BF Theory Description of Topological States of Matter - Firenze, Giugno 2013 Maria Cristina Diamantini

 
BF Theory Description of
Topological States of Matter
     Firenze, Giugno 2013
        Maria Cristina Diamantini
         Universitá di Perugia

                                    Firenze, Giugno 2013 – p. 1/18
Topologial States of Matter
Different states of matter are distinguished by their
internal structure ≡ orders.

                                                   Firenze, Giugno 2013 – p. 2/18
Topologial States of Matter
Different states of matter are distinguished by their
internal structure ≡ orders.
Orders are associated with symmetries (breaking of).

                                                   Firenze, Giugno 2013 – p. 2/18
Topologial States of Matter
Different states of matter are distinguished by their
internal structure ≡ orders.
Orders are associated with symmetries (breaking of).
1982 FQE states: strongly correlated, more rigid than a
solid ≡ incompressible, P and T breaking.

                                                   Firenze, Giugno 2013 – p. 2/18
Topologial States of Matter
Different states of matter are distinguished by their
internal structure ≡ orders.
Orders are associated with symmetries (breaking of).
1982 FQE states: strongly correlated, more rigid than a
solid ≡ incompressible, P and T breaking.
Fractionalization: quasiparticles carry only a fraction of
the unit charge of the electron; fractional statistic.

                                                   Firenze, Giugno 2013 – p. 2/18
Topologial States of Matter
Different states of matter are distinguished by their
internal structure ≡ orders.
Orders are associated with symmetries (breaking of).
1982 FQE states: strongly correlated, more rigid than a
solid ≡ incompressible, P and T breaking.
Fractionalization: quasiparticles carry only a fraction of
the unit charge of the electron; fractional statistic.
No symmetry breaking ⇒ new order.

                                                   Firenze, Giugno 2013 – p. 2/18
Topologial States of Matter
Different states of matter are distinguished by their
internal structure ≡ orders.
Orders are associated with symmetries (breaking of).
1982 FQE states: strongly correlated, more rigid than a
solid ≡ incompressible, P and T breaking.
Fractionalization: quasiparticles carry only a fraction of
the unit charge of the electron; fractional statistic.
No symmetry breaking ⇒ new order.
Topological order: new quantum order not based on
SSB (Wen); gapped in the bulk, gapless edge
excitations; ground state degeneracy, can break P and
T symmetries.

                                                   Firenze, Giugno 2013 – p. 2/18
Topologial States of Matter
Different states of matter are distinguished by their
internal structure ≡ orders.
Orders are associated with symmetries (breaking of).
1982 FQE states: strongly correlated, more rigid than a
solid ≡ incompressible, P and T breaking.
Fractionalization: quasiparticles carry only a fraction of
the unit charge of the electron; fractional statistic.
No symmetry breaking ⇒ new order.
Topological order: new quantum order not based on
SSB (Wen); gapped in the bulk, gapless edge
excitations; ground state degeneracy, can break P and
T symmetries.

                                                   Firenze, Giugno 2013 – p. 2/18
Low energy effective theory: topological field theory:
                      k
                        Z
             SCS =        d3 x bµ ǫµνα ∂ν bα ,
                     4π

Chern-Simons, jµ ∝ ǫµνα ∂ν bα = conserved charge
matter current, bµ U (1) gauge field.
  background independent,
  ground state degeneracy,
  quasiparticles have fractional statistics,
  presence of a gap without SSB, gapless edges
  modes,
  P and T breaking.

                                                  Firenze, Giugno 2013 – p. 3/18
Low energy effective theory: topological field theory:
                      k
                        Z
             SCS =        d3 x bµ ǫµνα ∂ν bα ,
                     4π

Chern-Simons, jµ ∝ ǫµνα ∂ν bα = conserved charge
matter current, bµ U (1) gauge field.
  background independent,
  ground state degeneracy,
  quasiparticles have fractional statistics,
  presence of a gap without SSB, gapless edges
  modes,
  P and T breaking.
(long distance properties can be explained by an
infinite-dimensional W1+∞ symmetry, Cappelli,
                                                   Firenze, Giugno 2013 – p. 3/18
Low energy effective theory: topological field theory:
                      k
                        Z
             SCS =        d3 x bµ ǫµνα ∂ν bα ,
                     4π

Chern-Simons, jµ ∝ ǫµνα ∂ν bα = conserved charge
matter current, bµ U (1) gauge field.
  background independent,
  ground state degeneracy,
  quasiparticles have fractional statistics,
  presence of a gap without SSB, gapless edges
  modes,
  P and T breaking.
(long distance properties can be explained by an
infinite-dimensional W1+∞ symmetry, Cappelli,
                                                   Firenze, Giugno 2013 – p. 3/18
Low energy effective theory: topological field theory:
                      k
                        Z
             SCS =        d3 x bµ ǫµνα ∂ν bα ,
                     4π

Chern-Simons, jµ ∝ ǫµνα ∂ν bα = conserved charge
matter current, bµ U (1) gauge field.
  background independent,
  ground state degeneracy,
  quasiparticles have fractional statistics,
  presence of a gap without SSB, gapless edges
  modes,
  P and T breaking.
(long distance properties can be explained by an
infinite-dimensional W1+∞ symmetry, Cappelli,
                                                   Firenze, Giugno 2013 – p. 3/18
Recently new topological phases of matter with
time-reversal symmetry have been discovered in 2 and
3D(Kane, Mele, Fu, Moore, Balents, Zhang, König):
topological insulators.

