Time-Reversal Symmetry and Kramers' Degeneracy

The Time-Reversal Operator in Quantum Mechanics

To understand the consequences of time-reversal symmetry, we must first establish how it is represented in quantum mechanics. A classical system is time-reversal invariant if, for any trajectory $x(t)$, the time-reversed trajectory $x(-t)$ is also a valid solution to the equations of motion. A simple example is a particle moving in a static potential, governed by Newton's second law, $m\ddot{x}=-\nabla V(x)$, which is second-order in time derivatives and thus insensitive to the sign of $t$.

In quantum mechanics, the dynamics are governed by the Schrödinger equation:

This equation is first-order in time. If we simply replace $t$ with $-t$, the left side changes sign while the right side does not. This means that $|\psi (-t)⟩$ is not, in general, a solution. However, if the Hamiltonian $H$ is a real operator (as it is for a spinless particle in a real potential), taking the complex conjugate of the entire equation gives:

Replacing $t$ with $-t$ in this new equation, we find:

This shows that the state $|{\psi }^{\ast }(-t)⟩$ evolves correctly according to the original Schrödinger equation. This suggests that the quantum operation for time reversal involves not only $t\to -t$ but also complex conjugation.

This leads us to define a time-reversal operator, denoted by $\Theta$, such that the time-reversed state $|{\psi }_r(t)⟩$ is given by $|{\psi }_r(t)⟩=\Theta |\psi (-t)⟩$. From our simple example, it's clear that $\Theta$ must involve complex conjugation. Such an operator cannot be linear; it belongs to a class of operators known as antilinear or antiunitary.

To delve deeper into the formal properties of $\Theta$, the following video provides an excellent introduction.

As the video explains, an antiunitary operator $\Theta$ has the following key properties:

1. Antilinearity: For any states $|{\psi }_1⟩,|{\psi }_2⟩$ and complex numbers $c_1,c_2$: $$\Theta (c_1|{\psi }_1⟩+c_2|{\psi }_2⟩)=c_1^{\ast }\Theta |{\psi }_1⟩+c_2^{\ast }\Theta |{\psi }_2⟩$$
2. Inner Product Transformation: It preserves the norm of states but conjugates inner products: $$⟨\Theta \psi |\Theta \varphi ⟩=⟨\varphi |\psi ⟩=⟨\psi |\varphi {⟩}^{\ast }$$
3. Action on Operators: Just as classical time reversal leaves position unchanged but flips momentum ($p\to -p$), the quantum operator $\Theta$ must transform the corresponding operators accordingly: $$\Theta \mathbf{x}{\Theta }^{-1}=\mathbf{x},\Theta \mathbf{p}{\Theta }^{-1}=-\mathbf{p}$$ This also implies that orbital angular momentum reverses sign: $\Theta \mathbf{L}{\Theta }^{-1}=-\mathbf{L}$. We postulate that spin angular momentum transforms in the same way: $\Theta \mathbf{S}{\Theta }^{-1}=-\mathbf{S}$.

A Hamiltonian is said to possess time-reversal symmetry if it is invariant under this transformation, meaning it commutes with $\Theta$:

This is the condition we will explore in the context of crystal Hamiltonians. Hamiltonians of the form $H={\mathbf{p}}^2/2m+V(\mathbf{x})$ are time-reversal invariant. The primary way to break this symmetry in a non-magnetic material is to apply an external magnetic field. Spin-orbit coupling, which we will see is crucial, is an internal effect and does not break time-reversal symmetry.

Consequences for Electronic Band Structures

Now, let's apply these ideas to the electronic states in a crystal. A Bloch state is an eigenstate of both the Hamiltonian $H$ and the translation operator, characterized by an energy $E_n(\vec{k})$ and a wavevector $\vec{k}$.

If the crystal's Hamiltonian is time-reversal invariant ($[H,\Theta ]=0$), and $|{\psi }_{n,\vec{k}}⟩$ is an eigenstate with energy $E_n(\vec{k})$, then $\Theta |{\psi }_{n,\vec{k}}⟩$ must also be an eigenstate with the same energy. What is the wavevector of this new state?

