The Anderson-Higgs Mechanism in Superconductors

The Anderson-Higgs Mechanism in Superconductors Department of Physics, Norwegian University of Science and Technology Summer School "Symmetries and Phase Transitions from Crystals and Superconductors to the Higgs particle and the Cosmos" Dresden 2016 beamer-tu-log

References P. W. Anderson, Phys. Rev. 130, 439 (1962) P. Higgs, Phys. Rev. Lett. 13, 508 (1964). Nobel Prize 2013. F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964). Nobel Prize 2013. G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, Phys. Rev. Lett. 13, 585 (1964) H. Kleinert, Gauge fields in Condensed Matter Physics, (World Scientific, Singapore (1989) A. K. Nguyen and A. Sudbø, Phys. Rev B 60, 15307 (1999) J. Hove and A. Sudbø, Phys. Rev. Lett. 84, 3426 (2000) E. Babaev, A. Sudbø, and N. W. Ashcroft, Nature 431, 666 (2004) J. Smiseth, E. Smørgrav, E. Babaev, and A. Sudbø, Phys. Rev. B 71, 214509 (2005) A. Sudbø, The Anderson Higgs Mechanism for the Meissner-effect in superconductors, in K. Fossheim: Superconductivity: Discovery and Discoverers. Ten Physics Nobel Laureates Tell Their Stories, Springer (2013) F. S. Nogueira, A. Sudbø, and I. Eremin, Phys. Rev. B 92, 224507 (2015) P. N. Galteland and A. Sudbø, Phys. Rev. B 94, 054518 (2016) beamer-tu-log

Outline Basic concepts Cooper-pairing 1 Basic concepts Cooper-pairing 2 Order parameter of SFs and SCs 3 Lattice gauge theory Current loops Vortex loops

Overview Basic concepts Cooper-pairing What is the Higgs mechanism? Definition of the superconducting state Theories of superconductors Higgs Mechanism I: Mean field theory physics Higgs Mechanism II: Fluctuation physics

Basic concepts Cooper-pairing What is the Higgs mechanism?

Basic concepts Cooper-pairing What is the Higgs mechanism? The Higgs mechanism is the dynamical generation of a mass of a gauge-boson (mediating interactions) resulting from the coupling to some matter field. In superconductors, it is the generation of a mass of the photon. Superconductors have radically different electrodynamics compared to non-superconductors. The Higgs mass therefore distinguishes a metallic state from a superconducting state.

Basic concepts Cooper-pairing Defining properties of a superconductor H. Kammerling-Onnes (1911). Nobel Prize 1913. But: SC is much more than just perfect conductivity! beamer-tu-log

Basic concepts Cooper-pairing Defining properties of a superconductor SC: Perfect diamagnet! Completely different from a perfect conductor! W. Meissner, R. Oechsenfeld (1933) beamer-tu-log

Cooper-pairing Basic concepts Cooper-pairing All theories of superconductivity somehow involve the concept of Cooper-pairing: Two electrons experience an effective attractive (!) interaction and form pairs. NB!! No more than pairs! (L. Cooper (1956); J. Bardeen, L. Cooper, R. J. Schrieffer (1957)). beamer-tu-log

Generalized BCS theory Basic concepts Cooper-pairing Effective BCS Hamiltonian H = ε k c k,σ c k,σ k,σ + V k,k c }{{} k,σ c k, σ c k, σc k,σ k,k,σ Effective interaction k k V k,k c k, σc k,σ H is invariant if c k,σ c k,σ e iθ. Theory is U(1)-symmetric. Note: Global U(1)-invariance! k 0 U(1)-symmetry is broken! If V k,k = V g k g k, then k = 0 (T )g k beamer-tu-log

Basic concepts Cooper-pairing Normal metal in zero electric field No net motion of electrons Electrons (almost) do not see each other

Normal metal in electric field Basic concepts Cooper-pairing Forced net motion of electrons Electrons (almost) do not see each other beamer-tu-log

Basic concepts Cooper-pairing Superconductors in zero electric field Unforced net motion of electrons Electrons states are co-operatively highly organized

Outline Order parameter of SFs and SCs 1 Basic concepts Cooper-pairing 2 Order parameter of SFs and SCs 3 Lattice gauge theory Current loops Vortex loops

Order parameter of SFs and SCs Building a theory of Superconductivity A theory of wave-function of Cooper-pairs Ψ(r) = Ψ(r) e iθ(r) This theory needs to be able to somehow distinguish (by an order parameter OP) the high-temperature normal metallic state from the low-temperature superconducting state. An OP is the expectation value of some operator that effectively distinguishes an ordered state from a disordered state, separated by a phase transition. Two states separated by a phase transition cannot be analytically continued from one to the other. What is the order parameter of a superconductor? beamer-tu-log

