behavior,
0.23Δ ⎞ TK ⎛⎜V2, V2, Δ⎞⎟ ∼ b 105 V23 exp ⎛⎜− ⎟. V22 ⎠ ⎝ ⎝ ⎠
(16)
3.3. Fermi liquid to non-Fermi liquid transition It is important to note that JFL is zero in the NFL regime (V1 = V2 ) and that JNFL equals 0 in the pure FL regime (V1 = 0 and V2 ≠ 0 or vice versa). The general expression for the FL to NFL transition has to reproduce Eqs. (13) and (16) properly. Fig. 3 shows the crossing over from the FL to the NFL behavior, corresponding to the case of V1 = 0.03 and 0 ⩽ V2 ⩽ V1 in the TCSIAM. Based on Wilson’s proposal, Eq. (9), we suggest the following expression for the Kondo temperature being valid in all three cases
− c4 1.5 ⎞. TK = ⎜⎛c1 JFL + c2 JNLF + c3 JFL JNLF ⎟⎞ exp ⎛ J ⎝ FL + JNLF ⎠ ⎝ ⎠ ⎜
⎟
(17)
In particular, Equation 3.3 represents a general case that fits the two curves with V1 = 0.03 in Fig. 3 by
−0.23Δ ∼ ⎞, TK ⎛⎜V1, V2, Δ⎞⎟ = D ⎛⎜V1, V2, Δ⎞⎟ exp ⎛ 2 ⎝ (V1 − V2) + V1 V2 ⎠ ⎝ ⎠ ⎝ ⎠
(18)
where ∼ D (V1, V2, Δ) = 105 [a (V1 − V2)2 + b (V1 V2)1.5 + c (V1 − V2)2 V1 V2)].
(19)
⎜
Fig. 4. Numerical and analytic results of the TCSIAM: The Kondo temperature TK in Kelvin as a function of V2 for different Δ values, (a) Δ/ D = 0.0001 and (b) Δ/ D = 0.0010 . D is the half bandwidth of conduction band (D = 5eV ). All points were obtained from Eq. (7) with NRG method. The continuous curves result from the regression analysis of Eq. (18).
⎟
fact, the analytic expression for the Kondo temperature TK deviates less than 4% from the simulation results and the residuals do not yield any systematic deviation for the three cases: FL, NFL and transition. Different values for the energy gap Δ between the ground and excited states are also in perfect agreement with the NRG results for the TCSIAM.
As before, regression analysis of the NRG numerical data reinforces our hypothesis. Table 3 shows the regression constants. Fig. 4 shows that Eq. (18) fits all simulation points from Fig. 2 in the distinct regimes. In Table 2 Fitting parameters of Eq. (15) in units of Kelvin K divided by different powers of the conduction band parameter D for diverse Δ values. Δ [D]
b1 [K / D1.5]
(b1/Δ1.5) [K / D3]
b1̃ [D]
3.4. Physical meaning of the Kondo temperature contributions
0.0001 0.0010 0.0035
0.9415 30.26 189.7
941500 956930 916147
0.2257 0.2284 0.2338
Eq. (18) gives a good fit to the simulation data of Fig. 2. This equation can be written as
5
Journal of Magnetism and Magnetic Materials 499 (2020) 166259
C.A. Philipps, et al.
−0.23Δ ⎤. TK ∼ [f (V 2) + g (V 3) + h (V 4 )] exp ⎡ 2 ⎢ ⎣ (V1 − V2) + V1 V2 ⎥ ⎦
(20)
[2]
The first term, f (V 2) → JLF = (V1 − V2)2 /Δ, represents the competition between each channel, to cancel spin magnetic impurity. Even when V1 ≠ V2 and both differ from zero, JLF is the dominant term. The nonzero antiferromagnetic coupling cancels the spin impurity at low temperatures. Thus, the system shows the Fermi liquid behavior. The second term g (V 3) is related to JNLF = (V1 V2)/Δ and is only dominant if the hybridization parameters are equal, V1 = V2 . Other mechanism than antiferromagnetic coupling may account for this. In this case, the Kondo temperature TK has the lowest values and is designated the non-Fermi liquid behavior. The third term h (V 4 ) is the most important one, since it allows the transition from the FL to the NFL behavior. The physical interpretation of the product of JLF JNLF is still been investigated. It is important to remark that other physical properties, as transport properties, can also be adopted to define the Kondo temperature. A particularly interesting result is pointed out by Darocca et al. [3] who used a similar idea to the WKM procedure. They fitted electronic conductance in quantum dot systems to find TK values and concluded that TK depends on the applied voltage. Furthermore, it is reasonable to suppose that the two procedures for the Kondo temperature determination, magnetic susceptibility and electronic conductance fitting, may have a relationship deserving a future work to gain a clarifying insight.