                                              Firenze, Giugno 2013 – p. 4/18
Recently new topological phases of matter with
time-reversal symmetry have been discovered in 2 and
3D(Kane, Mele, Fu, Moore, Balents, Zhang, König):
topological insulators.
Topological insulators are materials insulating in the
bulk but support conducting edge excitations.

                                                  Firenze, Giugno 2013 – p. 4/18
Recently new topological phases of matter with
time-reversal symmetry have been discovered in 2 and
3D(Kane, Mele, Fu, Moore, Balents, Zhang, König):
topological insulators.
Topological insulators are materials insulating in the
bulk but support conducting edge excitations.
Exp. observed in 2D ( Hsieh 2008) and 3D (Hasan
2009)

                                                  Firenze, Giugno 2013 – p. 4/18
Recently new topological phases of matter with
time-reversal symmetry have been discovered in 2 and
3D(Kane, Mele, Fu, Moore, Balents, Zhang, König):
topological insulators.
Topological insulators are materials insulating in the
bulk but support conducting edge excitations.
Exp. observed in 2D ( Hsieh 2008) and 3D (Hasan
2009)
3D, weak and strong topological insulators; strong:
topologically non-trivial, protected metallic surface,
quantized magnetoelectric polarizability.

                                                    Firenze, Giugno 2013 – p. 4/18
Recently new topological phases of matter with
time-reversal symmetry have been discovered in 2 and
3D(Kane, Mele, Fu, Moore, Balents, Zhang, König):
topological insulators.
Topological insulators are materials insulating in the
bulk but support conducting edge excitations.
Exp. observed in 2D ( Hsieh 2008) and 3D (Hasan
2009)
3D, weak and strong topological insulators; strong:
topologically non-trivial, protected metallic surface,
quantized magnetoelectric polarizability.
Which is the topological field theory that describes this
new phase of matter? Topological BF action

                                                    Firenze, Giugno 2013 – p. 4/18
2D CS Raction breaks P and T , but
     k
S = 2π  d3 x aµ ǫµνα ∂ν bα PT invariant if aµ is a vector;
mixed Chern-Simons, U (1) × U (1) .

                                                      Firenze, Giugno 2013 – p. 5/18
2D CS Raction breaks P and T , but
     k
S = 2π  d3 x aµ ǫµνα ∂ν bα PT invariant if aµ is a vector;
mixed Chern-Simons, U (1) × U (1) .
Natural generalization: jµ ∝ ǫµναβ ∂ν bαβ charge
fluctuations; φµν ∝ ǫµναβ ∂α aβ magnetic fluctuations.

                                                      Firenze, Giugno 2013 – p. 5/18
2D CS Raction breaks P and T , but
     k
S = 2π  d3 x aµ ǫµνα ∂ν bα PT invariant if aµ is a vector;
mixed Chern-Simons, U (1) × U (1) .
Natural generalization: jµ ∝ ǫµναβ ∂ν bαβ charge
fluctuations; φµν ∝ ǫµναβ ∂α aβ magnetic fluctuations.
Low energy theory:
       k
         R 4
SBF = 2π d x bµν ǫµναβ ∂α aβ , BF action, PT invariant.

                                                      Firenze, Giugno 2013 – p. 5/18
2D CS Raction breaks P and T , but
     k
S = 2π  d3 x aµ ǫµνα ∂ν bα PT invariant if aµ is a vector;
mixed Chern-Simons, U (1) × U (1) .
Natural generalization: jµ ∝ ǫµναβ ∂ν bαβ charge
fluctuations; φµν ∝ ǫµναβ ∂α aβ magnetic fluctuations.
Low energy theory:
       k
         R 4
SBF = 2π d x bµν ǫµναβ ∂α aβ , BF action, PT invariant.
gauge invariance:
aµ → aµ + ∂µ ξ , bµν → bµν + ∂µ ην − ∂ν ηµ .

                                                      Firenze, Giugno 2013 – p. 5/18
2D CS Raction breaks P and T , but
     k
S = 2π  d3 x aµ ǫµνα ∂ν bα PT invariant if aµ is a vector;
mixed Chern-Simons, U (1) × U (1) .
Natural generalization: jµ ∝ ǫµναβ ∂ν bαβ charge
fluctuations; φµν ∝ ǫµναβ ∂α aβ magnetic fluctuations.
Low energy theory:
       k
         R 4
SBF = 2π d x bµν ǫµναβ ∂α aβ , BF action, PT invariant.
gauge invariance:
aµ → aµ + ∂µ ξ , bµν → bµν + ∂µ ην − ∂ν ηµ .
              R 4 h 1                       i
ST M = SBF + d x 4e2 fµν f µν + 2g12 gµ g µ
fµν = ∂µ aν − ∂ν aµ ; gµ ≡ 16 ǫµναβ (∂ µ bνρ + ∂ ν bρµ + ∂ ρ bµν ),
[e2 ] = m−D+3 [g 2 ] = mD−1 , D = 3
                                                             Firenze, Giugno 2013 – p. 5/18
Topological mass m = keg2π , it plays the role of the gap
characterizing topological states of matter

                                                    Firenze, Giugno 2013 – p. 6/18
Topological mass m = keg2π , it plays the role of the gap
characterizing topological states of matter
Ground state degeneracy (Bergeron, Semenoff,
Szabo): k = k1 /k2 =⇒ |k1 k2 I|Np , I = determinant of the
intersection matrix; Np = 2g , g = genus, Np = 2 for the
torus.