The action of the time-reversal operator on a Bloch state is to take its complex conjugate. For a plane wave $e^{i\vec{k}\cdot \vec{x}}$, complex conjugation flips the sign of the wavevector: $(e^{i\vec{k}\cdot \vec{x}}{)}^{\ast }=e^{-i\vec{k}\cdot \vec{x}}$. A similar argument holds for a general Bloch state. This means that the time-reversed state $\Theta |{\psi }_{n,\vec{k}}⟩$ is a Bloch state with wavevector $-\vec{k}$. Since both states must have the same energy, we arrive at our first key constraint:

This relationship holds for any band structure in a material with time-reversal symmetry. Note that if the crystal also has inversion symmetry (a point group operation that sends $\vec{x}\to -\vec{x}$), then $E_n(\vec{k})=E_n(-\vec{k})$ is already guaranteed. Time-reversal symmetry ensures this relation holds even in crystals without inversion symmetry.

The following resource provides a concise mathematical derivation of this property.

Kramers' Degeneracy

The relation $E_n(\vec{k})=E_n(-\vec{k})$ is interesting, but the most dramatic consequences of time-reversal symmetry emerge when we consider its interplay with electron spin. To do this, we need to examine the operator ${\Theta }^2$, which corresponds to applying the time-reversal operation twice.

• For a spinless particle, the time-reversal operator is effectively just complex conjugation, $\Theta =K$. Applying it twice returns the original state: ${\Theta }^2=K^2=1$.
• For a particle with spin, the situation is different. As $\Theta$ must reverse the spin direction ($\Theta S_y{\Theta }^{-1}=-S_y$) and is antilinear, the operator for a single spin-1/2 particle takes the form $\Theta =e^{-i\pi S_y/\hslash }K=-i{\sigma }_{y}K$. Let's see what happens when we apply this twice: $${\Theta }^2|\psi ⟩=\Theta (-i{\sigma }_y{\psi }^{\ast })=(-i{)}^{\ast }{\sigma }_y(\Theta {\psi }^{\ast })=i{\sigma }_y(-i{\sigma }_y\psi )={\sigma }_y^2\psi =|\psi ⟩$$ Wait, let's be more careful. The operator $K$ conjugates everything to its right. $${\Theta }^2|\psi ⟩=(-i{\sigma }_{y}K)(-i{\sigma }_{y}K)|\psi ⟩=(-i{\sigma }_y)K(-i{\sigma }_y{\psi }^{\ast })=(-i{\sigma }_y)(i)(K{\sigma }_y{K}^{-1})(K{\psi }^{\ast })={\sigma }_y{\sigma }_y^{\ast }\psi$$ Since ${\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$, its complex conjugate is ${\sigma }_y^{\ast }=\left(\begin{array}{cc}0& i\\ -i& 0\end{array}\right)=-{\sigma }_y$.
  Therefore, $${\Theta }^2|\psi ⟩={\sigma }_y(-{\sigma }_y)\psi =-{\sigma }_y^2\psi =-I|\psi ⟩$$ So for a single spin-1/2 particle, ${\Theta }^2=-1$.

This remarkable result generalizes: for any quantum system, ${\Theta }^2=+1$ if the system contains an even number of half-integer spin particles (fermions), and ${\Theta }^2=-1$ if it contains an odd number of fermions. Since an electron is a fermion, any system with a single electron (or any odd number of electrons) will have ${\Theta }^2=-1$.

This leads to Kramers' theorem:

In any time-reversal invariant system with an odd number of electrons (or other half-integer spin particles), every energy level must be at least doubly degenerate.

