LIBERATION ON THE WALLS IN GAUGE THEORIES AND ANTI-FERROMAGNETS

Transcription

1 LIBERATION ON THE WALLS IN GAUGE THEORIES AND ANTI-FERROMAGNETS Tin Sulejmanpasic North Carolina State University Erich Poppitz, Mohamed Anber, TS Phys.Rev. D92 (2015) 2, and with Anders Sandvik, Hui Shao and M. Unsal In progress Recent Developments in Semiclassical Probes of Quantum Field Theories UMass Amherst 2016

2 INTRODUCTION: The vacuum structure of gauge theories

3 INTRODUCTION: The vacuum structure of gauge theories The vacuum structure 1st pass

4 INTRODUCTION: The vacuum structure of gauge theories The vacuum structure 1st pass - In pure gauge theorie one global symmetry is center symmetry and is unbroaken - No other global symmetries, no other order parameters - Vacuum is unique

5 INTRODUCTION: The vacuum structure of gauge theories The vacuum structure 1st pass - In pure gauge theorie one global symmetry is center symmetry and is unbroaken - No other global symmetries, no other order parameters - Vacuum is unique However in large N limit, on general grounds (Witten 1998)

6 INTRODUCTION: The vacuum structure of gauge theories The vacuum structure 1st pass - In pure gauge theorie one global symmetry is center symmetry and is unbroaken - No other global symmetries, no other order parameters - Vacuum is unique However in large N limit, on general grounds (Witten 1998) L = N 1 4g 2 N F 2 + i 16 2 N F F

7 INTRODUCTION: The vacuum structure of gauge theories The vacuum structure 1st pass - In pure gauge theorie one global symmetry is center symmetry and is unbroaken - No other global symmetries, no other order parameters - Vacuum is unique However in large N limit, on general grounds (Witten 1998) L = N t Hooft coupling 1 4g 2 N F 2 + i 16 2 N F F

8 INTRODUCTION: The vacuum structure of gauge theories The vacuum structure 1st pass - In pure gauge theorie one global symmetry is center symmetry and is unbroaken - No other global symmetries, no other order parameters - Vacuum is unique However in large N limit, on general grounds (Witten 1998) L = N t Hooft coupling 1 4g 2 N F 2 + i 16 2 N F F Keep fixed

9 INTRODUCTION: The vacuum structure of gauge theories The vacuum structure 1st pass - In pure gauge theorie one global symmetry is center symmetry and is unbroaken - No other global symmetries, no other order parameters - Vacuum is unique However in large N limit, on general grounds (Witten 1998) L = N t Hooft coupling 1 4g 2 N F 2 + i 16 2 N F F And vacuum energy Keep fixed

10 INTRODUCTION: The vacuum structure of gauge theories The vacuum structure 1st pass - In pure gauge theorie one global symmetry is center symmetry and is unbroaken - No other global symmetries, no other order parameters - Vacuum is unique However in large N limit, on general grounds (Witten 1998) L = N t Hooft coupling 1 4g 2 N F 2 + i 16 2 N F F And vacuum energy Keep fixed E( ) =E( +2 )

11 INTRODUCTION: The vacuum structure of gauge theories The vacuum structure 1st pass - In pure gauge theorie one global symmetry is center symmetry and is unbroaken - No other global symmetries, no other order parameters - Vacuum is unique However in large N limit, on general grounds (Witten 1998) L = N t Hooft coupling 1 4g 2 N F 2 + i 16 2 N F F And vacuum energy Keep fixed E( ) =E( +2 ) ) )

12 INTRODUCTION: The vacuum structure of gauge theories The vacuum structure 1st pass - In pure gauge theorie one global symmetry is center symmetry and is unbroaken - No other global symmetries, no other order parameters - Vacuum is unique However in large N limit, on general grounds (Witten 1998) L = N t Hooft coupling 1 4g 2 N F 2 + i 16 2 N F F And vacuum energy Keep fixed E( ) =E( +2 ) ) ) E( ) =N 2 f +2 k N

13 E( ) k=1 k=0 k=2 N=3 0 π 2π 3π 4π

14 E( ) k=1 k=0 k=2 N=3 0 π 2π 3π 4π So pure Yang-Mills has multiple vacua labeled by k, but they are non-degenerate except at θ=(2k+1)π

15 QCD ADJOINT

16 QCD ADJOINT L = 1 2g 2 trf 2 + n f X I=1 I µ D µ I! I -Weyl fermion in adjoint rep.

17 QCD ADJOINT L = 1 I! 2g 2 trf 2 + I e i n f X I=1 I I! U I J J,U 2 SU(N f ) µ D µ I! -Classical U(1) axial symmetry I -Weyl fermion in adjoint rep.

