Critical behavior of the QED 3 -Gross-Neveu-Yukawa model at four loops

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1 Critical behavior of the QED 3 -Gross-Neveu-Yukawa model at four loops Nikolai Zerf Institut für Physik Humboldt Universität zu Berlin DESY Zeuthen 2019 [arxiv: ] Nikolai Zerf (IP HUB) CB 4L DESYz / 28

2 Overview 1 The chiral Ising QED 3 -Gross-Neveu-Yukawa Model 2 Observables 3 Renormalization 4 Fixed Point Analysis 5 Results 6 Application Nikolai Zerf (IP HUB) CB 4L DESYz / 28

3 Motivation Obtain reliable & precise predictions In HEP the SM (QFT with L SM ) provides collider observables New physics L real = L SM +L X? New physics includes realization of SUSY? (MSSM, nmssm,...) Origin of L SM? Origin of QFTs? How to proceed? Build and run LHC to find new physics Calculate observables at higher order in PT Investigate on various QFTs (maybe no HEP relevancy) Nikolai Zerf (IP HUB) CB 4L DESYz / 28

4 Work in collaboration with... University of Alberta (Edmonton) Joseph Maciejko Rufus Boyack DESY Zeuthen Peter Marquard Nikolai Zerf (IP HUB) CB 4L DESYz / 28

5 The chiral Ising QED 3 -Gross-Neveu-Yukawa Model chiral Ising Gross-Neveu-model: L GN = Ψ / Ψ+ 1 2 g2 ( Ψ Ψ) 2. Ψ: Fermion with N flavours (closed fermion loops N) 4-Fermion interaction of Ising type (real singlet) Lorenzian Symmetry Renormalizable in d = 2 Nikolai Zerf (IP HUB) CB 4L DESYz / 28

6 The chiral Ising QED 3 -Gross-Neveu-Yukawa Model chiral Ising Gross-Neveu-model: L GN = Ψ / Ψ+ 1 2 g2 ( Ψ Ψ) 2. Ψ: Fermion with N flavours (closed fermion loops N) 4-Fermion interaction of Ising type (real singlet) Lorenzian Symmetry Renormalizable in d = 2 chiral Ising Gross-Neveu-Yukawa-model: L GNY = Ψ / Ψ+gφ Ψ Ψ ( µφ) m2 φ ! λ2 φ 4. Lorenzian Symmetry additional real scalar field φ UV complete in d = 4 (renormalizable) Nikolai Zerf (IP HUB) CB 4L DESYz / 28

7 The chiral Ising QED 3 -Gross-Neveu-Yukawa Model QED 3 : L QED = Ψ γ µ ( µ iea µ )Ψ+ 1 4 F µνf µν + 1 2ξ ( µa µ ) 2. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

8 The chiral Ising QED 3 -Gross-Neveu-Yukawa Model The chiral Ising QED 3 -Gross-Neveu-Yukawa model: L QED3 GNY = Ψ /DΨ+gφ Ψ Ψ ( µφ) m2 φ ! λ2 φ F µνf µν + 1 2ξ ( µa µ ) 2. Ψ: four-component fermion with N flavours UV complete in d = 4 (renormalizable) discrete, chiral Z2 symmetry: Ψ γ 5 Ψ, Ψ = Ψ γ 0 Ψγ 5, φ φ. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

9 Where to find such a QFT? In a condensed matter systems like in layers of graphene (d=2+1) [S.-S.Lee 07] Appearance as LowEnergyTheory in a mean field approximation Lorenzian Symmetry emergent at IR fixed point Band structure yields a linear fermionic dispersion relation at the Dirac-point: Nikolai Zerf (IP HUB) CB 4L DESYz / 28

10 The relevant physical Observables Critical exponents and Anomalous Dimenions at the non-trivial QuantumCricticalPoint in d = 2 + 1: where e, g,λ 0 holds η φ γ φ (e 2, g2,λ2 ), η φ 2 γ φ 2(e 2, g2,λ2 ), η A γ A (e 2, g2,λ2 ), ω... ν In d = 2+1: Strong dynamics Nikolai Zerf (IP HUB) CB 4L DESYz / 28

11 Methods FunctionalRenormalizationGroup MonteCarlo conformal Boot Strap Large N approximation Weak coupling expansion (PT) around d = 4 ǫ Nikolai Zerf (IP HUB) CB 4L DESYz / 28

