The Superfluid-Insulator transition

Transcription

1 The Superfluid-Insulator transition Boson Hubbard model M.P. A. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher, Phys. Rev. B 40, 546 (1989).

2 Superfluid-insulator transition Ultracold 87 Rb atoms - bosons M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

3 Insulator (the vacuum) at large U

4 Excitations:

5 Excitations:

6 Excitations of the insulator: S = d 2 rdτ τ ψ 2 + v 2 ψ 2 +(g g c ) ψ 2 + u 2 ψ 4

7 T S = d 2 rdτ τ ψ 2 + v 2 ψ 2 +(g g c ) ψ 2 + u 2 ψ 4 T KT Quantum critical ψ =0 ψ =0 Superfluid Insulator 0 g c CFT3 g

8 T Quantum critical T KT Superfluid Insulator 0 g c g

9 Classical vortices and wave oscillations of the condensate Dilute Boltzmann/Landau gas of particle and holes T T KT Quantum critical Superfluid Insulator 0 g c g

10 CFT at T>0 T Quantum critical T KT Superfluid Insulator 0 g c g

11 C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)

12 C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981)

13 Outline 1. Introduction to the Hubbard model Superexchange and antiferromagnetism 2. Coupled dimer antiferromagnet CFT3: the Wilson-Fisher fixed point 3. Honeycomb lattice: semi-metal and antiferromagnetism CFT3: Dirac fermions and the Gross-Neveu model 4. Quantum critical dynamics AdS/CFT and the collisionless-hydrodynamic crossover 5. Hubbard model as a SU(2) gauge theory Spin liquids, valence bond solids: analogies with SQED and SYM

14 Outline 1. Introduction to the Hubbard model Superexchange and antiferromagnetism 2. Coupled dimer antiferromagnet CFT3: the Wilson-Fisher fixed point 3. Honeycomb lattice: semi-metal and antiferromagnetism CFT3: Dirac fermions and the Gross-Neveu model 4. Quantum critical dynamics AdS/CFT and the collisionless-hydrodynamic crossover 5. Hubbard model as a SU(2) gauge theory Spin liquids, valence bond solids: analogies with SQED and SYM

15 Quantum critical transport Quantum perfect fluid with shortest possible relaxation time, τ R τ R k B T S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

16 Quantum critical transport Transport co-oefficients not determined by collision rate, but by universal constants of nature Electrical conductivity σ = e2 h [Universal constant O(1) ] K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

17 Quantum critical transport Transport co-oefficients not determined by collision rate, but by universal constants of nature Momentum transport η s viscosity entropy density = k B [Universal constant O(1) ] P. Kovtun, D. T. Son, and A. Starinets, Phys. Rev. Lett. 94, (2005)

18 Quantum critical transport Euclidean field theory: Compute current correlations on R 2 S 1 with circumference 1/T 1/T R 2

19 Quantum critical transport Euclidean field theory: Compute current correlations on R 2 S 1 with circumference 1/T 4πT Complex ω plane 2πT 2πT Direct 1/N or expansions for correlators at the Euclidean frequencies ω n =2πnT i ( n integer) or in the conformal collisionless regime, ω k B T.

20 Quantum critical transport Euclidean field theory: Compute current correlations on R 2 S 1 with circumference 1/T 4πT Complex ω plane 2πT 2πT Direct 1/N or expansions for correlators at the Euclidean frequencies ω n =2πnT i ( n integer) or in the conformal collisionless regime, ω k B T.

21 Density correlations in CFTs at T >0 Two-point density correlator, χ(k, ω) Kubo formula for conductivity σ(ω) = lim k 0 iω χ(k, ω) k2 For all CFT3s, at ω k B T χ(k, ω) = 4e2 h K k 2 v2 k 2 ω 2 ; σ(ω) = 4e2 h K where K is a universal number characterizing the CFT3, and v is the velocity of light.

22 Quantum critical transport Euclidean field theory: Compute current correlations on R 2 S 1 with circumference 1/T 4πT Complex ω plane 2πT 2πT Direct 1/N or expansions for correlators at the Euclidean frequencies ω n =2πnT i ( n integer) or in the conformal collisionless regime, ω k B T.

23 Quantum critical transport Euclidean field theory: Compute current correlations on R 2 S 1 with circumference 1/T 4πT Complex ω plane 2πT 2πT Strong coupling problem: Correlators at ω k B T, along the real axis, in the collision-dominated hydrodynamic regime.

24 Quantum critical transport Euclidean field theory: Compute current correlations on R 2 S 1 with circumference 1/T 4πT Complex ω plane 2πT 2πT Strong coupling problem: Correlators at ω k B T, along the real axis, in the collision-dominated hydrodynamic regime.

