Parity order as a fundamental driver of bosonic topology

TL;DR

This work demonstrates that parity order coupled to bond dimerization acts as a minimal mechanism to realize bosonic topology in one dimension, without density-density interactions or enlarged symmetries. Through DMRG studies and an effective spin-1 mapping, it reveals topological phases at $\rho=1/2$ (positive $V_p$) and $\rho=1$ (negative $V_p$), plus a dual topological phase at $\rho=3/2$, with clear edge-state signatures and winding-number analysis using twisted boundary conditions. The authors derive half- and unit-filling effective models that map to SSH-like physics and pair-hopping descriptions, respectively, and show robustness of the half-filled phase beyond a three-body constraint. These results establish parity order as a new organizing principle for correlation-driven bosonic topology and suggest experimental routes in ultracold-atom platforms to engineer such parity-driven topological states.

Abstract

Symmetry-protected topological (SPT) phases in interacting bosonic systems have been extensively studied, yet most realizations rely on fine-tuned interactions or enlarged symmetries. Here we show that a qualitatively different mechanism--parity order coupled to bond dimerization--acts as a fundamental driver of bosonic topology. Using density matrix renormalization group simulations, we identify two distinct topological phases absent in the purely dimerized model: an SPT phase at half filling stabilized by positive parity coupling, and a topological phase at unit filling stabilized by negative coupling that can be adiabatically connected to a trivial phase without breaking any symmetry. Our results establish parity order as a new organizing principle for correlation-driven bosonic topology.

Figures (8)

• Figure 1: (a) Schematic of the Hamiltonian in Eq. \ref{['eq:H']}, describing bosons on a dimerized chain with an onsite parity coupling: $V_p>0$ ($V_p<0$) energetically favors odd (even) site occupations. (b) Twisted phase winding number $\nu$ in the $t_1$--$V_p$ plane at half filling ($\rho=1/2$) for $L=8$ and $t_2=1$, showing a nontrivial phase with $\nu=1$ stabilized by positive parity coupling. The white star marks the BKT-type SF-BO transition point along the $t_1=0.2$ cut. (c) Corresponding winding number at unit filling ($\rho=1$), where a topological phase is stabilized for $V_p<0$. The white star indicates the crossover between BO and PBO phases along the $t_1=0.2$ cut. (d) Phase diagram in the chemical potential $\mu$--$V_p$ plane in the isolated-dimer limit ($t_1=1$, $t_2=0$), with phase boundaries obtained from Eq. \ref{['eq:mus']}, highlighting gapped phases at different fillings.
• Figure 2: Half filling ($\rho=1/2$) at $t_1=0.2$. (a) Charge gap under periodic boundary conditions as a function of $V_p$ for $L=20,40,60,80,100,120$, together with an extrapolation to the thermodynamic limit, signaling a BKT-type superfluid--bond order transition. (b) Bond energies $B_{1j}$ and $B_{2j}$ as functions of the bond index $j$ at $V_p=0$ (superfluid phase) and $V_p=2$ (bond order phase) for $L=120$. (c) Bond order parameter $O_{\mathrm{BO}}$ and pair bond order paarameter $O_{\mathrm{PBO}}$ as functions of $V_p$ for $L=40,60,80,100,120$. (d) Onsite densities $\langle n_j\rangle$ at $V_p=0$ and $V_p=2$ for $L=120$, illustrating the distinct density profiles associated with the bond order phase.
• Figure 3: Unit filling ($\rho=1$) at $t_1=0.2$. (a) Charge gap under periodic boundary conditions as a function of $V_p$ for $L=20,40,60,80,100,120$, together with an extrapolation to the thermodynamic limit, showing that the gap remains finite throughout. (b) Bond energies $B_{1j}$ and $B_{2j}$ as functions of the bond index $j$ at $V_p=0$ (bond order phase) and $V_p=-2$ (pair bond order phase) for $L=120$. (c) Bond order parameter $O_{\mathrm{BO}}$ and pair bond order parameter $O_{\mathrm{PBO}}$ as functions of $V_p$ for $L=40,60,80,100,120$. (d) Onsite densities $\langle n_j\rangle$ at $V_p=0$ and $V_p=-2$ for $L=120$, highlighting the distinct density profiles associated with the pair bond order phase.
• Figure 4: Results for $t_1=0.2$. (a) Parity order parameter $O_{\mathrm P}$ as a function of $V_p$ at half filling ($\rho=1/2$), decreasing from a finite value in the superfluid phase to zero upon entering the bond order phase. (b) Site-resolved expectation value of the parity operator $\langle (-1)^{\hat{n}_j} \rangle$ for $V_p=0$ and $V_p=2$ at $\rho=1/2$ for $L=120$. (c) Parity order parameter $O_{\mathrm P}$ at unit filling ($\rho=1$), decreasing from a large value in the pair bond order phase toward zero in the bond order phase. (d) Site-resolved expectation value of the parity operator $\langle (-1)^{\hat{n}_j} \rangle$ for $V_p=0$ and $V_p=-2$ at $\rho=1/2$ for $L=120$.
• Figure EM1: Ground-state energy of the two-site model at unit filling with $t_1=t_2=t=1$ as a function of the parity coupling $V_p$. Symbols show exact diagonalization (ED) results, while the solid line corresponds to the analytical expression in Eq. \ref{['eq:gs_energy']}. The inset displays the difference between numerical and analytical energies.