Aspects of geometric phases in QFT Vasilis Niarchos Durham University based on work with Marco Baggio and Kyriakos Papadodimas Annual Theory Meeting Durham, December 19, 2017
Parameter-spaces and geometric phases in QM (review I) 2
Frequently in QM the Hamiltonian/spectrum depends on continuous parameters —external couplings ( λ 1 , λ 2 , . . . ) can be components of a fixed or slowly varying background The space of λ ’s is endowed with rich geometric properties that encode important physical effects 3
Berry phase is one of the most characteristic examples under adiabatic cycles on space of λ ’s quantum states can pick up a phase R T 0 dt E Ψ ( t ) | Ψ ( ~ � ) i T = e i γ e − i | Ψ ( ~ � ) i 0 ~ Berry phase much like a frame dragged around a curved manifold 4
• Intrinsic property of the quantum system Phases depend only on path C in λ -space I d ~ � h Ψ ( ~ � | Ψ ( ~ � = i � ) | @ ~ � ) i C or Pancharatnam-Berry I connection d ~ � A ( ~ � = i � ) connection on a vector bundle C of Hilbert spaces • For D degenerate states the U(1) phase upgrades to a U(D) transform ⇒ Berry connection is non-abelian 5
• there is a corresponding curvature F µ ν = ∂ A ν ∂λ µ − ∂ A µ ∂λ ν + [ A µ , A ν ] • for which one can write a general spectral formula 1 ⇣ ⌘ X X F ( n ) ( E n � E m ) 2 h n, b | ∂ µ H | m, c i g cd ab = ( m ) h m, d | ∂ ν H | n, a i � ( µ $ ν ) µ ν m 6 = n c,d in sector with energy E n and degenerate states | n, a i g ab = h a | b i 6
Berry phase is a physically detectable effect with many applications… ⇣ ⌘ F ( n ) The curvature is a very interesting observable µ ν ab • typically hard to evaluate analytically, but • if known it helps constrain the parametric dependence of state overlaps and potentially other observables (examples soon) 7
Geometric phases in (continuum) QFT? 8
tt* equations: a beautiful example from the 90s (Cecotti-Vafa ’91) • 2d QFTs with 4 real supersymmetries • parameter space: complex (superpotential) couplings • topologically twisted, spectrum truncated to 1/2-BPS states (chiral - antichiral) corresponding operators close among themselves under the OPE (chiral ring) φ i ( x ) φ j (0) ∼ C k ij φ k (0) + regular 9
Cecotti & Vafa computed the Berry curvature of these ground states and found an expression that closes on the chiral ring � L K = [ C i , ¯ C j ] L � F i ¯ j K UV C L Q C ∗ ¯ ¯ N C ∗ ¯ ¯ Q ≡ C P RL − g K ¯ N iK g P ¯ R g U g ¯ j ¯ iV j ¯ ¯ • RHS is algebraic functional of OPE coefficients and 2-point functions L = h ¯ L | K i g K ¯ chiral-antichiral overlap → all SUSIES broken (topological-antitopological → tt*) 10
λ i • LHS is curvature, involves derivatives wrt complex couplings • In a set of conventions LHS is expressed only in terms of derivatives of the 2-point overlaps Leads to PDEs for coupling dependence of non-SUSY quantities which Cecotti & Vafa solved in several examples 11
Unfortunately these ideas were not pushed much further in the context of higher-dimensional QFTs • tt * equations in 4d QFTs were unknown until recently and results of analogous power were beyond reach • Berry phase as an intrinsic quantum property was not explored systematically in QFT 12
Parallel ideas in QFT (review II) 13
The idea to think about the properties of the space of parameters (theory space) seriously has been discussed over the years in (continuum) QFT in various contexts • Wilsonian RG • Zamolodchikov c-theorem in 2d QFTs Zamolodchikov metric g ij = x 4 h O i ( x ) O j (0) i � � x 2 = x 2 0 14
Early ‘90s: much interest in 2d CFTs / worldsheet description of strings In that context: 2d QFT couplings = spacetime fields Interest to understand the theory space of exactly marginal couplings (deformations that preserve conformal invariance) conformal manifolds It was understood (Kutasov, Sonoda, Zwiebach...) that besides the Zamolodchikov metric the conformal manifolds also possess natural notions of parallel transport, connection 15
Conformal perturbation theory: a connection in theory-space tells us how to compare operators in near-by CFTs A covariant derivative incorporates regularization prescriptions Z � d d +1 x h O µ ( x ) ϕ 1 ( z 1 ) . . . ϕ n ( z n ) i r µ h ϕ 1 ( z 1 ) · · · ϕ n ( z n ) i ⇠ regularised 16
