Higgs Bosons in Condensed Matter Muir Morrison Ph

Higgs Bosons in Condensed Matter Muir Morrison Ph 199 5/23/14

Outline • Spontaneous symmetry breaking & order parameters • Superfluids • Superconductors: detection of the Higgs… 30 years ago? • Generalities • Some speculations & links to high-energy

SSB & Order Parameters • Landau’s insight: phase transitions ⇔ broken symmetry • Order param: nonzero to zero across transtion • solid⇔liquid, ferromagnet⇔paramagnet, … • Superfluid & superconductors: condensate wavefunction (amplitude related to density) • Study fluctuations from ground state (“vacuum”)

Examples from J. Sethna, www. lassp. cornell. edu/sethna

Goldstone Modes • Spontaneously breaking symmetry creates new particles: degeneracy⇔zero-energy excitations (massless) • One Goldstone mode for each generator from Podolsky et. al. (2011)

Amplitude Mode • Amplitude mode has nonzero excitation energy (“mass”) • This is the “Higgs” mode from Podolsky et. al. (2011)

Superfluid • Weakly interacting bosons: • G. S. : • Expand & parametrize fluctuations: • Schrodinger Eq…

Superfluid (II) • Linearized: • Then • Dispersion relation: • No Higgs! What happened? • Phase/number: conjugate vars, not 2 modes

Is Higgs possible in superfluid? • Yes, with a relativistic dispersion • Tune interactions (r vs u) to near superfluid Mott-insulator transition => “massless” bosons • Drive amplitude mode by modulating lattice depth

Ultracold atom superfluid from physics. mcmaster. ca/people/faculty/O'Dell from Endres et al (2012) Drive amplitude mode by modulating lattice depth

Ultracold atom superfluid from Endres et al (2012)

Superconductors • Order param is topologically same as superfluid • BCS Hamiltonian: • Kinetic part is just like relativistic Dirac hamiltonian, even though this is nonrelativistic => particle-hole symm • Charge conserved, but particle number not

Superconductors (II) • So why don’t all superconductors have Higgs modes? • They do! Easy to miss: nearly impossible to directly observe local fluctuations in SC gap. • Need another observable tied to the Higgs mode • SC gap depends on Do. S at Fermi surface • => Any other modes that change Fermi Do. S will effectively couple to Higgs mode

Charge Density Waves • CDW: below some critical T, distorted lattice is favorable • Optical phonons modulate Do. S at Fermi surface from Littlewood (1981)

CDW & BCS coupling • SC gap • Electron-phonon int: • Interaction modifies phonon progagator. – Lowest order, ignore e-e int: nothing interesting. – Next order, include e-e “ladder diagrams” and…

CDW + BCS = Higgs • A new pole appears in phonon self-energy! • This is the Higgs (or, the coupling of lattice phonons to fluctuations in the superconducting gap)

Higgs detected, 1980 from Littlewood (1981)

CDW + BCS = Higgs • A new pole appears in phonon self-energy! • This is the Higgs (or, the coupling of lattice phonons to fluctuations in the superconducting gap) • More recent measurements (Measson et al, 2014) compare Nb. Se 2 with Nb. S 2 (1 st has CDW + SC, other only SC), proves CDW is necessary for Higgs. • (Aside: no one called it Higgs before ~2000…)

Enabling future experiments • Higgs modes are usually buried in noise. • When will Higgs modes in CMP be detectable or not? How best to search for them? • Short answer: measure scalar susceptibilities, not longitudinal from Podolsky et. al. (2011)

Higgs modes should be everywhere! • Defeats conventional wisdom that amplitude modes are strongly damped. They are not… • IF the right observable is measured. • Ex: in antiferromagnets, use Raman scattering (couples to order param squared), NOT neutron scattering (couples to projection of order param) • Podolsky et al evaluate many pages of RPA & ladder diagrams to prove this

Generalizing & Speculating • What if symmetry group is not U(1)? • E. g. , superfluid 3 He: symmetry group is SO(3)S x SO(3)L x U(1), order param is 3 x 3 matrix – 4 Goldstone modes & ≥ 14 Higgs modes! • Sum rules constrain various masses – Pick a symmetry group to generalize SM and predict new heavy Higgs partners • Volovik et al (2014) suggest there may be another Higgs in CDF & CMS data at 325 Ge. V?

Conclusion • Spontaneous symmetry breaking: not just for particle physics • Condensed matter can host Higgs in superfluids, superconductors, antiferromagnets, … • Higgs modes should be ubiquitous, with careful choice of measurements

References [1] P. Anderson, “Plasmons, gauge invariance, and mass, " Physical Review 130(1), pp. 439 -442, 1963. [2] J. Schwinger, “Gauge invariance and mass, " Physical Review 125(1), pp. 397{8, 1962. [3] C. Varma, “Higgs boson in superconductors, " Journal of low temperature physics 126(3 -4), pp. 901 -909, 2002. [4] P. Littlewood and C. Varma, “Gauge-invariant theory of the dynamical interaction of charge density waves and superconductivity, " Phys. Rev. Lett. 47(11), pp. 811 -14, 1981. [5] P. Littlewood and C. Varma, “Amplitude collective modes in superconductors and their coupling to charge-density waves, " Physical Review B 26(9), pp. 4883{4893, 1982. [6] M. -a. Measson, Y. Gallais, M. Cazayous, B. Clair, P. Rodiere, L. Cario, and A. Sacuto, “Amplitude Higgs mode in the 2 HNb. Se 2 superconductor, " Physical Review B 89, p. 060503, Feb. 2014. [7] M. Endres, T. Fukuhara, D. Pekker, M. Cheneau, P. Schauss, C. Gross, E. Demler, S. Kuhr, and I. Bloch, “The 'Higgs' amplitude mode at the two-dimensional superuid/Mott insulator transition, " Nature 487, pp. 454{8, July 2012. [8] D. Podolsky, A. Auerbach, and D. P. Arovas, “Visibility of the amplitude (Higgs) mode in condensed matter, " Physical Review B 84, p. 174522, Nov. 2011. [9] G. E. Volovik and M. a. Zubkov, “Higgs Bosons in Particle Physics and in Condensed Matter, " Journal of Low Temperature Physics 175, pp. 486 -497, Oct. 2014.