# Theory of Emergent Trionic Order in One-Dimensional Bose-Fermi Mixtures

## TL;DR

This paper develops an effective low-energy theory for a 1D Bose-Fermi mixture, revealing a phase transition to a trionic phase with bound states of two fermions and one boson, driven by a density resonance.

## Contribution

It introduces a microscopic model and analysis showing how trionic order emerges in 1D Bose-Fermi mixtures through a resonance mechanism and RG analysis.

## Key findings

- Identification of a phase transition to a trionic phase
- Derivation of an effective low-energy theory using bosonization
- Prediction of experimental signatures for trionic correlations

## Abstract

We study a one-dimensional (1D) lattice mixture of hard-core bosons and spinless fermions with attractive interspecies interaction and correlated fermion pair hopping. Using Schrieffer-Wolff (SW) transformation and bosonization, we derive an effective low-energy theory that reveals a density-induced resonance mechanism. When the filling ratio satisfies $\rho_{c0}:\rho_{b0} = 2:1$, a non-oscillatory cosine term emerges in the bosonized theory. This term favors the formation of trions, i.e., bound states of two fermions and one boson. By performing a renormalization-group (RG) analysis, we identify a phase transition from a gapless two-component Luttinger liquid to a partially gapped trionic phase, where trionic correlations exhibit dominant quasi-long-range order. Our findings provide a microscopic understanding of composite superfluidity and offer experimentally relevant signatures for Bose-Fermi mixtures in 1D lattices.