Properties of monopole operators in 3d gauge theories

Transcription

1 Properties of monopole operators in 3d gauge theories Silviu S. Pufu Princeton University Based on: arxiv: arxiv: (with Ethan Dyer and Mark Mezei) work in progress with Ethan Dyer, Mark Mezei, and Subir Sachdev KITP, January 31, 2014 Silviu Pufu (Princeton University) / 20

2 Abelian gauge theories in 3d Consider (compact) U(1) gauge theory in 3 dimensions: IR dynamics: L = 1 4e 2 F 2 µν + matter w/ integer U(1) charges. No matter = confinement [Polyakov]. Lots of matter = interacting CFT [Appelquist,... ]. Maxwell term is irrelevant. Can be studied in 1/N f expansion. Few matter fields: the theory may confine. Analog of χsb if we have few fermions [Pisarski; Vafa, Witten;... ] When should we expect an interacting CFT? What is the operator spectrum of this CFT? Silviu Pufu (Princeton University) / 20

3 Monopole operators A U(1) gauge theory in 3d has a global U(1) top symmetry with conserved current jµ top = 1 4π ɛ µνρf νρ. Monopole operators are local operators with non-zero U(1) top charge. Monopole operators are disorder operators that insert a monopole singularity in the gauge field A monopole of charge q Z/2 centered at the origin: S 2 F = 4πq. Monopole operator M q (0) satisfies OPE M q (0) F M q (0)4πq +. S 2 Lots of operators with given q. Silviu Pufu (Princeton University) / 20

4 Monopole operators A U(1) gauge theory in 3d has a global U(1) top symmetry with conserved current jµ top = 1 4π ɛ µνρf νρ. Monopole operators are local operators with non-zero U(1) top charge. Monopole operators are disorder operators that insert a monopole singularity in the gauge field A monopole of charge q Z/2 centered at the origin: S 2 F = 4πq. Monopole operator M q (0) satisfies OPE M q (0) F M q (0)4πq +. S 2 Lots of operators with given q. Silviu Pufu (Princeton University) / 20

5 Monopole operators in CFT In CFT: use state-operator correspondence and study the theory on S 2 R. Let M q (0) correspond to the ground state in the sector of monopole flux S 2 F = 4πq [Borokhov, Kapustin, Wu 02]. Silviu Pufu (Princeton University) / 20

6 Roles played by monopole operators Mechanism for confinement: monopole proliferation in theories where U(1) top is explicitly broken microscopically [Polyakov]. Key insight: failure of the Bianchi identity implies Wilson loop area law. If monopole operators are irrelevant = confinement (e.g. [Hermele, Senthil, Fisher, Lee, Nagaosa, Wen] ). Order parameters for continuous phase transitions between two ordered phases (i.e. which evade the Landau-Ginsburg-Wilson paradigm) [Sachdev, Read 89]. A discrete Z k subgroup of U(1) top SO(2) rot is a symmetry of the lattice Hamiltonian. Relevant monopole ops that transform under Z k : order parameters. Relevant monopole ops invariant under Z k : can cause confinement. Silviu Pufu (Princeton University) / 20

7 Roles played by monopole operators Mechanism for confinement: monopole proliferation in theories where U(1) top is explicitly broken microscopically [Polyakov]. Key insight: failure of the Bianchi identity implies Wilson loop area law. If monopole operators are irrelevant = confinement (e.g. [Hermele, Senthil, Fisher, Lee, Nagaosa, Wen] ). Order parameters for continuous phase transitions between two ordered phases (i.e. which evade the Landau-Ginsburg-Wilson paradigm) [Sachdev, Read 89]. A discrete Z k subgroup of U(1) top SO(2) rot is a symmetry of the lattice Hamiltonian. Relevant monopole ops that transform under Z k : order parameters. Relevant monopole ops invariant under Z k : can cause confinement. Silviu Pufu (Princeton University) / 20