[3]
[4]
[5]
[6]
[7] [8] [9]
[10]
[11]
4. Conclusion
[12]
Following K. G. Wilson’s procedure from reference [10] for the twochannel single impurity Anderson Model, an generalized analytic expression for the Kondo temperature TK was obtained. Numerical and analytic results agree with the Fermi liquid and the non-Fermi liquid behavior. This article shows, in the low temperature limit, that the twochannel single impurity Anderson model reproduces the TK ∼ V 2exp(−1/ V 2) characteristic in the FL regime, identical to the onechannel Anderson models. However, when hybridization parameters equal, V1 = V2 , the TCSIAM follows the TK ∼ V 3exp(−1/ V 2) dependency for T < 100 K showing the NFL behavior. Our proposed analytic expression, Eq. (18), is capable of covering these two cases and the transition from the Fermi liquid to the non-Fermi liquid regime.
[13]
[14]
[15]
[16]
[17]
CRediT authorship contribution statement [18]
C.A. Philipps: Software, Validation, Formal analysis, Writing - review & editing, Funding acquisition. P.G.B. Shiota: Methodology, Software, Investigation, Writing - original draft, Visualization. F.M. Zimmer: Conceptualization, Software, Writing - original draft, Funding acquisition. J.V.B. Ferreira: Conceptualization, Methodology, Software, Formal analysis, Resources, Supervision, Project administration.
[19]
[20]
Declaration of Competing Interest [21]
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. [22]
Acknowledgments [23]
F. M. Zimmer acknowledges support from CNPq/Brazil and C. A. Phillips thanks gratefully to CAPES/Brazil for Grant No. 001. [24]
References [25]
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Kondo temperature equation for the Fermi liquid and the non-Fermi liquid regime for a two-channel anderson model C.A. Philippsa, P.G.B. Shiotaa, F.M. Zimmerb, J.V.B. Ferreirab, a b
T
⁎
Programa de Pós-Graduaçāo em Ciências dos Materiais, Instituto de Física, Universidade Federal de Mato Grosso do Sul, Campo Grande-MS, Brazil Instituto de Física, Universidade Federal de Mato Grosso do Sul, Campo Grande-MS, Brazil
A R T I C LE I N FO
A B S T R A C T
Keywords: Kondo temperature Anderson model Kondo effect Fermi liquid Non-Fermi liquid
Multichannel Kondo resonance causes non-Fermi liquid behavior in many systems, as nanostructured metallic quantum dots and metallic alloys. The Kondo effect occurs at low temperatures and its indicating parameter is the Kondo temperature TK . This parameter can be used to renormalize thermodynamic and transport properties. The Kondo temperature is an important quantity to analyze the antiferromagnetic interaction between a localized spin and itinerant ones in a fermionic bath. There are many different definitions of the Kondo temperature. One of them is that it universalizes magnetic impurity susceptibility. This version was created for the onechannel Anderson model, but it is also valid in two-channel Anderson models, where a single magnetic impurity interacts with the conduction band electrons by two different channels. If the two coupling parameters are different, the model exhibits the well known Fermi liquid regime. When the coupling parameters are equal, the model displays the non-Fermi liquid regime. Each one shows different universal magnetic susceptibility behavior with appropriate TK value. In this article, Numerical Renormalization Group method is applied to study the Kondo temperature systematically. Different values of the coupling parameters in a two-channel Anderson model are examined. Finally, a single analytic expression for the Kondo temperature, valid in the Fermi and non-Fermi liquid regime, is presented.