18 QCD ADJOINT L = 1 I! 2g 2 trf 2 + I e i n f X I=1 I I! U I J J,U 2 SU(N f ) µ D µ I! -Classical U(1) axial symmetry I -Weyl fermion in adjoint rep. I 2Nn f anomaly instantons breaks U(1) to Z2Nnf

19 QCD ADJOINT L = 1 I! µ j µ 5 = 2g 2 trf 2 + I e i n f X I=1 I I! U I J J,U 2 SU(N f ) 2Nn f n f 16 2 tr adjf µ Fµ ) µ D µ I! -Classical U(1) axial symmetry I -Weyl fermion in adjoint rep. anomaly instantons breaks U(1) to Z2Nnf Q 5 =2n f NQ top

20 QCD ADJOINT L = 1 I! µ j µ 5 = 2g 2 trf 2 + I e i n f X I=1 I I! U I J J,U 2 SU(N f ) 2Nn f n f 16 2 tr adjf µ Fµ ) µ D µ I! -Classical U(1) axial symmetry I -Weyl fermion in adjoint rep. anomaly instantons breaks U(1) to Z2Nnf Q 5 =2n f NQ top Q mod 2Nn f conserved ) Z2Nnf remains

21 QCD ADJOINT L = 1 I! µ j µ 5 = 2g 2 trf 2 + I e i n f X I=1 I I! U I J J,U 2 SU(N f ) 2Nn f n f 16 2 tr adjf µ Fµ ) µ D µ I! -Classical U(1) axial symmetry I -Weyl fermion in adjoint rep. anomaly instantons breaks U(1) to Z2Nnf Q 5 =2n f NQ top Q mod 2Nn f conserved ) Z2Nnf remains Znf parti belongs to SU(nf) so the symmetry is SU(nf)xZ2N

22 I I 6=0) SU(n f ) Z 2N! SO(n f ) Z 2 Spontanous continuous chiral symmetry breaking

23 I I 6=0) SU(n f ) Z 2N! SO(n f ) Z 2 Spontanous continuous chiral symmetry breaking Z 2N! Z 2

24 I I 6=0) SU(n f ) Z 2N! SO(n f ) Z 2 Spontanous continuous chiral symmetry breaking Z 2N! Z 2 Therefore since the coset Z2N/Z2=ZN the theory has N isolated degenerate vacua

25 In Super YM the same as before, except there is no continuous symmetries, only descrete h i6=0) Z 2N! Z 2 Spontanous (discrete) chiral symmetry breaking

26 In Super YM the same as before, except there is no continuous symmetries, only descrete h i6=0) Z 2N! Z 2 Spontanous (discrete) chiral symmetry breaking In both QCD(adj) and its supersymmetric limit there is a spontanously broken ZN symmetry leading to N isolated, discrete vacua, labeled by k=0,, N-1

27 In Super YM the same as before, except there is no continuous symmetries, only descrete h i6=0) Z 2N! Z 2 Spontanous (discrete) chiral symmetry breaking In both QCD(adj) and its supersymmetric limit there is a spontanously broken ZN symmetry leading to N isolated, discrete vacua, labeled by k=0,, N-1 vacuum k

28 In Super YM the same as before, except there is no continuous symmetries, only descrete h i6=0) Z 2N! Z 2 Spontanous (discrete) chiral symmetry breaking In both QCD(adj) and its supersymmetric limit there is a spontanously broken ZN symmetry leading to N isolated, discrete vacua, labeled by k=0,, N-1 vacuum k vacuum k+1

29 In Super YM the same as before, except there is no continuous symmetries, only descrete h i6=0) Z 2N! Z 2 Spontanous (discrete) chiral symmetry breaking In both QCD(adj) and its supersymmetric limit there is a spontanously broken ZN symmetry leading to N isolated, discrete vacua, labeled by k=0,, N-1 Domain vacuum k wall vacuum k+1