8 Monopole operators as order parameters Square lattice antiferromagnet with SU(N) spins at each site. J-Q model: H = J ij S α β(i)s β α(j) + Q (four spins) S = d 2 r dτ [ ( µ ia µ )z α 2 + s z α 2 + u( z α 2 ) ] 4e 2 F µν 2 At the critical point e = and s is tuned to zero = same universality class as the CP N 1 model [Motrunich, Vishwanath 04; Senthil, Balents, Fisher, Sachdev, Vishwanath 04]. Néel order: z α 0; VBS order: M 1/2 0. [Sachdev, Read] Silviu Pufu (Princeton University) / 20

9 Spin liquids In other models, obtain compact QED with some number of fermions = spin liquids [Wen,... ]. L = 1 4e 2 F 2 µν + N f ψ α(i /D + /A)ψ α. a=1 Monopole operators that transform under lattice symmetries = VBS order parameters. Monopoles invariant under lattice symmetries can cause confinement. Non-trivial CFT exists only if they are irrelevant [Heremele, Senthil, Fisher, Lee, Nagaosa, Wen 04]. Silviu Pufu (Princeton University) / 20

10 Properties of monopole operators in CFT Task: determine quantum numbers under conformal group (scaling dimension, Lorentz spin) and under the flavor symmetry. Method: 1/N f expansion. Good because gauge field fluctuations are suppressed. In fermionic theory, for instance: Integrate out ψ: S = Z = d 3 x g N f ψ α(i /D + /A)ψ α. α=1 DA exp [ N f tr log(i /D + /A) ] Saddle point approximation A = A + a for some saddle A. At large N f, gauge field fluctuations are suppressed by 1/ N f. Silviu Pufu (Princeton University) / 20

11 Properties of monopole operators in CFT Task: determine quantum numbers under conformal group (scaling dimension, Lorentz spin) and under the flavor symmetry. Method: 1/N f expansion. Good because gauge field fluctuations are suppressed. In fermionic theory, for instance: Integrate out ψ: S = Z = d 3 x g N f ψ α(i /D + /A)ψ α. α=1 DA exp [ N f tr log(i /D + /A) ] Saddle point approximation A = A + a for some saddle A. At large N f, gauge field fluctuations are suppressed by 1/ N f. Silviu Pufu (Princeton University) / 20

12 A brief history of the 1/N f expansion The leading order epoch (treat gauge field as background): [Murthy, Sachdev 90] : bosonic theory, scaling dim s to order O(N f ). [Borokhov, Kapustin, Wu 02] : fermionic theory, scaling dim s to order O(N f ), flavor charges of M 1/2. [Metlitski, Hermele, Senthil, Fisher 08] : bosonic theory, scaling dim s to order O(N f ). Assumption: rotationally-invariant saddle. The subleading order epoch (fluctuations of the gauge field): [SSP 13] : fermionic theory, scaling dim of M 1/2 to order O(N 0 f ). [Dyer, Mezei, SSP 13] : fermionic theory, scaling dim s to order O(N 0 f ) for all M q; flavor symmetry charges; generalization to QCD. [Dyer, Mezei, SSP, Sachdev, in progress] : bosonic theory, scaling dim s to order O(N 0 f ). Silviu Pufu (Princeton University) / 20

13 A brief history of the 1/N f expansion The leading order epoch (treat gauge field as background): [Murthy, Sachdev 90] : bosonic theory, scaling dim s to order O(N f ). [Borokhov, Kapustin, Wu 02] : fermionic theory, scaling dim s to order O(N f ), flavor charges of M 1/2. [Metlitski, Hermele, Senthil, Fisher 08] : bosonic theory, scaling dim s to order O(N f ). Assumption: rotationally-invariant saddle. The subleading order epoch (fluctuations of the gauge field): [SSP 13] : fermionic theory, scaling dim of M 1/2 to order O(N 0 f ). [Dyer, Mezei, SSP 13] : fermionic theory, scaling dim s to order O(N 0 f ) for all M q; flavor symmetry charges; generalization to QCD. [Dyer, Mezei, SSP, Sachdev, in progress] : bosonic theory, scaling dim s to order O(N 0 f ). Silviu Pufu (Princeton University) / 20

