LIBERATION ON THE WALLS IN GAUGE THEORIES AND ANTI-FERROMAGNETS Tin - - PowerPoint PPT Presentation
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, 021701 and with Anders Sandvik, Hui Shao and M. Unsal In progress
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, 021701 and with Anders Sandvik, Hui Shao and M. Unsal — In progress
Recent Developments in Semiclassical Probes of Quantum Field Theories — UMass Amherst 2016
INTRODUCTION:
The vacuum structure of gauge theories
INTRODUCTION:
The vacuum structure of gauge theories The vacuum structure 1st pass
INTRODUCTION:
The vacuum structure of gauge theories The vacuum structure 1st pass
and is unbroaken
INTRODUCTION:
The vacuum structure of gauge theories The vacuum structure 1st pass
and is unbroaken
However in large N limit, on general grounds (Witten 1998)
INTRODUCTION:
The vacuum structure of gauge theories The vacuum structure 1st pass
and is unbroaken
However in large N limit, on general grounds (Witten 1998)
L = N ✓ 1 4g2N F 2 + i θ 16π2N F ˜ F ◆
INTRODUCTION:
The vacuum structure of gauge theories The vacuum structure 1st pass
and is unbroaken
However in large N limit, on general grounds (Witten 1998)
L = N ✓ 1 4g2N F 2 + i θ 16π2N F ˜ F ◆
t’Hooft coupling
INTRODUCTION:
The vacuum structure of gauge theories The vacuum structure 1st pass
and is unbroaken
However in large N limit, on general grounds (Witten 1998)
L = N ✓ 1 4g2N F 2 + i θ 16π2N F ˜ F ◆
t’Hooft coupling Keep fixed
INTRODUCTION:
The vacuum structure of gauge theories The vacuum structure 1st pass
and is unbroaken
However in large N limit, on general grounds (Witten 1998)
L = N ✓ 1 4g2N F 2 + i θ 16π2N F ˜ F ◆
t’Hooft coupling Keep fixed And vacuum energy
INTRODUCTION:
The vacuum structure of gauge theories The vacuum structure 1st pass
and is unbroaken
However in large N limit, on general grounds (Witten 1998)
L = N ✓ 1 4g2N F 2 + i θ 16π2N F ˜ F ◆
t’Hooft coupling Keep fixed
E(θ) = E(θ + 2π)
And vacuum energy
INTRODUCTION:
The vacuum structure of gauge theories The vacuum structure 1st pass
and is unbroaken
However in large N limit, on general grounds (Witten 1998)
L = N ✓ 1 4g2N F 2 + i θ 16π2N F ˜ F ◆
t’Hooft coupling Keep fixed
) ⇒
E(θ) = E(θ + 2π)
And vacuum energy
INTRODUCTION:
The vacuum structure of gauge theories The vacuum structure 1st pass
and is unbroaken
However in large N limit, on general grounds (Witten 1998)
L = N ✓ 1 4g2N F 2 + i θ 16π2N F ˜ F ◆
t’Hooft coupling Keep fixed
) ⇒
E(θ) = E(θ + 2π)
And vacuum energy
E(θ) = N 2f ✓θ + 2πk N ◆
k=0 k=2 k=1 N=3
θ E(θ)
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 k=0 k=2 k=1 N=3
θ E(θ)
π 2π 3π 4π
QCD ADJOINT
QCD ADJOINT
in adjoint rep.
L = 1 2g2 trF 2 +
nf
X
I=1
¯ λIσµDµλI !
λI
QCD ADJOINT
in adjoint rep.
L = 1 2g2 trF 2 +
nf
X
I=1
¯ λIσµDµλI !
λI
λI → λIeiα
λI → UI
JλJ , U ∈ SU(Nf)
QCD ADJOINT
in adjoint rep.
L = 1 2g2 trF 2 +
nf
X
I=1
¯ λIσµDµλI !
λI
anomaly—instantons breaks U(1) to Z2Nnf
I ∼ λ2Nnf
λI → λIeiα
λI → UI
JλJ , U ∈ SU(Nf)
QCD ADJOINT
in adjoint rep.