30 In N=2 Super-Yang mills softly broken to N=1 an entire microscopic picture is known For SU(2) the picture is: there are two degenerate vacua Monopoles condense Dyons condense (e,m)=(0,1) (e,m)=(1,-1)

31 In N=2 Super-Yang mills softly broken to N=1 an entire microscopic picture is known For SU(2) the picture is: there are two degenerate vacua Monopoles condense Dyons condense

32 In N=2 Super-Yang mills softly broken to N=1 an entire microscopic picture is known For SU(2) the picture is: there are two degenerate vacua So although there are no objects with unit fundamental charge, there is an excitation on the wall supporting unit fundamental charges

33 In N=2 Super-Yang mills softly broken to N=1 an entire microscopic picture is known For SU(2) the picture is: there are two degenerate vacua confining string can terminate Due to S.-J. Rey 1998 Explored by Witten in M-theory construction of N=1 SYM

34 - For N<2 no such statements can be made rigorous - The problem comes from the fact that monopoles while a feature of 4D N=2 theory, are very elusive in N=1 theories and theories with no supersymmetries (they require gauge fixing, assumptions of abelian dominance, etc.) - The potential implications in non-supersymmetric theories were mostly ignored.

35 SO HOW TO STUDY THESE THEORIES? - Non-abelian gauge theories in 4D do not have a small, tunable dimensionless parameter - There is a prescription by M. Unsal on how to analytically study confining phenomena in 4D - The prescription involves compacifying one direction in a way that prevents confinement/deconfinement transition - The theory obtains a dimensionless parameter LΛ which can be made arbitrarily small - It turns out that the theory is completely analytically calculable with semi-classical methods for LΛ<<1 - Note that this is NOT thermal compactification. In fact the thermal theory is not analytically tractable. - Also note that this is not a 3D YM theory.

36

37 S = 1 2g 2 Z d 4 x trf 2 µ! L g 2 Z d 3 tr F 2 ij +(D i A 0 ) 2

38 S = 1 2g 2 Z d 4 x trf 2 µ! L g 2 Z d 3 tr F 2 ij +(D i A 0 ) 2 If confinement is preserved, roughly ha 0 i6=0

39 S = 1 2g 2 Z d 4 x trf 2 µ! L g 2 Z d 3 tr F 2 ij +(D i A 0 ) 2 If confinement is preserved, roughly ha 0 i6=0 A 0 (compact) Higgs field in adjoint rep.

40 S = 1 2g 2 Z d 4 x trf 2 µ! L g 2 Z d 3 tr F 2 ij +(D i A 0 ) 2 If confinement is preserved, roughly ha 0 i6=0 A 0 (compact) Higgs field in adjoint rep. A i which do not commute with the Higgs are heavy and decouple from the low energy dynamics SU(N)! U(1) N 1

41 S = 1 2g 2 Z d 4 x trf 2 µ! L g 2 Z d 3 tr F 2 ij +(D i A 0 ) 2 If confinement is preserved, roughly ha 0 i6=0 A 0 (compact) Higgs field in adjoint rep. A i which do not commute with the Higgs are heavy and decouple from the low energy dynamics SU(N)! U(1) N 1 I will focus on SU(2) here for simplicity

42

43 abelian U(1) gauge theory

44 ~1/L SU(2) abelian U(1) gauge theory

45 ~1/L SU(2) abelian U(1) gauge theory But is not a free non-abelian gauge theory.

46 ~1/L SU(2) abelian U(1) gauge theory But is not a free non-abelian gauge theory. ~1/L SU(2) ~B ˆr r 2

47 ~1/L SU(2) abelian U(1) gauge theory But is not a free non-abelian gauge theory. ~1/L SU(2) U(1) magnetic monopoles ~B ˆr r 2

48

49 In fact due to the presence of the monopoles the theory (Unsal et. al to present)

50 In fact due to the presence of the monopoles the theory (Unsal et. al to present)

51 In fact due to the presence of the monopoles the theory (Unsal et. al to present) - Is in the confined phase regardless of the radius of compactification

52 In fact due to the presence of the monopoles the theory (Unsal et. al to present) - Is in the confined phase regardless of the radius of compactification - Is confining even upon introduction of quarks due to the interplay with the U(1) anomaly (not true in a genuinely 3D Affleck, Harvey, Witten 1992)

