Kramers' Degeneracy and Time-Reversal Symmetry

1. The Proof of Kramers' Theorem

The derivation of Kramers' theorem is an elegant proof by contradiction that flows directly from the properties of $\Theta$. The full argument is laid out concisely in the lecture notes from UC Berkeley.

Let's walk through the logic presented in the reading.

1. Setup: We consider a system with a time-reversal invariant Hamiltonian $H$, meaning $[\Theta ,H]=0$. The system contains an odd number of fermions (e.g., a single electron), so we know that ${\Theta }^2=-1$.

2. Energy Eigenstates: Let $|\psi ⟩$ be an eigenstate of $H$ with energy $E$:

   $$H|\psi ⟩=E|\psi ⟩$$

   Now, consider the time-reversed state, $|{\psi }_T⟩=\Theta |\psi ⟩$. Since $\Theta$ and $H$ commute, $|{\psi }_T⟩$ must also be an eigenstate with the same energy $E$:

   $$H|{\psi }_T⟩=H(\Theta |\psi ⟩)=\Theta H|\psi ⟩=\Theta (E|\psi ⟩)=E(\Theta |\psi ⟩)=E|{\psi }_T⟩$$

   Here we used the anti-linearity of $\Theta$ to pull out the real scalar $E$.

3. The Crucial Question: We have two states, $|\psi ⟩$ and $|{\psi }_T⟩$, with the same energy. The question is: are they the same state? That is, are they linearly dependent?

   • If they are linearly independent, then we have found two different states with the same energy. This means the energy level $E$ is degenerate, and the theorem is proven.
   • If they are linearly dependent, this would mean the level is non-degenerate. Let's explore this possibility.

4. The Contradiction: Assume the level is non-degenerate. Then $|\psi ⟩$ and $|{\psi }_T⟩$ must describe the same physical state, meaning they are linearly dependent:

   $$|{\psi }_T⟩=\Theta |\psi ⟩=c|\psi ⟩$$

   for some complex constant $c$. Now, let's apply $\Theta$ one more time:

   $${\Theta }^2|\psi ⟩=\Theta (\Theta |\psi ⟩)=\Theta (c|\psi ⟩)$$

   Because $\Theta$ is anti-linear, the scalar $c$ becomes its complex conjugate:

   $${\Theta }^2|\psi ⟩=c^{\ast }\Theta |\psi ⟩=c^{\ast }(c|\psi ⟩)=|c{|}^2|\psi ⟩$$

   Since $\Theta$ is also anti-unitary (${\Theta }^{†}\Theta =I$), it preserves the norm of states. This means $|c|$ must be 1, so $|c{|}^2=1$. The equation simplifies to:

   $${\Theta }^2|\psi ⟩=|\psi ⟩$$

   But this is a direct contradiction! Our initial premise was that we are dealing with a system of an odd number of fermions, for which we know ${\Theta }^2=-1$. The assumption of a non-degenerate state leads to the conclusion that ${\Theta }^2=+1$.

5. Conclusion: The assumption of non-degeneracy must be false. Therefore, $|\psi ⟩$ and $\Theta |\psi ⟩$ must be linearly independent, and every energy level in such a system must be at least doubly degenerate.

The pair of states $(|\psi ⟩,\Theta |\psi ⟩)$ is known as a Kramers pair. It's important to note that these two states are not just any two degenerate states; they are specifically related by the time-reversal operation.

2. Consequences in Condensed Matter Physics

This theorem is not just a mathematical curiosity. It has profound and directly observable consequences in solid-state systems, where electrons (fermions) move in the periodic potential of a crystal lattice.

The energy eigenvalues for an electron in a crystal are described by a band structure, $E(\mathbf{k})$, which gives the energy as a function of the crystal momentum (or wavevector) $\mathbf{k}$. Time-reversal symmetry imposes a fundamental constraint on these band structures. The time-reversal operator flips the sign of momentum, $\mathbf{p}\to -\mathbf{p}$, and since $\mathbf{p}=\hslash \mathbf{k}$, it also maps $\mathbf{k}\to -\mathbf{k}$. It also flips the electron's spin.

Therefore, Kramers' theorem implies a fundamental symmetry in the band structure:

Every state with momentum $\mathbf{k}$ and spin-up is degenerate with a state at momentum $-\mathbf{k}$ with spin-down.

Now consider special points in the crystal's momentum space (the Brillouin zone) known as Time-Reversal Invariant Momenta (TRIMs). These are points where $\mathbf{k}$ is equivalent to $-\mathbf{k}$ (e.g., $\mathbf{k}=0$, the $\Gamma$ point, or points on the zone boundary). At a TRIM, the state $|{\psi }_{\mathbf{k},\uparrow }⟩$ and its time-reversed partner $|{\psi }_{-\mathbf{k},\downarrow }⟩$ exist at the same momentum point. At these specific points, Kramers' theorem applies in its most direct form, guaranteeing that the spin-up and spin-down states are degenerate: $E({\mathbf{k}}_{\text{TRIM}},\uparrow )=E({\mathbf{k}}_{\text{TRIM}},\downarrow )$.

The image below illustrates the effect of this symmetry on a schematic band structure.

