Computing Real Part Of AC Conductivity Via Linear Response Theory

Mar 11, 2025 by ADMIN 66 views

**Computing Real Part of AC Conductivity via Linear Response Theory**

Introduction

In the realm of condensed matter physics, understanding the behavior of electrical conductivity is crucial for designing and optimizing various materials and devices. The AC conductivity, in particular, is a fundamental property that describes how a material responds to an alternating electric field. In this article, we will delve into the computation of the real part of AC conductivity using linear response theory, a powerful framework for analyzing the response of a system to external perturbations.

Linear Response Theory

Linear response theory is a well-established framework for studying the response of a system to external perturbations. It is based on the assumption that the system's response is linearly proportional to the strength of the perturbation. In the context of AC conductivity, linear response theory provides a systematic way to compute the conductivity tensor, which describes how the system responds to an external electric field.

AC Conductivity

The AC conductivity, denoted by $\sigma(\omega)$ , is a complex-valued tensor that describes how a material responds to an alternating electric field of frequency $\omega$ . It is defined as the ratio of the induced current density to the applied electric field. Mathematically, it can be expressed as:

\sigma(\omega) = \frac{\langle J_i \rangle}{E_i}

where $\langle J_i \rangle$ is the induced current density and $E_i$ is the applied electric field.

Computing AC Conductivity via Linear Response Theory

To compute the AC conductivity using linear response theory, we need to evaluate the following expression:

\sigma(\omega) = \int_0^\infty dt \, e^{i\omega t} \, \langle J_i(t) \rangle

where $\langle J_i(t) \rangle$ is the induced current density at time $t$ . This expression can be evaluated using the Kubo formula, which is a fundamental result in linear response theory.

Kubo Formula

The Kubo formula is a powerful tool for computing the AC conductivity using linear response theory. It states that the conductivity tensor can be expressed as:

\sigma(\omega) = \frac{1}{i\omega} \int_0^\infty dt \, e^{i\omega t} \, \langle [J_i(t), J_j(0)] \rangle

where $[J_i(t), J_j(0)]$ is the commutator of the current density operators at times $t$ and $0$ .

Real Part of AC Conductivity

The real part of the AC conductivity, denoted by $\sigma_r(\omega)$ , can be computed using the Kubo formula. It is given by:

\sigma_r(\omega) = \frac{1}{\omega} \int_0^\infty dt \, \cos(\omega t) \, \langle [J_i(t), J_j(0)] \rangle

This expression can be evaluated using various approximation schemes, such as the random phase approximation (RPA) or the dynamical mean-field theory (DMFT).

Random Phase Approximation (RPA)

The RPA is a simple approximation scheme that assumes that the current density operators at different times are uncorrelated. This leads to a significant simplification of the Kubo formula, which can be evaluated analytically.

Dynamical Mean-Field Theory (DMFT)

The DMFT is a more sophisticated approximation scheme that takes into account the correlations between the current density operators at different times. This leads to a more accurate description of the AC conductivity, but also requires more computational resources.

Numerical Implementation

To compute the real part of the AC conductivity using linear response theory, we need to implement the Kubo formula numerically. This can be done using various numerical methods, such as the Lanczos algorithm or the Chebyshev polynomial expansion.

Conclusion

In conclusion, computing the real part of the AC conductivity via linear response theory is a powerful tool for understanding the behavior of electrical conductivity in various materials and devices. The Kubo formula provides a systematic way to compute the conductivity tensor, and the RPA and DMFT approximation schemes can be used to simplify the calculation. By implementing the Kubo formula numerically, we can gain insights into the behavior of the AC conductivity and design more efficient materials and devices.

References

Kubo, R. (1957). Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. Journal of the Physical Society of Japan, 12(6), 570-586.
Mahan, G. D. (2000). Many-particle physics. Springer.
Altland, A., & Simons, B. (2010). Condensed matter field theory. Cambridge University Press.

Appendix

A.1 Kubo Formula Derivation

The Kubo formula can be derived using the following steps:

Start with the definition of the conductivity tensor:

\sigma(\omega) = \frac{\langle J_i \rangle}{E_i}

Use the Kubo formula to express the conductivity tensor in terms of the current density operators:

\sigma(\omega) = \frac{1}{i\omega} \int_0^\infty dt \, e^{i\omega t} \, \langle [J_i(t), J_j(0)] \rangle

Use the commutator identity:

[J_i(t), J_j(0)] = \langle J_i(t) J_j(0) \rangle - \langle J_i(0) J_j(0) \rangle

Simplify the expression using the assumption of uncorrelated current density operators:

\langle J_i(t) J_j(0) \rangle = \langle J_i(t) \rangle \langle J_j(0) \rangle

Evaluate the integral using the RPA approximation:

\sigma_r(\omega) = \frac{1}{\omega} \int_0^\infty dt \, \cos(\omega t) \, \langle [J_i(t), J_j(0)] \rangle

A.2 Numerical Implementation

Lanczos Algorithm

The Lanczos algorithm is a numerical method for computing the eigenvalues and eigenvectors of a matrix. It can be used to evaluate the Kubo formula numerically.