12 Methods FunctionalRenormalizationGroup MonteCarlo conformal Boot Strap Large N approximation Weak coupling expansion (PT) around d = 4 ǫ Nikolai Zerf (IP HUB) CB 4L DESYz / 28

13 Example: chiral Ising Gross-Neveu UC η φ for N = 8 [Ihrig,Mihaila,Scherer 18]: Nikolai Zerf (IP HUB) CB 4L DESYz / 28

14 Renormalization of χ I -QED 3 -GNY L QED3 GNY = Ψ /DΨ+gφ Ψ Ψ ( µφ) m2 φ ! λ2 φ F µνf µν + 1 2ξ ( µa µ ) 2. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

15 Renormalization of χ I -QED 3 -GNY L 0 QED 3 GNY = Ψ 0 /D 0 Ψ 0 + g 0 φ 0 Ψ0 Ψ ( µφ 0 ) (m2 ) 0 (φ 0 ) ! (λ0 ) 2 (φ 0 ) F0 µν(f 0 ) µν + 1 2ξ 0( µa 0 µ) 2. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

16 Renormalization of χ I -QED 3 -GNY L 0 QED 3 GNY = Ψ 0 /D 0 Ψ 0 + g 0 φ 0 Ψ0 Ψ ( µφ 0 ) (m2 ) 0 (φ 0 ) ! (λ0 ) 2 (φ 0 ) F0 µν(f 0 ) µν + 1 2ξ 0( µa 0 µ) 2. Field and coupling redefinitions (using MS and RS= µ): φ 0 = Z φ φ, Ψ 0 = Z Ψ Ψ, A 0 = Z A A, λ 0 =µ ǫ/2 Z λ λ, g 0 =µ ǫ/2 Z g g, e 0 =µ ǫ/2 Z e e, (m 2 ) 0 =Z m 2m 2 µ 2, ξ 0 =Z ξ ξ. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

17 Renormalization of χ I -QED 3 -GNY L 0 QED 3 GNY = Ψ 0 /D 0 Ψ 0 + g 0 φ 0 Ψ0 Ψ ( µφ 0 ) (m2 ) 0 (φ 0 ) ! (λ0 ) 2 (φ 0 ) F0 µν(f 0 ) µν + 1 2ξ 0( µa 0 µ) 2. Field and coupling redefinitions (using MS and RS= µ): φ 0 = Z φ φ, Ψ 0 = Z Ψ Ψ, A 0 = Z A A, λ 0 =µ ǫ/2 Z λ λ, g 0 =µ ǫ/2 Z g g, e 0 =µ ǫ/2 Z e e, (m 2 ) 0 =Z m 2m 2 µ 2, ξ 0 =Z ξ ξ. Use short hands: Z φ 4 =Z 2 λ Z2 φ, Z ΨΨφ =Z gz Ψ Zφ, Z ΨΨA =Z e Z Ψ ZA, Z φ 2 =Z m 2Z φ. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

18 Renormalization of χ I -QED 3 -GNY L 0 QED 3 GNY =Z Ψ Ψ / Ψ+iµ ǫ/2 Z ΨΨA e Ψ /AΨ+µ ǫ/2 Z ΨΨφ gφ Ψ Ψ + Z φ 1 2 ( µφ) 2 + Z φ m2 µ 2 φ 2 +µ ǫ Z φ 4 1 4! λ2 φ 4 + Z A 1 4 F µνf µν + 1 2ξ ( µa µ ) 2. Field and coupling redefinitions (using MS and RS= µ): φ 0 = Z φ φ, Ψ 0 = Z Ψ Ψ, A 0 = Z A A, λ 0 =µ ǫ/2 Z λ λ, g 0 =µ ǫ/2 Z g g, e 0 =µ ǫ/2 Z e e, (m 2 ) 0 =Z m 2m 2 µ 2, ξ 0 =Z ξ ξ. Use short hands: Z φ 4 =Z 2 λ Z2 φ, Z ΨΨφ =Z gz Ψ Zφ, Z ΨΨA =Z e Z Ψ ZA, Z φ 2 =Z m 2Z φ. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