25 Density correlations in CFTs at T >0 Two-point density correlator, χ(k, ω) Kubo formula for conductivity σ(ω) = lim k 0 iω χ(k, ω) k2 For all CFT3s, at ω k B T, we have the Einstein relation χ(k, ω) = e 2 χ c Dk 2 Dk 2 iω ; σ(ω) =e2 Dχ c = e2 h Θ 1Θ 2 where the compressibility, χ c, and the diffusion constant D obey χ = k BT (hv) 2 Θ 1 ; D = hv2 k B T Θ 2 with Θ 1 and Θ 2 universal numbers characteristic of the CFT3 K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

26 Density correlations in CFTs at T >0 In CFT3s collisions are phase randomizing, and lead to relaxation to local thermodynamic equilibrium. So there is a crossover from collisionless behavior for ω k B T,to hydrodynamic behavior for ω k B T. σ(ω) = e 2 e 2 h K, ω k BT h Θ 1Θ 2 σ Q, ω k B T and in general we expect K = Θ 1 Θ 2 (verified for Wilson- Fisher fixed point). K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

27 SU(N) SYM3 with N = 8 supersymmetry Has a single dimensionful coupling constant, e 0, which flows to a strong-coupling fixed point e 0 = e 0 in the infrared. The CFT3 describing this fixed point resembles critical spin liquid theories. This CFT3 is the low energy limit of string theory on an M2 brane. The AdS/CFT correspondence provides a dual description using 11-dimensional supergravity on AdS 4 S 7. The CFT3 has a global SO(8) R symmetry, and correlators of the SO(8) charge density can be computed exactly in the large N limit, even at T>0.

28 SU(N) SYM3 with N = 8 supersymmetry The SO(8) charge correlators of the CFT3 are given by the usual AdS/CFT prescription applied to the following gauge theory on AdS4: S = 1 4g 2 4D d 4 x gg MA g NB F a MNF a AB where a = labels the generators of SO(8). Note that in large N theory, this looks like 28 copies of an Abelian gauge theory.

29 Collisionless to hydrodynamic crossover of SYM3 Imχ(k, ω)/k 2 Im K k2 ω 2 P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

30 Collisionless to hydrodynamic crossover of SYM3 Imχ(k, ω)/k 2 Im Dχ c Dk 2 iω P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

31 Universal constants of SYM3 χ c = k BT (hv) 2 Θ 1 D = hv2 k B T Θ 2 σ(ω) = 2N 3/2 K, ω k B T Θ 1 Θ 2, ω k B T K = 3 Θ 1 = 8π2 2N 3/2 9 Θ 2 = 3 8π 2 C. Herzog, JHEP 0212, 026 (2002) P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, Phys. Rev. D 75, (2007)

32 Electromagnetic self-duality Unexpected result, K =Θ 1 Θ 2, and σ(ω)isfrequency independent. This is traced to a four-dimensional electromagnetic self-duality of the theory on AdS 4. In the large N limit, the SO(8) currents decouple into 28 U(1) currents with a Maxwell action for the U(1) gauge fields on AdS 4. The conductivity takes the self-dual value σ(ω) = 1/g4D 2.

33 Electromagnetic self-duality Unexpected result, K =Θ 1 Θ 2, and σ(ω)isfrequency independent. This is traced to a four-dimensional electromagnetic self-duality of the theory on AdS 4. In the large N limit, the SO(8) currents decouple into 28 U(1) currents with a Maxwell action for the U(1) gauge fields on AdS 4. The conductivity takes the self-dual value σ(ω) = 1/g4D 2. S = 1 4g 2 4D d 4 x gg MA g NB F a MNF a AB

34 Electromagnetic self-duality Unexpected result, K =Θ 1 Θ 2, and σ(ω)isfrequency independent. This is traced to a four-dimensional electromagnetic self-duality of the theory on AdS 4. In the large N limit, the SO(8) currents decouple into 28 U(1) currents with a Maxwell action for the U(1) gauge fields on AdS 4. The conductivity takes the self-dual value σ(ω) = 1/g4D 2. S = 1 4g 2 4D d 4 x gg MA g NB F a MNF a AB

35 Electromagnetic self-duality Unexpected result, K =Θ 1 Θ 2, and σ(ω)isfrequency independent. This is traced to a four-dimensional electromagnetic self-duality of the theory on AdS 4. In the large N limit, the SO(8) currents decouple into 28 U(1) currents with a Maxwell action for the U(1) gauge fields on AdS 4. The conductivity takes the self-dual value σ(ω) = 1/g4D 2. All these special features are generic to theories with a Maxwell-Einstein gravity dual. They are not expected to survive stringy 1/N 2 corrections

