Symmetry and Duality in Bosonization of Two-Dimensional Dirac Fermions

Recent work on a family of boson-fermion mappings has emphasized the interplay of symmetry and duality: Phases related by a particle-vortex duality of bosons (fermions) are related by time-reversal symmetry in their fermionic (bosonic) formulation. We present exact mappings for a number of concrete models that make this property explicit on the operator level. We illustrate the approach with one- and two-dimensional quantum Ising models and then similarly explore the duality web of complex bosons and Dirac fermions in ($2+1$) dimensions. We generalize the latter to systems with long-range interactions and discover a continuous family of dualities embedding the previously studied cases.

Popular Summary

Duality between two (or more) theories means that, despite appearances, they are in fact one and the same. In many famous cases, dualities relate systems that are hard to study to those that are much simpler, thus providing a window for understanding the former via the latter. Knowledge of dualities has proven to be extremely powerful, for example, for understanding fascinating states of particles that strongly interact with one another. A synergistic effort by the condensed-matter and high-energy-theory communities has recently culminated in the discovery of several new dualities for understanding how Dirac fermions—particles described by the celebrated Dirac equation—behave when confined to two dimensions (e.g., electrons in graphene). In field-theoretic approaches to dualities, constructing concrete models and keeping track of their symmetries is often difficult. We introduce an alternate approach that overcomes both challenges, leading to explicit models for which duality is exact at all scales.

Dualities for Dirac fermions relate, for example, topological electronic states to both quantum electrodynamics and quantum phase transitions of bosons. We generalize these dualities to include systems with long-range interactions and discover a continuous family of dualities that includes the previously known examples as special cases. In particular, our extended family includes members that, unlike the previous cases, are amenable to numerical simulations and can thus be used to test these dualities.

Our results suggest a strategy for discovering new dualities and may help elucidate symmetries in several other dualities that have recently attracted considerable interest both in condensed-matter physics and in string theory.

Figure 1

In our formulation, the fermion spin is interpreted as the relative orientation of the boson and vortex. Under fermionic time reversal, the spin is flipped, corresponding to replacing bosons $\leftrightarrow$ vortices, i.e., duality. This schematic picture will be made precise in the course of this paper.

Figure 2

Equation (4) describes a conformal map from the complex half-plane onto itself. The upper figure depicts (orthogonal) lines of fixed radius or angle. The lower figure shows the same lines in the transformed coordinate system. A number of important special cases are as follows: (i) Purely imaginary $z_{\text{boson}}$, corresponding to time-reversal-invariant boson systems, map onto the unit circle $|z_{\text{Dirac}}|=1$ for fermions, i.e., self-dual models. The limits $z_{\text{boson}}\to i\infty$ and $z_{\text{boson}}=i{0}^{+}$ correspond to $z_{\text{Dirac}}=+1$ and $z_{\text{Dirac}}=-1$, i.e., the fermion models with half-integer Chern-Simons described in Eq. (1). (ii) Self-dual boson models, $|z_{\text{boson}}|=1$, map onto purely imaginary $z_{\text{Dirac}}=i\tan [\arg (z_{\text{boson}})/2]$, i.e., time-reversal invariant fermions. (iii) The special point $z_{\text{boson}}=z_{\text{Dirac}}=i$ is invariant under the conformal map and corresponds to a model that is simultaneously self-dual and time-reversal invariant for both bosons and fermions.

