nLab Floreanini-Jackiw theory

Idea

The Floreanini-Jackiw Lagrangian (FL, due to Floreanini & Jackiw 1987) is a Lagrangian density for a scalar field in 2-dimensions whose equations of motion single out (essentially) just the “right moving” (“chiral”) half of solutions of the usual relativistic wave equation: the chiral boson. As such, the FJ-Lagrangian density is an approach, in low dimension, to the notoriously subtle problem of finding Lagrangian densities for self-dual higher gauge fields.

The FL-Lagrangian arises also as the effective boundary field theory of abelian Chern-Simons theory (Wen 1992 §2.5, 1995 §3.3, cf. Tong 2016 §6.1.2). This realizes the FL theory as the abelian case of Wess-Zumino-Witten theory (cf. CS/WZW correspondence) and witnesses it as the effective field theory of edge modes in fractional quantum Hall systems.

From abelian Chern-Simons theory

The Lagrangian density of abelian Chern-Simons theory with gauge field 1-form $a$ is

(1)$$L(a) = \tfrac{k}{4\pi} a \wedge \mathrm{d}a \mathrlap{\,.}$$

Consider this now on a globally hyperbolic 3D spacetime $X^{1,0} \times X^2$:

1. with vanishing first de Rham cohomology

   (2)$$H^1_{dR}\big(X^2\big) = 0 \mathrlap{\,,}$$

2. with spatial boundary

   $$X^{1,0} \times \partial X^2 \;\subset\; X^{1,0} \times X^2 \mathrlap{\,.}$$

Now (Wen 1992 above (2.62), 1995 above (3.44), Tong 2016 below (6.9), Lopez & Fradkin 2001 (3.4), Wolf, Read & Teh 2023 above (2.15))

1. choose temporal gauge

   (3)$$a_{\partial_{t'}} = 0 \mathrlap{\,,}$$

2. a potential 0-form $\phi$ solving the induced Gauss law constraint $\partial_i a_j - \partial_j a_i = 0$, by (2):

   (4)$$a_i = \mathrm{d}_i\phi \,,$$

   where $(x^i)_{i = 1}^{2}$ denote local coordinate functions on $X^2$.

Then, by Stokes' theorem, the CS Lagrangian density (1) becomes a boundary Lagrangian density for $\phi$ on $X^{1,0} \times \partial X^2$ as follows (indicated in Wen 1992 (2.62)):

$$\begin{array}{rll} \tfrac{k}{4\pi} \int_{X^{1,0} \times X^2} a \wedge \mathrm{d} a & = - \tfrac{k}{4\pi} \int_{X^{1,0} \times X^2} \big( a \wedge \partial_{t'} a \big) \mathrm{d}t' & \text{by}\;\text{(3)} \\ & = - \tfrac{k}{4\pi} \int_{X^{1,0} \times X^2} \epsilon^{i j} (\partial_i \phi) (\partial_{t'} \partial_j \phi) \mathrm{d}t' \wedge \mathrm{d} x^1 \wedge \mathrm{d}x^2 & \text{by}\;\text{(4)} \\ & = - \tfrac{k}{4\pi} \int_{X^{1,0} \times X^2} \epsilon^{i j} \partial_j \big( (\partial_i \phi) (\partial_{t'} \phi) \big) \mathrm{d}t' \wedge \mathrm{d}x^1 \wedge \mathrm{d}x^2 & \text{by Leibniz' rule} \\ & = \tfrac{k}{4\pi} \int_{X^{1,0} \times \partial X^2} \big( (\partial_i \phi) (\partial_{t'} \phi) \big) \mathrm{d}t' \wedge \mathrm{d}^i x & \text{by Stokes' theorem.} \end{array}$$

Often this is considered for $X^2$ the upper half plane with coordinate $x$ along the edge, in which case the boundary Lagrangian density is hence

$$L(\phi) = \tfrac{k}{4\pi} (\partial_x \phi) (\partial_{t'}\phi) \mathrlap{\,.}$$

Now, understanding (Wen ‘92 (2.64))

$$t' \coloneqq t - c x$$

as a lightcone gauge coordinate, this reads:

$$L(\phi) = \tfrac{k}{4\pi} (\partial_x \phi) \big( \partial_{t} - c \partial_x\phi \big) \mathrlap{\,.}$$

This is, up to normalization, the Floreanini-Jackiw Lagrangian density.

In applications to solid state physics, the “speed of light”, $c$, here is taken to be the propagation speed of edge mode currents and typically denoted “$v$”, understood to be a free parameter.