                                                    Firenze, Giugno 2013 – p. 6/18
Topological mass m = keg2π , it plays the role of the gap
characterizing topological states of matter
Ground state degeneracy (Bergeron, Semenoff,
Szabo): k = k1 /k2 =⇒ |k1 k2 I|Np , I = determinant of the
intersection matrix; Np = 2g , g = genus, Np = 2 for the
torus.
BF theory provides a generalization of fractional
statistics to arbitrary dimensions (Semenoff, Szabo),
(3+1): particles around vortex strings.

                                                    Firenze, Giugno 2013 – p. 6/18
Topological mass m = keg2π , it plays the role of the gap
characterizing topological states of matter
Ground state degeneracy (Bergeron, Semenoff,
Szabo): k = k1 /k2 =⇒ |k1 k2 I|Np , I = determinant of the
intersection matrix; Np = 2g , g = genus, Np = 2 for the
torus.
BF theory provides a generalization of fractional
statistics to arbitrary dimensions (Semenoff, Szabo),
(3+1): particles around vortex strings.
Supports edges excitations (Momen, Balachandran;
Hansson; Cho and Moore; Maggiore, Magnoli et al.)

                                                    Firenze, Giugno 2013 – p. 6/18
(2+1) dimensions
LT M = − 4e12 fµν f µν + 2π
                          k
                            aµ ǫµαν ∂α bν − 4g12 gµν g µν ;
fµν = (∂µ aν − ∂ν aµ ), gµν = (∂µ bν − ∂ν bµ ).

                                                              Firenze, Giugno 2013 – p. 7/18
(2+1) dimensions
LT M = − 4e12 fµν f µν + 2π
                          k
                            aµ ǫµαν ∂α bν − 4g12 gµν g µν ;
fµν = (∂µ aν − ∂ν aµ ), gµν = (∂µ bν − ∂ν bµ ).
JJA on a square lattice can be exactly mapped in the
model described by ST M with U (1) × U (1) gauge
theories ⇒ topological excitations, closed electric loops
and closed magnetic loops

                                                              Firenze, Giugno 2013 – p. 7/18
(2+1) dimensions
LT M = − 4e12 fµν f µν + 2π
                          k
                            aµ ǫµαν ∂α bν − 4g12 gµν g µν ;
fµν = (∂µ aν − ∂ν aµ ), gµν = (∂µ bν − ∂ν bµ ).
JJA on a square lattice can be exactly mapped in the
model described by ST M with U (1) × U (1) gauge
theories ⇒ topological excitations, closed electric loops
and closed magnetic loops
BF model as fundamental theory describing topological
phases (Sodano, Trugenberger MCD 1996)

                                                              Firenze, Giugno 2013 – p. 7/18
(2+1) dimensions
LT M = − 4e12 fµν f µν + 2π
                          k
                            aµ ǫµαν ∂α bν − 4g12 gµν g µν ;
fµν = (∂µ aν − ∂ν aµ ), gµν = (∂µ bν − ∂ν bµ ).
JJA on a square lattice can be exactly mapped in the
model described by ST M with U (1) × U (1) gauge
theories ⇒ topological excitations, closed electric loops
and closed magnetic loops
BF model as fundamental theory describing topological
phases (Sodano, Trugenberger MCD 1996)
Condensation (lack of) of topological excitations
determine the phase diagram

                                                              Firenze, Giugno 2013 – p. 7/18
(2+1) dimensions
LT M = − 4e12 fµν f µν + 2π
                          k
                            aµ ǫµαν ∂α bν − 4g12 gµν g µν ;
fµν = (∂µ aν − ∂ν aµ ), gµν = (∂µ bν − ∂ν bµ ).
JJA on a square lattice can be exactly mapped in the
model described by ST M with U (1) × U (1) gauge
theories ⇒ topological excitations, closed electric loops
and closed magnetic loops
BF model as fundamental theory describing topological
phases (Sodano, Trugenberger MCD 1996)
Condensation (lack of) of topological excitations
determine the phase diagram
Exactly reproduce the superconducting/insulator (SI)
quantum phase transition at T = 0 for e/g = 1.

                                                              Firenze, Giugno 2013 – p. 7/18
(2+1) dimensions
LT M = − 4e12 fµν f µν + 2π
                          k
                            aµ ǫµαν ∂α bν − 4g12 gµν g µν ;
fµν = (∂µ aν − ∂ν aµ ), gµν = (∂µ bν − ∂ν bµ ).
JJA on a square lattice can be exactly mapped in the
model described by ST M with U (1) × U (1) gauge
theories ⇒ topological excitations, closed electric loops
and closed magnetic loops
BF model as fundamental theory describing topological
phases (Sodano, Trugenberger MCD 1996)
Condensation (lack of) of topological excitations
determine the phase diagram
Exactly reproduce the superconducting/insulator (SI)
quantum phase transition at T = 0 for e/g = 1.
Degeneracy: (k1 k2 ) if k = k1 /k2 , but k = 1 for planar JJA
                                                              Firenze, Giugno 2013 – p. 7/18
SI transition in JJA is a quantum conformal critical point.