Figure 3

Left panel: Phase diagram of bosons with Chern-Simons term (${\gamma }_{\text{boson}}$), marginally long-range interactions (${\lambda }_{\text{boson}}$), and additional nonuniversal short-range interactions ($u_{\mathrm{short}\text{−}\mathrm{range}}$). Time-reversal symmetry is present at ${\gamma }_{\text{boson}}=0$. When additionally ${\lambda }_{\text{boson}}=0$, the bosons interact via a gauge field that obeys Maxwell dynamics. This corresponds to vortices with purely short-range interactions, which may either condense or become gapped. The transition between the two phases is governed by the Wilson-Fisher fixed point. In contrast, ${\gamma }_{\text{boson}}=0$ and ${\lambda }_{\text{boson}}=\infty$ formally corresponds to bosons with purely short-range interactions. These two limits are related by the standard boson-vortex duality of Eq. (1). Finite, nonzero ${\lambda }_{\text{boson}}$ interpolates between the two with self-duality realized at ${\lambda }_{\text{boson}}=1$, corresponding to Eq. (3). This point describes a quantum phase transition between the same phases as the Wilson-Fisher fixed point (either bosons or vortices condense) but is of a different universality class because of the marginally long-range interactions (strictly speaking, long-distance properties of the two phases also change qualitatively for finite nonzero ${\lambda }_{\mathrm{bos}}$). For nonzero ${\gamma }_{\text{bosons}}$, this point extends into a self-dual line of phase transitions at ${\lambda }_{\text{bosons}}^2+{\gamma }_{\text{bosons}}^2=1$; this includes the case of bosons with purely statistical interactions described by Eq. (2). In the figure, we assumed that this line of self-dual phase transitions lies in the $u_{\mathrm{short}\text{−}\mathrm{range}}=0$ plane, which is something we can realize in explicit wire models. Right panel: The same models can be transcribed into Dirac fermions with statistical (${\gamma }_{\text{Dirac}}$), marginally long-range (${\lambda }_{\text{Dirac}}$), and additional short-range interactions ($v_{\mathrm{short}\text{−}\mathrm{range}}$). The self-dual line of bosons maps onto time-reversal-invariant fermions ${\gamma }_{\text{Dirac}}=0$, which are critical. It separates gapped phases with opposite sign of the Dirac mass $m_{\text{Dirac}}$, which corresponds to either bosons or vortices condensing. The ${\gamma }_{\text{Dirac}}=0$ line includes the special cases of $N=1$ ${\mathrm{QED}}_3$ (${\lambda }_{\text{Dirac}}=0$), Dirac fermions with short-range interactions (${\lambda }_{\text{Dirac}}=\infty$), and self-dual Dirac fermions (${\lambda }_{\text{Dirac}}=1$). As for the case of bosons, the self-dual point extends to a self-dual line for ${\gamma }_{\text{Dirac}}\ne 0$. It includes the case of Dirac fermions with purely statistical interactions, ${\lambda }_{\text{Dirac}}=0$ and ${\gamma }_{\text{Dirac}}=1$, which is self-dual and for which short-range interactions $v_{\mathrm{short}\text{−}\mathrm{range}}$ can drive a phase transition in the Wilson-Fisher universality class.

Figure 4

Different sets of variables for representing the transverse-field Ising model. Microscopic Ising spins reside on sites labeled by integers. Dual Ising spins reside on the dual lattice sites denoted by half-integers. Majorana fermions are obtained by combining direct Ising spins with dual Ising spins and are thus most naturally associated with sites labeled by quarter-integers.

Figure 5

Schematic RG flow of the bosonic wire model defined in Eq. (21).

Figure 6

Bosons $e^{i\phi }$ reside on the direct lattice labeled by integers. Vortices $e^{i\tilde{\phi }}$ live on the dual lattice labeled by half-integers. Vortex hopping across wire $y$ is implemented by a phase slip $e^{-2i{\theta }_y}$ on that wire. Creating an isolated vortex requires a string of such operators emanating from infinity. Here, we define the vortex creation operator by running “half” of the string from $-\infty$ and the other half from $+\infty$.

Figure 7

Schematic RG flow for the array of counterpropagating chiral wires, viewed as arising on the surface of a 3D topological insulator. Systems that preserve the time-reversal symmetry defined in Eq. (50) are confined to the separatrix and flow to the free-Dirac-fermion fixed point.

Figure 8

Composite fermions are constructed by combing a boson with a vortex. With bosons residing on the direct lattice and vortices on the dual lattice, composite fermions are most naturally associated with new quarter-integer lattice. The fermion chirality depends on the relative position of the boson and vortex.

Figure 9

Energy spectra of the perturbed Ising models $H_0+\delta H$ and $H_0+\delta H^{\prime }$. The former preserves $T$ and is gapless along the self-dual line $J=h$. For small $u$, this line separates gapped paramagnetic and ferromagnetic phases. In contrast, for large $u$, the system enters an extended gapless phase with central charge 1, and self-duality is not required for the gaplessness. The model $H_0+\delta H^{\prime }$ preserves $T^{\prime }$ and is gapless only at self-duality.