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Nathan Seiberg QED3 Nathan Seiberg IAS Analyzing a Lagrangian QFT Semiclassical physics, mostly in the UV (reliable, straightforward, but can be subtle) • Global symmetry and its ‘t Hooft anomalies • Weakly coupled limits: flat directions, small parameters, … Quantum physics in the IR (mostly conjectural) • Consistency with the global symmetry (including ‘t Hooft anomalies) and the various semiclassical limits • Approximate methods: lattice, bootstrap, , 1/ , … • Integrability • String constructions 2 QED3 [Many references using various methods] Simple, characteristic example, demonstrating surprising phenomena. Many applications. • (1) gauge field • fermions with charges and masses • A bare Chern-Simons term. – Label the theory as 1 with a parameter . – When all the fermions are massive, at low energies a TQFT 1 with = + sign . 1 2 Since , 1 2 ∑ + . ∈ ℤ 2 2 3 � ∈ ℤ Global symmetries • Charge-conjugation : (with appropriate action on the fermions) → − • 1 , = 0 time-reversal : ( , ) , ( , ) , 0 0 0 (with appropriate action on the fermions )→ − → − − • Standard algebra on all the fundamental fields = = 1 2 = 1 • For equal charges and masses2 − more symmetries, e.g. − . 4 Global magnetic (topological) symmetry • 1 symmetry: = . 1 • The charged operators are monopole operators (like 2 a disorder operator). – Remove a point from spacetime and specify boundary conditions around it. • Massless fermions have zero modes, which can lead to “funny” quantum numbers. 5 Global magnetic (topological) symmetry In many applications, the magnetic symmetry is approximate or absent • In lattice constructions • When the gauge (1) is embedded at higher energies in a non-Abelian gauge group • When the gauge 1 is emergent • In the generalization of the gauge 1 2 to ( ) with higher (only a magnetic symmetry). ≅ ℤ2 It is natural to break 1 explicitly by adding a monopole operator to the Lagrangian. 6 Example: 1 , = 1 with = 1 1 + 2 • Since cannot vanish, ∈ isℤ violated (parity anomaly). • All gauge invariant polynomials in the fundamental fields are bosons. • All gauge invariant monopole operators are bosons. – 1 simplest monopole operator has spin 0 1 – 1 2 simplest monopole operator has spin 1 3 2 7 Example: 1 , = 2 with = 1 [Cordova, Hsin, NS] 0 • All gauge invariant polynomials in the fundamental fields are bosons with integer flavor (isospin) (2). • All gauge invariant monopole operators are bosons with flavor (2) isospin = mod 1. • On a single monopole = 1. More generally, 2 = 2 = 1 rather than the standard2 2−= = 1 = +1. − Related, but distinct statements2 in [Wang,2 T. Senthil; Metlitski, Fidkowski, Chen, Vishwanath; Witten] and− see also Hsin’s poster. 8 Examples: 1 , = 1 with = 2 [Cordova, Hsin, NS] 0 • All gauge invariant polynomials in the fundamental fields are bosons. • All gauge invariant monopole operators have spin = mod 1, i.e. 1 = 1 2 – In the previous example “fractional” isospin. Here “fractional” spin.− − • Standard -symmetry = 1 2 9 − Break the magnetic (1) symmetry [Cordova, Hsin, NS; Gomis, Komargodski, NS] • Add to the Lagrangian a charge 2 monopole operator. – Cannot add a charge 1 monopole – it is a fermion. • It breaks the – magnetic symmetry 1 : 1 – -symmetry → 2 − • But it preserves another time-reversal symmetry (a subgroup of the original symmetry) = ′ 2 10 Break the magnetic (1) symmetry [Cordova, Hsin, NS; Gomis, Komargodski, NS] = ′ 2 • Its algebra is non-standard = ( 1) ’ ′ = 1 ′ = 1 2’ =1 − 1 − − and do not commute.2 ≠ − is not a conventional time-reversal symmetry, but 풯 is. 풯 11 풯 What is the long distance behavior? • What are the phases? – Gapped? Topological? – Symmetry breaking? • What happens at the phase transitions? First or second order? – And if second order, free or interacting? • It is clear for large 1 1 − + ? < 0 > 0 12 CONJECTURES AHEAD 13 Vary at large | | or at large • Large . Essentially free fermions with a modified Gauss law constraint • Large . Second-order transition as a function of the fermion mass [Appelquist, Nash, Wijewardhana] • Conjecture that this is the case for all , , and . (1) (1) 1 2 1 2 − + < 0 2 >20 Second order transition at = 0 14 1 , = 1 with charge one ( = 1) 1 / flows to the (2) Wilson-Fisher fixed point [Chen, Fisher, Wu; Barkeshli, McGreevy; NS, T. Senthil, Wang, Witten;1 Karch2 , Tong] • The spin-zero monopole operator of the gauge theory is the order parameter of the (2) model. • Emergent -symmetry in the IR 1 flows to a fixed point with (3) symmetry / [Aharony, Benini, Hsin, NS; Benini, Hsin, NS] 3 2 • The spin-one monopole operator of the gauge theory becomes a conserved current in O 2 (3). • Several dual bosonic and fermionic descriptions → 15 1 , = 2 with = 1 [Xu, You; Karch, Tong; Hsin, NS; Benini, Hsin, NS; Wang, Nahum,0 Metlitski, Xu, T. Senthil] • Microscopically: global (2), (they do not commute), and -symmetry – and have a non-standard algebra = = 1 • Conjectured IR2 behavior:2 fixed point with enhanced − global (4) symmetry • Dual fermionic description: 1 with = 2 – Its (2) flavor symmetry includes the original 0 1 , and vice versa. 16 1 , = 2 with = 1 [Benini, Hsin, NS; Komargodski, NS] 0 Break the magnetic symmetry by adding to the Lagrangian a double monopole operator • This explicitly breaks the flavor 2 1 and the magnetic 1 . → • For some range of the2 remaining flavor 1 symmetry is spontaneously→ ℤ broken. – The UV theory is massive, but the IR theory is gapless! Goldstone boson < 0 > 0 (2) Wilson-Fisher fixed points 17 1 , = 1 with = 2 [Cordova, Hsin, NS] 0 • Flows to a free Dirac fermion and a decoupled 1 TQFT – Similar, but not identical to [Son; Wang, T. Senthil; 2 Metlitski, Vishwanath; NS, T. Senthil, Wang, Witten]. • The monopole operator in the UV becomes the free fermion in the IR. • The Wilson line with charge 1 in the UV is described as the Wilson line of 1 (semion). 2 18 Break the magnetic (1) symmetry [Cordova, Hsin, NS; Gomis, Komargodski, NS] Add to the Lagrangian a charge 2 monopole operator. = • It preserves with a nonstandard algebra ′ 2 • In the IR it splits the fixed point with a massless Dirac fermion to two points with massless Majorana fermions 1 1 1 2 2 2 < 0 > 0 Massless Majorana fermions 19 Summary QED3: 1 gauge theory with charged fermions. • Subtle global symmetry, especially in the action on monopoles – Fractional (global symmetry) charges and spins – Unusual algebras involving and • Conjectures about the IR behavior of these systems – Enhanced global symmetry – Dual descriptions This is a tiny part of a long story about the dynamics of 2+1 dimensional QFT. Most of it is still unknown… 20.
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