Effects of Topological Terms: Hopf, $\Theta$, Chern-Simons, WZW, Berry phase term

Question

What are the effects and the differences of Topological Terms?

For example, I had known and heard several of them are called Topological,

(1) Hopf term,

(2) $\Theta$ term,

(3) Chern-Simons term,

(4) WZW term

Are they all (5) Berry phase terms? Or are they not?

It looks that WZW term may go one higher dimension than the space time dimension, does that distinguish WZW term from other terms? Are some of these terms giving same or different contrary effects?

We can focus on 2 dimensional space and 1 dimensional time (2+1 d). Or 3+1 d if it illuminates more physics.

Answer 1

All the topological terms in quantum field theory can be obtained as Berry phases.

When we have access only up to a certain energy scale, the fast degrees of freedom are effectively integrated out, leaving only interactions in the effective action of the slow degrees of freedom. These interactions are known to be expressed by means effective gauge fields depending nonlinearly on the slow coordinates. The topologically nontrivial parts of these interactions are given by the topological terms. In certain cases, the effective gauge fields may dynamically develop kinetic terms making the effective theory a full gauge theory.

A popular method to see this phenomenon is to start from a fermionic theory, where the fermions which consist of the fast degrees are minimally coupled to external Yang-Mills fields and by means of Yukawa coupling to external scalar fields: $$ \mathcal{L} = i \hbar \bar{\psi}( \gamma^{\mu}(\partial_{\mu} –ieA_{\mu}-ig \gamma_5 B_{\mu}) + \phi + \gamma_5 \phi_5 ) \psi$$ The principle upon which the construction of the topological terms is to assume that the fermions are confined to their vacuum state since we do not have enough energy to excite them, then by the adiabatic theorem: $$ e^{i \mathrm{Berry \: Phase}} = \mathrm{Vaccum \: amplitude} = \:_{ \mathrm {in}}\langle 0 | 0 \rangle_{ \mathrm {out}}$$ ($ | 0\rangle$ denotes the fermion Fock vacuum). The right hand side can also be computed by means of an Euclidean path integral.

Using this principle, with Yukawa coupled fermions Stone obtained the Wess-Zumino term in 0+1 dimensions.

Again with exclusively Yukawa coupled fermions Abanov and Wiegmann obtained the theta term in 1+1 dimensions, the Wess-Zumino term in 2+1 dimensions and a $\mathbb{Z}_2$ valued theta term in 3+1 dimensions.

The coefficient of the Wess-Zumino term is quantized. It counts the number of Fermion species.

It is worthwhile to mention that it is not compulsory to start with continuum fermions. One can also start from lattice spins as fast degrees of freedom or lattice fermions.

Integrating out massive lattice fermions in odd space times dimensions vectorially minimally coupled to gauge fields produces the corresponding Chern-Simons terms with coefficients equal to the Chern numbers of the associated Berry phases.

Doing the same thing in even space times dimensions, we obtain the gauge field theta term, please see for example this thesis by Pavan Ramakrishna Hosur for the derivation of the Abelian theta term.

More deeply, when canonical quantization is applied to the effective theories after the integration of the fast degrees of freedom, the canonical momenta have the forms: $$ \pi_i(x) =g_{ij} \partial_t \phi_j(x) + A_j(\phi(x))$$ Where $\phi_i$ are the slow coordinates, $ \pi_i$ are the corresponding canonical momenta, $ g_{ij}$ a metric in the slow coordinate manifold and $A_j$ is an effective gauge potential in the coordinate manifold originating from the topological term.

Wu and Zee call this effective field an effective gauge structure. It is actually a functional gauge field in the infinite dimensional slow coordinate manifold: $$\mathcal{A} = \int dx A_j(\phi(x)) \delta \phi_j (x)$$ It has a corresponding gauge field $$ \mathcal{F} = \delta \mathcal{A} = \int dx F_{ij}(\phi(x)) \delta \phi_i(x)\delta \phi_j (x)$$

(It is worthwhile to mention that when the effective theory is anomalous, the current algebra recieves an extention given by the functional gauge field:

$$ [J_a(x), J_b(y) ] = i f_{ab}^c J_c(x)\delta(x-y) + F_{ij} \delta_a \phi^i \delta_b \phi^j$$

Therefore, this description clarifies the connection between the Berry phase and the chiral anomaly)

These functional fields offer a deeper understanding of the connections between the Berry phase and the topological terms.

The Berry phase can be obtained as the holonomy of a Berry connection: $$\phi_B = \int_{\gamma} A_B$$ When the value of the Berry phase depends on the integration path $\gamma$, the Berry phase is called geometrical. In this case, the surface integral of the Berry curvature $F_B = dA_B$ over a closed surface containing the integration path must be quantized. $$ \frac{1}{2\pi i}\int F_B = n$$ This means that the Berry connection $A_B$ describes a monopole, since its flex is non-vanishing. When the Berry phase does not depend on the integration path the Berry phase is called topological (as in the case of the Aharonov-Bohm effect). In this case we must have: $$ F_B = d A_B = 0$$ and the Berry connection is flat. In this case no quantization condition exists (since the holonomy is the magnetic flux in the Aharonov Bohm solenoid)

The same happens with the functional Abelian gauge structure, when its functional gauge field is non-vanishing, it describes a functional magnetic monopole and leads to a quantization condition, such as in the case of the Wess-Zumino term coefficient.

When the functional Abelian gauge structure is flat, it does not lead to a quantization condition such as in the case of the theta term.

Answer 2

The Berry phase $\Phi$ is defined through the amplitude $$ \langle \psi (t)|\psi (t +T)\rangle = e^{i\Phi}, $$ where $T$ is the period of closed adiabatical evolution of the external parameters $\{b\}$ of the hamiltonian of the given canonical system. It's non-trivialness is caused by the topology. Namely, assuming the n-dimensional hamiltonian (for the fixed momentum $\mathbf p$), the non-zero Berry phase occurs if the mapping $$ M \to CP^{n-1}, $$ of the manifold $M$ of parameters $\{b\}$ on the space of $n$ complex vectors defined up to be unit up the phase can't be continuously deformed to the mapping $$ M \to S^{2n-1} $$
of complex unit vectors with unit norm. The physical interpretation of such obstruction is level crossing.

Contrary to the Berry phase, the other terms from your question (at least the WZW term, the $\theta$-term and Chern-Simons term) arise from very different reasons (although related to topology). Although in particular cases (assuming the gauge fields to be changed adiabatically) there are some close parallels between the Berry phase and, say, the chiral anomaly, in fact the mentioned above terms has topological properties independently on the adiabaticity assumption, while the Berry phase is defined only within the adiabaticity assumption and - moreover, - as far as I know, it acquires topological interpretation only for very special assumptions of adiabaticity and periodicity of adiabatic evolution.

As for the WZW term, in fact its value depends only on the boundary of the $2n+1$-dimensional space-time, which is just the $2n$-dimensional space-time.