19 How to obtain Zs? Using DREG get poles in small ǫ = 4 d of 1-PI n-point functions at L = 1, 2, 3, 4 loops for any kinematics Z φ, Z φ 2 Z Ψ Z A Z φ 4 Z ΨΨφ Z ΨΨA Nikolai Zerf (IP HUB) CB 4L DESYz / 28

20 How to obtain Zs? Using DREG get poles in small ǫ = 4 d of 1-PI n-point functions at L = 1, 2, 3, 4 loops for any kinematics Z φ, Z φ 2 Z Ψ Z A Z φ 4 Z ΨΨφ Z ΨΨA Further we obtain Z mψ,1 and Z mψ,a flavour singlet & adjoint mass term: L 0 m 1 =µz Ψ1Ψ m 1 Ψ 1 Ψ, L 0 mta =µz ΨTA Ψ m T A Ψ TA Ψ. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

21 How to obtain Zs? Number of Feynman diagrams: Loops Nikolai Zerf (IP HUB) CB 4L DESYz / 28

22 How to obtain Zs? (technics) Chosing massiv tadpole kinematics Use single IR regulator mass M in every denominator Expand in small external momenta pi /M 1 All integrals are 1-scale tadpole integrals Use infrared rearrangement [Chetyrkin,Misiak,Munz] to consistently subtract artificial terms M 2 Nikolai Zerf (IP HUB) CB 4L DESYz / 28

23 How to obtain Zs? (technics) Chosing massiv tadpole kinematics Use single IR regulator mass M in every denominator Expand in small external momenta pi /M 1 All integrals are 1-scale tadpole integrals Use infrared rearrangement [Chetyrkin,Misiak,Munz] to consistently subtract artificial terms M 2 Fully automatized setup QGRAF [Nogueira] q2e [Seidensticker] exp [Seidensticker] FORM [Vermaseren & Co] CRUSHER [Marquard,Seidel] Nikolai Zerf (IP HUB) CB 4L DESYz / 28

24 19 Fully Massive Master Integrals 2 [Laporta][Czakon 04] Nikolai Zerf (IP HUB) CB 4L DESYz / 28

25 β- & γ- Functions Bare parameters do not depend on RS µ β g 2 = d g2 d lnµ β λ 2 = dλ2 d lnµ β e 2 = de2 d lnµ β g 2 = ǫg 2 +β (1L) g 2 β λ 2 = ǫλ 2 +β (1L) λ β e 2 = ǫe 2 +β (1L) e 2 +β (2L) g 2 +β (2L) λ +β (2L) e 2 γ x = d ln Z x /d lnµ x {A,φ,φ 2,...} γ A = γ (1L) A γ φ = γ (1L) φ γ φ 2 = γ (1L) φ 2 +γ (2L) A +γ (2L) φ +γ (2L) φ 2 +γ (3L) A +γ (3L) φ +γ (3L) φ 2 QED Ward-Identity: β e 2 = ( ǫ+γ A )e 2. +β (3L) g 2 +β (3L) λ +β (3L) e 2 +γ (4L) A, +β (4L) g 2, +β (4L) λ, +β (4L) e 2, +γ (4L) φ, +γ (4L) φ 2. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

26 Showcasing β s - & γ s β (1L) e 2 = 8N 3 e4, β (1L) g 2 = 12e 2 g 2 +2(2N +3)g 4, β (1L) λ 2 = 2Ng 4 +8Ng 2 λ 2 +72λ 4, γ (1L) φ = 4Ng 2, γ (1L) φ 2 = 24λ 2. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

27 Showcasing β s - & γ s β (2L) e = 8Ne 6 4Ne 4 g 2, 2 ( = 24N β (2L) g 2 ) g 6 + ( 40N 3 ) 6 e 4 g 2 +4(5N +12)e 2 g 4 96g 4 λ 2 +96g 2 λ 4, β (2L) λ 2 = 16Ng 6 +28Ng 4 λ 2 288Ng 2 λ λ 6 8Ne 2 g 4 +40Ne 2 g 2 λ 2, γ (2L) φ = 20Ne 2 g 2 10Ng 4 +96λ 4, γ (2L) φ 2 = 8Ng 4 +96Ng 2 λ λ 4. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

