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 𝜖𝜖 𝑁𝑁 • 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 𝑈𝑈 𝑎𝑎𝜇𝜇 • A bare Chern-Simons𝑖𝑖 term. 𝑁𝑁𝑓𝑓 𝜓𝜓 𝑞𝑞𝑖𝑖 𝑚𝑚𝑖𝑖 – Label the theory as 1 with a parameter . – When all the fermions are massive, at low energies 𝑈𝑈 𝑘𝑘 𝑘𝑘 a TQFT 1 with = + sign . 1 2 Since 𝑘𝑘𝑙𝑙𝑙𝑙𝑙𝑙, 𝑙𝑙𝑙𝑙𝑙𝑙 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑈𝑈 1 𝑘𝑘 𝑘𝑘 2 ∑ 𝑚𝑚 𝑞𝑞 𝑙𝑙𝑙𝑙𝑙𝑙 + . 𝑘𝑘 ∈ ℤ 2 2 3 𝑘𝑘 � 𝑞𝑞𝑖𝑖 ∈ ℤ 𝑖𝑖 Global symmetries • Charge-conjugation : (with appropriate action on the fermions) 𝒞𝒞 𝑎𝑎𝜇𝜇 → −𝑎𝑎𝜇𝜇 • 1 , = 0 time-reversal : ( , ) , ( , ) , 0 0 0 𝑈𝑈(with appropriate𝑚𝑚 action on the𝒯𝒯 fermions𝑎𝑎 𝑡𝑡 𝑥𝑥 )→ 𝑎𝑎 −𝑡𝑡 𝑥𝑥 𝑎𝑎𝑖𝑖 𝑡𝑡 𝑥𝑥 → −𝑎𝑎𝑖𝑖 −𝑡𝑡 𝑥𝑥 • Standard algebra on all the fundamental fields = = 1 𝒯𝒯2𝒯𝒯 = 𝒞𝒞𝒞𝒞1𝐹𝐹 • For equal charges and𝒯𝒯 masses2 − more𝐹𝐹 symmetries, e.g. 𝒞𝒞𝒞𝒞 − . 4 𝑆𝑆𝑆𝑆 𝑁𝑁𝑓𝑓 Global magnetic (topological) symmetry

• 1 symmetry: = . 𝜇𝜇 1 𝜇𝜇𝜇𝜇𝜇𝜇 • The charged𝑀𝑀 operators are monopole𝜈𝜈 𝜌𝜌 operators (like 𝑈𝑈 𝑗𝑗 2𝜋𝜋 𝜖𝜖 𝜕𝜕 𝑎𝑎 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 𝑈𝑈 • When the gauge 1 is emergent • In the generalization of the gauge 1 2 to ( ) with higher𝑈𝑈 (only a magnetic symmetry). 𝑈𝑈 ≅ 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑁𝑁 𝑁𝑁 ℤ2 It is natural to break 1 explicitly by adding a monopole operator to the Lagrangian. 𝑈𝑈 𝑀𝑀 6 Example: 1 , = 1 with = 1

𝑘𝑘 𝑓𝑓 1 𝑈𝑈 𝑁𝑁 + 𝑞𝑞 2 • Since cannot vanish,𝑘𝑘 ∈ isℤ violated (parity anomaly). • All gauge invariant polynomials in the fundamental fields 𝑘𝑘are bosons. 𝒯𝒯 • All gauge invariant monopole operators are bosons. – 1 simplest monopole operator has spin 0 1 – 𝑈𝑈 1 2 simplest monopole operator has spin 1 3 𝑈𝑈 2 7 Example: 1 , = 2 with = 1 [Cordova, Hsin, NS] 0 𝑓𝑓 • All gauge invariant𝑈𝑈 polynomials𝑁𝑁 in the fundamental𝑞𝑞 fields are bosons with integer flavor (isospin) (2). • All gauge invariant monopole operators are bosons with 𝑆𝑆𝑆𝑆 flavor (2) isospin = mod 1. 𝑀𝑀 • On a single monopole = 1. More generally, 𝑆𝑆𝑆𝑆 2 = 2 = 1 rather than the standard2 𝒯𝒯 2−= 𝑀𝑀= 1 = +1. 𝒯𝒯 𝒞𝒞𝒞𝒞 − Related, but distinct statements2 in [Wang,2 T. Senthil;𝐹𝐹 Metlitski, Fidkowski, Chen, Vishwanath;𝒯𝒯 𝒞𝒞 𝒞𝒞Witten] and− see also Hsin’s poster.

