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Register a new userAmplitude / Higgs Modes in Condensed Matter Physics
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The order parameter and its variations in space and time in many different states in condensed matter physics at low temperatures are described by the complex function $Psi({bf r}, t)$. These states include superfluids, superconductors, and a subclass of antiferromagnets and charge-density waves. The collective fluctuations in the ordered state may then be categorized as oscillations of phase and amplitude of $Psi({bf r}, t)$. The phase oscillations are the {it Goldstone} modes of the broken continuous symmetry. The amplitude modes, even at long wavelengths, are well defined and decoupled from the phase oscillations only near particle-hole symmetry, where the equations of motion have an effective Lorentz symmetry as in particle physics, and if there are no significant avenues for decay into other excitations. They bear close correspondence with the so-called {it Higgs} modes in particle physics, whose prediction and discovery is very important for the standard model of particle physics. In this review, we discuss the theory and the possible observation of the amplitude or Higgs modes in condensed matter physics -- in superconductors, cold-atoms in periodic lattices, and in uniaxial antiferromagnets. We discuss the necessity for at least approximate particle-hole symmetry as well as the special conditions required to couple to such modes because, being scalars, they do not couple linearly to the usual condensed matter probes.
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A massive Goldstone (MG) mode, often referred to as a Higgs amplitude mode, is a collective excitation that arises in a system involving spontaneous breaking of a continuous symmetry, along with a gapless Nambu-Goldstone mode. It has been known in the previous studies that a pure amplitude MG mode emerges in superconductors if the dispersion of fermions exhibits the particle-hole (p-h) symmetry. However, clear understanding of the relation between the symmetry of the Hamiltonian and the MG modes has not been reached. Here we reveal the fundamental connection between the discrete symmetry of the Hamiltonian and the emergence of pure amplitude MG modes. To this end, we introduce nontrivial charge-conjugation ($mathcal C$), parity ($mathcal P$), and time-reversal ($mathcal T$) operations that involve the swapping of pairs of wave vectors symmetrical with respect to the Fermi surface. The product of $mathcal{CPT}$ (or its permutations) represents an exact symmetry analogous to the CPT theorem in the relativistic field theory. It is shown that a fermionic Hamiltonian with a p-h symmetric dispersion exhibits the discrete symmetries under $mathcal C$, $mathcal P$, $mathcal T$, and $mathcal{CPT}$. We find that in the superconducting ground state, $mathcal T$ and $mathcal P$ are spontaneously broken simultaneously with the U(1) symmetry. Moreover, we rigorously show that amplitude and phase fluctuations of the gap function are uncoupled due to the unbroken $mathcal C$. In the normal phase, the MG and NG modes become degenerate, and they have opposite parity under $mathcal T$. Therefore, we conclude that the lifting of the degeneracy in the superconducting phase and the resulting emergence of the pure amplitude MG mode can be identified as a consequence of the the spontaneous breaking of $mathcal T$ symmetry but not of $mathcal P$ or U(1).
We study the Higgs mode in a Bardeen-Cooper-Schrieffer (BCS) superconductor. Motivated by the observation that U(1) symmetry of the BCS Hamiltonian is not essential for the Higgs mode, we study the Ising-like Hamiltonian in the pseudospin representation. We show that the Higgs mode emerges as the lowest excited state of the Ising-like Hamiltonian due to spontaneous breakdown of $mathbb{Z}_2$ symmetry under the time-reversal operation $mathcal T$ in the pseudospin space. We further predict the existence of multiple Higgs modes that have quantized energy $2(n+1)Delta_0$ ($0le nle N_{k_F}$), where $Delta_0$ is the superconducting gap, $n$ is an integer, and $N_{k_F}$ is the number of states on the Fermi surface.
Skyrmions were originally introduced in Particle Physics as a possible mechanism to explain the stability of particles. Lately the concept has been applied in Condensed Matter Physics to describe the newly discovered topologically protected magnetic configurations called the magnetic Skyrmions. This elementary review introduces the concept at a level suitable for beginning students of Physics.
By studying the 2-dimensional Su-Schrieffer-Heeger-Bose-Hubbard model, we show the existence of topological Higgs amplitude modes in the strongly interacting superfluid phase. Using the slave boson approach, we find that, in the large filling limit, the Higgs excitations and the Goldstone excitations above the ground state are well decoupled, and both of them exhibit nontrivial topology inherited from the underlying noninteracting bands. At finite fillings, they become coupled at high energies; nevertheless, the topology of these modes are unchanged. Moreover, based on an effective action analysis, we further provide a universal physical picture for the topological character of Higgs and Goldstone modes. Our discovery of the first realization of the topological Higgs mode opens the path to novel investigations in various systems such as superconductors and quantum magnetism.
Ideas from quantum field theory and topology have proved remarkably fertile in suggesting new phenomena in the quantum physics of condensed matter. Here Ill supply some broad, unifying context, both conceptual and historical, for the abundance of results reported at the Nobel Symposium on New Forms of Matter, Topological Insulators and Superconductors. Since they distill some most basic ideas in their simplest forms, these concluding remarks might also serve, for non-specialists, as an introduction.
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