Is Strongly Correlated Electron Matter Full of Unparticles? Kiaran dave Charlie Kane Brandon Langley
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1 Is Strongly Correlated Electron Matter Full of Unparticles? Kiaran dave Charlie Kane Brandon Langley
2 2=3
3
4 goal 2 = 3
5 Properties all particles share?
6 Properties all particles share? charge fixed mass!
7 Properties all particles share? charge fixed mass! conserved (gauge invariance)
8 Properties all particles share? charge fixed mass! conserved (gauge not conserved invariance)
9 mass sets a scale
10 mass sets a scale mass breaks scale invariance
11 what is scale invariance? invariance on all length scales
12 f(x) =x 2
13 f(x) =x 2 f(x/λ) =(x/λ) 2 scale change
14 f(x) =x 2 f(x/λ) =(x/λ) 2 scale change scale invariance
15 f(x) =x 2 f(x/λ) =(x/λ) 2 scale change scale invariance f(x) =x 2 λ 2 g(λ) 1{
16 f(x) =x 2 f(x/λ) =(x/λ) 2 scale change scale invariance f(x) =x 2 λ 2 g(λ) 1{ g(λ) =λ 2
17 f(x) =ax 2 + bx 3 x x/λ
18 f(x) =ax 2 + bx 3 x x/λ f(x) Λ 2 (ax 2 + bx/λ)
19 f(x) =ax 2 + bx 3 x x/λ f(x) Λ 2 (ax 2 + bx/λ) not scale invariant
20 free field theory L = 1 2 µ φ µ φ x x/λ φ(x) φ(x) L Λ 2 L
21 free field theory L = 1 2 µ φ µ φ x x/λ φ(x) φ(x) L Λ 2 L scale invariant
22 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2
23 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 x x/λ φ(x) φ(x)
24 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 x x/λ φ(x) φ(x) m 2 φ 2
25 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2
26 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2
27 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2 no scale invariance
28 how should we think about field theory? QFT
29 how should we think about field theory? g 7 g 6 g 5 g 4 g 3 g 2 g 1 QFT
30 how should we think about field theory? g7 g6 g5 g4 g3 g2 g1 QFT
31 g7 g6 g5 g4 g3 g2 g1 QFT
32 what if m depends on the scale? g7 g6 g5 g4 g3 g2 g1 QFT
33 what if m depends on the scale? QFT
34 what if m depends on the scale? Lmi+7 Lmi+6 Lmi+5 Lmi+4 Lmi+3 Lmi+2 Lmi+1 Lmi QFT
35 sum over all mass L = i L(m i )
36 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m)
37 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2
38 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass!
39 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2
40 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!!
41 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!! not particles
42 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!! not particles
43 scale-invariant stuff is weird
44 scale-invariant stuff is weird particles energy E = ε p
45 scale-invariant stuff is weird particles energy E = ε p Green function (propagator) 1 G(E) = E ε p particle=infinity
46 scale-invariant stuff is weird particles unparticles energy E = ε p Green function (propagator) 1 G(E) = E ε p particle=infinity
47 scale-invariant stuff is weird particles unparticles G (E ε p ) d U 2 d U > 2 energy E = ε p Green function (propagator) 1 G(E) = E ε p unparticles=zero propagator particle=infinity
48 n massless particles G (E ε p ) n 2 unparticles=fractional number of massless particles
49 strongly correlated matter does the number of particles=number of charged stuff?
50 if no, perhaps unparticles make up the difference
51 counting particles
52 counting particles 1
53 counting particles 2 1
54 counting particles 2 3 1
55 counting particles
56 counting particles is there a more efficient way?
57 µ
58 n = µ N(ω)dω µ
59 n = µ N(ω)dω µ IR UV
60 n = µ N(ω)dω µ IR UV can n be deduced entirely from the IR(lowenergy) scale?