28 Showcasing β s - & γ s β (3L) e 2 = 6Ne6 g 2 +2N(7N +6)e 4 g 4 4N 9 (22N +9)e8, β (3L) λ = N (628N ζ3)g8 4 + N 2 (1736N ζ3)g6 λ 2 +12N( 72N ζ 3)g 4 λ Ng 2 λ (145+96ζ 3)λ 8 +N(116N ζ 3)e 4 g 4 2N(32N ζ 3)e 4 g 2 λ 2 +2N( 11+96ζ 3)e 2 g 6 +6N( ζ 3)e 2 g 4 λ N( 17+16ζ [ 3)e 2 g 2 λ 4, β (3L) g = 32N 2 +N(49 432ζ 327 ] 3) ζ3 e 4 g 4 [ ] 560N N(23 24ζ 3) 258 e 6 g 2 27 [ ( ) N 2 +N ζ3 697 ] ζ3 g (5N +7)g 6 λ 2 +24( 30N +91)g 4 λ g 2 λ 6 +[2N( 79+48ζ 3) ζ 3]e 2 g 6 +96e 2 g 4 λ 2. γ (3L) φ = 3N(7+16ζ 3)e 2 g 4 + N (200N ζ3)g6 4 +N( 32N ζ 3)e 4 g Ng 4 λ 2 720Ng 2 λ λ 6. γ (3L) φ = 32N(4N 2 9+3ζ3)g6 +12N(24N ζ 3)g 4 λ Ng 2 λ λ 6 +32N( 7+9ζ 3)e 2 g 4 +72N(17 16ζ 3)e 2 g 2 λ 2. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

29 Showcasing β s - & γ s β (4L) e 2 = 2N 3 [N( ζ3) ζ3]e6 g 4 + 2N 9 (322N ζ3)e8 g 2 240Ne 4 g 4 λ Ne 4 g 2 λ 4 4N 243 [616N2 +36N( ζ3)+5589]e 10 4N 9 [27N2 +N(299 24ζ3)+105+9ζ3]e 4 g 6, β (4L) λ = N [ ] 64N 2 2 ( ζ3) 32N( ζ3 288ζ4 1920ζ5)+3( ζ3 3696ζ ζ5) 24 [ g 10 N ζ5) ] 4N 2 ( ζ3)+N( ζ ζ ζ5) 3( ζ3 984ζ4 g 8 λ 2 +N[2304N 2 ( 1+2ζ3) 8N( ζ3 3888ζ4) ζ ζ ζ5]g 6 λ 4 +64N[6N(263+72ζ3) ζ3 5184ζ ζ5]g 4 λ 6 576N( ζ3 1728ζ4)g 2 λ ( ζ3 864ζ ζ5)λ 10 4N 243 [16N2 ( ζ3) 81N(185+16ζ3 240ζ4) 81( ζ3 +432ζ4 +240ζ5)]e 6 g 4 [ ] 32N N ( ζ3)+8N( ζ3 +96ζ4) ζ3 864ζ4 2880ζ5 e 6 g 2 λ N 6 [8N( ζ3 +336ζ ζ5)+3( ζ ζ ζ5)]e4 g 6 + 2N 9 [2N( ζ3 144ζ ζ5) 9( ζ3 7920ζ ζ5)]e4 g 4 λ 2 12N[4N( ζ3 48ζ4)+3( ζ3 +432ζ4 +960ζ5)]e 4 g 2 λ 4 γ (4L) φ = N 243 [16N2 ( ζ3)+972N( ζ3 +96ζ4)+81( ζ3 1296ζ4 4320ζ5)]e 6 g 2 2N 3 [N2 ( ζ3)+N( ζ3 +108ζ4) ζ3 +90ζ4 +120ζ5]g 8 + N 9 [N( ζ ζ4 5760ζ5) 9( ζ3 1008ζ4 +480ζ5)]e4 g 4 16N(76N ζ3)g 6 λ 2 64N(3N ζ3)g 4 λ Ng 2 λ λ 8 + N 12 [32N(161+96ζ3 72ζ4) ζ3 3888ζ4 1440ζ5]e2 g 6 944Ne 2 g 4 λ N( 67+48ζ3)e 2 g 2 λ 4, N 3 [3N( ζ3 1200ζ4)+4( ζ3 1566ζ ζ5)]e2 g 8 + N 6 [16N( ζ3 2736ζ4) ζ ζ ζ5]e2 g 6 λ 2 8N[9N( ζ3 144ζ4) ζ3+1080ζ ζ5]e 2 g 4 λ N( ζ3)e [ 2 g 2 λ 6., ( ) β (4L) 88N 3 g = 2N ( ζ3 324ζ4)+N 2648ζ3 +552ζ4 1680ζ ζ3 +342ζ ] [ ( ) 32N 1720ζ5 g N (83 144ζ3)+ ( ζ3 162ζ4)+N 288ζ ζ ] [ 16N +1344ζ3 e 8 g 2 3 4N2 + ( ζ3) ( ζ3 7776ζ4) ( ) 35 +N ζ3 +912ζ ζ ] 7464ζ ζ ζ5 e 6 g [96N 2 8N( ζ3) 3( ζ3)]g 8 λ 2 8[24N 2 +4N( ζ3)+135(33+40ζ3)]g 6 λ 4 ( +576(8N ζ3)g 4 λ g 2 λ 8 + [N ) +928ζ3 +64ζ4 640ζ5 9 ( +N ) ζ3 528ζ ζ ] +6024ζ3 432ζ ζ5 e 4 g ( ) [ ( ) 14944N ζ3 e 4 g 4 λ 2 + N 2 ( ζ3 +96ζ4)+N +2056ζ3 756ζ4 1400ζ ] +3386ζ3 918ζ4 1460ζ5 e 2 g 8 +16[N( ζ3) ζ3]e 2 g 6 λ [6N( 67+48ζ3) 815]e 2 g 4 λ 4. γ (4L) φ 2 = N 2 [64N2 ( 11+18ζ3)+8N( ζ3 +108ζ4 +560ζ5) ζ3 2016ζ ζ5]g 8 3N[256N 2 ( 1+2ζ3) 8N( ζ3 240ζ4) ζ ζ4 5760ζ5]g 6 λ 2 96N[16N(11+3ζ3) ζ3 216ζ4]g 4 λ N(313+96ζ3)g 2 λ (187+18ζ3 +36ζ4)λ 8 2N 3 [4N( ζ3 72ζ4 +240ζ5) ζ ζ ζ5]e4 g 4 +4N[4N( ζ3 48ζ4)+3( ζ3 +432ζ4 +960ζ5)]e 4 g 2 λ 2 4N[3N(89+192ζ3 96ζ4) ζ3 720ζ4 200ζ5]e 2 g 6 +8N[N( ζ3 432ζ4) ζ3 504ζ ζ5]e 2 g 4 λ N( 49+48ζ3)e 2 g 2 λ 4. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