• Right Panel ("Single group"): This represents a hypothetical spinless particle. Without the ${\Theta }^2=-1$ property, there is no Kramers' theorem. The bands are non-degenerate.
• Middle Panel ("Vanishing spin-orbit"): This shows a system with spin, but with the spin-orbit coupling (SOC) interaction turned off. Without SOC, spin and momentum are decoupled, and the Hamiltonian has a higher symmetry that makes the spin-up and spin-down bands degenerate for all $\mathbf{k}$.
• Left Panel ("Finite spin-orbit"): This is the most realistic case for a crystal that lacks inversion symmetry. The spin-orbit interaction couples the electron's spin to its momentum. Away from the TRIM at Wavevector=0, this coupling lifts the degeneracy, splitting the bands. However, at the TRIM itself (Wavevector=0), the degeneracy is protected by time-reversal symmetry and must persist. This required degeneracy at high-symmetry points is a direct consequence of Kramers' theorem.

The lifting of degeneracy away from TRIMs is responsible for many important phenomena in condensed matter, such as the spin Hall effect and the physics of topological insulators. However, the fact that this lifting is constrained by Kramers' theorem at specific points is a crucial organizing principle.

3. A Deeper Look: The Frobenius-Schur Indicator

Your background in physics and mathematics, and your goal of understanding symmetry methods, makes it worthwhile to connect this physical argument to the more formal language of group theory. This provides a powerful, generalized framework for understanding when time-reversal forces degeneracies.

As we will explore more in later modules on representation theory, the degeneracies of energy levels correspond to the dimensions of the irreducible representations (irreps) of the system's symmetry group. When we add time reversal, we must consider how it acts on these irreps. The time-reversed basis states for an irrep $\Gamma$ will transform according to its complex conjugate irrep, ${\Gamma }^{\ast }$. This leads to three distinct scenarios, first classified by Wigner.

The following presentation slide introduces a powerful mathematical tool, the Frobenius-Schur indicator, which allows us to determine which of the three cases applies to any given irrep.

Let's summarize the key ideas. For a given irrep $\Gamma$ of the spatial symmetry group, we can calculate the Frobenius-Schur indicator:

where $|G|$ is the order of the group and $\chi (g^2)$ is the character of the square of each group element. The value of this indicator tells us about the nature of the irrep and the consequences of time-reversal symmetry. For a system with half-integer spin ($\eta =-1$):

• Case (b): FSI = 0. The irrep $\Gamma$ is complex and not equivalent to its conjugate ${\Gamma }^{\ast }$. Time reversal maps states in $\Gamma$ to states in a different space ${\Gamma }^{\ast }$. To describe the physics, we must consider both irreps together. This means the actual energy level will have a degeneracy equal to the sum of their dimensions, leading to a "time-reversal doubling" of the degeneracy.

• Case (a) & (c): FSI = $\pm 1$. The irrep is self-conjugate ($\Gamma$ is equivalent to ${\Gamma }^{\ast }$). Here, time reversal maps states within the same representation space. This is where ${\Theta }^2$ becomes the deciding factor.

  • Case (a) for fermions (FSI = -1): The irrep may be one-dimensional. Time reversal does not force an additional degeneracy.
  • Case (c) for fermions (FSI = +1): This is the situation that corresponds to Kramers' theorem. The Frobenius-Schur theory proves that any irrep falling into this class must itself have an even dimension. The degeneracy is already "baked into" the irrep. Time reversal pairs up the basis states within the representation, forming Kramers pairs.

This group-theoretic approach is more abstract, but it is also more powerful. It allows us to predict, just by analyzing the character table of a crystal's point group, which energy levels will exhibit Kramers degeneracy, which will be doubled by time reversal, and which will be unaffected.

Conclusion

In this lesson, we have derived Kramers' degeneracy theorem and explored its profound consequences. Starting from the simple-looking but deeply significant property ${\Theta }^2=-1$, we proved that degeneracy is not an accident but a necessity for fermions in time-reversal symmetric systems.

Key Takeaways:

• Kramers' Theorem: For a system with an odd number of fermions and a time-reversal invariant Hamiltonian, every energy eigenstate is at least doubly degenerate.
• The Kramers Pair: The two degenerate states are $|\psi ⟩$ and its time-reversed partner $\Theta |\psi ⟩$, which are linearly independent.
• Physical Manifestation: In solid-state physics, Kramers' theorem guarantees that electronic bands are degenerate at time-reversal invariant momenta (TRIMs) in the Brillouin zone, even in the presence of spin-orbit coupling.
• Lifting the Degeneracy: The degeneracy can be lifted by applying an external magnetic field, which breaks time-reversal symmetry.
• Group Theoretic View: The Frobenius-Schur indicator provides a formal method to classify irreducible representations and predict which ones will exhibit Kramers' degeneracy based on their character theory.

With this lesson, we conclude our exploration of discrete symmetries in quantum mechanics. We have seen how simple spatial and temporal transformations give rise to powerful conservation laws, selection rules, and fundamental constraints on the structure of energy spectra.

In our next module, we will shift our focus from discrete to continuous symmetries. We will begin by defining Lie groups, the mathematical framework for continuous transformations like rotations and translations, and see how their structure gives rise to the generators of motion you are familiar with from quantum mechanics, such as the angular momentum operators. This will open the door to classifying particles and interactions in modern physics.