The electromagnetic duality and the 3+1D O(6) non-linear sigma model with a level-1 Wess-Zumino-Witten term

Introduction Non-linear sigma (NL$\sigma$) models describe the dynamics of Goldstone mode. Due to the interaction between the Goldston bosons, NL$\sigma$ models can develop a mass instead of being gapless [1]. A well-known example occurs in 1+1 dimensions, where the $O(3)$ NL$\sigma$ model is always gapful. On a different front, topological terms in the NL$\sigma$ model can fundamentally modify the energy spectrum. For example, the $O(3)$ NL$\sigma$ model with $\theta$ term is gapful for $\theta=2n\pi$ while it is gapless for $\theta=(2n+1)\pi$ for $n\in\mathbb{Z}$ [2]. In addition to the topological $\theta$ term, other topological terms, such as the Wess-Zumino-Witten [3] and the Hopf terms [4], are also known to change the Goldstone mode spectrum.

The $O(6)$ NL$\sigma$ model with a level-1 Wess-Zumino-Witten (WZW) term in 3+1 space-time dimensions The action of this model read

For easy reference, we shall name the first three components of $\hat{\Omega}$ the Neel order parameters, and the last three components the valence bond solid (VBS) order parameters, i.e., $\hat{\Omega}=(n_{1},n_{2},n_{3},v_{1},v_{2},v_{3})$. These names are motivated by the fact that Eq.(LABEL:o6) governs the low-energy dynamics of a three-dimensional Mott insulator with competing Neel and valence-bond solid (VBS) order [9] (see supplemental material section I). However, for the majority of this paper, these are just names we use to refer to the components of the $\hat{\Omega}$.

The VBS hedgehog Consider the following $\tilde{\hat{\Omega}}$ configuration corresponding to a hedgehog in $\left(v_{1},v_{2},v_{3}\right)$

where $(r,\theta,\phi)$ are the coordinates of the three dimensional space and $\tau$ is the Euclidean time. In Eq.(3), $(n_{1},n_{2},n_{3})$ is a unit vector and $f(r)$ is a smooth function satisfying $f(r)=\pi/2$ for $r>r_{c}$ (hedgehog core size) and $f(r)=0$ at $r=0$. It is trivial to show that $\sum_{i=1}^{6}\tilde{\Omega}_{i}^{2}=1.$ Plug Eq.(3) into Eq.(LABEL:o6), it is straightforward to show that

Eq.(4) is the Berry’s phase in the coherent state path integral of a spin 1/2 in 0+1 dimension. Therefore the hedgehog of the VBS order fractionalizes the spins! In the supplemental material section II, we present the microscopic theory of the hedgehog core.

The statistics of VBS hedgehogs We can determine the statistics of the VBS hedgehogs by computing the Berry phase associated with the hedgehog exchange. Because our space-time is $S^{4}$, the exchange process must be embedded in (i) vacuum creation of two pairs of hedgehog-anti-hedgehog, (ii) exchanging the two hedgehogs, and (iii) annihilating the hedgehogs with the anti-hedgehogs. The Berry phase contains two contributions: 1) that due to exchange of the hedgehogs and 2) that associated with the spin 1/2 in the cores of hedgehog and anti-hedgehog. To isolate the Berry phase due to the exchange, we lock the core spins in, say, the positive $n_{3}$-direction.

The CP¹ theory of Neel order The NL$\sigma$ model action for the Neel order parameter reads

It is well known that the above action can be rewritten as[11]

where $a_{\mu}$ is a compact dynamic gauge field and $z$ is the two-component complex boson field satisfying Eq.(5) and Eq.(6). Because the action in Eq.(10) is at most quadratic in $a_{\mu}$ the saddle point is exact, namely,

In Eq.(10), the condensation of the $z$ boson Higgs the gauge field $a_{\mu}$, hence mods out the redundant local phase gauge of freedom $z\rightarrow z^{i\varphi}z$ in Eq.(6). The resulting phase exhibits the Neel long-range order. Combining Eq.(8) and the above discussion suggests that the field theory for the VBS hedgehog is given by

Here $z$ represents the boson field for the VBS hedgehog. Once again, we see that when supplemented with the WZW term, the conventional Ginzburg-Landau theory is equivalent to a theory with fractionalized particles and gauge field.

