Anomalous dimensions, Mottness, Bose metals, and holography. D. Dalidovich G. La Nave G. Vanacore. Wednesday, January 18, 17

Size: px

Start display at page:

Download "Anomalous dimensions, Mottness, Bose metals, and holography. D. Dalidovich G. La Nave G. Vanacore. Wednesday, January 18, 17"

Lora Clara Hubbard
5 years ago
Views:

1 Anomalous dimensions, Mottness, Bose metals, and holography D. Dalidovich G. La Nave G. Vanacore

2 Anomalous dimensions, Mottness, Bose metals, and holography 1 p 2 (p 2 ) d U d/2 Fermi liquids D. Dalidovich G. La Nave G. Vanacore

3 = d ln g(l) d ln L = d 2 no metals for d apple 2

4 = d ln g(l) d ln L = d 2 no metals for d apple 2 2D MIT 2D IST Kravchenko 1995 Goldman 1989

5 = d ln g(l) d ln L = d 2 no metals for d apple 2 2D MIT 2D IST? Kravchenko 1995 Goldman 1989

6 = d ln g(l) d ln L = d 2 no metals for d apple 2 2D MIT 2D IST? Kravchenko 1995 unresolved? Goldman 1989

7 insulator-superconductor transition = h/4e 2

8 insulator-superconductor transition insulator = h/4e 2

9 insulator-superconductor transition insulator = h/4e 2 superconductor

10 phase-only critical bosons X H = E C p 2 X + J i i hiji cos( i i! i g c g = E C /J

11 phase-only critical bosons X H = E C p 2 X + J i i hiji cos( i i! i g c hni i6=0 h i i6=0 g = E C /J

12 phase-only critical bosons X H = E C p 2 X + J i i hiji cos( i i! i g c hni i6=0 h i i6=0 g = E C /J vortex-particle duality (MPAF 1990)

phase-only critical bosons X H = E C p 2 X + J i i hiji cos( i j ) @ @

13 phase-only critical bosons X H = E C p 2 X + J i i hiji cos( i i! i g c hni i6=0 h i i6=0 g = E C /J vortex-particle duality (MPAF 1990)

14 insulator-superconductor transition =# h 4e 2

15 insulator-superconductor transition =# h 4e 2 g c g = E C /J

16 insulator-superconductor transition =# h 4e 2 #=1 g c =0 g = E C /J

17 insulator-superconductor transition =# h 4e 2 #=1 g c #=0 =0 g = E C /J =0

18 insulator-superconductor transition =# h 4e 2 #=1 #=1 g c #=0 =0 g = E C /J =0

19 insulator-superconductor transition =# h 4e 2 #=1 #=1 g c #=0 =0 g = E C /J G(k,!) =k d/2! F k z =0 anomalous dimension

20 does this theory really work?

21 non-universality of c?

22 is a metallic phase for bosons possible?

23 is a metallic phase for bosons possible?

24 is a metallic phase for bosons possible? 1989

25 is a metallic phase for bosons possible?

26 is a metallic phase for bosons possible?

27 metal below H c2 activated region shifts to lower T as H increases

28 metal below H c2 mason/kapitulnik activated region (2000) shifts to lower T as H increases

29 not a refrigeration artifact

30 not a refrigeration artifact bose metal

31 not a refrigeration artifact bose metal / (H H SM ) 3

32 phases disrupting superconductivity dissipation (Kapitulnik)

33 phases disrupting superconductivity dissipation (Kapitulnik) Bose-Hubbard model (disordered) disorder-localised insulator (shortrange hopping)

35 is a Bose metal possible?

36 is a Bose metal possible? dc =lim T!0 lim!!0 (!, T)

37 is a Bose metal possible? dc =lim T!0 lim!!0 (!, T) collision-dominated transport Damle/Sachdev (hydrodynamic regime)

38 X H = E C p 2 X + J i i hiji cos( i j ) T T BKT m / T m<t m T g = E c /J

39 X H = E C p 2 X + J i i hiji cos( i j ) T T BKT m / T =? m<t m T g = E c /J

40 qp collisions n / e m/t

41 qp collisions n / e m/t / e m/t

42 qp collisions n / e m/t / e m/t / n O(1)