L = 1 2g2 trF 2 +
nf
X
I=1
¯ λIσµDµλI !
λI
∂µjµ
5 =
nf 16π2 tradjF µν ˜ Fµν ⇒ ∆Q5 = 2nfNQtop
anomaly—instantons breaks U(1) to Z2Nnf
I ∼ λ2Nnf
λI → λIeiα
λI → UI
JλJ , U ∈ SU(Nf)
QCD ADJOINT
in adjoint rep.
L = 1 2g2 trF 2 +
nf
X
I=1
¯ λIσµDµλI !
λI
∂µjµ
5 =
nf 16π2 tradjF µν ˜ Fµν ⇒ ∆Q5 = 2nfNQtop
conserved ⇒ Z2Nnf remains
Q mod 2Nnf
anomaly—instantons breaks U(1) to Z2Nnf
I ∼ λ2Nnf
λI → λIeiα
λI → UI
JλJ , U ∈ SU(Nf)
QCD ADJOINT
in adjoint rep.
L = 1 2g2 trF 2 +
nf
X
I=1
¯ λIσµDµλI !
λI
∂µjµ
5 =
nf 16π2 tradjF µν ˜ Fµν ⇒ ∆Q5 = 2nfNQtop
conserved ⇒ Z2Nnf remains
Q mod 2Nnf
anomaly—instantons breaks U(1) to Z2Nnf
I ∼ λ2Nnf
λI → λIeiα
λI → UI
JλJ , U ∈ SU(Nf)
Znf parti belongs to SU(nf) so the symmetry is SU(nf)xZ2N
Spontanous continuous chiral symmetry breaking
⌦ λIλI↵ 6= 0 ) SU(nf) ⇥ Z2N ! SO(nf) ⇥ Z2
Spontanous continuous chiral symmetry breaking
⌦ λIλI↵ 6= 0 ) SU(nf) ⇥ Z2N ! SO(nf) ⇥ Z2 Z2N → Z2
Spontanous continuous chiral symmetry breaking
⌦ λIλI↵ 6= 0 ) SU(nf) ⇥ Z2N ! SO(nf) ⇥ Z2 Z2N → Z2
Therefore since the coset Z2N/Z2=ZN the theory has N isolated degenerate vacua
Spontanous (discrete) chiral symmetry breaking
hλλi 6= 0 ) Z2N ! Z2
In Super YM the same as before, except there is no continuous symmetries, only descrete
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 Spontanous (discrete) chiral symmetry breaking
hλλi 6= 0 ) Z2N ! Z2
In Super YM the same as before, except there is no continuous symmetries, only descrete
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 Spontanous (discrete) chiral symmetry breaking
hλλi 6= 0 ) Z2N ! Z2
In Super YM the same as before, except there is no continuous symmetries, only descrete
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 Spontanous (discrete) chiral symmetry breaking
hλλi 6= 0 ) Z2N ! Z2
In Super YM the same as before, except there is no continuous symmetries, only descrete
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 wall
vacuum k vacuum k+1 Spontanous (discrete) chiral symmetry breaking
hλλi 6= 0 ) Z2N ! Z2
In Super YM the same as before, except there is no continuous symmetries, only descrete
For SU(2) the picture is: there are two degenerate vacua Monopoles condense Dyons condense (e,m)=(0,1) (e,m)=(1,-1) 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 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 In N=2 Super-Yang mills softly broken to N=1 an entire microscopic picture is known
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 Due to S.-J. Rey 1998 Explored by Witten in M-theory construction of N=1 SYM confining string can terminate
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.)
theories were mostly ignored.
SO HOW TO STUDY THESE THEORIES?
dimensionless parameter
confining phenomena in 4D
prevents confinement/deconfinement transition
made arbitrarily small
semi-classical methods for LΛ<<1
theory is not analytically tractable.
YM theory.