1. Introduction The Kondo temperature TK indicates the energy scale where the Kondo effect becomes important in the system of a magnetic impurity coupled to an electronic reservoir [1,2]. For example, electronic conductance at low temperatures can be affected by Kondo correlations in different systems: (i) quantum dot system [3,4] (ii) metallic alloys with diluted 4f 2 orbitals [5] or Kondo entanglement [6]. The parameter TK can be used to universalize thermal properties, for instance, the magnetic susceptibility of the atomic impurity, which has been observed both in simple metals, and in metallic alloys [5]. In general, simple metals can be described by Landau Fermi liquid theory (FL) [7]. However, with FL theory many phenomena of metallic alloys can not be covered properly, since they exhibit the so called non-Fermi liquid behavior (NFL) [8]. Therefore, it is important to know how to determine the Kondo temperature for different single impurity magnetic models for the FL and the NFL regime. Usually, a magnetic atom in a non-magnetic solid is represented by single impurity Anderson models (SIAMs). These models show the Kondo effect due to the antiferromagnetic interactions
⁎
between the localized spin (atom) and the spins of the conduction band electrons. At first, SIAMs could only yield the FL behavior. These models stipulated that the localized spin was completely quenched by the itinerant ones at sufficiently low temperatures (T ⩽ TK ). The two wellknown models were the Kondo and the Anderson models [9,10]. In these models, the procedures to obtain TK derive from perturbation theory [11] or numerical methods [12]. In numerical methods, the Kondo temperature is obtained from universalizing the magnetic susceptibility temperature curve. It was realized that these models were oversimplified and were not able to describe metallic alloys that exhibited the NFL behavior. In order to make the Kondo model more realistic, P. Nozières and A. Blandin introduced a concept called channel [13]. It is a new quantum number considered in the SIAM Hamiltonian representing the universality and symmetry of the physical system, and intends to catch intricate effects occurring in metals, as the crystal field splitting [14], rare earth impurity and spin orbit coupling [15]. There are three different cases that can be achieved by changing the relation between the number of channels n and the impurity spin S
Corresponding author. E-mail address: [email protected] (J.V.B. Ferreira).
https://doi.org/10.1016/j.jmmm.2019.166259 Received 14 July 2019; Received in revised form 27 November 2019; Accepted 1 December 2019 Available online 16 December 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 499 (2020) 166259
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Unlike the Kondo model generalization, the Anderson model generalization is non-trivial [19] due to the more complex interaction between the impurity and the conduction band electrons. In the Kondo model the coupling only flips impurity spin, whereas in the Anderson model the coupling ionizes the impurity atom. Hence, the diagonalization of the TCSIAM Hamiltonian with NRG is a laborious task [20,23]. The following Hamiltonian defines the two-channel single impurity Anderson model.
[16]: the number of channels n = 2S is capable of fully compensating the localized spin leading to a spin singlet ground state and usually to the emergence of the Fermi-liquid behavior; the number of conduction electron channels n < 2S is not high enough to yield a singlet ground state with the impurity spin partially compensated; and the number of channels n > 2S is larger than required to compensate the impurity spin. These situations are known as compensated, underscreened and overscreened Kondo (Anderson) models, respectively. The last two cases are the cause of the non-Fermi-liquid properties. Nowadays, the original Anderson (Kondo) model is known as the one-channel Anderson (Kondo) model and others models, with n ⩾ 2 , as n-channel Anderson (Kondo) models [17,18]. It is important to emphasize that the one-channel Anderson (Kondo) model does not show the non-Fermi liquid regime, but the two-channel or higher channel models are capable of yielding this behavior [8]. D. L. Cox proposed a two-channel Anderson model (TCSIAM) with a single impurity to study the transition from the Fermi liquid to the nonFermi liquid behavior in thermal properties of a localized magnetic atom. Each channel is considered as a specific quantum mechanism to join the conduction band with the localized spin, using the two coupling parameters, V1 ( J1) and V2 ( J2 ), in Anderson (Kondo) notation. If V1 equals V2 , localized spins are never completely quenched, so physical properties differ from the FL paradigm. Metals with d- or f-orbital valence electrons belong to a class of materials that, at low energies, have conduction band electrons that do not behave essentially as a collection of weakly interacting quasi particles – violating basic Landau theory assumption of the FL behavior [7]. The two-channel single impurity Anderson model Hamiltonian was diagonalized using Numerical Renormalization Group (NRG) and its magnetic susceptibility versus temperature curves were calculated in the FL and the NFL regime [19,20]. Remarkably, even in the non-Fermi liquid behavior, it is possible to define the Kondo temperature as a parameter that universalizes magnetic susceptibility [21]. However, these numerical data are still lacking an analytical expression. We address exactly this point in the present work, finding a general relation between the coupling parameters V1 and V2 of the two-channel single impurity Anderson model and the Kondo temperature in the Fermi liquid and the non-Fermi liquid regime. Thus, this article shows a systematic study of the TK for the TCSIAM covering FL and NFL behavior. The TK is obtained using numerical renormalization theory and a method to universalize the magnetic susceptibility curve [21]. Finally, an analytical expression describing the dependency between the Kondo temperature TK and the physical parameters, V1 and V2 of the TCSIAM, is suggested. This expression is based on Wilson’s analytic and numerical arguments [10]. A brief presentation of the two-channel single impurity Anderson model is given. Then, it is shown how Numerical Renormalization Group can be used to solve its Hamiltonian. The relation between the TCSIAM and the two-channel Kondo model is explained via the Schrieffer-Wolff transformation. Afterwards, the numerical procedure to determine the TK (K. G. Wilson’s procedure) is explained. Finally, numerical data and K. G. Wilson theory is used to obtain a generalized expression for the Kondo temperature, being valid in the Fermi liquid and non-Fermi liquid regime.