30 Determination of e, g,λ IR fixed point is reached for µ : {β e 2,β g 2,β λ 2} T = 0. Solve for zeros in {e, g,λ} loop by loop in dep. of ǫ χ I -QED 3 -GNY fixed point (1-Loop): e 2 = 3 8N ǫ+o(ǫ2 ), g 2 = 2N + 9 4N(2N + 3) ǫ+o(ǫ2 ), λ 2 2N 15+X = 144N(2N + 3) ǫ+o(ǫ2 ), X 4N N N N. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

31 Determination of e, g,λ N = 1: [Janssen,He 17] 1.5 e 2 = 3ǫ/4 1.0 QED 3-GN g 2 /ǫ G WF λ/ǫ Nikolai Zerf (IP HUB) CB 4L DESYz / 28

32 Critical Exponents inverse correlation length exponent ν 1 : RG eigenvalue of (relevant) scalar mass term: ν 1 = d ln(m2 ) d lnµ = 2+η φ 2 η φ. (e 2,g 2,λ2 ) stability exponent ω: ω = min(ev[j]), β e 2 β e 2 β e 2 e 2 g 2 λ 2 β J = g 2 β g 2 β g 2 e 2 g 2 λ 2 β λ 2 β λ 2 β λ 2 e 2 g 2 λ 2 (e 2,g 2,λ 2 ). stable FP/CP: ω > 0. unstable FP/CP: ω < 0. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

33 Quantum Phasetransitions Phenomenology of Quantum Phasetransitions x = (X X c )/X c ξ x ν (1+C x ω +...) correlation length around Quantum Critical Point (QCP) ξ correlation length exponent ν subleading/stability exponent ω Nikolai Zerf (IP HUB) CB 4L DESYz / 28

34 Excursion: FS Fermion Bilinear During the calculation of fermion mass renormalization: Nikolai Zerf (IP HUB) CB 4L DESYz / 28