                                                   Firenze, Giugno 2013 – p. 8/18
SI transition in JJA is a quantum conformal critical point.
Quantum conformal critical point:
  the ground state wave functional of the quantum
  theory at critical point is conformally invariant
  (Fradkin et al.);
  universality class of quantum Lifshitz theory ( z = 2)
  example: quantum square lattice dimer model,
  ground state        functional
                λ     2
                  R
  Ψ[φ] = e 2 −      d   x ∂i φ∂i φ = exp −S[φ]

  equal-time quantum Rcorrelations
  hφ(x1 ) . . . φ(xn )i = Z1 Dφ Ψ̄[φ]φ(x1 ) . . . φ(xn )Ψ[φ]
                                        R 2
   1                                  −λ d x ∂i φ∂i φ
     R
  Z    Dφ   φ(x   1 ) . . . φ(x n ) e
  Gaussian model, c = 1 quantum Lifshitz theory.
                                                     Firenze, Giugno 2013 – p. 8/18
SI transition in JJA is a quantum conformal critical point.
Quantum conformal critical point:
  the ground state wave functional of the quantum
  theory at critical point is conformally invariant
  (Fradkin et al.);
  universality class of quantum Lifshitz theory ( z = 2)
  example: quantum square lattice dimer model,
  ground state        functional
                λ     2
                  R
  Ψ[φ] = e 2 −      d   x ∂i φ∂i φ = exp −S[φ]

  equal-time quantum Rcorrelations
  hφ(x1 ) . . . φ(xn )i = Z1 Dφ Ψ̄[φ]φ(x1 ) . . . φ(xn )Ψ[φ]
                                        R 2
   1                                  −λ d x ∂i φ∂i φ
     R
  Z    Dφ   φ(x   1 ) . . . φ(x n ) e
  Gaussian model, c = 1 quantum Lifshitz theory.
                                                     Firenze, Giugno 2013 – p. 8/18
SI transition in JJA is a quantum conformal critical point.
Quantum conformal critical point:
  the ground state wave functional of the quantum
  theory at critical point is conformally invariant
  (Fradkin et al.);
  universality class of quantum Lifshitz theory ( z = 2)
  example: quantum square lattice dimer model,
  ground state        functional
                λ     2
                  R
  Ψ[φ] = e 2 −      d   x ∂i φ∂i φ = exp −S[φ]

  equal-time quantum Rcorrelations
  hφ(x1 ) . . . φ(xn )i = Z1 Dφ Ψ̄[φ]φ(x1 ) . . . φ(xn )Ψ[φ]
                                        R 2
   1                                  −λ d x ∂i φ∂i φ
     R
  Z    Dφ   φ(x   1 ) . . . φ(x n ) e
  Gaussian model, c = 1 quantum Lifshitz theory.
                                                     Firenze, Giugno 2013 – p. 8/18
Canonical quantization of LT M in the limit in which
magnetic permittivity is much larger than electric
permeability (magnetic term absent from the action)

                                                Firenze, Giugno 2013 – p. 9/18
Canonical quantization of LT M in the limit in which
magnetic permittivity is much larger than electric
permeability (magnetic term absent from the action)
Hodge decomposition
ai = ∂i ξ + ǫij ∂j φ , bi = ∂i λ + ǫij ∂j ψ ;

                                                Firenze, Giugno 2013 – p. 9/18
Canonical quantization of LT M in the limit in which
magnetic permittivity is much larger than electric
permeability (magnetic term absent from the action)
Hodge decomposition
ai = ∂i ξ + ǫij ∂j φ , bi = ∂i λ + ǫij ∂j ψ ;
∆φ ∝ j0 ; ∆ψ ∝ φ0 , Coulomb potentials for vortices and
charges respectively

                                                 Firenze, Giugno 2013 – p. 9/18
Canonical quantization of LT M in the limit in which
magnetic permittivity is much larger than electric
permeability (magnetic term absent from the action)
Hodge decomposition
ai = ∂i ξ + ǫij ∂j φ , bi = ∂i λ + ǫij ∂j ψ ;
∆φ ∝ j0 ; ∆ψ ∝ φ0 , Coulomb potentials for vortices and
charges respectively
Ground state waveR functional:
Ψ0 [ai , bi ] = exp −ik    d2 x [ψ∆ξ + φ∆λ] ×
                    h 4π                 i
× exp −k         2 x g (∂ φ)2 + e (∂ ψ)2
              R
         4π     d     e i         g i

                                                 Firenze, Giugno 2013 – p. 9/18
Canonical quantization of LT M in the limit in which
magnetic permittivity is much larger than electric
permeability (magnetic term absent from the action)
Hodge decomposition
ai = ∂i ξ + ǫij ∂j φ , bi = ∂i λ + ǫij ∂j ψ ;
∆φ ∝ j0 ; ∆ψ ∝ φ0 , Coulomb potentials for vortices and
charges respectively
Ground state waveR functional:
Ψ0 [ai , bi ] = exp −ik    d2 x [ψ∆ξ + φ∆λ] ×
                    h 4π                 i
× exp −k         2 x g (∂ φ)2 + e (∂ ψ)2
              R
         4π     d     e i         g i

(doubled) Gaussian model with central charge c = 1.
The universality class of the quantum SI critical point
(e/g = 1) critical point is thus that of quantum Lifshitz
theory.                                             Firenze, Giugno 2013 – p. 9/18
Canonical quantization of LT M in the limit in which
magnetic permittivity is much larger than electric
permeability (magnetic term absent from the action)
Hodge decomposition
ai = ∂i ξ + ǫij ∂j φ , bi = ∂i λ + ǫij ∂j ψ ;
∆φ ∝ j0 ; ∆ψ ∝ φ0 , Coulomb potentials for vortices and
charges respectively
Ground state waveR functional:
Ψ0 [ai , bi ] = exp −ik    d2 x [ψ∆ξ + φ∆λ] ×
                    h 4π                 i
× exp −k         2 x g (∂ φ)2 + e (∂ ψ)2
              R
         4π     d     e i         g i

(doubled) Gaussian model with central charge c = 1.
The universality class of the quantum SI critical point
(e/g = 1) critical point is thus that of quantum Lifshitz
theory.                                             Firenze, Giugno 2013 – p. 9/18
What happen in presence of magnetic frustration?