8 Examples: 1 , = 1 with = 2 [Cordova, Hsin, NS] 𝑈𝑈 0 𝑁𝑁𝑓𝑓 𝑞𝑞 • All gauge invariant polynomials in the fundamental fields are bosons. • All gauge invariant monopole operators have spin = mod 1, i.e. 𝑀𝑀 1 = 1 2 – In the previous example𝑀𝑀 “fractional”𝐹𝐹 isospin. Here “fractional” spin.− − • Standard -symmetry = 1 𝒯𝒯 2 𝐹𝐹 9 𝒯𝒯 − Break the magnetic (1) symmetry [Cordova, Hsin, NS; Gomis, Komargodski, NS] 𝑈𝑈 • 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 𝑀𝑀 𝑈𝑈 𝑀𝑀 → 𝒁𝒁2 − • But it preserves another time-reversal symmetry (a 𝒯𝒯 subgroup of the original symmetry) = 𝑖𝑖𝑖𝑖 ′ 2 𝑀𝑀 𝒯𝒯 𝒯𝒯𝑒𝑒 10 Break the magnetic (1) symmetry [Cordova, Hsin, NS; Gomis, Komargodski, NS] = 𝑈𝑈 𝑖𝑖𝑖𝑖 ′ 2 𝑀𝑀 • Its algebra is non-standard𝒯𝒯 𝒯𝒯𝑒𝑒

= ( 1) ’ ′ = 1 ′ = 𝑀𝑀 1 𝒯𝒯2’𝒞𝒞 =𝒞𝒞1𝒞𝒞 𝑀𝑀− 1 𝐹𝐹 𝒞𝒞𝒞𝒞 − − and do not commute.2 𝐹𝐹 𝒯𝒯 ≠ − is not a conventional time-reversal symmetry, but 𝒞𝒞 𝒯𝒯풯 is. 𝒯𝒯′ 11 𝒞𝒞𝒞𝒞′ 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 𝑚𝑚 1

− + 𝑈𝑈 𝑘𝑘𝑙𝑙𝑙𝑙𝑙𝑙 ? 𝑈𝑈 𝑘𝑘𝑙𝑙𝑙𝑙𝑙𝑙 < 0 > 0 12

𝑚𝑚 𝑚𝑚 CONJECTURES AHEAD

13 Vary at large | | or at large

𝑓𝑓 • Large . Essentially𝑚𝑚 free fermions𝑘𝑘 with a modified𝑁𝑁 Gauss law constraint 𝑘𝑘 • Large . Second-order transition as a function of the fermion mass [Appelquist, Nash, Wijewardhana] 𝑁𝑁𝑓𝑓 • Conjecture that this is the case for all , , and . 𝑚𝑚 𝑘𝑘 𝑞𝑞 𝑁𝑁𝑓𝑓 (1) (1)

1 2 1 2 𝑈𝑈 𝑘𝑘− 𝑁𝑁𝑓𝑓𝑞𝑞 𝑈𝑈 𝑘𝑘+ 𝑁𝑁𝑓𝑓𝑞𝑞 < 0 2 >20 Second order transition at = 0 𝑚𝑚 𝑚𝑚 14 𝑚𝑚 1 , = 1 with charge one ( = 1)