43 qp collisions n / e m/t / e m/t / n O(1) = 2 e 2 h

44 the insulator is a metal

45 the insulator is a metal but it is fragile 1! 1 +

46 the insulator is a metal but it is fragile 1! 1 +! 0

47 dissipation Z Z apple r + ie F [ ] = d 2 r d apple r ie ~ ~ A(~r, ) (~r, ) ~ ~ A(~r, ) (~r, ) +apple (~r, ) 2 + m 2 (~r, ) 2o + L dis 127,000 hits

48 dissipation Z Z apple r + ie F [ ] = d 2 r d apple r ie ~ ~ A(~r, ) (~r, ) ~ ~ A(~r, ) (~r, ) +apple (~r, ) 2 + m 2 (~r, ) 2o + L dis L dis = X ~k,!n! n ( ~ k,! n ) 2 ohmic dissipation 127,000 hits

49 but conductivity diverges at low T 4 =(e 2 /h)exp UT applet < /apple

50 but conductivity diverges at low T 4 =(e 2 /h)exp UT applet < /apple dissipation alone is not enough

51 disorder H = E C i 2 X hi,ji J ij cos( i j ) P (J ij )=1/ p 2 J 2 exp (J ij J 0 ) 2 /2J 2

52 disorder H = E C i 2 X hi,ji J ij cos( i j ) P (J ij )=1/ p 2 J 2 exp (J ij J 0 ) 2 /2J 2 3-phases

53 disorder H = E C i 2 X hi,ji J ij cos( i j ) P (J ij )=1/ p 2 J 2 exp (J ij J 0 ) 2 /2J 2 3-phases phase glass paramagnet superconductor

54 ln[z] = lim n!0 ([Z n ] 1)/n S i = (cos i, sin i )

55 ln[z] = lim n!0 ([Z n ] 1)/n S i = (cos i, sin i ) Q ab µ ( ~ k, ~ k 0,, 0 )=hs a µ( ~ k, )S b ( ~ k 0, 0 )i D( 0 1 )= lim n!0 Mn hqaa µµ( ~ k, ~ k 0,, 0 )i Edwards-Anderson order parameter

56 ln[z] = lim n!0 ([Z n ] 1)/n S i = (cos i, sin i ) Q ab µ ( ~ k, ~ k 0,, 0 )=hs a µ( ~ k, )S b ( ~ k 0, 0 )i D( 0 1 )= lim n!0 Mn hqaa µµ( ~ k, ~ k 0,, 0 )i Edwards-Anderson order parameter a µ( ~ k, ) =hs a µ( ~ k, )i SC order

57 F[,Q]=F SG (Q)+ X 1 applet Z d d x Z d 1 d 2 a,b,µ, free energy (k 2 +! n 2 + m 2 ) a,µ,k,! n X a µ(x, 1 ) +U Z d X a,µ a µ( ~ k,! n ) 2 b (x, 2 )Q ab µ (x, 1, 2 ) aµ (x, ) a 2 µ(x, ) Q ab µ ( ~ k,! 1,! 2 )= (2 ) d d (k) µ [D(! 1 )!1 +! 2,0 ab +!1,0! 2,0q ab. D(!) =! /apple

58 F[,Q]=F SG (Q)+ X 1 applet Z d d x Z d 1 d 2 a,b,µ, free energy (k 2 +! n 2 + m 2 ) a,µ,k,! n X a µ(x, 1 ) +U Z d X a,µ a µ( ~ k,! n ) 2 b (x, 2 )Q ab µ (x, 1, 2 ) [ aµ (x, ) new term a 2 µ(x, ) Q ab µ ( ~ k,! 1,! 2 )= (2 ) d d (k) µ [D(! 1 )!1 +! 2,0 ab +!1,0! 2,0q ab. D(!) =! /apple

59 similar problem F gauss = X a, ~ k,! n (k 2 +! 2 n +! n + m 2 ) a ( ~ k,! n ) 2 q X a,b, ~ k,! n! n,0 a ( ~ k,! n )[ b ( ~ k,! n )] new term propagator is replica offdiagonal G (0) ab (~ k,! n )=G 0 ( ~ k,! n ) ab + G 2 0( ~ k,! n )q!n,0

60 conductivity (! =0,T! 0) = 2 3 q EA m 4 e 2 h

61 conductivity (! =0,T! 0) = 2 3 q EA m 4 e 2 h / (g g c ) 2z

62 conductivity (! =0,T! 0) = 2 3 q EA m 4 e 2 h / (g g c ) 2z experiments: / (H H SM ) 3

63 is a phase glass stiff? F / s k 2?

