# Quantum phase diagram of two-dimensional transverse field Ising model: unconstrained tree tensor network and mapping analysis

## TL;DR

This study maps out the phase diagram of the frustrated 2D transverse field Ising model on a checkerboard lattice using an advanced tensor network method, revealing phase transitions and a mapped effective theory.

## Contribution

It introduces an unconstrained tree tensor network approach for accurate phase diagram analysis and maps the checkerboard lattice model to an equivalent square lattice model.

## Key findings

- Identifies a second order phase transition at $ ext{J}_2= ext{J}_1$ with critical field $ ext{Γ}_c=0.28$.
- Verifies the stability of the plaquette-VBS phase at low magnetic fields.
- Maps the highly frustrated point to an emergent string-VBS phase on the square lattice.

## Abstract

We investigate the ground-state phase diagram of the frustrated transverse field Ising (TFI) model on the checkerboard lattice (CL), which consists of N\'{e}el, collinear, quantum paramagnet and plaquette-valence bond solid (VBS) phases. We implement a numerical simulation that is based on the recently developed unconstrained tree tensor network (TTN) ansatz, which systematically improves the accuracy over the conventional methods as it exploits the internal gauge selections. At the highly frustrated region ($J_2=J_1$), we observe a second order phase transition from plaquette-VBS state to paramagnet phase at the critical magnetic field, $\Gamma_{c}=0.28$, with the associated critical exponents $\nu=1$ and $\gamma\simeq0.4$, which are obtained within the finite size scaling analysis on different lattice sizes $N=4\times 4, 6\times 6, 8\times8$. The stability of plaquette-VBS phase at low magnetic fields is examined by spin-spin correlation function, which verifies the presence of plaquette-VBS at $J_2=J_1$ and rules out the existence of a N\'{e}el phase. In addition, our numerical results suggest that the transition from N\'{e}el (for $J_2<J_1$) to plaquette-VBS phase is a deconfined phase transition. Moreover, we introduce a mapping, which renders the low-energy effective theory of TFI on CL to be the same model on $J_1-J_2$ square lattice (SL). We show that the plaquette-VBS phase of the highly frustrated point $J_2=J_1$ on CL is mapped to the emergent string-VBS phase on SL at $J_2=0.5J_1$.