                                              Firenze, Giugno 2013 – p. 10/18
What happen in presence of magnetic frustration?
Magnetic frustration: f = Φ/Φ0 , Φ = magnetic flux
piercing the plaquette, Φ0 = 2π/κ quantum of magnetic
flux.

                                              Firenze, Giugno 2013 – p. 10/18
What happen in presence of magnetic frustration?
Magnetic frustration: f = Φ/Φ0 , Φ = magnetic flux
piercing the plaquette, Φ0 = 2π/κ quantum of magnetic
flux.
For particular fractional values of the frustration f , the
array behaves in a similar way as with zero magnetic
field (van der Zant et al.);
    the position of the quantum SI transition is displaced
    from the self-dual point.

                                                   Firenze, Giugno 2013 – p. 10/18
What happen in presence of magnetic frustration?
Magnetic frustration: f = Φ/Φ0 , Φ = magnetic flux
piercing the plaquette, Φ0 = 2π/κ quantum of magnetic
flux.
For particular fractional values of the frustration f , the
array behaves in a similar way as with zero magnetic
field (van der Zant et al.);
    the position of the quantum SI transition is displaced
    from the self-dual point.
Observed fractions (experimentally and numerically)
f = 1/2, 2/5, 1/3, 2/7, 1/4 , for f ∈ [0, 1/2]. f = 1/2: the
critical parameter EC /EJ lowered by 0.7 (error margin
of 10%).

                                                     Firenze, Giugno 2013 – p. 10/18
What happen in presence of magnetic frustration?
Magnetic frustration: f = Φ/Φ0 , Φ = magnetic flux
piercing the plaquette, Φ0 = 2π/κ quantum of magnetic
flux.
For particular fractional values of the frustration f , the
array behaves in a similar way as with zero magnetic
field (van der Zant et al.);
    the position of the quantum SI transition is displaced
    from the self-dual point.
Observed fractions (experimentally and numerically)
f = 1/2, 2/5, 1/3, 2/7, 1/4 , for f ∈ [0, 1/2]. f = 1/2: the
critical parameter EC /EJ lowered by 0.7 (error margin
of 10%).
The quantum critical point is conformal even in
                                                     Firenze, Giugno 2013 – p. 10/18
Conjecture: shift in the central charge

                            f 2g
                ⇒c=1→ c=1−6
                             4e

                                          Firenze, Giugno 2013 – p. 11/18
Conjecture: shift in the central charge

                            f 2g
                ⇒c=1→ c=1−6
                             4e
Unitary minimal models are given by:
                2  e   m
            f=    , =     , m≥3
               m+1 g  m+1

                                          Firenze, Giugno 2013 – p. 11/18
Conjecture: shift in the central charge

                            f 2g
                ⇒c=1→ c=1−6
                             4e
Unitary minimal models are given by:
                 2  e   m
             f=    , =     , m≥3
                m+1 g  m+1

The first few predicted frustrations are thus

               f = 1/2, 2/5, 1/3, 2/7, 1/4 . . .

                                                   Firenze, Giugno 2013 – p. 11/18
Conjecture: shift in the central charge

                            f 2g
                ⇒c=1→ c=1−6
                             4e
Unitary minimal models are given by:
                 2  e   m
             f=    , =     , m≥3
                m+1 g  m+1

The first few predicted frustrations are thus

               f = 1/2, 2/5, 1/3, 2/7, 1/4 . . .

Critical coupling for the fully frustrated model with
f = 1/2 is e/g = 3/4 = 0.75 well within the 10% error
margin of the experimental value 0.7.
                                                   Firenze, Giugno 2013 – p. 11/18
Conjecture: shift in the central charge

                            f 2g
                ⇒c=1→ c=1−6
                             4e
Unitary minimal models are given by:
                 2  e   m
             f=    , =     , m≥3
                m+1 g  m+1

The first few predicted frustrations are thus

               f = 1/2, 2/5, 1/3, 2/7, 1/4 . . .

Critical coupling for the fully frustrated model with
f = 1/2 is e/g = 3/4 = 0.75 well within the 10% error
margin of the experimental value 0.7.
                                                   Firenze, Giugno 2013 – p. 11/18
(3+1) dimensions
                       h                                 
                              1      2        1       2
ST M = SBF + d4 x
                 R
                                 f
                            4e2 λ ij     +        f
                                             4e2 η 0i         +
                     i
    1      2+ 1 g 2
  2g η
     2 g 0   2g 2 λ i

                                                                  Firenze, Giugno 2013 – p. 12/18
(3+1) dimensions
                       h                                 
                              1      2        1       2
ST M = SBF + d4 x
                 R
                                 f
                            4e2 λ ij     +        f
                                             4e2 η 0i         +
                     i
    1      2+ 1 g 2
  2g η
     2 g 0   2g 2 λ i

What are λ and η ? Coupling to an external e.m. field
                                R 4 1 1 2 1 2i
                                      h
iejµ Aµ , induced action: S(A) = d x 2 λ E + η B ⇒
     1
ǫ=   λ   electric permittivity; µ = η magnetic permeability