Lagrangian and Equations of motion

For $\phi$ a scalar field on $\mathbb{R}^2$ (the latter equipped with its canonical coordinate functions, here denoted $t$ for time and $x$ for space), the FL Lagrangian density is (FL ‘87 (20)):

$$L(\phi) \;\coloneqq\; \tfrac{1}{2} (\partial_x\phi) \big( \partial_t \phi - c \partial_x \phi \big) \mathrlap{\,,}$$

where $c \in \mathbb{R} \setminus \{0\}$ is a constant which sets the velocity (the speed of light, often set to $c = 1$).

The corresponding Euler-Lagrange equation of motion is

$$\big( \partial_t - c \partial_x \big) \partial_x \phi = 0 \mathrlap{\,.}$$

The general solutions to this equation are of the form

(5)$$\phi(t,x) = \phi_c(x + c t) + g(t)$$

where $g$ is an arbitrary (smooth) function of just the temporal coordinate $t$ (hence is a “spatial zero mode”), while $\phi_c$ is a solution of the chiral/right-moving wave equation

$$\big( \partial_t - c \partial_x \big) \phi_c = 0 \mathrlap{\,.}$$

As effective theory of FQH edge modes

When the the FL theory is understood as an effective field theory for edge modes of fractional quantum Hall liquids (Wen 1992 §2.5, review in Tong 2016 p. 207), it is the spatial derivative

$$\rho \coloneqq \partial_x \phi$$

which represents the current density on the edge, and by (5) this again satisfies the chiral wave equation

$$\big( \partial_t - c \partial x \big) \rho = 0 \mathrlap{\,.}$$

References

General

The original article:

• Roberto Floreanini, Roman Jackiw: Self-dual fields as charge-density solitons, Phys. Rev. Lett. 59 (1987) 1873 [doi:10.1103/PhysRevLett.59.1873]

  (the FR-Lagrangian appears in equation (20))

A manifestly Lorentz group-invariant re-formulation:

• Paul K. Townsend: Manifestly Lorentz invariant chiral boson action, Phys. Rev. Lett. 124 (2020) 101604 [doi:10.1103/PhysRevLett.124.101604, arXiv:1912.04773]

As EFT for FQH edge modes

Interpretation as the boundary field theory of abelian Chern-Simons theory, in the context of effective field theory for edge modes of fractional quantum Hall systems:

• Jürg Fröhlich, Anthony Zee; §5 of: Large scale physics of the quantum hall fluid, Nuclear Physics B 364 3 (1991) 517-540 [doi:10.1016/0550-3213(91)90275-3]

• Michael Stone; pp. 45 in: Edge waves in the quantum Hall effect, Annals of Physics 207 1 (1991) 38-52 [doi:10.1016/0003-4916(91)90177-A]

• Xiao-Gang Wen; §2.5 in: Theory of Edge States in Fractional Quantum Hall Effects, International Journal of Modern Physics B 06 10 (1992) 1711-1762 [doi:10.1142/S0217979292000840, pdf]

• Andrea Cappelli, Gerald V. Dunne, Carlo A. Trugenberger, G. R. Zemba; §5 of: Conformal Symmetry and Universal Properties of Quantum Hall States, Nucl. Phys. B 398 (1993) 531-567 [doi:10.1016/0550-3213(93)90603-M, arXiv:hep-th/9211071]

• Xiao-Gang Wen; §3.3 in: Topological order in rigid states and edge excitations in fractional quantum Hall states, Advances in Physics 44 5 (1995) 405-437 [arXiv:cond-mat/9506066, doi;10.1080/00018739500101566]

reviewed in:

• David Tong; §6.1.2 in: The Quantum Hall Effect, lecture notes (2016) [arXiv:1606.06687, course webpage, pdf, pdf]

In the generality of multi-component (K-matrix) FQH systems:

• Ana Lopez, Eduardo Fradkin; equation (3.10) in: Effective field theory for the bulk and edge states of quantum Hall states in unpolarized single layer and bilayer systems, Phys. Rev. B 63 (2001) 085306 [doi:10.1103/PhysRevB.63.085306, arXiv:cond-mat/0008219]

• Michael Levin; (22) in: Protected edge modes without symmetry, Phys. Rev. X 3 021009 (2013) [doi:10.1103/PhysRevX.3.021009, arXiv:1301.7355]

Including non-relativistic corrections (Newton-Cartan geometry):

• William J. Wolf, James Read, Nicholas Teh; (2.15) of: Edge modes and dressing fields for the Newton-Cartan quantum Hall effect, Found Phys 53, 3 (2023) [arXiv:2111.08052, doi:10.1007/s10701-022-00615-4]