H = Hcb + Himp + Hhyb Hcb =
(1)
† ∊k ckσα ckσα
∑
(2)
k, σ , α
Himp= E0
∑
m, σ
m, σ +
σ
Hhyb =
⎛ ⎞ ⎜E0 + Δ⎟ ⎝ ⎠
∑
⎛ ⎞ ⎜E0 + Δ⎟ ⎝ ⎠
∑
∑ k, σ , α
m − 1, α
m − 1, α +
m + 1, α
m + 1, α
α
α
⎛ ⎞ Vα ⎜f−†1, σ , α ckσα + h. c. ⎟ ⎝ ⎠
(3)
(4)
Hcb, Himp and Hhyb represent the conduction band, the magnetic impurity and the band-impurity coupling respectively. Hcb represents a fermionic bath, depicted by a half filled metallic conduction band, isotropic and energy width equal to 2D (D is an adjustable parameter, in this article D = 5 eV is used) [24]. The magnetic localized impurity is modeled by six quantum states. The ground state is a double degenerate one, that has m electrons, energy E0 and spin S = 1/2 . The z component of spin is represented by the quantum number σ = ± 1/2 , therefore, the two magnetic states are m , σ . Following Nozières precept [13], the impurity in its ground states is allowed to interact with the conduction band via two different mechanisms, called channels, described by quantum number α (α = 1 or 2) each one having its own hybridization parameter Vα . After the interaction, the impurity will have gained or lost an electron, so that its spin is now zero and the state is not magnetic anymore. This new quantum state has energy E0 + Δ (Δ > 0 ), it is four times degenerated and is represented by the quantum states m ± 1, α , where the quantum number α indicates the channel that allowed the † impurity to gain or lose an electron. The operator ckσα creates a free → conduction band electron with momentum k , spin σ = ± 1/2 , channel α = 1 or 2 and energy εk . Operator f−†1, σ , α is defined in such a way that it represents the transition between the ground and the excited states (α = −1 → α = 2 ) and vice versa (α = −2 → α = 1), as well as retaining spin and channel symmetry in quantum states. f−†1, σ , α = m , σ m − 1, −α + (2σ ) m + 1, α m , −σ
(5)
2. Theoretical concepts
The behavior of the TCSIAM is determined by the physical parameters Δ, V1 and V2 . Δ corresponds to the energy difference between the ground and the excited states. V1 and V2 are hybridization parameters between the impurity and the conduction band. It is important to note that if V1 = 0 and V2 ≠ 0 , or vice versa, the TCSIAM reproduces the results of the traditional one-channel Anderson model.
2.1. Two-channel single impurity Anderson model – TCSIAM
2.2. Schrieffer-Wolff Transformation
The original Anderson and Kondo models are defined as onechannel models since they consider magnetic impurity and conduction band electrons with only one coupling mechanism between them, indicated by parameter V or J respectively. Metal alloys and new materials require a model with more degrees of freedom [13,17] to explain the non-Fermi liquid behavior [22]. D. L. Cox proposed a two-channel model which is a generalization of the one-channel Anderson model.
The Schrieffer-Wolff Transformation relates the Hamiltonian of the one-channel Anderson model to the one of the one-channel Kondo model. The relation shows that the Anderson model is equivalent to the Kondo model in the strong coupling regime [25,26]. Similar equivalence exists between the two-channel models at low excitation energies. The relation between the coupling parameters is Jα ∝ Vα2/Δ , where Jα is the hybridization parameter in the two-channel Kondo model and Vα in 2
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the two-channel Anderson model [21]. 2.3. Numerical Renormalization Group – NRG The Numerical Renormalization Group was developed by K. G. Wilson to diagonalize the one-channel Kondo model [10] and afterwards applied by him, H. R. Krishna-murthy and J. W. Wilkins to the one-channel Anderson model [12]. It transforms the Hamiltonian into a semi-infinite series of terms in decreasing order of energy scales. Only the first term takes the coupling with the impurity into account. The eigenstates are built recursively, allowing a truncation of states based on relevant energy scales, with minor effects on impurity properties. Thus we have a non-perturbative numerical procedure from which the energy spectrum of the model is obtained. Physical properties are calculated using the NRG with interleaving discretization [27,28] and the two-step transformation [23]. Fig. 1. The magnetic susceptibility χ for a null external magnetic field as function of the temperature T / TK . The black curve is the universal behavior of the one-channel Anderson model (FL). The other curves are identified by values of V1 = V2 = 0.01D and different Δ values showing the NFL behavior. Adapted from reference [21].