S = 1 2g2 Z d4x trF 2
µν → L
g2 Z d3tr
ij + (DiA0)2
S = 1 2g2 Z d4x trF 2
µν → L
g2 Z d3tr
ij + (DiA0)2
hA0i 6= 0 If confinement is preserved, roughly
S = 1 2g2 Z d4x trF 2
µν → L
g2 Z d3tr
ij + (DiA0)2
hA0i 6= 0 If confinement is preserved, roughly A0 —(compact) Higgs field in adjoint rep.
S = 1 2g2 Z d4x trF 2
µν → L
g2 Z d3tr
ij + (DiA0)2
hA0i 6= 0 If confinement is preserved, roughly A0 —(compact) Higgs field in adjoint rep. Ai which do not commute with the Higgs are heavy and decouple from the low energy dynamics SU(N) → U(1)N−1
S = 1 2g2 Z d4x trF 2
µν → L
g2 Z d3tr
ij + (DiA0)2
hA0i 6= 0 If confinement is preserved, roughly A0 —(compact) Higgs field in adjoint rep. Ai 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
abelian U(1) gauge theory
abelian U(1) gauge theory
SU(2)
~1/L
abelian U(1) gauge theory
SU(2)
~1/L But is not a free non-abelian gauge theory.
abelian U(1) gauge theory
SU(2)
~1/L But is not a free non-abelian gauge theory.
SU(2)
~1/L ~ B ∼ ˆ r r2
abelian U(1) gauge theory
SU(2)
~1/L But is not a free non-abelian gauge theory.
SU(2)
~1/L ~ B ∼ ˆ r r2 U(1) magnetic monopoles
In fact due to the presence of the monopoles the theory (Unsal et.
In fact due to the presence of the monopoles the theory (Unsal et.
In fact due to the presence of the monopoles the theory (Unsal et.
compactification
In fact due to the presence of the monopoles the theory (Unsal et.
compactification
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.
compactification
interplay with the U(1) anomaly (not true in a genuinely 3D — Affleck, Harvey, Witten 1992)
LΛ<<1
In fact due to the presence of the monopoles the theory (Unsal et.
compactification
interplay with the U(1) anomaly (not true in a genuinely 3D — Affleck, Harvey, Witten 1992)
LΛ<<1
not change in the regime LΛ>1 implying that systematic semi- classical expansion valid at LΛ <<1 can be used to reconstruct all
In fact due to the presence of the monopoles the theory (Unsal et.
compactification
interplay with the U(1) anomaly (not true in a genuinely 3D — Affleck, Harvey, Witten 1992)
LΛ<<1
not change in the regime LΛ>1 implying that systematic semi- classical expansion valid at LΛ <<1 can be used to reconstruct all
Dunne et. al.)
The effective action at L/Λ<<1: Fij — U(1) gauge theory L = L g2(L)F2
ij
+ monopoles i = 0, 1, 2 time space
The effective action at L/Λ<<1: Fij — U(1) gauge theory L = L g2(L)F2
ij
+ monopoles i = 0, 1, 2 time space @i ∼ ✏ijkFjk —Abelian duality (Polyakov 1977) σ ≡ σ + 2π —compact scalar field
The effective action at L/Λ<<1: Fij — U(1) gauge theory L = L g2(L)F2
ij
+ monopoles i = 0, 1, 2 time space L = g2(L) 2L(2π)2 ⇥ (∂iσ)2 − m2 cos σ ⇤ Due to monopole(-instantons) Massgap @i ∼ ✏ijkFjk —Abelian duality (Polyakov 1977) σ ≡ σ + 2π —compact scalar field
SOURCES:
SOURCES:
duality: Fij = 1
2✏ijk@k
SOURCES:
Q Stationary source: duality: Fij = 1
2✏ijk@k
SOURCES:
S