The Neel hedgehog A parallel discussion with “VBS” and “Neel” switched implies that the Neel hedgehog carries the Berry phase of the VBS order parameter. Specifically after the switch Eq.(4) becomes

If we define a two-component complex boson field

it is simple to show that

This implies the VBS hedgehog current $K_{\mu}$ couples to a gauge field, namely,

The CP¹ theory of VBS order The NL$\sigma$ model action for the VBS order parameter read

Similar to Eq.(10) we can rewrite Eq.(17) as

where $b_{\mu}$ is a compact dynamic gauge field and $w$ is the two-component complex boson field satisfying Eq.(14). In this case

Analogous to the discussion after Eq.(10), the condensation of the $w$ bosons induces the VBS long-range order. The above discussion suggests that the field theory for the Neel hedgehog is given by

Here $w$ represents the boson field for the Neel hedgehog.¹¹1When Eq.(LABEL:Neelhedgehog) is used to describe the acutal VBS order of a Mott insulator, there is an additional term $$S_{\rm aniso}[w^{\dagger}\vec{\sigma}w]$$ which should be added to Eq.(LABEL:Neelhedgehog). This term describes the anisotropy imposed by the lattice of the Mott insulator. However, for our purpose the term Neel and VBS are just names we assign to the components of $\hat{\Omega}$, therefore the above anisotropy term is absent.

The monopoles One can ask what do the Neel and VBS hedgehog correspond to in terms of the fields in Eq.(12) and Eq.(LABEL:Neelhedgehog). Consider a static Neel hedgehog at the spatial origin, namely,

where $r,\theta,\phi$ are the spatial spherical coordinates. Through $\hat{n}=z^{\dagger}\vec{\sigma}z$, the corresponding $z$ is²²2Here due to the singular configuration of $\hat{n}$ we need to take two patches to avoid the singularity around $\theta=0$ and $\theta=\pi$. Specifically the northern hemisphere patch is $$z^{N}(r,\theta,\phi)=\begin{pmatrix}\cos{\theta\over 2}\\ \sin{\theta\over 2}e^{i\phi}\\ \end{pmatrix},$$ and the southern hemisphere patch is $$z^{S}(r,\theta,\phi)=\begin{pmatrix}\cos{\theta\over 2}e^{-i\phi}\\ \sin{\theta\over 2}\\ \end{pmatrix}.$$ These two patches are related by a phase transformation $z^{N}=e^{i\phi}z^{S}$. In terms of $a_{j}={1\over i}z^{\dagger}\partial_{j}z$ the phase transformation is a gauge transformation. However in the following we shall compute the field strength $f_{ij}=\partial_{i}a_{j}-\partial_{j}a_{i}$ which is independent of the gauge. Therefore we can use either gauge to do the calculation.

Eq.(11) implies the corresponding gauge field (one-form) is $a={1\over i}z^{\dagger}dz$ whose field strength (two-form) is

If we integrate Eq.(21) over any closed surface enclosing the hedgehog the result is $2\pi$. Hence the Neel hedgehog is a monopole in $a_{\mu}$. The same calculation goes through if we change Neel to VBS and $a_{\mu}$ to $b_{\mu}$, i.e., a VBS hedgehog is a monopole in $b_{\mu}$.

The electromagnetic duality These discussions suggest the monopole in $b_{\mu}$ is the charge of $a_{\mu}$ and the monopole in $a_{\mu}$ is the charge of $b_{\mu}$. We summarize such relation in Table 1.

The Neel to VBS phase transition In the above duality, the condensation of, e.g., the $b_{\mu}$ monopole will confine the $b_{\mu}$ charge. According to Table 1 the $b_{\mu}$ charge is the $a_{\mu}$ monopole. Putting it differently, since the $b_{\mu}$ monopole is the $a_{\mu}$ charge, condensation of it will Higgs $a_{\mu}$ hence confine the $a_{\mu}$ monopole. Of course, the statement with $a_{\mu}$ and $b_{\mu}$ exchanged is true. If there is a direct continuous phase transition between the Neel (condensation of $b_{\mu}$ monopole or $a_{\mu}$ charge ) and VBS (condensation of $b_{\mu}$ charge or $a_{\mu}$ monopole) order, which is equivalent to a direct transition from the confinement to the Higgs phase in Eq.(12) and Eq.(LABEL:Neelhedgehog). At the critical point, neither the VBS nor the Neel hedgehog can condense but they can be both gapless. We conjecture that this happens at the self-dual point of the electromagnetic duality discussed in Table 1. The possibility of such a direct, continuous phase transition is supported by the Monte-Carlo simulation of the lattice version of Eq.(12) [12].

Spin liquids In Ref.[12], it is also shown that in the plane spanned by $g_{1}$-$g_{2}$ in Eq.(12), there exists a deconfined Coulomb phase where $a_{\mu}$ gauge bosons are gapless. In this phase the $a_{\mu}$ charge or $b_{\mu}$ monopole is massive, but finite energy, excitations. Since the $a_{\mu}$ charge is the spinon, this phase is a spin liquid. A similar argument applies to Eq.(LABEL:Neelhedgehog). In the deconfined phase, the $b_{\mu}$ gauge bosons are gapless, and the $b_{\mu}$ charge is a finite energy excitation. Since the $b_{\mu}$ charge fractionalizes the VBS order parameter, this phase is a “VBS” liquid. The spin liquid phase discussed above is analogous to that found in Ref.[13] for O(3) NL$\sigma$ model in (2+1)-D after the hedgehog suppression. In the phase diagram, these two fractionalized phases should sit symmetrically about the self-dual point.