Conventional phase transitions Order parameter of SFs and SCs Conventional phase transitions are well described by the Landau-Ginzburg-Wilson (LGW) paradigm L. D. Landau V. Ginzburg K. G. Wilson F = 1 2 ( M) 2 + r 2 M 2 + u 8 ( M 2 ) 2 r T T c r < 0 = M 0 beamer-tu-log

Conventional phase transitions Order parameter of SFs and SCs Conventional phase transitions are well described by the Landau-Ginzburg-Wilson (LGW) paradigm M M = 0 M = 0 T beamer-tu-log c

Order parameter of SFs and SCs Superconductor Superfluid

Order parameter of SFs and SCs Superconductor Superfluid Zero resistivity Zero viscosity beamer-tu-log

Order parameter of SFs and SCs Superconductor Superfluid Phase transition beamer-tu-log

Order parameter of SFs and SCs Superconductor Superfluid Order parameter beamer-tu-log

Order parameter of SFs and SCs Superconductor Superfluid Meissner effect

Order parameter of SFs and SCs Type I superconductor Type II superconductor Meissner effect Mixed phase

Vortices suppress Ψ Order parameter of SFs and SCs

Effective models Order parameter of SFs and SCs Construct effective models Complex matter field: ψ(r) = ψ(r) e iθ(r) Neglect microscopic details Valid near the phase transition

Effective models Order parameter of SFs and SCs Construct effective models Complex matter field: ψ(r) = ψ(r) e iθ(r) Neglect microscopic details Valid near the phase transition Ginzburg-Landau model for superconductors { g H[ψ, A] = d d r 2 [ iea(r)]ψ(r) 2 + αt 2 ψ(r) 2 + u 4! ψ(r) 4 + 1 } 2 [ A(r)]2 beamer-tu-log

Effective models Order parameter of SFs and SCs Construct effective models Complex matter field: ψ(r) = ψ(r) e iθ(r) Neglect microscopic details Valid near the phase transition Gross-Pitaevskii model for superfluids [ 1 H[ψ] = d d r 2m ψ(r) 2 + V (r) ψ(r) 2 + U ] 0 2 ψ(r) 4

Order parameter of a superfluid Order parameter of SFs and SCs Gross-Pitaevskii model for superfluids [ 1 H[ψ] = d d r 2m ψ(r) 2 + V (r) ψ(r) 2 + U ] 0 2 ψ(r) 4 Z = e βf = DΨDΨ e βh O = 1 DΨDΨ O(Ψ, Ψ ) e βh Z Candidate OPs Ψ(r) 2 = Ψ(r) e iθ(r) 2 Ψ 2 e iθ(r) 2 (Local) Υ = 2 F θ 2 (Non local) beamer-tu-log

Order parameter of a superfluid Order parameter of SFs and SCs V ( Ψ ) = α ( ) T Tc 2T c Ψ 2 + u 4! Ψ 4 For the Higgs-mechanism to work at all, we need Ψ 0!

Order parameter of a superfluid Order parameter of SFs and SCs A. K. Nguyen and A. Sudbø, Phys. Rev B 60, 15307 (1999). beamer-tu-log

Order parameter of SFs and SCs Order parameter of a superconductor Ginzburg-Landau model for superconductors { g H[Ψ, A] = d d r 2 [ iea(r)]ψ(r) 2 + αt 2 Ψ(r) 2 + u 4! Ψ(r) 4 + 1 } 2 [ A(r)]2 O = 1 DΨDΨ O(Ψ, Ψ ) e βh Z Ψ(r) 2 = Ψ(r) e iθ(r) 2 Ψ 2 e iθ(r) 2 (Local) Υ = 2 F θ 2 (Non local) beamer-tu-log

Order parameter of SFs and SCs Order parameter of a superconductor This will not work! Ψ(r) = Ψ(r) e iθ(r) Ψ 0 e iθ(r) H[Ψ, A] = { d d g Ψ0 2 r ( θ ea(r)) 2 + 1 } 2 2 [ A(r)]2 Ψ(r) 2 = 0! Υ = 0! Elitzur s theorem: The expectation value of a gauge-invariant local operator can never acquire a non-zero expectation value in a gauge-theory. In a gauge-theory, the Goldstone modes are "eaten up" by the gauge-field Υ = 0. beamer-tu-log