Q Stationary source: duality: Fij = 1
2✏ijk@k
SOURCES:
I
S
Fij✏ijkdSk = 2⇡Q
S
Gauss law: Q Stationary source: duality: Fij = 1
2✏ijk@k
SOURCES:
I
S
Fij✏ijkdSk = 2⇡Q
S
I
S
dxi∂iσ = 2πQ
Gauss law: Q Stationary source: duality: Fij = 1
2✏ijk@k
SOURCES:
I
S
Fij✏ijkdSk = 2⇡Q
S
I
S
dxi∂iσ = 2πQ
Gauss law: σ Winds Q times around the source Q Stationary source: σ—COMPACT SCALAR duality: Fij = 1
2✏ijk@k
CONFINING STRINGS
σ winds by 2π
CONFINING STRINGS
σ
winds by 2π Net winding by 2π
σ winds by 2π
CONFINING STRINGS
σ
winds by 2π Net winding by 2π forces
σmin = 2πk
V (σ) ∝ − cos(σ)
σ winds by 2π
CONFINING STRINGS
σ
winds by 2π Net winding by 2π forces
σmin = 2πk
V (σ) ∝ − cos(σ)
σ = 0 σ = 2π
σ winds by 2π
CONFINING STRINGS
σ
winds by 2π forces
σmin = 2πk
V (σ) ∝ − cos(σ)
Kink
σ = 0 σ = 2π
winding is localized on the string
σ winds by 2π
CONFINING STRINGS
σ
winds by 2π forces
σmin = 2πk
V (σ) ∝ − cos(σ)
Kink
σ = 0 σ = 2π
winding is localized on the string
σ winds by 2π
CONFINING STRINGS
q
¯ q
σ
winds by 2π forces
σmin = 2πk
V (σ) ∝ − cos(σ)
Kink Thickness of the string ~ scale of the density of monopoles
σ = 0 σ = 2π
4D Qtop=1 instantons
4D Qtop=1 instantons monopole with Qtop=1/2 anti-monopole with Qtop=1/2
4D Qtop=1 instantons monopole with Qtop=1/2 anti-monopole with Qtop=1/2
eiσ+i θ
2
M1~
e−iσ+i θ
2
M2~ instanton: +
4D Qtop=1 instantons monopole with Qtop=1/2 anti-monopole with Qtop=1/2
eiσ+i θ
2
M1~
e−iσ+i θ
2
M2~ instanton: +
e−iσ−i θ
2
M1~
eiσ−i θ
2
M2~ anti-instanton: +
4D Qtop=1 instantons monopole with Qtop=1/2 anti-monopole with Qtop=1/2
eiσ+i θ
2
M1~
e−iσ+i θ
2
M2~ instanton: +
e−iσ−i θ
2
M1~
eiσ−i θ
2
M2~ anti-instanton: +
Veff = (. . . ) cos(σ) cos(θ/2)
Veff = (. . . ) cos(σ) cos(θ/2)
Veff = (. . . ) cos(σ) cos(θ/2)
* But in the presence of fermions no θ dependence in the vacuum energy
should exist.
Veff = (. . . ) 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
Veff = (. . . ) 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
Veff = (. . . ) 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
Veff = (. . . ) 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
Veff = (. . . ) cos(2σ)
σ winds by
q
¯ q
forces σ=0 or 2π
V (σ) ∝ − cos(σ)
σ = 0 σ = 2π
CONFINING STRINGS
Veff = (. . . ) cos(2σ) σ winds by
q
¯ q
forces σ=0,π
σ = 0 σ = 2π
CONFINING STRINGS
σ = π
0.2 0.4
0.6
0.8
1.0
DW DW
σ = 0
σ = π vac
2) 1) 3)
LIBERATION OF QUARKS ON THE WALL
SPECULATION ABOUT 4D
vacuum 1 vacuum 2
SPIN ANTI-FERROMAGNETS AND VALENCE BOND SOLIDS
(in progress: Anders Sandvik, Hui Shao and Mithat Unsal)
Neel state — ferromagnetic order
pictures from Kaul, Melko, Sandvik Annu.Rev.Cond.Matt.Phys.4(1)179 (2013)
Valence Bond Solid singlets—dimer have long range crystaline order
VALENCE BOND SOLID VACUA
1 2 3 4
VALENCE BOND SOLID VACUA
1 2 3 4
SPINON
UNPAIRED SPINS ARE CONFINED
UNPAIRED SPINS ARE CONFINED
THE J-Q MODEL
THE J-Q MODEL
H = J X
hiji