64 energy landscape

65 energy landscape s 6=0

66 energy landscape s 6=0

67 energy landscape s 6=0 s =0

69 No

70 F gauss = X bose metal a, ~ k,! n (k 2 +! 2 n +! n + m 2 ) a ( ~ k,! n ) 2 q X a,b, ~ k,! n! n,0 a ( ~ k,! n )[ b ( ~ k,! n )] glassy physics

71 IST G(k,!) =k d/2 F! k z anomalous dimension

72 disorder holographically IST G(k,!) =k d/2 F! k z anomalous dimension

73 disorder holographically IST G(k,!) =k d/2 F! k z Z S = S 0 + d d xg(x)o(x) anomalous dimension

74 disorder holographically P [g(x)] / e 1 2f R d d xg(x) 2 IST G(k,!) =k d/2 F! k z Z S = S 0 + d d xg(x)o(x) anomalous dimension

75 disorder holographically P [g(x)] / e 1 2f R d d xg(x) 2 IST G(k,!) =k d/2 F! k z Z S = S 0 + d d xg(x)o(x) anomalous dimension 1 2 (@ µ ) 2 + m 2 2

76 disorder holographically P [g(x)] / e 1 2f R d d xg(x) 2 IST G(k,!) =k d/2 F! k z Z S = S 0 + d d xg(x)o(x) anomalous dimension Z d d x 0 (x)o(x) 1 2 (@ µ ) 2 + m 2 2

77 disorder holographically P [g(x)] / e 1 2f R d d xg(x) 2 IST G(k,!) =k d/2 F! k z Z S = S 0 + d d xg(x)o(x) anomalous dimension Z d d x 0 (x)o(x) 1 2 (@ µ ) 2 + m 2 2 [O] =d/2+

78 disorder holographically P [g(x)] / e 1 2f R d d xg(x) 2 IST G(k,!) =k d/2 F! k z Z S = S 0 + d d xg(x)o(x) anomalous dimension Z d d x 0 (x)o(x) 1 2 (@ µ ) 2 + m 2 2 [O] =d/2+

79 disorder holographically P [g(x)] / e 1 2f R d d xg(x) 2 IST G(k,!) =k d/2 F! k z Z S = S 0 + d d xg(x)o(x) anomalous dimension Z d d x 0 (x)o(x) 1 2 (@ µ ) 2 + m 2 2 [O] =d/2+ [O] <d Harris criterion

80 disorder holographically P [g(x)] / e 1 2f R d d xg(x) 2 IST G(k,!) =k d/2 F! k z Z S = S 0 + d d xg(x)o(x) anomalous dimension Z d d x 0 (x)o(x) 1 2 (@ µ ) 2 + m 2 2 [O] =d/2+ [O] <d Harris criterion are these anomalous dimensions related?

81 G(k,!) =k d/2 F! k µ

82 G(k,!) =k d/2 F! k µ anomalous dimension O/@ µ

83 use Caffarelli-Silvestre extension theorem (2006) g(x, 0) = f(x) xg + a zg 2 zg lim za z!0 = C d, ( r) f = 1 a 2

84 lim za z!0 C d, ( r) f x

85 lim za z!0 C d, ( r) f x

86 lim za z!0 C d, ( r) f x g(z =0,x)=f(x) = 1 a 2

87 BDHM(P) apple (x 1 ) (x n ) Z bulk[ ] =0 lim z n h (x 1,z) (x n,z)i bulk z!0 z d non-normalizable mode

88 BDHM(P) apple (x 1 ) (x n ) Z bulk[ ] =0 lim z n h (x 1,z) (x n,z)i bulk z!0 z d non-normalizable mode Heemskerk/Polchinski holographic renormalization z (x, `) = (x) IR UV z = ` z = x

89 Z bulk [ ]= Z D Z D z>`e S z>` Z D z<`e S z<`

90 Z Z Z bulk [ ]= D Z Z bulk [ ]= D z>`e S z>` } Z D z<`e S z<` } D IR[ ; `] UV[, ;, `]

91 Z Z Z bulk [ ]= D Z Z bulk [ ]= D z>`e S z>` } Z D z<`e S z<` } D IR[ ; `] UV[, ;, `] n-point function n-insertions of lim `!0