Order parameter of SFs and SCs Order parameter of a superconductor { H[Ψ, A] = d d g Ψ0 2 r ( θ ea(r)) 2 + 1 } 2 2 [ A(r)]2 { = d d g Ψ0 2 r (( θ) 2 2e θ A(r) + e 2 A 2) 2 + 1 } 2 [ A(r)]2 = { d d g Ψ0 2 r ( θ) 2 2 + g Ψ 0 2 e 2 }{{ 2 } m 2 A 2 + 1 2 [ A(r)]2 } j A {}}{ g Ψ 0 2 θ A(r) beamer-tu-log

Order parameter of SFs and SCs Order parameter of a superconductor Consider in more detail the gauge-part of the theory (introducing Fourier-transformed fields) H A = m 2 A q A q + 1 2 q2 A q A q 1 2 (S qa q + S q A q ) = (m 2 + 1 2 q2 ) A q A q 1 }{{} 2 (S qa q + j q A q ) = D 0 (q) D 0 (q) {( A q 1 ) ( 2 D 1 0 (q)s q A q 1 ) 2 D 1 0 (q)s q 1 4 D 1 0 S q S q S q g Ψ 0 2 e 2 F( θ) 2 beamer-tu-log

Order parameter of SFs and SCs Order parameter of a superconductor Integrate out the shifted gauge fields Ãq = A q 1 2 D 1 0 (q)s q: H eff (Ψ) = m2 e 2 q 2 2m 2 + q 2 S q S q m 2 g Ψ 0 2 e 2 Define a momentum-dependent phase-stiffness Υ q : 2 Υ q = d 2 H eff ds q ds q = m2 e 2 q 2 2m 2 + q 2 Interested in long-distance physics q 0. beamer-tu-log

Order parameter of SFs and SCs Order parameter of a superconductor lim Υ m 2 q 2 q = lim q 0 q 0 e 2 2m 2 + q 2 = g Ψ 0 2 ; e = 0 2 lim Υ m 2 q 2 q = lim q 0 q 0 e 2 2m 2 + q 2 = 0; e 0

Order parameter of SFs and SCs Order parameter of a superconductor lim Υ m 2 q 2 q = lim q 0 q 0 e 2 2m 2 + q 2 = g Ψ 0 2 ; e = 0 2 lim Υ m 2 q 2 q = lim q 0 q 0 e 2 2m 2 + q 2 = 0; e 0 The superfluid density is NOT an order parameter for a superconductor! It IS an order parameter for a superfluid!

Outline Lattice gauge theory Current loops Vortex loops 1 Basic concepts Cooper-pairing 2 Order parameter of SFs and SCs 3 Lattice gauge theory Current loops Vortex loops

Lattice gauge theory Current loops Vortex loops Ginzburg-Landau lattice model ( ) Z = DA Dψ α α e S where the action is { S = β d 3 1 r ( iea(r)) ψ α (r) 2 2 α } + V ({ ψ α (r) }) + 1 2 ( A(r))2

Lattice gauge theory Current loops Vortex loops Ginzburg-Landau lattice model London-approximation Ψ(r) Ψ 0 e iθ(r) S = β r { µ,α cos ( µ θ r,α ea r,µ ) + 1 2 ( A r) 2 }

Current-loop formulation Lattice gauge theory Current loops Vortex loops e β cos γ = I b (β)e ibγ b= ( ) Z = DA Dθ α r,µ,α α b r,µ,α= I br,µ,α (β)e ibr,µ,α( µθr,α ear,µ) r e β ( Ar )2 2 P. N. Galteland and A. S., Phys. Rev B94, 054518 (2016). beamer-tu-log

Current-loop formulation Lattice gauge theory Current loops Vortex loops Z θ = r,α θ-integration gives: 2π 0 dθ r,α e iθr,α( µ µbr,µ,α) b r,α = 0 r, α The partition function is then given by Z J = D(A) {b} r,α e δ br,α,0 r,µ,α I br,µ,α (β) S {}} J [ { ie ] b r,α A r + J A + β 2 ( A r) 2 α beamer-tu-log r P. N. Galteland and A. S., Phys. Rev B94, Asle 054518 Sudbø (2016).

Current-loop formulation Lattice gauge theory Current loops Vortex loops Integrating out the A-field gives Z = δ br,α,0 I br,α,µ (β) }{{} {b} r,α r,µ,α r,r e e (b r,α,µ) 2 /2β e2 2β α,β br,α b r,β D(r r ) }{{} 1/ r r This is a theory of supercurrents b r,α, forming closed loops, interacting via the potential D(r r ) mediated by the gauge-field A, as well as contact interaction. P. N. Galteland and A. S., Phys. Rev B94, 054518 (2016). beamer-tu-log

Lattice gauge theory Current loops Vortex loops Gauge-field correlations and Higgs mass A µ qa ν q = 1 δ 2 Z J Z 0 δj q,µ δj q,ν = 1 Z 0 {b,m} r,α ( δ2 S J δj qδj µ q ν J=0 δ br,α,0 r,µ,α I br,α,µ (β) δs J δs J δj q µ δjq ν ) e S J J=0