Si · Sj — minimal quantum anti-ferromagnet — Generically in the Neel state
THE J-Q MODEL
H = J X
hiji
Si · Sj — minimal quantum anti-ferromagnet — Generically in the Neel state
HJQ = J X
hiji
Si · Sj − Qx X
hijixhklix
PijPkl − Qy X
hikiyhjliy
PijPkl
Pij = Si · Sj − 1/4
—Singlet projector
THE J-Q MODEL
H = J X
hiji
Si · Sj — minimal quantum anti-ferromagnet — Generically in the Neel state
HJQ = J X
hiji
Si · Sj − Qx X
hijixhklix
PijPkl − Qy X
hikiyhjliy
PijPkl
Pij = Si · Sj − 1/4
—Singlet projector Q-terms introduce singlet attractions If Qx=Qy: 4 vacua, otherwise only 2
1 2 3 4 1 2 3 4
1 2 3 4
Different vacua
quark anti- quark Different vacua Different vacua Domain walls carying 1/2 electric flux
A deconfined quark
A deconfined spin
V ALENCE BOND SOLID GAUGE THEORY
J-Q model with a domain wall along y-direction with J=0, Qy=1 Spinon distance System size
ANTI-FERROMAGNET IN CONTINUUM
In continuum Valid for large S where finite differences can be approximated well by derivatives SE = S 4 Z d2x dt 1 vs (∂xn)2 + vs(∂tn)2
ANTI-FERROMAGNET IN CONTINUUM
H = J X
hiji
Si · Sj + . . . —staggered phase Si = ηiS ni , ηi = ±1 In continuum Valid for large S where finite differences can be approximated well by derivatives SE = S 4 Z d2x dt 1 vs (∂xn)2 + vs(∂tn)2
HEDGEHOGS AS MONOPOLES
HEDGEHOGS AS MONOPOLES
time space
HEDGEHOGS AS MONOPOLES
time space space-time hedgehog
underlying lattice n-field
HEDGEHOGS AS MONOPOLES
time space space-time hedgehog
underlying lattice The hedgehogs have different Berry phases depending on the the sub-lattice they belong to (Haldane 1988) 1 e
iπ 2
e− iπ
2 eiπ
n-field
ALTERNATIVE DESCRIPTION OF SPIN
ni = u†
iσui ,
σ = (σ1, σ2, σ3) ui — SU(2) doublet at each site with u†
iui = 1
ui → eiφiui — Local symmetry, i.e. gauge symmetry
ALTERNATIVE DESCRIPTION OF SPIN
ni = u†
iσui ,
σ = (σ1, σ2, σ3) ui — SU(2) doublet at each site with u†
iui = 1
ui → eiφiui — Local symmetry, i.e. gauge symmetry n-field
SU(2)
~1/L ~ B ∼ ˆ r r2 U(1) magnetic monopoles
1
e
iπ 2
e− iπ
2 eiπ
1
e
iπ 2
e− iπ
2 eiπ
Berry phases cancel each other and the first allowed term is cos(4σ)
1
e
iπ 2
e− iπ
2 eiπ
Berry phases cancel each other and the first allowed term is cos(4σ)
1
e
iπ 2
e− iπ
2
eiπ
Berry phases cancel each other and the first allowed term is cos(4σ)
1
e
iπ 2
e− iπ
2
eiπ
Berry phases cancel each other and the first allowed term is cos(4σ)
1
e
iπ 2
e− iπ
2 eiπ
Berry phases cancel each other and the first allowed term is cos(4σ)
1
e
iπ 2
e− iπ
2
eiπ
Berry phases cancel each other and the first allowed term is cos(4σ)
1
e
iπ 2
e− iπ
2
eiπ
Berry phases cancel each other and the first allowed term is cos(4σ)
1
e
iπ 2
e− iπ
2
eiπ
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.
Phases:
theory is that of goldstones
monopoles
interfere allowing only multiple monopole events to contribute.
for cos(σ) term.
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.