Abstract
Quantum fluctuations of the N\'eel state of the square lattice antiferromagnet are usually described by a $\mathbb{CP}^1$ theory of bosonic spinons coupled to a U(1) gauge field, and with a global SU(2) spin rotation symmetry. Such a theory also has a confining phase with valence bond solid (VBS) order, and upon including spin-singlet charge 2 Higgs fields, deconfined phases with $\mathbb{Z}_2$ topological order possibly intertwined with discrete broken global symmetries. We present dual theories of the same phases starting from a mean-field theory of fermionic spinons moving in $\pi$-flux in each square lattice plaquette. Fluctuations about this $\pi$-flux state are described by 2+1 dimensional quantum chromodynamics (QCD$_3$) with a SU(2) gauge group and $N_f=2$ flavors of massless Dirac fermions. It has recently been argued by Wang et al. (arXiv:1703.02426) that this QCD$_3$ theory describes the N\'eel-VBS quantum phase transition. We introduce adjoint Higgs fields in QCD$_3$, and obtain fermionic dual descriptions of the phases with $\mathbb{Z}_2$ topological order obtained earlier using the bosonic $\mathbb{CP}^1$ theory. We also present a fermionic spinon derivation of the monopole Berry phases in the U(1) gauge theory of the VBS state. The global phase diagram of these phases contains multi-critical points, and our results imply new boson-fermion dualities between critical gauge theories of these points.
Highlights
Spin-liquid states of the square lattice antiferromagnet, with global SU(2) spin rotation symmetry, have long been recognized as important ingredients in the theory of the cuprate high temperatures superconductors [1,2,3,4]
One of the purposes of the present paper is to present a unified theory of the three Z2 spin liquids noted above, but using the fermionic spinon approach [17,25,26]
In Appendix B, we review Wen’s classification scheme [17] and identify the lattice projective symmetry group” (PSG) corresponding to the two U(1) spin liquids as well as the five symmetric Z2 spin liquids
Summary
Spin-liquid states of the square lattice antiferromagnet, with global SU(2) spin rotation symmetry, have long been recognized as important ingredients in the theory of the cuprate high temperatures superconductors [1,2,3,4]. Chatterjee et al [22] proposed examining spin liquids which are proximate to the magnetically ordered Neel state These proximate states are reachable by continuous (or nearly continuous) quantum phase transitions involving the longwavelength excitations of the antiferromagnet. They used a CP1 theory of quantum fluctuations of the Neel state, expressed in terms of bosonic spinons zα to argue for the importance of three possible Z2 spin-liquid states. We obtain a phase diagram of the states proximate to the Neel state using the fermionic spinon approach, and propose critical theories of the phase transitions involving massless. Given the strongly coupled nature of the critical theories, this conclusion is based upon circumstantial, rather than firm, evidence
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