35 Excursion: FS Fermion Bilinear During the calculation of fermion mass renormalization: Nikolai Zerf (IP HUB) CB 4L DESYz / 28

36 Excursion: FS Fermion Bilinear During the calculation of fermion mass renormalization: Nikolai Zerf (IP HUB) CB 4L DESYz / 28

37 Excursion: FS Fermion Bilinear During the calculation of fermion mass renormalization: L m1 mixes (in d = 4) with: L φ 3 = 1 3! hφ3 Nikolai Zerf (IP HUB) CB 4L DESYz / 28

38 Excursion: FS Fermion Bilinear During the calculation of fermion mass renormalization: L m1 mixes (in d = 4) with: L φ 3 = 1 3! hφ3 flavour singlet fermion mass term L m1 breaks chiral Z 2 Symmetry Coupled system for β h and β m1 Nikolai Zerf (IP HUB) CB 4L DESYz / 28

39 Excursion: FS Fermion Bilinear During the calculation of fermion mass renormalization: L m1 mixes (in d = 4) with: L φ 3 = 1 3! hφ3 flavour singlet fermion mass term L m1 breaks chiral Z 2 Symmetry Coupled system for β h and β m1 Nikolai Zerf (IP HUB) CB 4L DESYz / 28

40 Excursion: FS Fermion Bilinear During the calculation of fermion mass renormalization: L m1 mixes (in d = 4) with: L φ 3 = 1 3! hφ3 flavour singlet fermion mass term L m1 breaks chiral Z 2 Symmetry Coupled system for β h and β m1 Its Eigenvalues at fixed point reproduce corresponding large N results expanded up to O(ǫ 4 ) Nikolai Zerf (IP HUB) CB 4L DESYz / 28

41 Excursion: FS Fermion Bilinear During the calculation of fermion mass renormalization: L m1 mixes (in d = 4) with: L φ 3 = 1 3! hφ3 flavour singlet fermion mass term L m1 breaks chiral Z 2 Symmetry Coupled system for β h and β m1 Its Eigenvalues at fixed point reproduce corresponding large N results expanded up to O(ǫ 4 ) L ma does not mix (tr[t A ] = 0) Nikolai Zerf (IP HUB) CB 4L DESYz / 28

42 Fermion Bilinear Scaling β-function for both masses and φ 3 coupling: x {1, T A } FS operator mixing yields: β mx = d m x d lnµ = ( 1 γ ΨxΨ +γ Ψ)m x, β h = d h d lnµ = ( ǫ γ φ γ φ)h. (β m1,β h ) T = M (m 1, h) T, Λ ± = 1 2 (M 11 + M 22 )± 1 4 (M 11 M 22 ) 2 + M 12 M 21. Scaling dimension of ΨxΨ and φ 3 at fixed point: ΨT A Ψ =d 1 η ΨT A Ψ +η Ψ, Ψ1Ψ =d+ Λ (e 2,g 2,λ2 ), φ 3 =d+ Λ + (e 2,g 2,λ2 ). Nikolai Zerf (IP HUB) CB 4L DESYz / 28

43 Checks Full agreement with indep. calc. at 1-Loop [Janssen,He 17] & 3-Loop [Ihrig,Mihaila,Scherer 18] e = 0, g = 0 φ 4 Ising Universality Class [Vladimirov,Kazakov,Tarasov 79] g = 0, λ = 0 QED [Gorishny,Kataev,Larin,Surguladze 91] e = 0 χ I GNY [Zerf,Mihaila,Marquard,Herbut,Scherer, 17] large N calculations [Manashov,Strohmaier,18][Gracey 92 18] ξ dependence in γ Ψ only linear and only in γ (1L) Ψ [Gracey, 7][Kißler,Kreimer 17] Nikolai Zerf (IP HUB) CB 4L DESYz / 28

44 Results At 1-Loop: [Janssen,He 17] η φ = 2N + 9 2N + 3 ǫ+o(ǫ2 ), ν 1 = 2 10N2 + 39N + X ǫ+o(ǫ 2 ), 6N(2N + 3) ω = ǫ+o(ǫ 2 ), ( ) 2N + 6 ΨxΨ = 3 ǫ+o(ǫ 2 ). 2N + 3 Nikolai Zerf (IP HUB) CB 4L DESYz / 28

