LIBERATION ON THE WALLS IN GAUGE THEORIES AND ANTI-FERROMAGNETS Tin - PowerPoint PPT Presentation

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 L Λ which can be made arbitrarily small - It turns out that the theory is completely analytically calculable with semi-classical methods for L Λ <<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.

1 Z µ ν → L Z d 4 x tr F 2 d 3 tr F 2 ij + ( D i A 0 ) 2 � � S = 2 g 2 g 2

1 Z µ ν → L Z d 4 x tr F 2 d 3 tr F 2 ij + ( D i A 0 ) 2 � � S = 2 g 2 g 2 If confinement is preserved, roughly h A 0 i 6 = 0

1 Z µ ν → L Z d 4 x tr F 2 d 3 tr F 2 ij + ( D i A 0 ) 2 � � S = 2 g 2 g 2 If confinement is preserved, roughly h A 0 i 6 = 0 A 0 —(compact) Higgs field in adjoint rep.

1 Z µ ν → L Z d 4 x tr F 2 d 3 tr F 2 ij + ( D i A 0 ) 2 � � S = 2 g 2 g 2 If confinement is preserved, roughly h A 0 i 6 = 0 A 0 —(compact) Higgs field in adjoint rep. which do not commute with the Higgs are heavy A i and decouple from the low energy dynamics SU ( N ) → U (1) N − 1

1 Z µ ν → L Z d 4 x tr F 2 d 3 tr F 2 ij + ( D i A 0 ) 2 � � S = 2 g 2 g 2 If confinement is preserved, roughly h A 0 i 6 = 0 A 0 —(compact) Higgs field in adjoint rep. which do not commute with the Higgs are heavy A i and decouple from the low energy dynamics SU ( N ) → U (1) N − 1 I will focus on SU(2) here for simplicity

abelian U(1) gauge theory

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

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

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

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

In fact due to the presence of the monopoles the theory (Unsal et. al. 2007 to present)

In fact due to the presence of the monopoles the theory (Unsal et. al. 2007 to present)

In fact due to the presence of the monopoles the theory (Unsal et. al. 2007 to present) - Is in the confined phase regardless of the radius of compactification

In fact due to the presence of the monopoles the theory (Unsal et. al. 2007 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)

In fact due to the presence of the monopoles the theory (Unsal et. al. 2007 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 
 L Λ <<1

In fact due to the presence of the monopoles the theory (Unsal et. al. 2007 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 
 L Λ <<1 - No phase transition implies that the microscopic structure does not change in the regime L Λ >1 implying that systematic semi- classical expansion valid at L Λ <<1 can be used to reconstruct all observables in this regime

In fact due to the presence of the monopoles the theory (Unsal et. al. 2007 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 
 L Λ <<1 - No phase transition implies that the microscopic structure does not change in the regime L Λ >1 implying that systematic semi- classical expansion valid at L Λ <<1 can be used to reconstruct all observables in this regime - This is the idea of resurgent trans-series construction (Unsal/ Dunne et. al.)

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

The effective action at L/ Λ <<1: L g 2 ( L ) F 2 + monopoles L = i = 0 , 1 , 2 ij time space — U(1) gauge theory F ij @ i � ∼ ✏ ijk F jk —Abelian duality (Polyakov 1977) —compact scalar field σ ≡ σ + 2 π

The effective action at L/ Λ <<1: L g 2 ( L ) F 2 + monopoles L = i = 0 , 1 , 2 ij time space — U(1) gauge theory F ij @ i � ∼ ✏ ijk F jk —Abelian duality (Polyakov 1977) —compact scalar field σ ≡ σ + 2 π g 2 ( L ) ( ∂ i σ ) 2 − m 2 cos σ ⇥ ⇤ L = 2 L (2 π ) 2 Massgap Due to monopole(-instantons)

SOURCES:

SOURCES: duality: F ij = 1 2 ✏ ijk @ k �

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

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

SOURCES: duality: F ij = 1 2 ✏ ijk @ k � Stationary source: I F ij ✏ ijk dS k = 2 ⇡ Q Gauss law: S Q S

SOURCES: duality: F ij = 1 2 ✏ ijk @ k � Stationary source: I F ij ✏ ijk dS k = 2 ⇡ Q Gauss law: S I dx i ∂ i σ = 2 π Q Q S S

SOURCES: duality: F ij = 1 2 ✏ ijk @ k � Stationary source: I F ij ✏ ijk dS k = 2 ⇡ Q Gauss law: S I σ —COMPACT 
 dx i ∂ i σ = 2 π Q SCALAR Q S S σ Winds Q times around the source

CONFINING STRINGS

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

CONFINING STRINGS σ winds by 2 π V ( σ ) ∝ − cos( σ ) forces Net winding σ min = 2 π k by 2 π winds by 2 π σ

CONFINING STRINGS σ winds by 2 π V ( σ ) ∝ − cos( σ ) σ = 0 forces Net winding σ min = 2 π k by 2 π σ = 2 π winds by 2 π σ

CONFINING STRINGS σ winds by 2 π V ( σ ) ∝ − cos( σ ) σ = 0 forces Kink σ min = 2 π k σ = 2 π winds by 2 π σ

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

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

4D Q top =1 instantons

monopole with Q top =1/2 4D Q top =1 instantons anti-monopole with Q top =1/2

monopole with Q top =1/2 4D Q top =1 instantons anti-monopole with Q top =1/2 e i σ + i θ e − i σ + i θ M 2 ~ instanton: + M 1 ~ 2 2

monopole with Q top =1/2 4D Q top =1 instantons anti-monopole with Q top =1/2 e i σ + i θ e − i σ + i θ M 2 ~ instanton: + M 1 ~ 2 2 e − i σ − i θ e i σ − i θ anti-instanton: + M 1 ~ M 2 ~ 2 2

monopole with Q top =1/2 4D Q top =1 instantons anti-monopole with Q top =1/2 e i σ + i θ e − i σ + i θ M 2 ~ instanton: + M 1 ~ 2 2 e − i σ − i θ e i σ − i θ anti-instanton: + M 1 ~ M 2 ~ 2 2 V eff = ( . . . ) cos( σ ) cos( θ / 2)

V eff = ( . . . ) cos( σ ) cos( θ / 2)

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

V eff = ( . . . ) cos( σ ) cos( θ / 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

V eff = ( . . . ) cos( σ ) cos( θ / 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

V eff = ( . . . ) cos( σ ) cos( θ / 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( σ ) cos( θ / 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 σ )

CONFINING STRINGS σ winds by V ( σ ) ∝ − cos( σ ) σ = 0 forces σ =0 or 2 π q ¯ q σ = 2 π

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

1.0 0.8 DW σ = π vac 0.6 DW 0.4 0.2 σ = 0

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

σ =0 σ = π

σ =0 σ = π

SPECULATION ABOUT 4D vacuum 1 vacuum 2

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

VALENCE BOND SOLID VACUA 2 1 3 4

VALENCE BOND SOLID VACUA 2 1 3 SPINON 4

UNPAIRED SPINS ARE CONFINED

UNPAIRED SPINS ARE CONFINED

THE J-Q MODEL

THE J-Q MODEL S i · S j — minimal quantum anti-ferromagnet X H = J h ij i — Generically in the Neel state