53 In fact due to the presence of the monopoles the theory (Unsal et. al to present) - Is in the confined phase regardless of the radius of compactification - Is confining even upon introduction of quarks due to the interplay with the U(1) anomaly (not true in a genuinely 3D Affleck, Harvey, Witten 1992) - Allow for a study of confining dynamics microscopically at LΛ<<1

54 In fact due to the presence of the monopoles the theory (Unsal et. al to present) - Is in the confined phase regardless of the radius of compactification - Is confining even upon introduction of quarks due to the interplay with the U(1) anomaly (not true in a genuinely 3D Affleck, Harvey, Witten 1992) - Allow for a study of confining dynamics microscopically at LΛ<<1 - No phase transition implies that the microscopic structure does not change in the regime LΛ>1 implying that systematic semiclassical expansion valid at LΛ <<1 can be used to reconstruct all observables in this regime

55 In fact due to the presence of the monopoles the theory (Unsal et. al to present) - Is in the confined phase regardless of the radius of compactification - Is confining even upon introduction of quarks due to the interplay with the U(1) anomaly (not true in a genuinely 3D Affleck, Harvey, Witten 1992) - Allow for a study of confining dynamics microscopically at LΛ<<1 - No phase transition implies that the microscopic structure does not change in the regime LΛ>1 implying that systematic semiclassical expansion valid at LΛ <<1 can be used to reconstruct all observables in this regime - This is the idea of resurgent trans-series construction (Unsal/ Dunne et. al.)

56

57 The effective action at L/Λ<<1: L = F ij L g 2 (L) F ij 2 + monopoles i =0, 1, 2 time space U(1) gauge theory

58 The effective action at L/Λ<<1: L = F ij L g 2 (L) F ij 2 + monopoles i =0, 1, 2 time space U(1) gauge i ijk F jk Abelian duality (Polyakov 1977) +2 compact scalar field

59 The effective action at L/Λ<<1: L = F ij L g 2 (L) F ij 2 + monopoles i =0, 1, 2 time space U(1) gauge i ijk F jk Abelian duality (Polyakov 1977) +2 compact scalar field L = g2 (L) (@i 2L(2 ) 2 ) 2 m 2 cos Massgap Due to monopole(-instantons)

60 SOURCES:

61 SOURCES: duality: F ij = 1 2 ijk@ k

62 SOURCES: duality: F ij = 1 2 ijk@ k Stationary source: Q

63 SOURCES: duality: F ij = 1 2 ijk@ k Stationary source: Q S

64 SOURCES: duality: F ij = 1 2 ijk@ k Stationary source: I Gauss law: S F ij ijk ds k =2 Q Q S

65 SOURCES: duality: F ij = 1 2 ijk@ k Stationary source: I Gauss law: I S F ij ijk ds k =2 Q Q S S dx i =2 Q

66 SOURCES: duality: F ij = 1 2 ijk@ k Stationary source: I Q S Gauss law: σ COMPACT SCALAR I S S F ij ijk ds k =2 Q dx i =2 Q σ Winds Q times around the source

67 CONFINING STRINGS

68 CONFINING STRINGS winds by 2π Net winding by 2π winds by 2π

69 CONFINING STRINGS winds by 2π Net winding by 2π V ( ) / cos( ) forces min =2 k winds by 2π

70 CONFINING STRINGS winds by 2π =0 V ( ) / cos( ) Net winding by 2π forces min =2 k =2 winds by 2π

71 CONFINING STRINGS winds by 2π =0 Kink V ( ) / cos( ) forces min =2 k =2 winds by 2π

72 CONFINING STRINGS winds by 2π =0 Kink V ( ) / cos( ) forces min =2 k =2 winding is localized on the string winds by 2π

73 CONFINING STRINGS winds by 2π =0 Kink V ( ) / cos( ) forces min =2 k q q =2 winds by 2π winding is localized on the string Thickness of the string ~ scale of the density of monopoles

74

75 4D Qtop=1 instantons

76 4D Qtop=1 instantons monopole with Qtop=1/2 anti-monopole with Qtop=1/2

77 4D Qtop=1 instantons monopole with Qtop=1/2 anti-monopole with Qtop=1/2 e i +i 2 M1~ e i +i instanton: + M2~ 2