2.4. Computational details Computational data was obtained with the following machine. Intel (R) Core(TM) i7 CPU 860 2.80 GHz 8 processors RAM 16 GB. The operation system was Linux 2.6.26-2-amd64 x86-64 GNU/Linux. The NRG programs were implemented using C++ language and double floating point precision. Auxiliary programs to do Interleaving procedure and parallel routines were done using Script language. Interleaving procedure was done with 8 values for the discretization of the conduction band. In general, each Kondo temperature value TK took 0.75 hours of run time and used 250 MB of memory. About 3200 TK values were obtained.
On the contrary, the TCSIAM shows Fermi and non-Fermi liquid behavior, even though it is not obvious that, in the NFL case, Tχ (T ) solely depends on T / TK . However, WKM procedure still works. Fig. 1 shows Tχ (T ) scaled by T / TK . The hybridization parameters Vα change the magnetic susceptibility and therefore alter the Kondo temperature. Nevertheless, all curves have the same value at low temperatures in a rescaled graph. If both coupling parameters are equal, (V1 = V2 ), the NFL behavior occurs but TK remains defined by WKM procedure. At low temperatures, these curves also present a universal behavior, although different from the one-channel Anderson model. Both universal curves, the FL and the NFL, satisfy the condition given from WKM procedure. Until now, a mathematical expression describing the Kondo temperature for the TCSIAM has not been found yet. In the next section, we propose an expression based on NRG, WKM and analytic approximations.
2.5. Kondo temperature – TK Firstly, we will present a chronological description of the Kondo temperature concept in the one-channel Kondo model. In 1964, J. Kondo designated T0 as the temperature value at which the resistivity expansion diverges [9]. Later in 1975, K. G. Wilson denoted this parameter as Kondo temperature TK [10]. He used it in scaling formulas by combining results from NRG and analytic calculations. For example, he showed that the product of the thermal energy and the magnetic susceptibility kB Tχ (T ) depends only on T / TK in the range of 0.5 TK < T < 16 TK , whereas χ (T ) is the magnetic susceptibility of the impurity. V. Melnikov in 1982 and P. Wiegmann in 1983 [29], refined this formula, leading to
kB Tχ (T ) 0.17 ≈ T (gμB )2 1 + 1.4 T
We present numerical results obtained from the TCSIAM for several sets of parameters {Δ, V1, V2 }. The hybridization parameters are related to the Fermi liquid behavior for V1 ≠ V2 and to the non-Fermi liquid in the case of V1 = V2 . In order to obtain the Kondo temperature, we adopt the following procedure. For a given set of Δ, V1 and V2 , the TCSIAM Hamiltonian is diagonalized using NRG technique, interleaving discretization and the two-step transformation methods. The magnetic susceptibility dependence of the temperature is evaluated and the TK value is obtained from WKM procedure via Eq. (7).
−1
( ) K
3. Results and discussion
(6)
where kB is the Boltzmann constant, g is the electron gyromagnetic ratio and μB is the Bohr magneton. Eq. (6) is used until today to obtain an universalized expression for the magnetic susceptibility of lattice impurities. Following the same concepts, other researchers analyzed the onechannel Anderson model. In 1980, Krishna-murthy, Wilkins and Wilson calculated the magnetic susceptibility of the Anderson model using NRG and second order perturbation theory [12]. They fitted their results to the Curie–Weiss law using Eq. (6) and showed the universal behavior of the magnetic susceptibility. Interestingly, they obtained χ (T ) via NRG using this equation and making it possible to determine TK in the case of T = TK . The magnetic susceptibility dependence of the temperature is evaluated and TK is obtained from
kB TK χ /(gμB )2 ≈ 0.07.
3.1. NRG data The Kondo temperature dependency on the hybridization parameters V1 and V2 for two different Δ values is shown in Fig. 2. The references of this article show that the Kondo Effect is only perceptible in systems with low temperatures, meaning T = 100 K and lower. For a given value of V1 (or V2 ), the TK value is smaller when V1 = V2 which corresponds to the non-Fermi liquid quantum states. When V1 ≠ V2 , namely the Fermi-liquid quantum states, the TK has higher values and they are completely symmetric in relation to the principal vertical axis defined by the V1 = V2 plane. Another important result is the decrease of the Kondo temperature with increasing energy difference Δ between the ground and excited states.