                                                                  Firenze, Giugno 2013 – p. 12/18
(3+1) dimensions
                       h                                 
                              1      2        1       2
ST M = SBF + d4 x
                 R
                                 f
                            4e2 λ ij     +        f
                                             4e2 η 0i         +
                     i
    1      2+ 1 g 2
  2g η
     2 g 0   2g 2 λ i

What are λ and η ? Coupling to an external e.m. field
                                R 4 1 1 2 1 2i
                                      h
iejµ Aµ , induced action: S(A) = d x 2 λ E + η B ⇒
     1
ǫ=   λ   electric permittivity; µ = η magnetic permeability
Add a T-breaking term: iφφµν Fµν , Fµν = ∂µ Aν − ∂ν Aµ :

                      iθ                        4 iθ
               Z                             Z
                   4             µν
S(A) → S(A) + d x        2
                           Fµν F̃ = S(A) + d x 2 E · B ,
                     16π                           4π
     keφ                      keφ
θ=    2    T-invariance ⇒     2π    = integer

                                                                  Firenze, Giugno 2013 – p. 12/18
(3+1) dimensions
                       h                                 
                              1      2        1       2
ST M = SBF + d4 x
                 R
                                 f
                            4e2 λ ij     +        f
                                             4e2 η 0i         +
                     i
    1      2+ 1 g 2
  2g η
     2 g 0   2g 2 λ i

What are λ and η ? Coupling to an external e.m. field
                                R 4 1 1 2 1 2i
                                      h
iejµ Aµ , induced action: S(A) = d x 2 λ E + η B ⇒
     1
ǫ=   λ   electric permittivity; µ = η magnetic permeability
Add a T-breaking term: iφφµν Fµν , Fµν = ∂µ Aν − ∂ν Aµ :

                      iθ                        4 iθ
               Z                             Z
                   4             µν
S(A) → S(A) + d x        2
                           Fµν F̃ = S(A) + d x 2 E · B ,
                     16π                           4π
     keφ                      keφ
θ=    2    T-invariance ⇒     2π    = integer
The model describes strong topological insulator
(Moore) with magnetoelectric polarizability.                      Firenze, Giugno 2013 – p. 12/18
(3+1) dimensions
                       h                                 
                              1      2        1       2
ST M = SBF + d4 x
                 R
                                 f
                            4e2 λ ij     +        f
                                             4e2 η 0i         +
                     i
    1      2+ 1 g 2
  2g η
     2 g 0   2g 2 λ i

What are λ and η ? Coupling to an external e.m. field
                                R 4 1 1 2 1 2i
                                      h
iejµ Aµ , induced action: S(A) = d x 2 λ E + η B ⇒
     1
ǫ=   λ   electric permittivity; µ = η magnetic permeability
Add a T-breaking term: iφφµν Fµν , Fµν = ∂µ Aν − ∂ν Aµ :

                      iθ                        4 iθ
               Z                             Z
                   4             µν
S(A) → S(A) + d x        2
                           Fµν F̃ = S(A) + d x 2 E · B ,
                     16π                           4π
     keφ                      keφ
θ=    2    T-invariance ⇒     2π    = integer
The model describes strong topological insulator
(Moore) with magnetoelectric polarizability.                      Firenze, Giugno 2013 – p. 12/18
(3+1) dimensions
                       h                                 
                              1      2        1       2
ST M = SBF + d4 x
                 R
                                 f
                            4e2 λ ij     +        f
                                             4e2 η 0i         +
                     i
    1      2+ 1 g 2
  2g η
     2 g 0   2g 2 λ i

What are λ and η ? Coupling to an external e.m. field
                                R 4 1 1 2 1 2i
                                      h
iejµ Aµ , induced action: S(A) = d x 2 λ E + η B ⇒
     1
ǫ=   λ   electric permittivity; µ = η magnetic permeability
Add a T-breaking term: iφφµν Fµν , Fµν = ∂µ Aν − ∂ν Aµ :

                      iθ                        4 iθ
               Z                             Z
                   4             µν
S(A) → S(A) + d x        2
                           Fµν F̃ = S(A) + d x 2 E · B ,
                     16π                           4π
     keφ                      keφ
θ=    2    T-invariance ⇒     2π    = integer
The model describes strong topological insulator
(Moore) with magnetoelectric polarizability.                      Firenze, Giugno 2013 – p. 12/18
U (1) × U (1) gauge symmetry ⇒ the dual field strengths
contain singularities (Polyakov):
   chargeons (electric topological defects);
   spinons (magnetic topological defects).

                                                Firenze, Giugno 2013 – p. 13/18
U (1) × U (1) gauge symmetry ⇒ the dual field strengths
contain singularities (Polyakov):
   chargeons (electric topological defects);
   spinons (magnetic topological defects).
The quantum phase structure is governed by three
parameters that drive the condensation of topological
defects: k , ǫ and η ; T = 0 quantum phase structure:
   chargeons condensation phase → top.
   superconductor
   no condensation → top. insulator
   spinons condensation phase → top. confinement

                                                Firenze, Giugno 2013 – p. 13/18
BF versus conventional superconductors
    BF: gap arises from topological mechanism.