the curvature of this connection involves integrated correlation functions Roughly: Z Z d d +1 x d d +1 y h O [ µ ( x ) O ν ] ( y ) ϕ 1 ( z 1 ) ϕ 2 ( z 2 ) i [ r µ , r ν ] 12 ⇠ } antisymmetry integrations lead to divergences regularisation leads to non-vanishing commutators ⇒ curvature 17
Ranganathan-Sonoda-Zwiebach (’93) prescription (cut out small balls around operators, do not allow collisions, remove divergent pieces) 4-point operator formula for the curvature on R 4 1 Z Z d 4 x d 4 y h ϕ l ( 1 ) O [ µ ( x ) O ν ] ( y ) ϕ k (0) i ( F µ ν ) kl = (2 π ) 4 | x | ≤ 1 | y | ≤ 1 18
Connection in conformal perturbation theory vs Berry connection in QM They are obviously related concepts… Q: Can one go from one to the other? Q: Can one import the geometric phases of QM systematically & more generally in QFT? Computable? Q: Will this lead to new lessons? 19
Berry phases in (non-perturbative) QFT (old meets modern) 20
Technical approach Baggio-VN-Papadodimas ’17 Berry phase by quantizing QFT in the Hamiltonian framework Obvious issues: • UV divergences: ☞ renormalize • IR issues: e.g. continuous spectra ☞ put theory on a hypercylinder : , compact R × Q Q 21
Example 1 : Berry phase of photons • Consider electromagnetism with a θ interaction e.g. Wilczek, PRL ‘89 θ L = − 1 4 e 2 F µ ν F µ ν + 64 π 2 F µ ν ˜ F µ ν • θ has non-trivial implications if it varies in spacetime, or if there are boundaries/walls • interested in adiabatic changes of θ in time 22
• this theory is free: as we change e and θ adiabatically the spectrum is unchanged, but the energy eigenstates can rotate • Hamiltonian deformations Z Z @ e 2 H = 1 1 ⇣ B 2 ⌘ E 2 − ~ ~ d 3 x ~ E · ~ d 3 x @ θ H = , B e 4 8 ⇡ 2 23
R × T 3 • Place on and quantize in Coulomb gauge s ~ e 2 ⇣ x + ¯ x ⌘ k, ✏ e − i ! k t + i ~ k, ✏ e i ! k t − i ~ X X e ✏ ( ~ e ✏ ( ~ k ) a † ~ k · ~ k · ~ A ( t, ~ x ) = k ) a ~ ~ ~ ~ 2 ! k V ✏ = ± ~ k k i = 2 ⇡ n i ! k = c | ~ V = R 3 k | , , n i ∈ Z , R • Evaluate the spectral sum in the formula for Berry curvature The curvature has non-vanishing components only for identical external states 24
• for a general multi-photon state with | n + , n − i positive helicity photons, negative helicity photons n + n − 1 τ ) ( n + , n − ) = n + − n − ( F τ ¯ (Im τ ) 2 8 τ = θ 2 π + i 4 π e 2 ☞ a non-trivial Berry phase for photons ☞ independent of momentum 25
☞ Implications: linearly polarized light changes polarization under ( e, θ ) variation 1 ⇥ ⇤ p | p z , ˆ x i = | p z , + i + | p z , �i 2 1 | p z , ˆ e i φ | p z , + i + e − i φ | p z , �i ⇥ ⇤ p φ i = 2 ☞ experimentally visible? (e.g. topological insulators…) e.g. Essin-Moore-Vanderbilt, PRL ’09 26
Example 2: conformal manifolds, conformal pert. theory • In CFT there is a natural correspondence between states and operators OPERATOR-STATE correspondence R × S d in radial quantization or on | I i ⇠ O I (0) | 0 i • The Berry connection for states should map to a natural connection for operators in Conformal Perturbation Theory 27
• Two seemingly different expressions for curvature Berry double d-dim integral 1 X X ( ∆ I � ∆ n ) 2 h J | ∂ µ H | n, a i g ab ( F µ ν ) IJ = ( n ) h n, b | ∂ ν H | I i � ( µ $ ν ) a,b ∈ H n n/ ∈ H I CFT Z Z d d +1 x d d +1 y h O J ( 1 ) O µ ( x ) O ν ( y ) O I (0) i � ( µ $ ν ) ( F µ ν ) IJ = | x | ≤ 1 | y | ≤ 1 double (d+1)-dim integral These expressions compute the same object ! Baggio-VN-Papadodimas ’17 28
Example 3: SUPERconformal manifolds, tt * equations Conformal manifolds are rare in non-SUSY QFTs In SUSY QFTs they are abundant: superconformal manifolds Depending on amount of SUSY, their existence can be argued (non- perturbatively) in QFT, their dimension is known etc... Also examples in large-N limit in AdS/CFT from (super)gravity solutions 29
Here focus on 4d N=2 SCFTs --8 Poincare supersymmetries (very similar statements for 2d N=(2,2) --4 Poincare SUSIES) 30
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