78 4D Qtop=1 instantons monopole with Qtop=1/2 anti-monopole with Qtop=1/2 e i +i 2 M1~ e i +i instanton: + M2~ 2 e i i 2 anti-instanton: M1~ + e i i M2~ 2

79 4D Qtop=1 instantons monopole with Qtop=1/2 anti-monopole with Qtop=1/2 e i +i 2 M1~ e i +i instanton: + M2~ 2 e i i 2 anti-instanton: M1~ + e i i M2~ 2 V eff =(...) cos( ) cos( /2)

80

81 V eff =(...) cos( ) cos( /2)

82 V eff =(...) cos( ) cos( /2) * But in the presence of fermions no θ dependence in the vacuum energy should exist.

83 V eff =(...) cos( ) cos( /2) * But in the presence of fermions no θ dependence in the vacuum energy should exist. * Technically this is because monopoles have fermion zero modes, and the first allowed term which couples to the σ-field is composed out of topologically trivial configurations composed out of 1-2 monopole anti-monopole pair

84 V eff =(...) cos( ) cos( /2) * But in the presence of fermions no θ dependence in the vacuum energy should exist. * Technically this is because monopoles have fermion zero modes, and the first allowed term which couples to the σ-field is composed out of topologically trivial configurations composed out of 1-2 monopole anti-monopole pair * Alternatively the same effect can be achieved by setting θ=π

85 V eff =(...) cos( ) cos( /2) * But in the presence of fermions no θ dependence in the vacuum energy should exist. * Technically this is because monopoles have fermion zero modes, and the first allowed term which couples to the σ-field is composed out of topologically trivial configurations composed out of 1-2 monopole anti-monopole pair * Alternatively the same effect can be achieved by setting θ=π * Either way we have

86 V eff =(...) cos( ) cos( /2) * But in the presence of fermions no θ dependence in the vacuum energy should exist. * Technically this is because monopoles have fermion zero modes, and the first allowed term which couples to the σ-field is composed out of topologically trivial configurations composed out of 1-2 monopole anti-monopole pair * Alternatively the same effect can be achieved by setting θ=π * Either way we have V eff =(...) cos(2 )

87 CONFINING STRINGS σ winds by V ( ) / cos( ) =0 forces σ=0 or 2π q =2 q

88 CONFINING STRINGS σ winds by q =0 = q V eff =(...) cos(2 ) forces σ=0,π =2

89 DW = vac 0.6 DW 0.4 =0 0.2

90 LIBERATION OF QUARKS ON THE WALL 1) 2) 3)

91 σ=0 σ=π

92 σ=0 σ=π

93 SPECULATION ABOUT 4D vacuum 1 vacuum 2

94 SPIN ANTI-FERROMAGNETS AND VALENCE BOND SOLIDS (in progress: Anders Sandvik, Hui Shao and Mithat Unsal) Neel state ferromagnetic order Valence Bond Solid singlets dimer have long range crystaline order pictures from Kaul, Melko, Sandvik Annu.Rev.Cond.Matt.Phys.4(1)179 (2013)

95 VALENCE BOND SOLID VACUA

96 VALENCE BOND SOLID VACUA SPINON 4

97 UNPAIRED SPINS ARE CONFINED

98 UNPAIRED SPINS ARE CONFINED

99 THE J-Q MODEL

100 THE J-Q MODEL H = J X hiji S i S j minimal quantum anti-ferromagnet Generically in the Neel state

101 THE J-Q MODEL H = J X hiji S i S j minimal quantum anti-ferromagnet Generically in the Neel state H JQ = J X X S i S j Q x P ij P kl hiji hiji x hkli x Q y X hiki y hjli y P ij P kl P ij = S i S j 1/4 Singlet projector

102 THE J-Q MODEL H = J X hiji S i S j minimal quantum anti-ferromagnet Generically in the Neel state H JQ = J X X S i S j Q x P ij P kl hiji hiji x hkli x Q y X hiki y hjli y P ij P kl P ij = S i S j 1/4 Singlet projector Q-terms introduce singlet attractions If Qx=Qy: 4 vacua, otherwise only 2