                                                 Firenze, Giugno 2013 – p. 14/18
BF versus conventional superconductors
    BF: gap arises from topological mechanism.
    BCS: gap arises from SSB U (1) → ZN , Z2 for Cooper
    pairs:

                                                  Firenze, Giugno 2013 – p. 14/18
BF versus conventional superconductors
    BF: gap arises from topological mechanism.
    BCS: gap arises from SSB U (1) → ZN , Z2 for Cooper
    pairs:
    residual Aharonov-Bohm (AB) interaction between
    charges and vortices (Bais et al.)

                                                  Firenze, Giugno 2013 – p. 14/18
BF versus conventional superconductors
    BF: gap arises from topological mechanism.
    BCS: gap arises from SSB U (1) → ZN , Z2 for Cooper
    pairs:
    residual Aharonov-Bohm (AB) interaction between
    charges and vortices (Bais et al.)
    AB interaction ≡BF theory with k = N ;.

                                                  Firenze, Giugno 2013 – p. 14/18
BF versus conventional superconductors
    BF: gap arises from topological mechanism.
    BCS: gap arises from SSB U (1) → ZN , Z2 for Cooper
    pairs:
    residual Aharonov-Bohm (AB) interaction between
    charges and vortices (Bais et al.)
    AB interaction ≡BF theory with k = N ;.
    N 2 ground state degeneracy.

                                                  Firenze, Giugno 2013 – p. 14/18
BF versus conventional superconductors
    BF: gap arises from topological mechanism.
    BCS: gap arises from SSB U (1) → ZN , Z2 for Cooper
    pairs:
    residual Aharonov-Bohm (AB) interaction between
    charges and vortices (Bais et al.)
    AB interaction ≡BF theory with k = N ;.
    N 2 ground state degeneracy.
    k = 2 for Cooper pairs and ground state degeneracy
    k 2 = 4 on the torus (Hansson et al.).

                                                  Firenze, Giugno 2013 – p. 14/18
In conventional superconductors, photons acquire a
mass through SSB. What is the corresponding effective
action in topological superconductors?

                                              Firenze, Giugno 2013 – p. 15/18
In conventional superconductors, photons acquire a
mass through SSB. What is the corresponding effective
action in topological superconductors?
Need to compute the effective electromagnetic action
induced by the condensation of the topological defects.

                                                Firenze, Giugno 2013 – p. 15/18
In conventional superconductors, photons acquire a
mass through SSB. What is the corresponding effective
action in topological superconductors?
Need to compute the effective electromagnetic action
induced by the condensation of the topological defects.
Julia-Toulouse mechanism (Quevedo and
Trugenberger): the condensation of topological defects
in solid state media generates new hydrodynamical
modes for the low-energy effective theory

                                                Firenze, Giugno 2013 – p. 15/18
In conventional superconductors, photons acquire a
mass through SSB. What is the corresponding effective
action in topological superconductors?
Need to compute the effective electromagnetic action
induced by the condensation of the topological defects.
Julia-Toulouse mechanism (Quevedo and
Trugenberger): the condensation of topological defects
in solid state media generates new hydrodynamical
modes for the low-energy effective theory
These new modes are the long wavelength fluctuations
of the continuous distribution of topological defects that
get promoted to a continuous two-form antisymmetric
field Bµν

                                                  Firenze, Giugno 2013 – p. 15/18
In conventional superconductors, photons acquire a
mass through SSB. What is the corresponding effective
action in topological superconductors?
Need to compute the effective electromagnetic action
induced by the condensation of the topological defects.
Julia-Toulouse mechanism (Quevedo and
Trugenberger): the condensation of topological defects
in solid state media generates new hydrodynamical
modes for the low-energy effective theory
These new modes are the long wavelength fluctuations
of the continuous distribution of topological defects that
get promoted to a continuous two-form antisymmetric
field Bµν

                                                  Firenze, Giugno 2013 – p. 15/18
Chargeons condensation phase
Spinons are diluted; chargeons get promoted to a
continuous two-form antisymmetric field Bµν :

                                       1
              Z
        TS        4
       Seff = d x iπk Bµν ǫµναβ Fαβ + Fµν Fµν
                                       4

                       1
                   +     2
                           Hµνα Hµνα
                     12Λ

                                              Firenze, Giugno 2013 – p. 16/18
Chargeons condensation phase
Spinons are diluted; chargeons get promoted to a
continuous two-form antisymmetric field Bµν :

                                       1
              Z
        TS        4
       Seff = d x iπk Bµν ǫµναβ Fαβ + Fµν Fµν
                                       4

                        1
                    +     2
                            Hµνα Hµνα
                      12Λ
Λ is a new mass scale describing, essentially, the
average density of the condensed charges

                                                 Firenze, Giugno 2013 – p. 16/18
Chargeons condensation phase
Spinons are diluted; chargeons get promoted to a
continuous two-form antisymmetric field Bµν :

                                       1
              Z
        TS        4
       Seff = d x iπk Bµν ǫµναβ Fαβ + Fµν Fµν
                                       4

                        1
                    +     2
                            Hµνα Hµνα
                      12Λ
Λ is a new mass scale describing, essentially, the
average density of the condensed charges
This action describes a topologically massive photon
with quantized mass m = 4πkΛ ⇒ no SSB

                                                 Firenze, Giugno 2013 – p. 16/18
Chargeons condensation phase
Spinons are diluted; chargeons get promoted to a
continuous two-form antisymmetric field Bµν :

                                       1
              Z
        TS        4
       Seff = d x iπk Bµν ǫµναβ Fαβ + Fµν Fµν
                                       4

                        1
                    +     2
                            Hµνα Hµνα
                      12Λ
Λ is a new mass scale describing, essentially, the
average density of the condensed charges
This action describes a topologically massive photon
with quantized mass m = 4πkΛ ⇒ no SSB
Mass arises as a consequence of quantum mechanical
condensation of topological excitations: mechanism of
topological superconductivity (Allen, Bowick, Lahiri)
                                                 Firenze, Giugno 2013 – p. 16/18
In Progress
(2+1) dimensions: CS term induced by radiative
quantum effect, e.g. in 2 + 1 dimensional QED, CS
induced by one-loop fermion effective action.

