Kramers Degeneracy

Statement

If an antiunitary time-reversal operator $T$ satisfies

$$T^2=-I$$

and commutes with the Hamiltonian,

$$THT^{-1}=H,$$

then every nondegenerate energy eigenstate is forbidden. Each eigenstate $\lvert\psi\rangle$ has a distinct orthogonal partner $T\lvert\psi\rangle$ with the same energy.

Assumptions

• $T$ is antiunitary.
• $T^2=-I$ on the Hilbert space sector considered.
• The Hamiltonian is time-reversal invariant.
• The eigenstate is a normalizable discrete eigenstate or lies in a setting where the eigenspace statement is meaningful.
• Perturbations that break the relevant time-reversal symmetry can lift the degeneracy.

Why the Partner Has the Same Energy

If

$$H\lvert\psi\rangle=E\lvert\psi\rangle,$$

then, using $THT^{-1}=H$,

$$H(T\lvert\psi\rangle) = E(T\lvert\psi\rangle).$$

The partner is orthogonal to $\lvert\psi\rangle$ because $T^2=-I$ is incompatible with a one-dimensional invariant subspace.

Quantum-Mechanical Meaning

Kramers degeneracy is central for systems with an odd number of spin-$1/2$ degrees of freedom and time-reversal symmetry. It underlies Kramers doublets in atomic, molecular, and condensed-matter settings.

The result is antiunitary. It is not the same as an ordinary unitary symmetry degeneracy, and its assumptions are sensitive to magnetic fields and other time-reversal-breaking terms.

Canonical Links

• Antiunitary Symmetries
• Time Reversal
• Kramers Degeneracy, Working Explanation
• Time-Reversal Operator
• Wigner Theorem
• Spin-Half Hilbert Space

Common Mistakes

• Applying Kramers degeneracy when $T^2=+I$.
• Treating antiunitary time reversal like an ordinary unitary operator.
• Forgetting that a magnetic field generally breaks time-reversal symmetry.
• Assuming every degeneracy in a time-reversal-invariant system is a Kramers degeneracy.
• Ignoring which Hilbert-space sector carries $T^2=-I$.

Quick Check

Why can a state not be proportional to its own Kramers partner when $T^2=-I$?

Solution

If $T\lvert\psi\rangle=c\lvert\psi\rangle$, antiunitarity gives $T^2\lvert\psi\rangle=c^*T\lvert\psi\rangle=\lvert c\rvert^2\lvert\psi\rangle$. This cannot equal $-\lvert\psi\rangle$. Thus $T\lvert\psi\rangle$ is a distinct state.

References

• H. A. Kramers, “Théorie générale de la rotation paramagnétique dans les cristaux,” Proceedings of the Royal Netherlands Academy of Arts and Sciences 33, 959-972, 1930.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Physics of Condensed Matter, Springer, 2008.