Fermionic Spinon Theory of Square Lattice Spin Liquids near the Néel State

Quantum fluctuations of the Néel 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 (${\mathrm{QCD}}_3$) with a SU(2) gauge group and $N_f=2$ flavors of massless Dirac fermions. It has recently been argued by Wang et al. [Deconfined Quantum Critical Points: Symmetries and Dualities, Phys. Rev. X 7, 031051 (2017).] that this ${\mathrm{QCD}}_3$ theory describes the Néel-VBS quantum phase transition. We introduce adjoint Higgs fields in ${\mathrm{QCD}}_3$ 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 multicritical points, and our results imply new boson-fermion dualities between critical gauge theories of these points.

Popular Summary

High-temperature superconductivity—where electrical current flows with zero resistance at relatively high temperatures—was discovered in 1987 in a class of copper-based crystals known as cuprates. When researchers vary the density of mobile electrons in the cuprates (by doping with impurity ions), these materials transform from magnetic insulators to high-temperature superconductors. A central open question has been the precise relationship between the magnetism and superconductivity. A certain class of magnetism, found in spin liquids with topological order (a material where the electron spins constantly fluctuate and never align), has been viewed as central to understanding the nature of the superconductivity. In particular, a large class of spin-liquid candidates has been proposed as relevant to the physics of cuprates. We present a unified theory of spin liquids that are of particular relevance to the cuprates.

One of our main results is a subtle duality between descriptions of these spin liquids using particles with Bose and Fermi statistics. While these spin liquids may appear distinct because their representation uses bosons or fermions, their fully renormalized quasiparticle excitations are shown to be equivalent. A large class of bosonic spin liquids has been proposed using a theory of fluctuating antiferromagnetism, and we show that these liquids are equivalent to fermionic spin liquids proposed from a theory of electrons localizing into Mott insulators.

This unification opens the way to a comprehensive understanding of the disparate physical properties of the doped cuprates in the “pseudogap” regime found at temperatures above the superconducting critical temperature.

Figure 1

(a) Schematic phase diagram of the ${\mathbb{CP}}^1$ theory in Eq. (1.4) as a function of $g$ and $s_1$ [$s_2$ in Eq. (1.7) is large and positive]; Eq. (1.2) describes the deconfined critical Néel-VBS transition at a critical $g=g_c$. (b) Schematic phase diagram of the SU(2) ${\mathrm{QCD}}_3$ theory with $N_f=2$ flavors of massless Dirac fermions in Eq. (1.6) as a function of $s$ and ${\overline{s}}_1$ [${\overline{s}}_2$ in Eq. (1.8) is large and positive]. The “Wen” labels refer to the naming scheme in Ref. [17]. The ${\mathbb{Z}}_2$ spin liquids $A_b$ and $A_f$ in (a) and (b) are argued to be topologically identical, as are the confining states with VBS order. The critical spin liquids in (b) are argued to be unstable to the corresponding phases with magnetic order in (a), with the critical SU(2) spin liquid surviving only at the Néel-VBS transition. All ${\mathbb{Z}}_2$ spin liquids are shown shaded in all figures.

Figure 2

(a) Schematic phase diagram of the ${\mathbb{CP}}^1$ theory in Eqs. (1.4) and (1.7) as a function of $s_1$ and $s_2$ (for large $g$). (b) Schematic phase diagram of the SU(2) ${\mathrm{QCD}}_3$ theory with $N_f=2$ flavors of massless Dirac fermions in Eqs. (1.6) and (1.8) as a function of ${\overline{s}}_1$ and ${\overline{s}}_2$ (for $s<0$ and $|s|$ large). All four phases in (a) and (b) are argued to be topologically identical. So for the ${\mathbb{Z}}_2$ spin liquids, $A_b=A_f$, $B_b=B_f$, and $C_b=C_f$. Phases $B_f$ and $C_f$ do not appear in Wen’s classification [17] because they break global symmetries.

Figure 3

How $\mathrm{tr}({\sigma }^a\overline{X}{T}^{j}X)$ transform under the physical symmetries. $T^j=\{{\mu }^y,{\sigma }^a,{\mu }^x{\sigma }^a,{\mu }^z{\sigma }^a\}$ are the ten generators of SO(5).