Lattice gauge theory Current loops Vortex loops Gauge-field correlations and Higgs mass A q A q 1 q 2 + m 2 A Excitations Higgs mass A q A q = 1 2 β q 2 e2 β q 2 b q,α b q,β αβ ma 2 = lim 2 q 0 β A q A q beamer-tu-log

Lattice gauge theory Current loops Vortex loops Gauge-field correlations and Higgs mass e 2 lim q 0 2β B q B q (1 C 2 (T ))q 2, T > T C. q 2 C 3 (T )q 2+η A, T = T C. q 2 C 4 (T )q 4, T < T C. m 2 A { 0, T TC. 1 C 4 (T ), T < T C. Fluctuating supercurrents B q α b q,α create a Higgs mass!

Vortex-loop formulation Lattice gauge theory Current loops Vortex loops Go back to vortices

Vortex-loop formulation Lattice gauge theory Current loops Vortex loops Z = D(A) δ br,α,0 I br,µ,α (β) {b} r,α r,µ,α e [ie α br,α Ar + β 2 ( Ar )2 ] Solve constraint b r,α = 0 r, α b r,α = K r,α r

Vortex-loop formulation Lattice gauge theory Current loops Vortex loops Z = D(A) δ br,α,0 I br,µ,α (β) {b} r,α r,µ,α e [ie α br,α Ar + β 2 ( Ar )2 ] r Use Poisson summation formula e i2πn h = δ( h K ) n= e i2πm h = K = m= K = δ(h K ) m = n ( m = 0!) beamer-tu-log

Vortex-loop formulation Lattice gauge theory Current loops Vortex loops Z = D(A) δ br,α,0 I br,µ,α (β) {b} r,α r,µ,α e [ie α br,α Ar + β 2 ( Ar )2 ] r Integrate out A. Z = D(A) D(h) {m} I hr,µ,α (β) r,µ,α e [i2πm h+ie α hr,α Ar + β 2 ( Ar )2 ] r beamer-tu-log

Vortex-loop formulation Lattice gauge theory Current loops Vortex loops Z = D(A) δ br,α,0 I br,µ,α (β) {b} r,α r,µ,α e [ie α br,α Ar + β 2 ( Ar )2 ] Z = D(h) I hr,α (β) e {m} r,α r,α r [ ] i2πm r,α h r,α+ e2 2 h2 r,α h is a massive (dual) gauge-field mediating interactions between the m s. Integrate out h. beamer-tu-log

Vortex-loop formulation Lattice gauge theory Current loops Vortex loops Z = D(A) δ br,α,0 I br,µ,α (β) {b} r,α r,µ,α e [ie α br,α Ar + β 2 ( Ar )2 ] r Z = δ mr,α,0 e 4π2 β r,r mr,α m r,α D(r r ) {m} r,α 1 D(q) = q 2 + e 2 β

Lattice gauge theory Current loops Vortex loops Gauge-field correlations and Higgs mass A µ qa ν q 1 q 2 + m 2 A A q A q = 1 4π 2 e 2β2 β q 2 1 + + e 2 β q 2 ( q 2 m q,α m q,β + e 2 β) αβ }{{} Destroys Higgs mass ma 2 = lim 2 q 0 β A q A q beamer-tu-log

Lattice gauge theory Current loops Vortex loops Gauge-field correlations and Higgs mass Vortex-loop blowout

Lattice gauge theory Current loops Vortex loops Gauge-field correlations and Higgs mass Vortex correlator G(q) = m q m q J. Hove and A. Sudbø, Phys. Rev. Lett. 84, 3426 (2000).

Lattice gauge theory Current loops Vortex loops Physical interpretation of the Higgs mass Gauge-field correlator A µ qa ν q 1 q 2 + m 2 A m A 1 λ λ: London penetration length. Anderson-Higgs mechanism yields the Meissner-effect, the defining property of a superconductor. Manifestation of a broken local (gauge) U(1)-invariance.

Lattice gauge theory Current loops Vortex loops 3D superconductors and superfluids are dual Vortices in a three-dimensional superconductor behave like supercurrents in a superfluid, and vice versa, with inverted temperature axes. 3D superfluids and superconductors are dual to each other. The field theory of one is the field theory of the topoligical defects (vortices) of the matter-field of the other. Order parameter for a superfluid is the superfluid density (non-local) or the condensate fraction (local). OP for a superconductor is the Higgs mass (non-local). Higgs mass is created by supercurrent-loop proliferation, and is destroyed by vortex-loop proliferation.