61 Luttinger s `theorem
62 Each particle has an energy
63 Each particle has an energy particle count is deducible from the Green function
64 Luttinger s Theorem Green function (propagator) G(E) = G(E) 1 E ε p ε p E
65 Luttinger s Theorem Green function (propagator) G(E) = G(E) 1 E ε p E>ε p ε p E
66 Luttinger s Theorem Green function (propagator) G(E) = G(E) 1 E ε p E>ε p ε p E E<ε p
67 Luttinger s Theorem Green function (propagator) G(E) = G(E) 1 E ε p E>ε p ε p E E<ε p particle density= number of sign changes of G
68 Counting sign changes?
69 Counting sign changes? Θ(x)
70 Counting sign changes? Θ(x) = { 0 x<0
71 Counting sign changes? Θ(x) = { 0 x<0 1 x>0
72 Counting sign changes? Θ(x) = { 0 x<0 1 x>0 1/2 x =0
73 Counting sign changes? Θ(x) = { 0 x<0 1 x>0 1/2 x =0 counts sign changes
74 Counting sign changes? Θ(x) = { 0 x<0 1 x>0 1/2 x =0 counts sign changes n =2 p Θ(G(E =0, p)) Luttinger Theorem for electrons
75 Is this famous theorem from 1960 correct?
76 How do functions change sign? E>ε p ε p E E<ε p
77 How do functions change sign? E>ε p ε p E E<ε p divergence
78
79 Is there another way?
80
81 Yes
82 zero-crossing ε p E
83 zero-crossing ε p E no divergence is necessary
84 closer look at Luttinger s theorem n =2 p Θ(G(E =0, p))
85 closer look at Luttinger s theorem n =2 p Θ(G(E =0, p)) divergences+ zeros
86 closer look at Luttinger s theorem n =2 p Θ(G(E =0, p)) divergences+ zeros how can zeros affect the particle count?
87
88 zeros arise from unparticles
89 so what gives?
90 A model with zeros but Luttinger fails
91 A model with zeros but Luttinger fails e t t t t t t t t
92 A model with zeros but Luttinger fails e t t t t t t t t
93 A model with zeros but Luttinger fails e t t t t t t t t no hopping=> no propagation (zeros)
94 A model with zeros but Luttinger fails N flavors e t t t t t of e- t t t no hopping=> no propagation (zeros)
95 e spin SU(2)
96 e spin SU(2) generalization N flavors of spin SU(N)
97 e spin SU(2) generalization N =5 N flavors of spin SU(N)
98 e spin SU(2) generalization N =5 N flavors of spin SU(N)
99 e spin SU(2) generalization 2E 25 N = N flavors of spin SU(N)
100 e spin SU(2) generalization 2E N flavors of spin SU(N) 25 N = H = U 2 (n 1 + n N ) 2
101 SU(2) U
102 SU(2) U G = # ω + U/2 + # ω U/2
103 SU(N) for N=3: no particle-hole symmetry
104 SU(N) for N=3: no particle-hole symmetry lim µ(t ) T 0
105 G αβ (ω = 0) = δ αβ K(n + 1) K(n) 2n N N
106 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N
107 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N
108 < 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N
109 < 0 > 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N
110 a zero must exist < 0 > 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N
111 Luttinger s theorem n = NΘ(2n N)
112 Luttinger s theorem n = NΘ(2n N) { 0, 1, 1/2
113 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2
114 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 2=3
115 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 2=3
116 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3
117 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3 odd
118 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3 odd no solution
119 Luttinger s theorem
120
121 Problem
122 Problem G=0
123 Problem G=0 G = 1 E ε p =0??
124 Problem G=0 G = 1 E ε p =0?? G αβ (ω = 0) = δ αβ K(n + 1) K(n) 2n N N
125 Problem G=0 G = 1 E ε p =0?? G αβ (ω = 0) = δ αβ K(n + 1) K(n) 2n N N must diverge
126 G = 1 E ε p Σ G=0
127 self energy diverges G = 1 E ε p Σ G=0
128 if Σ is infinite
129 if Σ is infinite lifetime of a particle vanishes
130 if Σ is infinite lifetime of a particle Σ < p vanishes
131 if Σ is infinite lifetime of a particle Σ < p vanishes no particle
132 zeros and particles are incompatible
133 zeros and particles are incompatible Luttinger theorem is wrong!
134 how to count particles?
135 how to count particles? some charged stuff has no particle interpretation
136 G(p, ω) = 0 UV no physical quark or electron states G(p L,ω = 0) = 0 IR G BCS (ω, p) = ω + v( p p F ) ω 2 2 v 2 ( p p F ) 2 G 1+1QCD = ik m 2 g 2 /π +2k + k + g 2 k /πλ i
137 where else do we expect to see zeros?
