nLab abelian Chern-Simons theory

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Context

∞\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

Quantum field theory

Topological physics

Contents

Idea

By abelian Chern-Simons theory one means Chern-Simons theory with abelian gauge group (typically the circle group or a torus-product of copies of these).

One major application of abelian Chern-Simons theory is as an effective field theory of the fractional quantum Hall effect.

Properties

Space of quantum states

For abelian Chern-Simons theory with NN gauge fields (A i) i=1 N(A_i)_{i = 1}^N and Lagrangian density of the form K ijA i∧dA jK^{i j} A_i \wedge \mathrm{d} A_j (for KK an N×NN \times N symmetric matrix and using Einstein summation convention), the dimension of the Hilbert space of quantum states ℋ g\mathscr{H}_g (obtained by geometric quantization, cf. quantization of D=3 Chern-Simons theory) over an orientable surface of genus gg is the absolute value of the determinant of KK raised to the ggth power:

dim(ℋ g)=|det(K)| g. dim(\mathscr{H}_g) \;=\; \left\vert det(K)\right\vert^g \,.

(for N=1N=1 see Manoliu 1998a p 40, for general NN cf. Belov & Moore 2005 p 26)

For the modular group-action on these state spaces see at integer Heisenberg group the section Modular automorphisms (there for g=1g=1).

For a non-orientable surface with kk crosscaps, it is

dim(ℋ k)=|det(K)| k−1. dim(\mathscr{H}_k) \;=\; \left\vert det(K)\right\vert^{k-1} \,.

(e.g. arXiv:1509.03920 (73))


Abelian Chern-Simons as effective QFT for FQH systems

The following is a streamlined digest of the traditional argument and Ansatz [originally culminating in Zee 1995, Wen 1995, more recently highlighted by Witten 2016 pp 30, Tong 2016 §5] for abelian Chern-Simons theory at level k∈ℕk \in \mathbb{N} as an effective field theory for fractional quantum Hall systems at filling fraction ν=1/k\nu = 1/k.


Preliminaries

Consider a spacetime Σ\Sigma of dimension 1+21 + 2, to be thought of as the worldvolume of the conducting sheet that hosts the fractional quantum Hall system.

On this spacetime, the electric current density is a differential 2-form JJ.

Note that the corresponding electric current vector field J→\vec J is characterized by its contraction into the given volume form dvoldvol being equal to the density 2-form

J=dvol(J→,⋯), J \;=\; dvol\big(\, \vec J, \cdots \big) \,,

and that the conservation law for J→\vec J, hence the vanishing of its divergence is equivalent to the density 2-form being closed differential form:

(1)divJ→=0⇔dJ=0. div \vec J \;=\; 0 \;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\; \mathrm{d} J \;=\; 0 \,.

We write AA for the vector potential 1-form on Σ\Sigma which encodes the background electromagnetic field, and F=dAF \,=\, \mathrm{d}A for its field strength, the Faraday tensor. Since we regard this as a fixed background field, we do not (need to) consider the Maxwell Lagrangian density L YM≔F∧⋆FL_{YM} \,\coloneqq\, F \wedge \star F.

But we do (need to) consider the interaction term between the electromagnetic field and the electric current density, which is

(2)L int≔A∧J. L_{int} \,\coloneqq\, A \wedge J \,.

Here we assume that AA is globally defined, in fact we shall assume that Σ\Sigma has vanishing de Rham cohomology in degree=2,

(3)H dR 2(Σ)=0. H^2_{dR}(\Sigma) \;=\; 0 \,.

This means we are focused on the local nature of fields, not on their global properties, as (for better or worse) usual for Lagrangian field theory.


The Ansatz

The auxiliary gauge potential. With the assumption (3) also the conserved current density (1) admits a coboundary, a differential 1-form to be denoted aa:

J=da. J \,=\, \mathrm{d}a \,.

The central Ansatz of the approach is to think of this 1-form as an effective dynamical gauge field for the FQH dynamics.


The Lagrangian density. This in turn suggests that the effective Lagrangian density should be the sum of

  1. the interaction term (2) already mentioned, which thus specializes to

    L int=A∧J=A∧da, L_{int} \;=\; A \wedge J \;=\; A \wedge \mathrm{d}a \,,

  2. a further interaction term for aa, now regarded as a gauge potential, to its own current density jj, to be thought of as the current of FQH quasi-particles (or quasi-holes)

    L quas=a∧j L_{quas} \;=\; a \wedge j

  3. a dynamical term for aa itself, where the only sensible choice is the Chern-Simons form at level kk

    L CS=k12a∧da. L_{CS} \;=\; k \tfrac{1}{2} a \wedge \mathrm{d}a \,.