Proof of Kramers' Theorem:
Let $|\psi ⟩$ be an energy eigenstate of a time-reversal invariant Hamiltonian $H$.
Then $|\varphi ⟩=\Theta |\psi ⟩$ is also an eigenstate with the same energy.
Now, let's assume, for the sake of contradiction, that this energy level is non-degenerate. This would mean that $|\varphi ⟩$ and $|\psi ⟩$ must describe the same state, so they can only differ by a phase factor:

Let's apply $\Theta$ a second time:

But for a system with an odd number of electrons, we know ${\Theta }^2=-1$. This gives us:

This is a contradiction. Our initial assumption of non-degeneracy must be false. Therefore, $|\psi ⟩$ and $\Theta |\psi ⟩$ must be linearly independent, meaning the energy level is at least doubly degenerate. This guaranteed pair of degenerate states, $|\psi ⟩$ and $\Theta |\psi ⟩$, is called a Kramers pair.

The following reading provides a detailed derivation of both the ${\Theta }^2=-1$ property and Kramers' theorem itself.

Kramers' Degeneracy in Band Structures

How does this apply to band structures?
For an electron in a crystal, the states are labeled by both wavevector $\vec{k}$ and spin $s$ (e.g., $|{\psi }_{n,\vec{k},\uparrow }⟩$). The time-reversal operator flips both the wavevector and the spin:

Kramers' theorem tells us that the states $|{\psi }_{n,\vec{k},\uparrow }⟩$ and $\Theta |{\psi }_{n,\vec{k},\uparrow }⟩$ are distinct, orthogonal states with the same energy. This implies a degeneracy:

This holds at every point $\vec{k}$ in the Brillouin zone.

A special case occurs at time-reversal invariant momenta (TRIMs), which are points in the Brillouin zone where $\vec{k}$ is equivalent to $-\vec{k}$. This means $\vec{k}=-\vec{k}+\vec{K}$ for some reciprocal lattice vector $\vec{K}$. The most obvious TRIM is the $\Gamma$ point ($\vec{k}=0$). Other TRIMs exist at the center and corners of the Brillouin zone boundary.

At a TRIM, $|{\psi }_{n,\vec{k},\uparrow }⟩$ and its time-reversed partner $\Theta |{\psi }_{n,\vec{k},\uparrow }⟩=|{\psi }_{n^{\prime },\vec{k},\downarrow }⟩$ are both at the same $\vec{k}$-point. Kramers' theorem thus guarantees that at these specific points, every energy level must be at least doubly degenerate. Each band must "stick together" with another band at these points.

This degeneracy is a direct consequence of time-reversal symmetry for spin-1/2 particles and cannot be broken unless time-reversal symmetry itself is broken (e.g., by applying a magnetic field). Spin-orbit coupling, an interaction that links an electron's spin to its motion, can split spin-up and spin-down bands, but it respects time-reversal symmetry. Therefore, even in materials with strong spin-orbit coupling, Kramers' degeneracy remains at the TRIMs.

The following video discusses spin-orbit interaction and explicitly mentions the resulting Kramers doublets.

Conclusion

In this lesson, we have moved beyond spatial symmetries to explore the profound impact of time-reversal symmetry on the electronic states in solids.

Key Takeaways:

• Time reversal in quantum mechanics is represented by an antiunitary operator $\Theta$ that involves complex conjugation.
• For a system with spin-1/2 particles like electrons, applying the time-reversal operator twice gives ${\Theta }^2=-1$.
• Kramers' theorem states that for any time-reversal invariant system with an odd number of electrons, every energy level must be at least doubly degenerate.
• In the context of band structures, this theorem guarantees that $E_n(\vec{k})=E_n(-\vec{k})$ throughout the Brillouin zone.
• At special time-reversal invariant momenta (TRIMs) like the $\Gamma$ point, Kramers' theorem requires that every energy band be degenerate with another, forming what are known as Kramers pairs. This degeneracy is robust and persists even in the presence of strong spin-orbit coupling.

These constraints are fundamental to our modern understanding of electronic materials. The protection of degeneracies at specific points in the Brillouin zone is a cornerstone of the theory of topological insulators, where these principles guarantee the existence of exotic electronic states.

In our next and final lesson of this module, we will combine our knowledge of both spatial and time-reversal symmetries to understand how they dictate the selection rules for optical transitions between electronic bands.