Constructing tensor network wavefunction for a generic two-dimensional quantum phase transition via thermofield double states

TL;DR

This paper introduces a tensor network wavefunction approach using thermofield double states to analyze generic two-dimensional quantum phase transitions, linking quantum criticality to three-dimensional statistical models.

Contribution

It generalizes Rokhsar-Kivelson wavefunctions by incorporating thermofield double states and expresses the resulting entropies as three-dimensional statistical models.

Findings

Derived the partition function of a 3D $Z_2$ lattice gauge-Higgs model.

Linked quantum phase transitions to 3D universality classes.

Applied framework to the toric code model with magnetic fields.

Abstract

The most important feature of two-dimensional quantum Rokhsar-Kivelson (RK) type models is that their ground state wavefunction norms can be mapped into the partition functions of two-dimensional statistical models so that the quantum phase transitions become the thermal phase transitions of the corresponding statistical models. For a generic quantum critical point, we generalize the framework of RK wavefunctions by introducing the concept of the thermofield double (TFD) state, which is a purification of the equilibrium density operator. Moreover, by expressing the TFD state in terms of the projected entangled pair state, its $N$-order of R\'{e}nyi entropy results in a three-dimensional statistical model in Euclidian spacetime, describing the generic quantum phase transitions. Using the toric code model with two parallel magnetic fields as an example, we explain these ideas and derive the partition function of the three-dimensional $Z_2$ lattice gauge-Higgs model, where the phase transitions are characterized by the three-dimensional universality classes.