(7)
We call this the WKM procedure in this article. The product Tχ (T ) depends only on T / TK and is well defined for the one-channel Kondo and Anderson models. It is important to note that the one-channel models show only Fermi liquid behavior. 3
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This makes the TCSIAM behave as the one-channel Anderson model and displays the FL quantum behavior. The second plane is restricted by V1 = 0.03D leading to an impurity interaction with the conduction band via two channels. In channel α = 2 , the coupling value varies from zero to V1 (0 ⩽ V2 ⩽ V1). The two curves of this case represent the TCSIAM going through a transition from the Fermi liquid to the non-Fermi liquid behavior. Furthermore, the numerical results confirm the symmetry of the band-impurity coupling properly. Changing the values of the two coupling constants V1 and V2 leads to the same Kondo temperature, as the dashed lines show in Fig. 3, meaning that the computational error of the NRG is sufficiently low. The third plane is determined by the restriction V1 = V2 . In this case the TCSIAM is entirely in the NFL quantum state. 3.2. Analytic expressions for the Kondo temperature Unfortunately, the NRG procedure gives only numerical results instead of a mathematical expression for the Kondo temperature as function of the model parameters V1, V2 and Δ . However, following K. G. Wilson’s idea, it is possible via regression analysis to fit expressions obtained from perturbation theory to the numerical results. Wilson proposed an expression for the Kondo temperature for the one-channel Kondo model (Equation IX. 55 and Equation IX. 94 from reference [10]). Here we use a similar notation,
∼ D (ρJ ) = D [c0 + c1 ρJ + c2 (ρJ )2 + O ((ρJ )3)]
(8)
∼ TK (ρJ ) ∼ D (ρJ ) exp(−1/ ∣2ρJ ∣)
(9)
where ci are constants, ρ is the electronic state density near the Fermi level simplified by ρ = 1/2D , J is the hybridization parameter and ∼ D (ρJ ) is a power series expansion in the dimensionless product of ρJ . Krishna-murthy, Wilkins and Wilson proposed a similar expression for the one-channel Anderson model [12], Fig. 2. Numerical results of the TCSIAM: The Kondo temperature surface TK in Kelvin as a function of the hybridization parameters, V1 and V2 , for two energy differences between the ground and excited states, (a) Δ/ D = 0.0001 and (b) Δ/ D = 0.001. 2D is the adjustable conduction bandwidth.
TK ∼ D∣2ρJ ∣1/2 exp(−1/ ∣2ρJ ∣),
(10)
V 2/Δ
is obtained by the Schrieffer-Wolff transformation where J ∝ [25]. Note that the term J 1/2 is a generalization to non-integer exponents of the polynomial ansatz in Eq. (8). 3.2.1. TK expression for the Fermi liquid behavior The TCSIAM can reproduce the one-channel Kondo and Anderson models when V1 = 0 and V2 ≠ 0 (or vice versa), where the Kondo and Anderson coupling parameters are related by Jα ∝ Vα2/Δ. To find an expression for the Kondo temperature that reproduces the data of Figs. 2 and 3 we use Eq. (9) as starting point. Thus,
TK (V1, V2, Δ) = c (2ρJFL) exp(− c /(2 ̃ ρJFL)),
(11)
)2 /Δ
is a parameter to account for the FL behavior, where JFL = (V1 − V2 and fitting constants, c and c ,̃ are introduced. Simplifying in the case of single channel interaction, V1 = 0 , Eq. (11) can be written as
a ̃Δ TK ⎛⎜0, V2, Δ⎞⎟ = a1 ⎛⎜V22/Δ⎞⎟ exp ⎛⎜− 1 2 ⎞⎟, V2 ⎠ ⎝ ⎝ ⎠ ⎝ ⎠
(12)
where a1 = 2cρ . Regression parameters are shown in Table 1, obtained by fitting this expression to the numerical data from Fig. 3. A further simulation with the value of Δ = 0.005 is included.
Fig. 3. Numerical results of the TCSIAM: The Kondo temperature TK in Kelvin as function of V2 for constant Δ/ D values of 0.0001 and 0.001. Blue and yellow curve correspond to V1 = 0.03D , green and red to V1 = 0 , purple and brown to V1 = V2 . These curves are 2-dimensional cut-outs from Fig. 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 1 Fitting parameters of Eq. (12) in units of Kelvin K divided by different powers of the conduction band parameter D for diverse Δ values.