45 Results At 4-Loop (N = 1 numeric rep.): η φ 2.2ǫ ǫ ǫ ǫ 4 +O(ǫ 5 ), ν ǫ ǫ ǫ ǫ 4 +O(ǫ 5 ), ω ǫ+0.3ǫ ǫ ǫ 4 +O(ǫ 5 ), Ψ1Ψ 3 1.6ǫ ǫ ǫ ǫ 4 +O(ǫ 5 ), ΨTAΨ 3 1.6ǫ+1.987ǫ ǫ ǫ 4 +O(ǫ 5 ). Nikolai Zerf (IP HUB) CB 4L DESYz / 28

46 Results At 4-Loop (N = 1 numeric rep.): η φ 2.2ǫ ǫ ǫ ǫ 4 +O(ǫ 5 ), ν ǫ ǫ ǫ ǫ 4 +O(ǫ 5 ), ω ǫ+0.3ǫ ǫ ǫ 4 +O(ǫ 5 ), Ψ1Ψ 3 1.6ǫ ǫ ǫ ǫ 4 +O(ǫ 5 ), ΨTAΨ 3 1.6ǫ+1.987ǫ ǫ ǫ 4 +O(ǫ 5 ). naive extrapolation to d = 3 means ǫ = 1! Nikolai Zerf (IP HUB) CB 4L DESYz / 28

47 Results At 4-Loop (N = 1 numeric rep.): η φ 2.2ǫ ǫ ǫ ǫ 4 +O(ǫ 5 ), ν ǫ ǫ ǫ ǫ 4 +O(ǫ 5 ), ω ǫ+0.3ǫ ǫ ǫ 4 +O(ǫ 5 ), Ψ1Ψ 3 1.6ǫ ǫ ǫ ǫ 4 +O(ǫ 5 ), ΨTAΨ 3 1.6ǫ+1.987ǫ ǫ ǫ 4 +O(ǫ 5 ). naive extrapolation to d = 3 means ǫ = 1! Use Padé / Padé-Borel approximants for extrapolation Nikolai Zerf (IP HUB) CB 4L DESYz / 28

48 Application Conjectured duality in d = 3 at QCP [Wang,Nahum,Metlitski,Xu,Senthil 17]: χ I -QED 3 -GNY(N = 1) SU(2)NCCP 1. SU(2)NCCP 1 : L CP 1 = α=1,2 D b z α 2 ( z z 2 2 ) 2. zα : complex bosonic spinon field with SU(2) flavour Db : covariant derivative containing NC U(1) gauge field b µ EFT for Néel VBS phase transitions by spin 1/2 magnets on square lattice Emergent SO(5) symmetry at the QCP (seen in num. calc. [Nahum,Serna,Chalker,. 15]) Nikolai Zerf (IP HUB) CB 4L DESYz / 28

49 Web of Dualities CP 1 z σ z z η Néel = η VBS M b QED 3 GN η Néel = η φ φ η VBS = η φ ν CP 1 = ν QED3 GN φ 2 z z [ Ψσ z Ψ]=3 1/ν QED3 GN 1/ν CP 1 =3 [ Ψσ z Ψ] Ψσ z Ψ [Ihrig,Mihaila,Scherer 18] Nikolai Zerf (IP HUB) CB 4L DESYz / 28

50 Web of Dualities Duality/SO(5) symmetry implies relation within χ I -QED 3 -GNY(N = 1): ΨT A Ψ = 3 ν 1. Results for η φ can be directly compared: η φ = η Néel = η VBS. Nikolai Zerf (IP HUB) CB 4L DESYz / 28

51 Padée Extrapolation ( ΨTA Ψ = 3 ν 1.) 3.0 _ T A (N=1) [0/2] [1/1] [0/3] [1/2] [2/1] [1/3] [2/2] [3/1] 1/ (N=1) Nikolai Zerf (IP HUB) CB 4L DESYz / 28 [0/2] [1/1] [0/3] [1/2] [2/1] [1/3] [2/2] [3/1]

