# Analytical solutions of inhomogeneous transverse field Ising models

## TL;DR

This paper provides analytical solutions for inhomogeneous transverse field Ising models, exploring how inhomogeneities influence quantum phase transitions and offering a systematic approach for models with multiple impurities.

## Contribution

It introduces a generalizable analytical method to diagonalize inhomogeneous Ising models with multiple impurities and junctions, enhancing understanding of their quantum critical behavior.

## Key findings

- Inhomogeneity affects the quantum phase transition point.
- Correlation functions and magnetization reveal critical behavior.
- Systematic approach applicable to arbitrary impurity configurations.

## Abstract

The inhomogeneous transverse field Ising models mainly impurity based and the joint chain are analysed analytically using Jordan-Wigner transformations. The effects of inhomogeneities on the phase transition have been discussed in detail. We constructed an ansatz to diagonalize the two models which are taken into consideration. The inhomogeneity is quantified by a coupling parameter, which can be tuned to control the occurrence of quantum phase transition in these models. We have shown a systematic setup using which we can generalise the solution to a system with an arbitrary number of impurity sites and junctions, which are separated by at least two lattice sites. We have analysed the quantum critical point by calculating the correlation functions, transverse magnetization and the gap between the ground state and the first excited state.