92 Z apple lim `!0 D IR (x 1 ) (x n ) UV[ ] =0

93 Z apple lim `!0 D IR (x 1 ) (x n ) UV[ ] =0 lim ` n `!0 Z D IR (x 1 ) (x n ) UV[ ] =0

94 Z apple lim `!0 D IR (x 1 ) (x n ) UV[ ] =0 lim ` n `!0 Z D IR (x 1 ) (x n ) UV[ ] =0 operator identity (P): O = C O lim z!0 z (x, z)

95 O = C O lim z!0 z (x, z)

96 use conjugate momentum O = C O lim z!0 z (x, z) = C O lim z!0 z z (x, z)

97 use conjugate momentum O = C O lim z!0 z (x, z) = C O lim z!0 z z (x, z) + m 2 =0 = Fz d 2 + Gz d 2 +, F,G 2C 1 (H), F = 0 + O(z 2 ), G = g 0 + O(z 2 )

98 use conjugate momentum O = C O lim z!0 z (x, z) = C O lim z!0 z z (x, z) + m 2 =0 = Fz d 2 + Gz d 2 +, F,G 2C 1 (H), F = 0 + O(z 2 ), G = g 0 + O(z 2 ) O(x) =2 g 0

99 solves massive scalar eom

100 solves massive scalar eom g = z d/2 solves CS extension problem p d2 +4m 2 := 2

101 solves massive scalar eom g = z d/2 solves CS extension problem p d2 +4m 2 := 2 O =( ) 0 the O for massive scalar field

102 independent of interactions Z (`)= ( )= Z S bulk [ ] <y<`= D d d x Z ` <y<` e S bulk[ ] <y<` dy y d+1 y 2 2 (@ y ) 2 + r x 2 + V ( )

103 independent of interactions Z (`)= ( )= Z S bulk [ ] <y<`= D d d x Z ` <y<` e S bulk[ ] <y<` dy y d+1 y 2 2 (@ y ) 2 + r x 2 + V ( ) z = y/`

104 independent of interactions Z (`)= ( )= Z S bulk [ ] <y<`= D d d x Z ` <y<` e S bulk[ ] <y<` dy y d+1 y 2 2 (@ y ) 2 + r x 2 + V ( ) S bulk [ ] /`<z<1 = 1`d Z d d x Z 1 /` dz z d+1 z = y/` z 2 2 (@ z ) 2 z2`2 + 2 r x 2 + V ( )

105 independent of interactions Z (`)= ( )= Z S bulk [ ] <y<`= D d d x Z ` <y<` e S bulk[ ] <y<` dy y d+1 y 2 2 (@ y ) 2 + r x 2 + V ( ) S bulk [ ] /`<z<1 = 1`d Z d d x Z 1 /` dz z d+1 z = y/` z 2 2 (@ z ) 2 z2`2 + 2 r x 2 + V ( )

106 independent of interactions Z (`)= ( )= Z S bulk [ ] <y<`= D d d x Z ` <y<` e S bulk[ ] <y<` dy y d+1 y 2 2 (@ y ) 2 + r x 2 + V ( ) S bulk [ ] /`<z<1 = 1`d Z d d x Z 1 /` dz z d+1 z = y/` z 2 2 (@ z ) 2 z2`2 + 2 r x 2 + V ( )

107 independent of interactions Z (`)= ( )= Z S bulk [ ] <y<`= D d d x Z ` <y<` e S bulk[ ] <y<` dy y d+1 y 2 2 (@ y ) 2 + r x 2 + V ( ) S bulk [ ] /`<z<1 = 1`d Z d d x Z 1 /` dz z d+1 z = y/` z 2 2 (@ z ) 2 z2`2 + 2 r x 2 + V ( ) saddle point is exact

108 independent of interactions Z (`)= ( )= Z S bulk [ ] <y<`= D d d x Z ` <y<` e S bulk[ ] <y<` dy y d+1 y 2 2 (@ y ) 2 + r x 2 + V ( ) S bulk [ ] /`<z<1 = 1`d Z d d x Z 1 /` dz z d+1 z = y/` z 2 2 (@ z ) 2 z2`2 + 2 r x 2 + V ( ) saddle point is exact = Fy d 2 + Gy d 2 +, F = 0 + O(y 2 ), G =( 0) + O(y 2 )