1 / flows𝑘𝑘 to𝑓𝑓 the (2) Wilson-Fisher fixed point [Chen𝑈𝑈, Fisher, Wu;𝑁𝑁 Barkeshli, McGreevy; NS, T. Senthil,𝑞𝑞 Wang, 𝑈𝑈Witten;1 Karch2 , Tong] 𝑂𝑂 • The spin-zero monopole operator of the gauge theory is the order parameter of the (2) model. • Emergent -symmetry in the IR 𝑂𝑂 1 flows to a fixed point with (3) symmetry / 𝒯𝒯 [Aharony, Benini, Hsin, NS; Benini, Hsin, NS] 3 2 •𝑈𝑈 The spin-one monopole operator𝑆𝑆𝑆𝑆 of the gauge theory becomes a conserved current in O 2 (3). • Several dual bosonic and fermionic descriptions → 𝑆𝑆𝑆𝑆 15 1 , = 2 with = 1 [Xu, You; Karch, Tong; Hsin, NS; Benini, Hsin, NS; Wang, 𝑈𝑈Nahum,0 Metlitski,𝑁𝑁𝑓𝑓 Xu, T. Senthil]𝑞𝑞 • Microscopically: global (2), (they do not commute), and -symmetry 𝑈𝑈 𝒞𝒞 – and have a non-standard algebra 𝒯𝒯 = = 1 𝒯𝒯 𝒞𝒞𝒞𝒞 • Conjectured IR2 behavior:2 fixed point𝑀𝑀 with enhanced 𝒯𝒯 𝒞𝒞𝒞𝒞 − global (4) symmetry • Dual fermionic description: 1 with = 2 𝑂𝑂 – Its (2) flavor symmetry includes the original 𝑈𝑈 0 𝑁𝑁𝑓𝑓 1 , and vice versa. 𝑆𝑆𝑆𝑆 16 𝑈𝑈 𝑀𝑀 1 , = 2 with = 1 [Benini, Hsin, NS; Komargodski, NS] 𝑈𝑈 0 𝑁𝑁𝑓𝑓 𝑞𝑞 Break the magnetic symmetry by adding to the Lagrangian a double monopole operator • This explicitly breaks the flavor 2 1 and the magnetic 1 . 𝑆𝑆𝑆𝑆 → 𝑈𝑈 • For some range𝑀𝑀 of the2 remaining flavor 1 symmetry𝑈𝑈 is spontaneously→ ℤ broken. 𝑚𝑚 𝑈𝑈 – The UV theory is massive, but the IR theory is gapless! Goldstone boson < 0 > 0

(2) Wilson-Fisher fixed points 17 𝑚𝑚 𝑚𝑚 𝑂𝑂 1 , = 1 with = 2 [Cordova, Hsin, NS] 𝑈𝑈 0 𝑁𝑁𝑓𝑓 𝑞𝑞 • Flows to a free Dirac fermion and a decoupled 1 TQFT 𝜒𝜒 – Similar, but not identical to [Son; Wang, T. Senthil; 𝑈𝑈 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 𝜒𝜒 as the Wilson line of 1 (semion).

2 𝑈𝑈 18 Break the magnetic (1) symmetry [Cordova, Hsin, NS; Gomis, Komargodski, NS] 𝑈𝑈 Add to the Lagrangian a charge 2 monopole operator. = • It preserves 𝑖𝑖𝑖𝑖 with a nonstandard algebra ′ 2 𝑀𝑀 • In the IR it splits𝒯𝒯 the𝒯𝒯 fixed𝑒𝑒 point with a massless Dirac fermion to two points with massless Majorana fermions

1 1 1

𝑈𝑈 2 𝑈𝑈 2 𝑈𝑈 2 < 0 > 0

𝑚𝑚 Massless Majorana fermions 𝑚𝑚 19 Summary

QED3: 1 gauge theory with charged fermions.

• Subtle global𝑘𝑘 symmetry, especially in the action on monopoles𝑈𝑈 – Fractional (global symmetry) charges and spins – Unusual algebras involving and • Conjectures about the IR behavior of these systems 𝒞𝒞 𝒯𝒯 – 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