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

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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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