In order to facilitate the analysis, Fig. 3 shows curves when vertical planes intersect the two axes of Fig. 2. Three types of planes are considered. One plane determined by the condition V1 = 0 . In this case, the impurity only interacts with the conduction band via channel α = 2 . 4
Δ [D]
a1 [K / D]
a1/Δ [K / D 2]
a1̃ [D]
0.0001 0.0010 0.0050
13.68 133.0 693.5
136800 133000 138700
0.2270 0.2290 0.2317
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Table 1 shows that (a1/Δ) ∼ (1.35 × 105) K / D 2 and a1̃ ∼ 0.23D for different Δ . This aspect allows to further simplify Eq. (12). It can be written, with the new constant a, as
0.23Δ ⎞ TK ⎛⎜0, V2, Δ⎞⎟ = a 105 V22 exp ⎛⎜− ⎟. V22 ⎠ ⎝ ⎝ ⎠
Table 3 Fitting parameters of Eq. (19) for the two curves with V1 = 0.03 of Fig. 3 in units of Kelvin K divided by different powers of the conduction band parameter D for diverse Δ values. Δ[D]
a [K / D 2]
b [K / D3]
c [K / D 4]
(13)
This is the Kondo temperature expression for the TCSIAM in the case of V1 = 0 and V2 ≠ 0 corresponding to the pure Fermi liquid behavior. The value of the constant a will be investigated later on. The same expression is used when V2 = 0 and V1 ≠ 0 , simply changing V1 and V2 .
0.0001
1.39
9.41
3.10·103
0.0010
1.33
9.48
3.66·103
3.2.2. TK expression for the non-Fermi liquid behavior Although Eq. (9) was not designed for the non-Fermi liquid behavior, V1 = V2 ≠ 0 , we adopt a similar expression starting with
TK (V1, V2, Δ) = c (2ρJNLF )1.5 exp(− c /(2 ̃ ρJNLF )),
(14)
where JNLF = (V1 V2)/Δ , is a parameter to denote the NFL behavior and c and c ̃ are regression constants. Considering V1 = V2 , Eq. (14) turns into
⎛ TK ⎜V2, V2, ⎝
b ̃Δ ⎞ Δ⎟ = b1 (V22/Δ)1.5exp ⎛⎜− 1 2 ⎞⎟. ⎝ V2 ⎠ ⎠
(15)
/Δ1.5
is approxiAnalysis of the regression parameters shows that b1 mately b105K / D3 and b1̃ ∼ 0.23D for different Δ values, see Table 2. This aspect is remarkable, since it allows further simplification, leading to the following equation as the Kondo temperature expression for the TCSIAM in the case of the non-Fermi liquid behavior,
0.23Δ ⎞ TK ⎛⎜V2, V2, Δ⎞⎟ ∼ b 105 V23 exp ⎛⎜− ⎟. V22 ⎠ ⎝ ⎝ ⎠
(16)
3.3. Fermi liquid to non-Fermi liquid transition It is important to note that JFL is zero in the NFL regime (V1 = V2 ) and that JNFL equals 0 in the pure FL regime (V1 = 0 and V2 ≠ 0 or vice versa). The general expression for the FL to NFL transition has to reproduce Eqs. (13) and (16) properly. Fig. 3 shows the crossing over from the FL to the NFL behavior, corresponding to the case of V1 = 0.03 and 0 ⩽ V2 ⩽ V1 in the TCSIAM. Based on Wilson’s proposal, Eq. (9), we suggest the following expression for the Kondo temperature being valid in all three cases
− c4 1.5 ⎞. TK = ⎜⎛c1 JFL + c2 JNLF + c3 JFL JNLF ⎟⎞ exp ⎛ J ⎝ FL + JNLF ⎠ ⎝ ⎠ ⎜
⎟
(17)
In particular, Equation 3.3 represents a general case that fits the two curves with V1 = 0.03 in Fig. 3 by
−0.23Δ ∼ ⎞, TK ⎛⎜V1, V2, Δ⎞⎟ = D ⎛⎜V1, V2, Δ⎞⎟ exp ⎛ 2 ⎝ (V1 − V2) + V1 V2 ⎠ ⎝ ⎠ ⎝ ⎠
(18)
where ∼ D (V1, V2, Δ) = 105 [a (V1 − V2)2 + b (V1 V2)1.5 + c (V1 − V2)2 V1 V2)].
