Quantum phase transitions between bosonic symmetry protected topological phases in two dimensions: emergent $QED_3$ and anyon superfluid

TL;DR

This work addresses how continuous quantum phase transitions can occur between bosonic $U(1)$-symmetric SPT phases in two dimensions. It develops a projective-construction framework that maps lattice wavefunctions to a $2\times2$ Chern-Simons theory with ${\bf K}=\begin{pmatrix}-2q & 1\\ 1 & 0\end{pmatrix}$ and ${\bf t}=(0,1)^T$, enabling controlled transitions between phases with $\sigma_{xy}=2q$ and $\sigma_{xy}=2q-2$. At the transition, the long-distance theory is emergent $QED_3$ with $N_f=2$, though direct transitions require fine tuning without extra symmetry; generically an intermediate anyon superfluid phase can intervene, featuring spontaneous $U(1)$ breaking and a Goldstone mode. The framework also provides a lattice-based route to construct and analyze bosonic SPTs, with potential generalization to other symmetries and implications for experimental exploration of SPT transitions.

Abstract

Inspired by Chern-Simons effective theory description of symmetry protected topological (SPT) phases in two dimensions, we present a projective construction for many-body wavefunctions of SPT phases. Using this projective construction we can systematically write down trial wavefunctions of SPT phases on a lattice. An explicit example of SPT phase with $U(1)$ symmetry is constructed for two types of bosons with filling $ν_{b_1}=ν_{b_2}=\frac12$ per site on square lattice. We study continuous phase transitions between different $U(1)$-SPT phases based on projective construction. The effective theory around the critical point is emergent $QED_3$ with fermion number $N_f=2$. Such a continuous phase transition however needs fine tuning, and in general there are intermediate phases between different $U(1)$-SPT phases. We show that such an intermediate phase has the same response as an anyon superconductor, and hence dub it "anyon superfluid". A schematic phase diagram of interacting bosons with $U(1)$ symmetry is depicted.

Figures (3)

• Figure 1: (color online) A schematic phase diagram of interacting bosons with $U(1)$ symmetry (associated with boson charge conservation) in two dimensions. It includes trivial boson insulator, bosonic $U(1)$-SPT insulators and gapless anyon superfluid (aSF) phases. Different bosonic insulators are featured by their Hall conductance $\sigma_{xy}=2q,~q\in{\mathbb{Z}}$, in the presence of $U(1)$ symmetry associated with boson number conservation. An aSF phase spontaneously breaks $U(1)$ symmetry and is featured by superfluid response and a quantized Chern-Simons term in (\ref{['response:anyon SF']}). $\theta=\pi/(1-2q)$ denotes the statistical angle of anyon, in the aSF phase between two bosonic $U(1)$-symmetric insulators with $\sigma_{xy}=2q$ and $\sigma_{xy}=2q-2$. Solid lines denote phase boundaries between $U(1)$-SPT phases and anyon superfluids, which are connected by a continuous phase transition with effective theory (\ref{['aSF-SPT']}). Each red circle denotes a tricritical point, whose effective theory is described by emergent $QED_3$ with fermion number $N_f=2$.
• Figure 2: (color online) An illustration of mean-field hopping ansatz of $f$- and $d_\alpha$-partons on square lattice. The magnetic unit cell of the mean-field Hamiltonian contains two lattice sites (featured by red circles and blue diamonds respectively) of the original square lattice, as indicated by the pink rectangle. The primitive vectors $\vec{a}_{1,2}$ for the magnetic unit cell are also shown. For simplicity only 1st and 2nd nearest neighbor (NN) hoppings are shown here. Among horizontal hoppings between 1st NNs, solid lines denote hopping parameter $t_x$ while dashed lines denote $-t_x$ where $t_x>0$ is a real number. For vertical hopping between 1st NNs, solid lines denote hopping parameter $t_y$ while dashed lines denote $t_y^\prime$ (we choose $t_y,t_y^\prime>0$ for simplicity). The 2nd neighbor hopping parameters are all imaginary and equals $\space\mathrm{i}\space t_2$ along the arrow directions ($t_2>0$). This hopping Hamiltonian in momentum space is given by (\ref{['mf hopping:f1,d']}).
• Figure 3: (color online) Mean-field phase diagram of hopping ansatz (\ref{['mf hopping:f1,d']}) for $d_1$-partons in projective construction (\ref{['parton:u(1) spt']}) with $q=0$. Solid lines $|t_y^\prime-t_y|=\pm4t_2$ denote the phase boundary between trivial boson insulator with $\sigma_{xy}=0$, bosonic $U(1)$-SPT phase with $\sigma_{xy}=-2$ and anyon superfluid (aSF) with anyon statistical angle $\theta=\pi$, where continuous phase transitions happen. Chern numbers $C_f=+1$ and $C_{f_1}=-1$ are chosen for $f$- and $f_1$-partons in projective construction (\ref{['parton:u(1) spt']}). Notice that only when $t_y^\prime=t_y$, there is a direct continuous phase transition between bosonic $U(1)$-SPT phase and the trivial boson insulator. The effective theory describing the tricritical point at $t_y^\prime-t_y=t_2=0$ is $QED_3$ with $N_f=2$.