Chebyshev Polynomial Expansion

The Chebyshev polynomial expansion is a numerical method for approximating a function using a series of Chebyshev polynomials. It can be used to evaluate the Kubo formula numerically.

Code Implementation

Here is an example code implementation of the Kubo formula using the Lanczos algorithm:

import numpy as np
from scipy.linalg import eigh
def kubo_formula(N, omega, J):
# Initialize the matrix
M = np.zeros((N, N))
for i in range(N):
for j in range(N):
M[i, j] = J[i, j] * np.exp(-1j * omega * (i - j))
# Compute the eigenvalues and eigenvectors
eigenvalues, eigenvectors = eigh(M)

# Compute the conductivity tensor
sigma = np.zeros((N, N))
for i in range(N):
    for j in range(N):
        sigma[i, j] = 1 / (1j * omega) * np.sum(eigenvectors[:, i] * np.conj(eigenvectors[:, j]))

return sigma

N = 100
omega = 1.0
J = np.random.rand(N, N)
sigma = kubo_formula(N, omega, J)
print(sigma)

Q: What is the purpose of linear response theory in computing AC conductivity?

A: Linear response theory is a powerful framework for analyzing the response of a system to external perturbations. In the context of AC conductivity, it provides a systematic way to compute the conductivity tensor, which describes how the system responds to an external electric field.

Q: What is the Kubo formula, and how is it used to compute AC conductivity?

A: The Kubo formula is a fundamental result in linear response theory that expresses the conductivity tensor in terms of the current density operators. It is used to compute the AC conductivity by evaluating the integral of the commutator of the current density operators at different times.

Q: What is the random phase approximation (RPA), and how is it used to simplify the Kubo formula?

A: The RPA is a simple approximation scheme that assumes that the current density operators at different times are uncorrelated. This leads to a significant simplification of the Kubo formula, which can be evaluated analytically.

Q: What is the dynamical mean-field theory (DMFT), and how is it used to improve the accuracy of the Kubo formula?

A: The DMFT is a more sophisticated approximation scheme that takes into account the correlations between the current density operators at different times. This leads to a more accurate description of the AC conductivity, but also requires more computational resources.

Q: How is the real part of the AC conductivity computed using linear response theory?

A: The real part of the AC conductivity is computed using the Kubo formula, which is evaluated using the RPA or DMFT approximation schemes.

Q: What are some common numerical methods used to implement the Kubo formula?

A: Some common numerical methods used to implement the Kubo formula include the Lanczos algorithm and the Chebyshev polynomial expansion.

Q: Can you provide an example code implementation of the Kubo formula using the Lanczos algorithm?

A: Yes, here is an example code implementation of the Kubo formula using the Lanczos algorithm:

import numpy as np
from scipy.linalg import eigh
def kubo_formula(N, omega, J):
# Initialize the matrix
M = np.zeros((N, N))
for i in range(N):
for j in range(N):
M[i, j] = J[i, j] * np.exp(-1j * omega * (i - j))
# Compute the eigenvalues and eigenvectors
eigenvalues, eigenvectors = eigh(M)

# Compute the conductivity tensor
sigma = np.zeros((N, N))
for i in range(N):
    for j in range(N):
        sigma[i, j] = 1 / (1j * omega) * np.sum(eigenvectors[:, i] * np.conj(eigenvectors[:, j]))

return sigma


N = 100
omega = 1.0
J = np.random.rand(N, N)
sigma = kubo_formula(N, omega, J)
print(sigma)

This code implementation uses the Lanczos algorithm to compute the eigenvalues and eigenvectors of the matrix, and then uses these results to compute the conductivity tensor.

Q: What are some common applications of linear response theory in computing AC conductivity?

A: Some common applications of linear response theory in computing AC conductivity include:

Designing and optimizing materials and devices for various applications, such as electronics and optoelectronics.
Studying the behavior of complex systems, such as superconductors and superfluids.
Developing new numerical methods and approximation schemes for computing AC conductivity.

Q: What are some challenges and limitations of linear response theory in computing AC conductivity?

A: Some challenges and limitations of linear response theory in computing AC conductivity include:

The need for accurate and efficient numerical methods for evaluating the Kubo formula.
The difficulty of incorporating non-linear effects and correlations into the theory.
The need for more sophisticated approximation schemes to improve the accuracy of the results.

Q: What are some future directions for research in linear response theory and AC conductivity?

A: Some future directions for research in linear response theory and AC conductivity include:

Developing new numerical methods and approximation schemes for computing AC conductivity.
Incorporating non-linear effects and correlations into the theory.
Studying the behavior of complex systems, such as superconductors and superfluids.