109 independent of interactions Z (`)= ( )= Z S bulk [ ] <y<`= D d d x Z ` <y<` e S bulk[ ] <y<` dy y d+1 y 2 2 (@ y ) 2 + r x 2 + V ( ) S bulk [ ] /`<z<1 = 1`d Z d d x Z 1 /` dz z d+1 z = y/` z 2 2 (@ z ) 2 z2`2 + 2 r x 2 + V ( ) saddle point is exact = Fy d 2 + Gy d 2 +, F = 0 + O(y 2 ), G =( 0) + O(y 2 ) same asymptotics!

110 boundary `action = Z 0( ) 0 + S int S int = f Z d d x nx O i (x)! 2 + Z d d x nx (O i (x)) 2 i=1 i=1 Fujita, et al. 2008

111 boundary `action Z = 0( ) 0 + S int S int = f Z d d x nx O i (x)! 2 + Z d d x nx (O i (x)) 2 i=1 i=1 ho(k)iho( k)i = fg(k) G(k)) 2 Fujita, et al glassy order

112 boundary `action Z = 0( ) 0 + S int S int = f Z d d x nx O i (x)! 2 + Z d d x nx (O i (x)) 2 i=1 i=1 ho(k)iho( k)i = fg(k) G(k)) 2 Fujita, et al glassy order

113 IMT Phys. Rev. Lett. 109, (2012)

114

115

116 holographic construction strong interaction +disorder (Mott physics)

117 holographic construction strong interaction +disorder (Mott physics) probe fermions [ (r, k,!)] = m k

118 holographic construction strong interaction +disorder (Mott physics) probe fermions [ (r, k,!)] = m k

119 Mott-like physics S probe (, ) = Z d d x p gi ( M D M m + )

120 Mott-like physics S probe (, ) = Z d d x p gi ( M D M m + ) what is hidden here?

121 Mott-like physics S probe (, ) = Z d d x p gi ( M D M m + ) what is hidden here? consider p gi (D m ipf µ µ )

122 Mott-like physics S probe (, ) = Z d d x p gi ( M D M m + ) what is hidden here? one possibility consider p gi (D m ipf µ µ )

123 AdS 2 ±(r,!, k) / r ±m kl p changes scaling dimension m 2 k = m 2 + pe d ± kl 2 r 0 e d =1/ p 2d(d 1) increasing p should change the spectral weight at 0

124 How is the spectrum modified? P=0 Fermi surface peak

125 How is the spectrum modified? P P=0 Fermi surface peak

126 How is the spectrum modified? P P= <p< > kf > 1/2 <! / k k F =! / (k k F ) 2 k F `Fermi Liquid Fermi surface peak

127 How is the spectrum modified? P P=0 Fermi surface peak

128 How is the spectrum modified? P P=0 p = 0.53 kf =1/2 MFL 0.53 <p<1/ p 6 1/2 > kf > 0 <! = =! / (k k F ) 1/(2 k F ) NFL Fermi surface peak

129 How is the spectrum modified? ST P P=0 p = 0.53 kf =1/2 MFL 0.53 <p<1/ p 6 1/2 > kf > 0 <! = =! / (k k F ) 1/(2 k F ) NFL Fermi surface peak

130 How is the spectrum modified? ST P P=0 Fermi surface peak

131 How is the spectrum modified? ST P>4.2 P P=0 Fermi surface peak

132 How is the spectrum modified? ST P>4.2 P P=0 Fermi surface peak Edalati,Leigh, PP PRL, 106 (2011)

133 How is the spectrum modified? ST P>4.2 P P=0 Fermi surface peak Edalati,Leigh, PP PRL, 106 (2011) dynamical spectral weight transfer

134 Schwarzschild/AdS G. Vanacore, PRD 2014

135 chiral symmetry and Pauli term

136 chiral symmetry and Pauli term! e i 5

137 chiral symmetry and Pauli term! e i 5 X breaks chiral symmetry if { 5,X}6=0

138 chiral symmetry and Pauli term! e i 5 X breaks chiral symmetry if { 5,X}6=0 { 5, µ F µ }6=0 Pauli term breaks chiral symmetry