14 Scaling dimensions in fermionic theory The IR scaling dimension of M q are [Borokhov, Kapustin, Wu 02; SSP 13; Dyer, Mezei, SSP 13] : q scaling dimension [M q ] 0 0 1/ N f O(1/N f ) N f O(1/N f ) 3/ N f O(1/N f ) N f O(1/N f ) 5/ N f O(1/N f ) M 1/2 is irrelevant, provided: [M 1/2 ] > 3 = N f 12. So we expect the theory does not confine whenever N f 12. F-theorem = confinement impossible for N f 12. Also [Grover]. Silviu Pufu (Princeton University) / 20

15 Scaling dimensions in fermionic theory The IR scaling dimension of M q are [Borokhov, Kapustin, Wu 02; SSP 13; Dyer, Mezei, SSP 13] : q scaling dimension [M q ] 0 0 1/ N f O(1/N f ) N f O(1/N f ) 3/ N f O(1/N f ) N f O(1/N f ) 5/ N f O(1/N f ) M 3/2 is irrelevant, provided: [M 3/2 ] > 3 = N f 4. So we expect the theory does not confine whenever N f 4 if M 1/2 and M 1 transform non-trivially under lattice symmetries. Silviu Pufu (Princeton University) / 20

16 Flavor quantum numbers in fermionic theory N f fermions in U(1) gauge theory have SU(N f ) flavor symmetry. How do monopole operators transform in representations of SU(N f )? Step 1: Ground state of N f fermions on S 2 R in the presence of uniform magnetic flux F = q sin θdθ dφ. Step 2: Take into account the effect of having a dynamical gauge field. Silviu Pufu (Princeton University) / 20

17 Step 1: Fermions on S 2 in uniform magnetic flux The Dirac equation (i /D + /A)ψ α = 0 has solutions with energy ± (j + 1/2) 2 q 2 transforming in the spin-j irrep of SU(2) rot. 2 q N f zero-energy modes with j = q 1/2 (for each flavor there are 2j + 1 zero-energy modes). Creation and annihilation operators c α jm and cα jm, respectively. Ground state: non-zero energy modes are not excited. Ground state Hilbert space G has dimension 2 2 q N f. It is spanned by Ω, c α jm Ω, cα 1 jm 1 c α 2 jm 2 Ω, etc. Silviu Pufu (Princeton University) / 20

18 Step 1: Fermions on S 2 in uniform magnetic flux The Dirac equation (i /D + /A)ψ α = 0 has solutions with energy ± (j + 1/2) 2 q 2 transforming in the spin-j irrep of SU(2) rot. 2 q N f zero-energy modes with j = q 1/2 (for each flavor there are 2j + 1 zero-energy modes). Creation and annihilation operators c α jm and cα jm, respectively. Ground state: non-zero energy modes are not excited. Ground state Hilbert space G has dimension 2 2 q N f. It is spanned by Ω, c α jm Ω, cα 1 jm 1 c α 2 jm 2 Ω, etc. Silviu Pufu (Princeton University) / 20

19 Step 2: Dynamical gauge field At IR fixed point, ignore Maxwell field. The action is S = d 3 x N f ψ α(i /D + /A + /a)ψ α. α=1 The path integral over a µ imposes j µ = N f α=1 ψ αγ µ ψ α = 0. The space of physical ground states G phys consists of those states of G that satisfy j µ (x) χ = 0. Silviu Pufu (Princeton University) / 20

