From QED$_3$ to Self-Dual Multicriticality in the Fradkin-Shenker Model

Abstract

We consider the Fradkin-Shenker ${\mathbb Z}_2$ gauge-Higgs lattice model in 2+1 dimensions, i.e. the toric code deformed by an in-plane magnetic field. Its phase diagram contains a multicritical CFT with gapless, mutually non-local electric and magnetic particles, exchanged by a ${\mathbb Z}_2^{\mathsf{D}}$ self-duality symmetry. We introduce a staggered generalization of the model in which these particles carry global $U(1)_e$ and $U(1)_m$ charges, respectively, and we propose a continuum QFT description in terms of QED$_3$ with $N_f = 2$ Dirac fermion flavors and a charge-two Higgs field with Yukawa couplings. The conjectured phase diagram harbors a multicritical CFT with $(O(2)_e \times O(2)_m)\rtimes\mathbb{Z}_2^\mathsf{D}$ symmetry, some of which is emergent in the QFT description. We compute the scaling dimensions of some operators using a large-$N_f$ expansion and find agreement with the emergent selection rules. The staggered model admits a deformation to the original Fradkin-Shenker model, which maps to unit-charge monopole operators in Higgs-Yukawa-QED$_3$ that break the $U(1)_e \times U(1)_m$ symmetry. We show explicitly that this deformation reproduces all features of the Fradkin-Shenker phase diagram. Finally, we propose a multicritical duality between Higgs-Yukawa-QED$_3$ and the easy-plane $\mathbb{ CP}^1$ model (i.e. two-flavor scalar QED$_3$ with a suitable potential), which describes spin-1/2 anti-ferromagnets on a square lattice. This duality implies a first-order line of Néel-VBS transitions ending in a deconfined quantum multicritical point, described by the same $O(2)_e \times O(2)_m$ symmetric CFT that arises in the staggered Fradkin-Shenker model, which separates it from a gapped ${\mathbb Z}_2$ spin liquid phase.

Figures (15)

• Figure 1: Relation between the $2+1$d models in this paper, with clickable links, and their global symmetries. On the left are two lattice models: the Fradkin-Shenker (FS) model, which refers to the toric code in an in-plane field, and the Staggered Fradkin-Shenker (SFS) model, which is a $U(1)_e \times U(1)_m$-symmetric variant we introduce. When tuned to their self-dual multicritical points, we expect these lattice models to flow to conformal field theories, which we refer to as the FS CFT and the SFS CFT, respectively. On top, we propose two continuum QFTs that flow to the SFS CFT at their multicritical point: Higgs-Yukawa-QED$_3$ (HYQED) and the easy-plane $\mathbb{CP}^1$ (EP${\mathbb C}\hbox{$\mathbb P$}^1$) model. In both, a $\mathbb{Z}_2$ symmetry (referred to as mirror and duality, respectively) emerges in the IR, which is required to match the global symmetry of the SFS CFT. From the point of view of the continuum Lagrangians, the symmetry-breaking deformation from the SFS model to the FS model is given by certain unit-charge monopole operators. These deformations preserve a $\mathbb{Z}_2^{\mathsf{C}_e} \times \mathbb{Z}_2^{\mathsf{C}_m} \subset O(2)_e \times O(2)_m$ symmetry corresponding to certain lattice rotations; it is not known if they act on the FS CFT in the deep IR, and for this reason we have not included them in the figure (see section \ref{['FSviaMonos']} for more detail).
• Figure 2: Schematic phase diagram of the Fradkin-Shenker (FS) model \ref{['eq:Hfsintro']}, reproduced from Wu:2012cj, as a function of $h_e,h_m \geq 0$. The $\mathbb{Z}_2^{\sf D}$ duality symmetry acts as ${\sf D}: h_e \leftrightarrow h_m$ and is preserved along the self-dual diagonal $h_e = h_m$. In the lower left region there is the toric code phase, described at long distances by an emergent $\mathbb{Z}_2$ gauge theory, which couples electrically and magnetically to dynamical, massive $e$ and $m$ anyons that are exchanged by $\sf D$. These become massless on the second-order $\mathbb{Z}_2$-gauged Ising lines (indicated in blue and labeled Ising$^*_e$ and Ising$^*_m$, respectively), which describe electric and magnetic Higgs transitions into the trivial Higgs/confined phase. The Ising$^*_{e,m}$ lines meet at a multicritical point (indicated in red) on the self-dual diagonal, which is described by the FS CFT. Continuing further along this diagonal into the Higgs/confined phase, we find a first-order (dotted) line on which $\sf D$ is spontaneously broken, and which ends at a conventional Ising critical point (indicated in green).
• Figure 3: Conjectured phase diagram of Higgs-Yukawa-QED as a function of the $\sf D$-even Higgs mass $m_\phi^2$ and the $\sf D$-odd fermion mass $m_3$ in \ref{['intro:masses']}. The multicritical point at the origin, described by the SFS CFT, is shown in red. The phase to its right ($m_3 > 0$) spontaneously breaks $U(1)_m$, leading to an $S^1_m$ circle of vacua. Its $\sf D$-reflection at $m_3 < 0$ spontaneously breaks $U(1)_e$, with an $S^1_e$ circle of vacua. These circles coexist on the $m_\phi^2 > 0$ axis (indicated by the dotted black line), where the vacua are $S_e^1 \sqcup S_m^1$ and the duality symmetry ${\sf D} : S_e^1 \leftrightarrow S_m^1$ is spontaneously broken. The phase at $m_\phi^2 < 0$ is gapped, with a $\mathbb{Z}_2$ TQFT in the IR, whose $e$ and $m$ anyons are exchanged by $\sf D$. The transitions from the TQFT phase to the symmetry-breaking phases (indicated by blue lines) occur when either the $e$ or the $m$ anyon of the $\mathbb{Z}_2$ TQFT becomes massless. The resulting second-order transitions are described by $\mathbb{Z}_2$-gauged versions of the $O(2)$ Wilson-Fisher CFT, which we denote by $O(2)_{e,m}^*$, ending at the multicritical point.
• Figure 4: The phase diagram of Higgs-Yukawa-QED$_3$ (see figure \ref{['figU1phasediagram']}), deformed by monopole operators of $U(1)_{\mathcal{M}}$ charge $q_{\mathcal{M}} = 4$, reinterpreted in the Néel-VBS context via the multicritical duality with the EP${\mathbb C}\hbox{$\mathbb P$}^1$ model in section \ref{['intro:easy']}. The $U(1)_e$ SSB phase corresponds to the easy-plane Néel phase, where the $U(1)_e$ spin-rotation symmetry is spontaneously broken. The $U(1)_m$ symmetry of HYQED is explicitly broken to the $\mathbb{Z}_4^\text{rot.}$ lattice rotation group by the monopole operators, leading to a gapped phase with four vacua and $\mathbb{Z}_4^\text{rot.}$ SSB. The monopole operators are irrelevant on the second-order $O(2)_{e,m}^*$ lines and at the deconfined quantum multicritical point described by the SFS CFT. The $\mathbb{Z}_2$ TQFT phase is interpreted as a gapped $\mathbb{Z}_2$ spin liquid, where all symmetries are unbroken, and the monopole deformations have no effect in the IR.
• Figure 5: The $e$ and $m$ string operators can be represented as curves on the lattice and its dual, respectively. Where the red lattice curve crosses a link, we apply $Z_\ell$, and where the blue dual lattice curve intersects a link, we apply $X_\ell$. These anticommute at each point where they intersect, showing the $-1$ braiding of the $e$ and $m$ anyons.