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

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

3. 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”)

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

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

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

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

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

11. 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

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

13. Ultracold atom superfluid from Endres et al (2012)

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

15. 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 DoS at Fermi surface => Any other modes that change Fermi DoS will effectively couple to Higgs mode

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

17. 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…

18. 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)

19. Higgs detected, 1980 from Littlewood (1981)

20. 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 NbSe 2 with NbS 2 (1st has CDW + SC, other only SC), proves CDW is necessary for Higgs. (Aside: no one called it Higgs before ~2000…)

21. 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)

22. 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

23. 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 3x3 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 GeV?

24. 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

25. 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 2HNbSe2 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.