52 Padée Extrapolation η φ 4 (N=1) [1/1] [2/1] [1/3] [2/2] [3/1] Nikolai Zerf (IP HUB) CB 4L DESYz / 28

53 Comparing with Literature N = 1 1/ν 3 1/ν η φ ΨTAΨ [0/2] [0/2] PB [1/1] [1/1] PB [0/3] [0/3] PB [1/2] [1/2] PB 2.14 [2/1] [2/1] PB [0/4] [0/4] PB [1/3] [1/3] PB [2/2] [2/2] PB 1.74 [3/1] [3/1] PB NCCP 1 χ I -QED 3 -GNY (N = 1) η Néel 0.26(3) [1] η φ 2.1(1) [2] 0.35(3) [3] 1.3(9) [4] 0.30(5) [5] 0.22 [6] 0.259(6) [7] η VBS 0.28(8) [5] 0.25(3) [7] 3 1/ν 1.72(5) [1] 3 1/ν 2.33(1) [2] 1.53(9) [3] 2.7(4) [4] 1.15(19) [5] 1.21 [6] ΨTAΨ 2.12(50) [2] 1 [7] 1.8(5) [4] 0.76(4) [8] [1] [Sandvik 07] [2] [Ihrig,Janssen,Mihaila,Scherer 18] [3] [Melko,Kaul 08] [4] [Janssen,He 17] [5] [Pujari,Damle,Alet 13] [6] [Bartosch 13] [7] [Nahum,Serna,Chalker,. 15] [8] [Shao,Guo,Sandvik 16] Nikolai Zerf (IP HUB) CB 4L DESYz / 28

54 Summary Renormalization of χ I -QED 3 -GNY around d = 4 performed for general N at the four loop level Fixed point analysis of η φ,ν 1,ω, ΨTA Ψ, Ψ1Ψ via ǫ expansion Padé extrapolation to d = 3 Inconclusive numerical results concerning the duality for N = 1 χ I -QED 3 -GNY(N = 1) SU(2)NCCP 1. Five loop analysis? Nikolai Zerf (IP HUB) CB 4L DESYz / 28

55 To backup slides Nikolai Zerf (IP HUB) CB 4L DESYz

56 Pure QED 3 at 5-Loop (d=3) QED3 _ N [0/2] [1/1] [0/3] [2/1] [1/3] [2/2] [3/1] [0/5] [3/2] [4/1] Nikolai Zerf (IP HUB) CB 4L DESYz

57 Padé Approximants Padé: [m/n](ǫ) m i=0 a iǫ i 1+ n j=1 b jǫ j, Borel: (ǫ) = k k ǫ k, B (ǫ) k (ǫ) = 0 k k! ǫk, dt e t B (ǫt). Nikolai Zerf (IP HUB) CB 4L DESYz

58 References [1] A. W. Sandvik, Evidence for Deconfined Quantum Criticality in a Two- Dimensional Heisenberg Model with Four-Spin Interactions, Phys. Rev. Lett. 98, (2007). [2] B. Ihrig, L. Janssen, L. N. Mihaila and M. M. Scherer, Deconfined criticality from the QED 3 -Gross-Neveu model at three loops, Phys. Rev. B 98, no. 11, (2018) [3] R. G. Melko and R. K. Kaul, Scaling in the Fan of an Unconventional Quantum Critical Point, Phys. Rev. Lett. 100, (2008). [4] L. Janssen and Y.-C. He, Critical behavior of the QED 3 -Gross-Neveu model: Duality and deconfined criticality, Phys. Rev. B 96, (2017). [5] S. Pujari, K. Damle, and F. Alet, Néel-State to Valence-Bond-Solid Transition on the Honeycomb Lattice: Evidence for Deconfined Criticality, Phys. Rev. Lett. 111, (2013). [6] L. Bartosch, Corrections to scaling in the critical theory of deconfined criticality, Phys. Rev. B 88, (2013). [7] A. Nahum, J. T. Chalker, P. Serna, M. Ortuño, and A. M. Somoza, Deconfined Quantum Criticality, Scaling Violations, and Classical Loop Models, Phys. Rev. X 5, (2015). [8] H. Shao, W. Guo, and A. W. Sandvik, Quantum criticality with two length scales, Science 352, 213 (2016). Nikolai Zerf (IP HUB) CB 4L DESYz

Theory of Elementary Particles homework VIII (June 04)

Theory of Elementary Particles homework VIII June 4) At the head of your report, please write your name, student ID number and a list of problems that you worked on in a report like II-1, II-3, IV- ).