20 Step 2: Dynamical gauge field The current is: j µ (x) = ( c α jm cα jm N ) f 2 δ mm S q,jm (θ, φ)γµ S q,jm (θ, φ) +. m,m Integrate j τ (x) against Y 00 (θ, φ) = total charge constraint: m,α cα jm cα jm χ = q N f χ. Integrate j τ (x) against Y 1m (θ, φ) = total spin vanishes. Integrate j τ (x) against Y lm (θ, φ), l > 1 = more complicated constraints. All l 1 constraints: invariance under the SU(2 q ) that rotates the m index of c α jm. Silviu Pufu (Princeton University) / 20

21 Fun with group theory Package all c α jm into a column vector of length 2 q N f and let SU(2 q N f ) act on it. Clearly, SU(2 q N f ) SU(N f ) SU(2 q ). Group theory question: find SU(2 q ) singlets under the decomposition of the rank- q N f anti-symmetric tensor irrep of SU(2 q N f ). Answer: only one SU(2 q ) singlet transforming in the SU(N f ) irrep with Young diagram { N f /2 } {{ } 2 q Silviu Pufu (Princeton University) / 20

22 Fun with group theory Package all c α jm into a column vector of length 2 q N f and let SU(2 q N f ) act on it. Clearly, SU(2 q N f ) SU(N f ) SU(2 q ). Group theory question: find SU(2 q ) singlets under the decomposition of the rank- q N f anti-symmetric tensor irrep of SU(2 q N f ). Answer: only one SU(2 q ) singlet transforming in the SU(N f ) irrep with Young diagram { N f /2 } {{ } 2 q Silviu Pufu (Princeton University) / 20

23 Quantum numbers of monopole operators in fermionic theory M q transforms under SU(N f ) as [Dyer, Mezei, SSP 13] : { N f /2 } {{ } 2 q These operators are Lorentz scalars b/c SU(2) rot SU(2 q ). Scaling dimensions q scaling dimension [M q ] 0 0 1/ N f O(1/N f ) N f O(1/N f ) 3/ N f O(1/N f ) N f O(1/N f ) 5/ N f O(1/N f ) Silviu Pufu (Princeton University) / 20

24 Generalization to U(N c ) QCD w/ N f fundamentals Monopole operator with lowest scaling dimension: = 0.265N f (N c 1) O(N 2 c /N f ). Predictions: U(1) deconfines when N f 12, U(2) when N f 14, U(3) when N f 16, etc. More complicated monopole operators are parameterized by {q a Z/2}, a = 1, 2,..., N c. They transform as { N f /2 } {{ } 2 Nc a=1 qa Not all {q a } yield independent operators!! [Dyer, Mezei, SSP 13] Silviu Pufu (Princeton University) / 20

25 Open questions Can the properties of monopole operators I described be verified through lattice QED/QCD or through the conformal bootstrap? Do monopole operators play any role in non-supersymmetric (bosonization) dualities? U(N) k + fundamental boson: k if k N. U(N) k + fundamental fermion: k 3/2 if k N. Silviu Pufu (Princeton University) / 20

26 Open questions Can the properties of monopole operators I described be verified through lattice QED/QCD or through the conformal bootstrap? Do monopole operators play any role in non-supersymmetric (bosonization) dualities? U(N) k + fundamental boson: k if k N. U(N) k + fundamental fermion: k 3/2 if k N. Silviu Pufu (Princeton University) / 20

27 Tentative comments on the critical bosonic theory Assuming spherically symmetric flux through S 2 [Sachdev, Murthy 90; Metlitski, Hermele, Senthil, Fisher 08; Dyer, Mezei, SSP, Sachdev, in progress] : q scaling dimension [M q ] 0 0 1/ N b O(1/N b ) N b O(1/N b ) 3/2 negative mode.. Lorentz scalars invariant under SU(N) global symmetry. M 1/2 is irrelevant, provided: [M 1/2 ] > 3 = N b 24. So we expect the theory does not confine whenever N b 24. Silviu Pufu (Princeton University) / 20