In total, the traditional Ansatz for the effective Lagrangian density for “single layer” FQH systems at filling fraction 1/k1/k is hence:

(4)L≔k12a∧da−A∧da⏟J+a∧j. L \,\coloneqq\, k \tfrac{1}{2}\, a\wedge \mathrm{d}a \,-\, A \wedge \underset{J}{ \underbrace{ \mathrm{d}a } } \,+\, a \wedge j \,.

[Wen 1995 (2.11)]


The equation of motion. The Euler-Lagrange equation of motion for the effective gauge field aa induced by (4) is simply

(5)δLδA=0⇔J=1k(F−j). \frac{\delta L}{\delta A} \;=\; 0 \;\;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\;\; J \,=\, \tfrac{1}{k}\Big( F - j \Big) \,.

(This means in particular that if, on topologically non-trivial Σ\Sigma, one were to insist (which is not a logical necessity) to subject the effective gauge field aa to Dirac charge quantization (as AA is), then the ordinary electromagnetic field would have to be assumed to carry magnetic charge being a multiple of kk.)


Reproducing FQH phenomena

Recovering the Hall conductivity. The first plausibility check on the effective model is to observe that its equation of motion (5) implies the correct Hall conductivity relation for a fractional quantum Hall system at filling fraction 1/k1/k:

Namely assuming that the electric field is oriented in the yy-direction, so that the Faraday tensor is of the form (cf there):

F=E ydt∧dy+Bdx∧dy F \;=\; E_y \mathrm{d}t \wedge \mathrm{d}y + B \mathrm{d}x\wedge \mathrm{d}y

and assuming that the quasi-particles are stationary, so that

j∝dvol(∂ 0)∝dx∧dy, j \;\propto\; dvol(\partial_0) \;\propto\; \mathrm{d}x \wedge \mathrm{d}y \mathrlap{\,,}

the temporal component of the equation of motion (5) becomes

J x=1kE y J_x \;=\; \tfrac{1}{k} E_y

which is the correct form of the Hall field E yE_y for given longitudinal current J XJ_X at filling fraction ν=1/k\nu = 1/k – see there.

(Following Zee 1995 (4.5), later authors [Witten 2016 §2.3, Tong 2016 p 161] deduce this in a more roundabout way, by first inserting the equation of motion back into the Lagrangian density and then varying that with respect to AA. It seems that this unnecessary step is brought about by first tacitly renaming “JJ” to “ff” and then forgetting that this was just a renaming.)


Recovering the fractional quasi-particles. Second, the spatial component of the equation of motion (5) says that (1.) the magnetic flux quanta and (2.) the quasi-particles contribute their kkth fraction to the total charge density:

J 0=1k(B−j 0). J_0 \;=\; \tfrac{1}{k}(B - j_0) \,.

Here

  • statement (1) reflects exactly the composite fermion model, where at kkth filling fraction each electron is bound to kk quanta of magnetic flux

  • statement (2) is the desired property of quasi-holes to have kkth fractional charge.


This largely concludes the traditional justification for the effective abelian Chern-Simons theory (4).


References

General

Many general reviews of Chern-Simons theory have a section focused on the abelian case, for instance:

  • Gregory Moore, §2 in: Introduction to Chern-Simons Theories, TASI lecture notes (2019) [pdf, pdf]

  • David Grabovsky, §1 in: Chern–Simons Theory in a Knotshell (2022) [pdf, pdf]

On the light-cone quantization of abelian Chern-Simons theory:

On boundary conditions and line-defects in abelian Chern-Simons theory:

Discussion via locally covariant algebraic quantum field theory:

On lattice formulation of abelian Chern-Simons theory:

and its canonical quantization:

In relation to the dilogarithm:

Abelian Chern-Simons for frational quantum Hall effect

The idea of abelian Chern-Simons theory as an effective field theory exhibiting the fractional quantum Hall effect (abelian topological order) goes back to

and was made more explicit in:

Early review:

Further review and exposition:

For discussion of the fractional quantum Hall effect via abelian but noncommutative (matrix model-)Chern-Simons theory

On edge modes:

Further developments:

Last revised on January 22, 2025 at 09:30:33. See the history of this page for a list of all contributions to it.

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