103

104

105 Different vacua

106 quark Different vacua Domain walls carying 1/2 electric flux antiquark A deconfined quark on a domain wall GAUGE THEORY VALENCE BOND SOLID A deconfined spin on a domain wall Different vacua

107 J-Q model with a domain wall along y-direction with J=0, Qy=1 Spinon distance System size

108 ANTI-FERROMAGNET IN CONTINUUM In continuum S E = S Z d 2 xdt 4 apple 1 v s (@ x n) 2 + v s (@ t n) 2 + (Berry phase) Valid for large S where finite differences can be approximated well by derivatives

109 ANTI-FERROMAGNET IN CONTINUUM H = J X hiji S i S j +... S i = i S n i, i = ±1 staggered phase In continuum S E = S Z d 2 xdt 4 apple 1 v s (@ x n) 2 + v s (@ t n) 2 + (Berry phase) Valid for large S where finite differences can be approximated well by derivatives

110 HEDGEHOGS AS MONOPOLES

111 HEDGEHOGS AS MONOPOLES time space

112 HEDGEHOGS AS MONOPOLES time space n-field space-time hedgehog - singular in continuum - possible because of the underlying lattice

113 HEDGEHOGS AS MONOPOLES n-field time space-time hedgehog - singular in continuum - possible because of the underlying lattice space 1 e i 2 e i 2 e i The hedgehogs have different Berry phases depending on the the sub-lattice they belong to (Haldane 1988)

114 ALTERNATIVE DESCRIPTION OF SPIN n i = u i u i, =( 1, 2, 3) u i SU(2) doublet at each site with u i u i =1 u i! e i i u i Local symmetry, i.e. gauge symmetry

115 ALTERNATIVE DESCRIPTION OF SPIN n i = u i u i, =( 1, 2, 3) u i SU(2) doublet at each site with u i u i =1 u i! e i i u i Local symmetry, i.e. gauge symmetry n-field, ~1/L SU(2) ~B ˆr r 2 U(1) magnetic monopoles

116 1 e i 2 e i 2 e i

117 1 e i 2 e i 2 e i e ±i Berry phases cancel each other and the first allowed term is cos(4σ)

118 1 e i 2 e i 2 e i e ±i Berry phases cancel each other and the first allowed term is cos(4σ)

119 e i 2 1 e i e i 2 e ±i Berry phases cancel each other and the first allowed term is cos(4σ)

120 e i 2 e i e ±i( /2) 1 e i 2 Berry phases cancel each other and the first allowed term is cos(4σ)

121 1 e i 2 e i 2 e i e ±i Berry phases cancel each other and the first allowed term is cos(4σ)

122 e i 2 1 e i e i 2 e ±i Berry phases cancel each other and the first allowed term is cos(4σ)

123 e i 2 e i e ±i( /2) 1 e i 2 Berry phases cancel each other and the first allowed term is cos(4σ)

124 e i 2 e i e ±i( /2) 1 e i 2 Berry phases cancel each other and the first allowed term is cos(4σ) Under the Q-deformation only π rotations are a symmetry, so cos(2σ) is allowed.

125 Phases: - Neel - u-field condenses breaking SU(2) symmetry - u-condensate acts like a Higgs field and the effective theory is that of goldstones - VBS - u-field is massive and can be integrated out - Effective theory is the 3D U(1) gauge theory with monopoles - Confining phase - Monopoles (hedgehogs) couple to Berry phases which interfere allowing only multiple monopole events to contribute. - The different Qx and Qy terms allow for cos(2σ) but not for cos(σ) term. - Effective action is the same as in gauge theories

126 CONCLUSION Gauge theories with multiple vacua have an incredibly curious confining string structure They generically exhibit features that suggest strings are made out of domain walls Immediate consequences are: liberation of charges on the domain walls, strings ending on domain walls, charges are intersections of domain walls. Not a supersymmetric phenomena, as is folklore. QCD(adj) confining with N vacua (discrete chiral symmetry breaking). θ=π confining with 2 vacua (CP symmetry breaking) In non-degenerate vacua, there may be residual effects from this considerations (i.e. strings are composed out of Witten k-vacua) Spin-antiferromagnets in the VBS phase exhibit the same phenomenon Domain walls as spin-guides? So far deconfined spinons were only proposed in at a critical point between the Neel and the VBS state, but it may be possible to achieve this on the domain wall without being at the critical point.