Figure 4

How $\mathrm{tr}({\sigma }^a\overline{X}{\mathrm{\Gamma }}^j{\gamma }^{\mu }X)$ transform under the physical symmetries. ${\mathrm{\Gamma }}^j=\{{\mu }^x,{\mu }^z,{\mu }^y{\sigma }^a\}$ transform under the vector representation of the emergent SO(5).

Figure 5

Symmetry transformation properties of bilinears of the form $\mathrm{tr}({\sigma }^a\overline{X}i{\partial }_{\mu }X)$, $\mathrm{tr}({\sigma }^a\overline{X}{\mathrm{\Gamma }}^{j}i{\partial }_{\mu }X)$, and $\mathrm{tr}({\sigma }^a\overline{X}{T}^j{\gamma }^{\mu }i{\partial }_{\nu }X)$, which do not transform under spin. The operators that can couple to a Higgs field in a gapped symmetric spin ${\mathbb{Z}}_2$ spin liquid are colored; entries with the same color transform into one another under $R_{\pi /2}$.

Figure 6

All symmetric PSGs associated with symmetric ${\mathbb{Z}}_2$ spin liquids in which $⟨{\mathrm{\Phi }}^x⟩\ne 0$, where $\mathrm{\Phi }$ couples to $\mathrm{tr}({\sigma }^a\overline{X}{\mu }^{y}X)$. These are listed as a function of the operator $\mathrm{tr}({\sigma }^a\overline{X}MX)$ which ${\mathrm{\Phi }}_1$ couples to. We assume that only $⟨{\mathrm{\Phi }}_1^y⟩\ne 0$.

Figure 7

Nematic PSGs associated with order parameters of the form ${\mathrm{\Phi }}^a\mathrm{tr}({\sigma }^a\overline{X}{\mu }^{y}X)+{\mathrm{\Phi }}_{2i}^a\mathrm{tr}({\sigma }^a\overline{X}{M}^{i}X)$. We do not include $\mathrm{tr}({\sigma }^a\overline{X}{\partial }_{0}X)$ since this operator is invariant under the action of $R_{\pi /2}$ and already accounted for as $s\mathrm{PSG}5$. The labels $x$, $y$ are simply a convenient notation and do not necessarily signify a physical direction.

Figure 8

Symmetries broken depending on the orientation in gauge space taken by the Higgs condensates. The fields couple to the bilinears as $\mathrm{tr}(\mathrm{\Phi }\overline{X}{\mu }^{y}X)+\mathrm{tr}({\mathrm{\Phi }}_1\overline{X}i{\partial }_{0}X)+\mathrm{tr}({\mathrm{\Phi }}_{2x}\overline{X}i{\partial }_{x}X)$.

Figure 9

The columns labeled “$s\mathrm{PSG}1\text{−}5$” list the symmetry fractionalizations of the gapped, symmetric ${\mathbb{Z}}_2$ spin liquids given in Fig. 7. The corresponding bosonic symmetry fractionalization numbers are obtained by multiplying the $s\mathrm{PSG}$ numbers with the those given in the Vison and Twist columns. We see that $s$PSG5 corresponds to the ${\mathbb{Z}}_2[0,0]$ state of Ref. [23]. No bosonic counterparts to $s$PSG1–4 are present in Ref. [23].

Figure 10

Spin liquids according to the labeling scheme given in Ref. [17] and reviewed in Appendix pp2-s4. All of the spin liquids listed are found to be proximate to the $\pi$-flux phase $\mathrm{SU}2n0$ though not necessarily $\mathrm{U}1\mathrm{C}n0n1$. While the symmetry fractionalization of $s\mathrm{PSG}1$ and $s\mathrm{PSG}5$ corresponds to multiple fermionic PSGs, the two that are italicized ($\mathrm{Z}2\mathrm{B}xx23$ and $\mathrm{Z}2\mathrm{B}xx2z$) are not proximate to $\mathrm{U}1\mathrm{C}n0n1$ and therefore cannot represent the Higgs phases we obtain (see Appendix pp2-s4).

Figure 11

Symmetry fractionalization of nematic PSGs ($n\mathrm{PSGs}$) for spin liquids listed in Fig. 7. $n\mathrm{PSG}7x$ (highlighted in blue) corresponds to the fermionic PSG determined in Ref. [21]. The columns labeled “V” and “T” list the vison fractionalization numbers and the twist factors, respectively.

Figure 12

Fermion bubble to calculate flux response.