The Kondo scale ${T}_{\mathrm{K}}$ for impurity systems is expected to guarantee universal scaling of physical quantities. However, in practice, not every definition of ${T}_{\mathrm{K}}$ 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 ${T}_{\mathrm{K}}$ 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 ${T}_{\mathrm{K}}^{\mathrm{sc}}$ 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 ${T}_{\mathrm{K}}^{\mathrm{sc}},\mathrm{B}$ for universal scaling differs, yet becomes equivalent to ${T}_{\mathrm{K}}^{\mathrm{sc}}$ 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 ${T}_{\mathrm{K}}\ensuremath{\lesssim}0.01\phantom{\rule{0.16em}{0ex}}D$ 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.The Kondo scale ${T}_{\mathrm{K}}$ for impurity systems is expected to guarantee universal scaling of physical quantities. However, in practice, not every definition of ${T}_{\mathrm{K}}$ 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 ${T}_{\mathrm{K}}$ 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 ${T}_{\mathrm{K}}^{\mathrm{sc}}$ 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 ${T}_{\mathrm{K}}^{\mathrm{sc}},\mathrm{B}$ for universal scaling differs, yet becomes equivalent to ${T}_{\mathrm{K}}^{\mathrm{sc}}$ 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 ${T}_{\mathrm{K}}\ensuremath{\lesssim}0.01\phantom{\rule{0.16em}{0ex}}D$ 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.

arXiv:1309.3757Physical Review BFeb 21, 201429 citations

Dynamical scaling

Springer eBooksJan 25, 20060 citations

Universal features of the order-parameter fluctuations: Reversible and irreversible aggregation

Robert S. Botet, Marek Płoszajczak

We discuss the universal scaling laws of order-parameter fluctuations in any system in which a second-order critical behavior can be identified. These scaling laws can be derived rigorously for equilibrium systems when combined with a finite-size scaling analysis. The relation between the order parameter, the criticality, and the scaling law of fluctuations has been established, and the connection between the scaling function and the critical exponents has been found. We give examples in out-of-equilibrium aggregation models such as the Smoluchowski kinetic equations, or at-equilibrium Ising and percolation models.We discuss the universal scaling laws of order-parameter fluctuations in any system in which a second-order critical behavior can be identified. These scaling laws can be derived rigorously for equilibrium systems when combined with a finite-size scaling analysis. The relation between the order parameter, the criticality, and the scaling law of fluctuations has been established, and the connection between the scaling function and the critical exponents has been found. We give examples in out-of-equilibrium aggregation models such as the Smoluchowski kinetic equations, or at-equilibrium Ising and percolation models.

arXiv:cond-mat/0004003Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsAug 1, 200054 citations

The mean-field scaling function of the universality class of absorbing phase transitions with a conserved field

S. Lübeck, Alfred Hucht

We consider two mean-field like models which belong to the universality class of absorbing phase transitions with a conserved field. In both cases we analytically derive the order parameter as a function of the control parameter and of an external field conjugated to the order parameter. This allows us to calculate the universal scaling function of the mean-field behaviour. The obtained universal function is in perfect agreement with recently obtained numerical data of the corresponding five- and six-dimensional models, showing that four is the upper critical dimension of this particular universality class.We consider two mean-field like models which belong to the universality class of absorbing phase transitions with a conserved field. In both cases we analytically derive the order parameter as a function of the control parameter and of an external field conjugated to the order parameter. This allows us to calculate the universal scaling function of the mean-field behaviour. The obtained universal function is in perfect agreement with recently obtained numerical data of the corresponding five- and six-dimensional models, showing that four is the upper critical dimension of this particular universality class.

arXiv:cond-mat/0204634Journal of Physics A Mathematical and GeneralMay 31, 200218 citations

Universality

WORLD SCIENTIFIC eBooksNov 1, 20130 citations

The Kondo scale

{T}_{\mathrm{K}}

for impurity systems is expected to guarantee universal scaling of physical quantities. However, in practice, not every definition of

{T}_{\mathrm{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}_{\mathrm{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}_{\mathrm{K}}^{\mathrm{sc}}

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}_{\mathrm{K}}^{\mathrm{sc}},\mathrm{B}

for universal scaling differs, yet becomes equivalent to

{T}_{\mathrm{K}}^{\mathrm{sc}}

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 ${T}_{\mathrm{K}}\ensuremath{\lesssim}0.01\phantom{\rule{0.16em}{0ex}}D$ 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.