The critical phenomena of a single defect

TL;DR

This paper investigates how a single point defect affects critical phenomena in isotropic systems, revealing that thermodynamic variations follow specific heat exponents and diverge logarithmically or linearly at the critical point.

Contribution

It establishes a general relation between defect-induced thermodynamic variations and pure system exponents using renormalization group theory, supported by numerical analysis of the 2D Ising model.

Findings

Internal energy variation diverges logarithmically with lattice size at criticality.

Heat capacity variation diverges linearly with lattice size at criticality.

Near the critical point, energy variation behaves as ln|t| and heat capacity as |t|^{-1}.

Abstract

We consider the critical system with a point defect and study the variation of thermodynamic quantities, which are the differences between those with and without the defect. Within renormalization group theory, we show generally that the critical exponent of the internal energy variation is the specific heat exponent of a pure system, and the critical exponent of the heat capacity variation is that for the temperature derivative of specific heat of a pure system. This conclusion is valid for the isotropic systems with a short-range interaction. As an example we solve the two dimensional Ising model with a point defect numerically. The variations of the free energy, internal energy and specific heat are calculated with bond propagation algorithm. At the critical point, the internal energy variation diverges with the lattice size logarithmically and the heat capacity variation diverges with size linearly. Near the critical point, the internal energy variation behaves as $\ln |t|$ and the heat capacity variation behaves as $|t|^{-1}$, where $t$ is the reduced temperature.