Quantum phase transitions between bosonic symmetry-protected topological phases in two dimensions: Emergent ${\mathrm{QED}}_3$ and anyon superfluid

Inspired by the Chern-Simons effective theory description of symmetry protected topological (SPT) phases in two dimensions, we present a projective construction for many-body wave functions of SPT phases. Using this projective construction, we can systematically write down trial wave functions 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 ${\nu }_{b_1}={\nu }_{b_2}=\frac{1}{2}$ per site on a 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 an emergent ${\mathrm{QED}}_3$ 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.

Figure 1

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 (40). $\theta =\pi /(1-2q)$ 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 (42). Each red circle denotes a tricritical point, whose effective theory is described by emergent ${\mathrm{QED}}_3$ with fermion number $N_f=2$.

Figure 2

An illustration of mean-field hopping ansatz of $f$ and $d_{\alpha }$ partons on a 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 first- and second-nearest-neighbor (NN) hoppings are shown here. Among horizontal hoppings between first 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 first 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 second-neighbor hopping parameters are all imaginary and equals $\mathrm{i}{t}_2$ along the arrow directions ($t_2>0$). This hopping Hamiltonian in momentum space is given by (31).

Figure 3

Mean-field phase diagram of hopping ansatz (31) for $d_1$ partons in projective construction (27) with $q=0$. Solid lines $|t_y^{\prime }-t_y|=\pm 4t_2$ 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 (27). 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 ${\mathrm{QED}}_3$ with $N_f=2$.