(19)
⎜
Fig. 4. Numerical and analytic results of the TCSIAM: The Kondo temperature TK in Kelvin as a function of V2 for different Δ values, (a) Δ/ D = 0.0001 and (b) Δ/ D = 0.0010 . D is the half bandwidth of conduction band (D = 5eV ). All points were obtained from Eq. (7) with NRG method. The continuous curves result from the regression analysis of Eq. (18).
⎟
fact, the analytic expression for the Kondo temperature TK deviates less than 4% from the simulation results and the residuals do not yield any systematic deviation for the three cases: FL, NFL and transition. Different values for the energy gap Δ between the ground and excited states are also in perfect agreement with the NRG results for the TCSIAM.
As before, regression analysis of the NRG numerical data reinforces our hypothesis. Table 3 shows the regression constants. Fig. 4 shows that Eq. (18) fits all simulation points from Fig. 2 in the distinct regimes. In Table 2 Fitting parameters of Eq. (15) in units of Kelvin K divided by different powers of the conduction band parameter D for diverse Δ values. Δ [D]
b1 [K / D1.5]
(b1/Δ1.5) [K / D3]
b1̃ [D]
3.4. Physical meaning of the Kondo temperature contributions
0.0001 0.0010 0.0035
0.9415 30.26 189.7
941500 956930 916147
0.2257 0.2284 0.2338
Eq. (18) gives a good fit to the simulation data of Fig. 2. This equation can be written as
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Journal of Magnetism and Magnetic Materials 499 (2020) 166259
C.A. Philipps, et al.
−0.23Δ ⎤. TK ∼ [f (V 2) + g (V 3) + h (V 4 )] exp ⎡ 2 ⎢ ⎣ (V1 − V2) + V1 V2 ⎥ ⎦
(20)
[2]
The first term, f (V 2) → JLF = (V1 − V2)2 /Δ, represents the competition between each channel, to cancel spin magnetic impurity. Even when V1 ≠ V2 and both differ from zero, JLF is the dominant term. The nonzero antiferromagnetic coupling cancels the spin impurity at low temperatures. Thus, the system shows the Fermi liquid behavior. The second term g (V 3) is related to JNLF = (V1 V2)/Δ and is only dominant if the hybridization parameters are equal, V1 = V2 . Other mechanism than antiferromagnetic coupling may account for this. In this case, the Kondo temperature TK has the lowest values and is designated the non-Fermi liquid behavior. The third term h (V 4 ) is the most important one, since it allows the transition from the FL to the NFL behavior. The physical interpretation of the product of JLF JNLF is still been investigated. It is important to remark that other physical properties, as transport properties, can also be adopted to define the Kondo temperature. A particularly interesting result is pointed out by Darocca et al. [3] who used a similar idea to the WKM procedure. They fitted electronic conductance in quantum dot systems to find TK values and concluded that TK depends on the applied voltage. Furthermore, it is reasonable to suppose that the two procedures for the Kondo temperature determination, magnetic susceptibility and electronic conductance fitting, may have a relationship deserving a future work to gain a clarifying insight.
[3]
[4]
[5]
[6]
[7] [8] [9]
[10]
[11]
4. Conclusion
[12]
Following K. G. Wilson’s procedure from reference [10] for the twochannel single impurity Anderson Model, an generalized analytic expression for the Kondo temperature TK was obtained. Numerical and analytic results agree with the Fermi liquid and the non-Fermi liquid behavior. This article shows, in the low temperature limit, that the twochannel single impurity Anderson model reproduces the TK ∼ V 2exp(−1/ V 2) characteristic in the FL regime, identical to the onechannel Anderson models. However, when hybridization parameters equal, V1 = V2 , the TCSIAM follows the TK ∼ V 3exp(−1/ V 2) dependency for T < 100 K showing the NFL behavior. Our proposed analytic expression, Eq. (18), is capable of covering these two cases and the transition from the Fermi liquid to the non-Fermi liquid regime.
[13]
[14]
[15]
[16]
[17]
CRediT authorship contribution statement [18]
C.A. Philipps: Software, Validation, Formal analysis, Writing - review & editing, Funding acquisition. P.G.B. Shiota: Methodology, Software, Investigation, Writing - original draft, Visualization. F.M. Zimmer: Conceptualization, Software, Writing - original draft, Funding acquisition. J.V.B. Ferreira: Conceptualization, Methodology, Software, Formal analysis, Resources, Supervision, Project administration.
[19]
[20]
Declaration of Competing Interest [21]
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. [22]
Acknowledgments [23]
F. M. Zimmer acknowledges support from CNPq/Brazil and C. A. Phillips thanks gratefully to CAPES/Brazil for Grant No. 001. [24]
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