139 chiral symmetry and Pauli term! e i 5 X breaks chiral symmetry if { 5,X}6=0 { 5, µ F µ }6=0 Pauli term breaks chiral symmetry mass generation scenario

140 Reissner-Nordstrom/AdS hep-th: Alsup,Siopsis, Eletherios p! p poles! zeros

141 Reissner-Nordstrom/AdS hep-th: Alsup,Siopsis, Eletherios p! p poles! zeros

142 Flow equations u 2p f(u)@ u ± = 2(mL)u ± +[v (u) k]+[v + (u) ± k] 2 ±, v ± (u) = 1 p! + Qq(1 u 2 d ) ± Qpu 2 d. f(u)

143 Flow equations u 2p f(u)@ u ± = 2(mL)u ± +[v (u) k]+[v + (u) ± k] 2 ±, v ± (u) = 1 p! + Qq(1 u 2 d ) ± Qpu 2 d. f(u) ±! ± 1/ ± p! k! p k

144 Flow equations u 2p f(u)@ u ± = 2(mL)u ± +[v (u) k]+[v + (u) ± k] 2 ±, v ± (u) = 1 p! + Qq(1 u 2 d ) ± Qpu 2 d. f(u) ±! ± 1/ ± p! k! p k u 2p f(u)@ u ± = +2(mL)u ± [v (u) k] [v + (u) ± k] 2 ±,

145 Flow equations u 2p f(u)@ u ± = 2(mL)u ± +[v (u) k]+[v + (u) ± k] 2 ±, v ± (u) = 1 p! + Qq(1 u 2 d ) ± Qpu 2 d. f(u) ±! ± 1/ ± p! k! p k u 2p f(u)@ u ± = +2(mL)u ± [v (u) k] [v + (u) ± k] 2 ±, Green functions are -inverses of one another!!

146 Flow equations u 2p f(u)@ u ± = 2(mL)u ± +[v (u) k]+[v + (u) ± k] 2 ±, v ± (u) = 1 p! + Qq(1 u 2 d ) ± Qpu 2 d. f(u) ±! ± 1/ ± p! k! p k u 2p f(u)@ u ± = +2(mL)u ± [v (u) k] [v + (u) ± k] 2 ±, Green functions are -inverses of one another!! 1 DetG R (!, k; m, p) = DetG R (!, k; m, p)

147 Schwarzschild/AdS

148 Q q

149 why is this an insulator? Q q

150 why is this an insulator? Q q who stole the charge?

151 why is this an insulator? Q q who stole the charge? p F µ µ

152 why is this an insulator? Q q who stole the charge? p F µ µ fermion back reaction is important!!

153 Q q

154 Q q two forces

155 Q q two forces Coulomb repulsion between Q, q gravity

156

157

158

159

160 q b q c q d

161 Q q

162 if q<q c gravity wins Q q

163 if Q q<q c q<q c Q q gravity wins infall to black hole

164 Mott Insulator if Q q<q c q<q c Q q gravity wins infall to black hole

165 Mott Insulator if Fermi surfaces exist Q q<q c q<q c Q q gravity wins infall to black hole

166 Mott Insulator if Fermi surfaces exist Q q<q c q<q c Q q gravity wins infall to black hole n charge = n rh + n q

167 Mott Insulator if Fermi surfaces exist Q q<q c q<q c Q q gravity wins infall to black hole n charge = n rh + n q zeros: composite stuff

168 Mott Insulator if Fermi surfaces exist Q q<q c q<q c Q q gravity wins infall to black hole n charge = n rh + n q zeros: composite stuff poles

169 Mott Insulator if Fermi surfaces exist Q q<q c q<q c Q q gravity wins infall to black hole n charge = n rh + n q No Luttinger `Theorem zeros: composite stuff poles

170 disorder include random electric field perturbatively in metric

171 disorder include random electric field perturbatively in metric?

`Anomalous dimensions for conserved currents from holographic dilatonic models to superconductivity. Gabriele La Nave. Thanks to: NSF, EFRC (DOE)

`Anomalous dimensions for conserved currents from holographic dilatonic models to superconductivity Thanks to: NSF, EFRC (DOE) Gabriele La Nave Kridsangaphong Limtragool Pippard s problem J s 6= c 4 2