                                              Firenze, Giugno 2013 – p. 17/18
In Progress
(2+1) dimensions: CS term induced by radiative
quantum effect, e.g. in 2 + 1 dimensional QED, CS
induced by one-loop fermion effective action.
Can we induce a BF term computing one loop effective
action in (3+1) dimensionns?

                                              Firenze, Giugno 2013 – p. 17/18
In Progress
(2+1) dimensions: CS term induced by radiative
quantum effect, e.g. in 2 + 1 dimensional QED, CS
induced by one-loop fermion effective action.
Can we induce a BF term computing one loop effective
action in (3+1) dimensionns?
Start from: L = ψ̄γ µ (∂µ − ieAµ ) ψ + me ψ̄ψ + igBµν J µν +
− 41 Fµν F µν + 12
                1
                   Hµνα H µνα

                                                     Firenze, Giugno 2013 – p. 17/18
In Progress
(2+1) dimensions: CS term induced by radiative
quantum effect, e.g. in 2 + 1 dimensional QED, CS
induced by one-loop fermion effective action.
Can we induce a BF term computing one loop effective
action in (3+1) dimensionns?
Start from: L = ψ̄γ µ (∂µ − ieAµ ) ψ + me ψ̄ψ + igBµν J µν +
− 41 Fµν F µν + 12
                1
                   Hµνα H µνα
To couple to the psudotensor Bµν and maintain P and T
we take the current:

   J µν = iµB ∂α ψ̄γ 5 {γ α , σ µν }ψ = −3µB ∂α ψ̄γ 5 γ [α γ µ γ ν] ψ

[. . .] ≡ total antisymmetrization, µB = e/2me Bohr
magneton, S µ,αβ = (1/4)ψ̄{γ µ , σ αβ }ψ spin current.
                                                               Firenze, Giugno 2013 – p. 17/18
iBµν J µν is invariant under gauge transformations
Bµν → Bµν + ∂µ λν − ∂ν λµ , (first attempt ( Leblanc et al
1993), no gauge invariance).

                                                   Firenze, Giugno 2013 – p. 18/18
iBµν J µν is invariant under gauge transformations
Bµν → Bµν + ∂µ λν − ∂ν λµ , (first attempt ( Leblanc et al
1993), no gauge invariance).
J 0i = −µB ǫijk ∂j ψ † diag(σk , −σk )ψ non-relativistic limit ⇒
curl of the spin density ≡ spin current density jS in the
non-relativistic Pauli equation

                                                        Firenze, Giugno 2013 – p. 18/18
iBµν J µν is invariant under gauge transformations
Bµν → Bµν + ∂µ λν − ∂ν λµ , (first attempt ( Leblanc et al
1993), no gauge invariance).
J 0i = −µB ǫijk ∂j ψ † diag(σk , −σk )ψ non-relativistic limit ⇒
curl of the spin density ≡ spin current density jS in the
non-relativistic Pauli equation
Integrate over fermion to show (to be confirmed!):
   spin-spin interactions are mediated by a scalar
   gauge boson in its antisymmetric tensor formulation;
   BF term is induced ⇒ gauge invariant, topological
   mass for the photons, leading to the Meissner effect;
   the one-loop effective equations of motion for the
   charged spin gauge boson are the London
   equations.
                                                        Firenze, Giugno 2013 – p. 18/18
iBµν J µν is invariant under gauge transformations
Bµν → Bµν + ∂µ λν − ∂ν λµ , (first attempt ( Leblanc et al
1993), no gauge invariance).
J 0i = −µB ǫijk ∂j ψ † diag(σk , −σk )ψ non-relativistic limit ⇒
curl of the spin density ≡ spin current density jS in the
non-relativistic Pauli equation
Integrate over fermion to show (to be confirmed!):
   spin-spin interactions are mediated by a scalar
   gauge boson in its antisymmetric tensor formulation;
   BF term is induced ⇒ gauge invariant, topological
   mass for the photons, leading to the Meissner effect;
   the one-loop effective equations of motion for the
   charged spin gauge boson are the London
   equations.
                                                        Firenze, Giugno 2013 – p. 18/18
iBµν J µν is invariant under gauge transformations
Bµν → Bµν + ∂µ λν − ∂ν λµ , (first attempt ( Leblanc et al
1993), no gauge invariance).
J 0i = −µB ǫijk ∂j ψ † diag(σk , −σk )ψ non-relativistic limit ⇒
curl of the spin density ≡ spin current density jS in the
non-relativistic Pauli equation
Integrate over fermion to show (to be confirmed!):
   spin-spin interactions are mediated by a scalar
   gauge boson in its antisymmetric tensor formulation;
   BF term is induced ⇒ gauge invariant, topological
   mass for the photons, leading to the Meissner effect;
   the one-loop effective equations of motion for the
   charged spin gauge boson are the London
   equations.
                                                        Firenze, Giugno 2013 – p. 18/18
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