138 where else do we expect to see zeros? strongly correlated systems
139 where else do we expect to see zeros? strongly correlated systems Mott insulators
140 NiO insulates d 8? What is a Mott Insulator?
141 NiO insulates d 8? What is a Mott Insulator? EMPTY STATES= METAL
142 NiO insulates d 8? What is a Mott Insulator? EMPTY STATES= METAL band theory fails!
143 NiO insulates d 8? perhaps this costs energy What is a Mott Insulator? EMPTY STATES= METAL band theory fails!
144 Mott Problem: NiO (Band theory failure) (N rooms N occupants) t U t
145
146
147 U/t = 10 1 interactions dominate: Strong Coupling Physics
148 photoemission
149 photoemission particles energy E = ε p
150 photoemission particles energy E = ε p p 2
151 photoemission particles energy E = ε p p 2
152 photoemission particles energy E = ε p p 2 two crossings: closed surface of excitations
153 what is seen experimentally? Fermi arcs: no double crossings (PDJ,JCC,ZXS)
154 what is seen experimentally? Fermi arcs: no double crossings (PDJ,JCC,ZXS) seen infinities not seen zeros E F k
155 zeros do not affect the particle density
156 zeros do not affect the particle density no Luttinger theorem
157 zeros do not affect the particle density experimental data no Luttinger theorem
158 where do these zeros come from?
159 classical (atomic) limit
160 warm-up no hopping: x holes density of states 1+x 1 x } }
161 warm-up no hopping: x holes density of states 1+x 1 x } 1-x 2x }
162 warm-up no hopping: x holes density of states 1+x 1 x } 1-x 2x }
163 warm-up no hopping: x holes density of states 1+x 1 x } 1-x 2x } spectral weight: x- dependent G = 1+x ω + µ + U/2 + 1 x ω + µ U/2
164 quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ )
165 quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ ) double occupancy in ground state!!
166 quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ ) double occupancy in ground state!! W PES > 1 + x
167 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α
168 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer
169 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x
170 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x # of charge e states
171 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x # of charge e states # of electron states in lower band
172 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x # of charge e states # of electron states in lower band not exhausted by counting electrons alone?
173 Wilsonian analysis density of states d(uhb) zeros n< conserved charge
174 Wilsonian analysis density of states 1 + x + α(t/u, x) charge 2e stuff d(uhb) zeros n< conserved charge
175 do particles +unparticles=infinities+zeros?
176 what really is the summation over mass? Lmi+7 Lmi+6 Lmi+5 Lmi+4 Lmi+3 Lmi+2 Lmi+1 Lmi
177 mass=energy
178 high energy (UV) QFT low energy (IR) related to sum over mass
179 high energy (UV) QFT low energy (IR) dg(e) = β(g(e)) dlne locality in energy related to sum over mass
180 implement E-scaling with an extra dimension QFT low energy (IR) dg(e) = β(g(e)) dlne locality in energy high energy (UV) related to sum over mass
181 implement E-scaling with an extra dimension QFT dg(e) dlne = β(g(e)) locality in energy related to sum over mass
182 gauge-gravity duality (Maldacena, 1997) implement E-scaling with an extra dimension UV IR gravity QF T QFT dg(e) dlne = β(g(e)) locality in energy related to sum over mass
183 what s the geometry?
184 what s the geometry? dg(e) dlne = β(g(e)) =0 scale invariance (continuous)
185 what s the geometry? dg(e) dlne = β(g(e)) E λe x µ x µ /λ =0 scale invariance (continuous)
186 what s the geometry? dg(e) dlne = β(g(e)) E λe x µ x µ /λ =0 scale invariance (continuous) solve Einstein equations
187 what s the geometry? dg(e) dlne = β(g(e)) E λe x µ x µ /λ =0 scale invariance (continuous) solve Einstein equations ds 2 = u L 2 ηµν dx µ dx ν + L E 2 de 2 anti de-sitter space
188 `Holography r 0 ds 2 = L2 r 2 dt 2 + dx 2 + dr 2 AdS d+1 β(g) is local geometrize RG flow r = L 2 /E symmetry: {t, x, r} {λt, λx, λr}
189 `Holography finite density: RN black hole r 0 ds 2 = L2 r 2 dt 2 + dx 2 + dr 2 AdS d+1 β(g) is local geometrize RG flow r = L 2 /E symmetry: {t, x, r} {λt, λx, λr}
190 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2
191 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 classical gravity in a d+1 curved spacetime
192 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 classical gravity in a d+1 curved spacetime gauge theory in AdS 4
193 High T_c unparticles variable mass UPt_3
194 emergent gravity High T_c unparticles variable mass UPt_3
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