How to distinguish between a topological state from from a non-topological one?

How to distinguish between a topological state from from a non-topological one? Is there any standard procedure for identifying the topological features of a given hamiltonian? In general what are the types of interaction (like spin-orbit coupling, spin-spin interaction etc.) that can lead to a topologically non trivial state and what is the natural way of thinking about excitations in these states?

Concepts of Berry curvature and Chern number are key to make a difference between what is topological and what it is not.

Let's consider an hamiltonian $\mathcal{H}$ which is a function of N time-dependant parameters $(\Gamma_1(t)\,...\,\Gamma_j(t)...\,\Gamma_N(t))\equiv\mathbf{\Gamma}(t)$. The evolution of such system is given by the Schrödinger equation : $$\mathrm{i}\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=\mathcal{H}(\mathbf{\Gamma}(t))|\Psi(t)\rangle$$ At each instant $t$, one can expand the solution along the eigen basis of $\mathcal{H}$, which supposed to be known : $$|\Psi(t)\rangle=\sum_k\alpha_k(t)|\phi_k(\mathbf{\Gamma}(t))\rangle$$ Suppose that $\mathbf{\Gamma}(t)$ varies soomthly in time, and given the initial condition $|\Psi(t=0)\rangle=\alpha_i|\phi_i\rangle$, then the system follows adiabatically the $|\phi_i\rangle$ initial state, so that : $$|\Psi(t)\rangle\approx\alpha_i|\phi_i(\mathbf{\Gamma}(t))\rangle$$ Then the Schrödinger equation reads : $$\mathrm{i}\hbar\,\left[\dot{\alpha_i}+\dot{\mathbf{\Gamma}}\cdot\langle\phi_i(\mathbf{\Gamma})|\mathbf{\nabla}\phi_i(\mathbf{\Gamma})\rangle\right]=E_i(t)\,\alpha_i$$ This expression highlights a key notion, the Berry connexion : $$\mathbf{\mathcal{A}}_i(\mathbf{\Gamma})=\mathrm{i}\hbar\,\langle\phi_i(\mathbf{\Gamma})|\mathbf{\nabla}\phi_i(\mathbf{\Gamma})\rangle$$ The solution $\alpha_i(t)$ then reads as : $$\alpha_i(t)=e^{\mathrm{i}(\varphi_B+\varphi_d(t))}\alpha_i(0)$$ where $\varphi_d(t)=-\frac{1}{\hbar}\int^t_0\mathrm{d}s\,E_i(s)$ is the usual dynimical phase, and $\varphi_B$ is known to be the Berry phase of the system : $$\varphi_B=\frac{1}{\hbar}\int^t_0\mathrm{d}s\,\dot{\mathbf{\Gamma}}(s)\cdot\mathbf{\mathcal{A}}_i(\mathbf{\Gamma}(s))=\frac{1}{\hbar}\int^{\mathbf{\Gamma}(t)}_{\mathbf{\Gamma}(0)}\mathrm{d}\mathbf{\Gamma}\,\cdot\mathbf{\mathcal{A}}_i(\mathbf{\Gamma})$$ Now, suppose that the time evolution of $\mathbf{\Gamma}$ describes a cyclic process such that $\mathbf{\Gamma}(0)=\mathbf{\Gamma}(T)$ where $T$ is thus the period it takes to the system to perform one cycle on a given path $C$.

Then, $$\varphi_B=\frac{1}{\hbar}\oint_C\mathrm{d}\mathbf{\Gamma}\,\cdot\mathbf{\mathcal{A}}_i(\mathbf{\Gamma})=\frac{1}{\hbar}\oint_\Sigma\mathrm{d}\mathbf{\Sigma}\,\cdot\mathbf{\mathcal{B}}_i$$ using Stokes'theorem on a closed surface $\mathbf{\Sigma}$, where $\mathbf{\mathcal{B}}_i=\mathbf{\nabla}\cdot\mathbf{\mathcal{A}}_i$ is the Berry curvature.

Then, if you want to know if your system is topologically non trivial, then you just have to compute $\varphi_B$, or more precisely, the associated Chern number : $$\eta=\frac{\varphi_B}{2\pi}\in\mathbb{Z}$$ If $\eta=0$, then your system is topologically trivial.

Note the above approach is very general, especially when it comes to the $\mathbf{\Gamma}$ parameters, which can be anything (magnetic field, spin-orbit coupling, laser beam radiation, etc) as long as the adiabaticity is preserved.

• Would you please relate what you introduced (Berryology) to the topological features of the Hamiltonian ? – FraSchelle Feb 22 '16 at 19:29
• Said differently, why should a system with non-zero Berry phase be non-trivial topologically speaking ? – FraSchelle Mar 15 '17 at 21:27
• This only works for Chern insulators. A nontrivial TR invariant system will have a Chern number of $0$. – honey.mustard Apr 9 '17 at 7:04