36 Lessons from AdS/CFT Strongly interacting quantum-critical systems are nearly perfect fluids. The AdS description of such perfect fluids suggests that All continuous global symmetries are (approximately) self dual. We can define a conductivity σ(ω) for each global symmetry: σ(ω) is (nearly) ω independent The value of σ(ω) is close to the self dual value

37 Lessons from AdS/CFT Strongly interacting quantum-critical systems are nearly perfect fluids. The AdS description of such perfect fluids suggests that All continuous global symmetries are (approximately) self dual. We can define a conductivity σ(ω) for each global symmetry: σ(ω) is (nearly) ω independent The value of σ(ω) is close to the self dual value These features are found in experimental and numerical studies

38 Resistivity of Bi films Conductivity σ σ Superconductor (T 0) = σ Insulator (T 0) = 0 σ Quantum critical point (T 0) 4e2 h Self-dual value = 4e 2 /h D. B. Haviland, Y. Liu, and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989) M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)

39 Quantum critical transport in graphene σ(ω) = e 2 h e 2 hα 2 (T ) π 2 + O O 1 ln(λ/ω) 1 ln(α(t )), ω k B T, ω k B T α 2 (T ) η s = k B α 2 (T ) where the fine structure constant is α(t )= α 1+(α/4) ln(λ/t ) T 0 4 ln(λ/t ) L. Fritz, J. Schmalian, M. Müller and S. Sachdev, Physical Review B 78, (2008) M. Müller, J. Schmalian, and L. Fritz, Physical Review Letters 103, (2009)

40 Outline 1. Introduction to the Hubbard model Superexchange and antiferromagnetism 2. Coupled dimer antiferromagnet CFT3: the Wilson-Fisher fixed point 3. Honeycomb lattice: semi-metal and antiferromagnetism CFT3: Dirac fermions and the Gross-Neveu model 4. Quantum critical dynamics AdS/CFT and the collisionless-hydrodynamic crossover 5. Hubbard model as a SU(2) gauge theory Spin liquids, valence bond solids: analogies with SQED and SYM

41 Outline 1. Introduction to the Hubbard model Superexchange and antiferromagnetism 2. Coupled dimer antiferromagnet CFT3: the Wilson-Fisher fixed point 3. Honeycomb lattice: semi-metal and antiferromagnetism CFT3: Dirac fermions and the Gross-Neveu model 4. Quantum critical dynamics AdS/CFT and the collisionless-hydrodynamic crossover 5. Hubbard model as a SU(2) gauge theory Spin liquids, valence bond solids: analogies with SQED and SYM

42 t ε k H 0 = i<j t ij c iα c iα k ε k c kα c kα k Begin with free electrons.

43 t ε k H 0 = i<j t ij c iα c iα k ε k c kα c kα k Begin with free electrons. Add local antiferromagnetism with order parameter ϕ H sdw = i ϕ (r i )( 1) i c iα σ αβc iβ

44 t ε k H 0 = i<j t ij c iα c iα k ε k c kα c kα k Begin with free electrons. Add local antiferromagnetism with order parameter ϕ H sdw = i ϕ (r i )( 1) i c iα σ αβc iβ The phase with ϕ = 0 is an insulator with a gap between conduction and valence bands.

45 Z. Y. Meng et al., Nature 464, 847 (2010).

46 R z (x, τ) Néel Perform SU(2) rotation R z on filled band of electrons: c ψ+ c = z z z z ψ ε k ψ ± states empty ψ ± states filled k

47 Quantum disordering magnetic order c = z z ψ+ c z z ψ

48 Quantum disordering magnetic order c = z z ψ+ c z z ψ SU(2) spin

49 Quantum disordering magnetic order c = z z ψ+ c z z ψ U(1) charge

50 Quantum disordering magnetic order c = z z ψ+ c z z ψ U U 1 SU(2) s;gauge S. Sachdev, M. A. Metlitski, Y. Qi, and S. Sachdev Phys. Rev. B 80, (2009)

51 Quantum disordering magnetic order c = z z ψ+ c z z ψ The Hubbard model can be written exactly as a lattice gauge theory with a SU(2) s;g SU(2) spin U(1) charge invariance. The SU(2) s;g is a gauge invariance, while SU(2) spin U(1) charge is a global symmetry

52 Matter context of SU(2) s;g SU(2) spin U(1) charge theory Fundamental fermions ψ transforming as (2, 1, 1), Fundamental scalar z transforming as ( 2, 2, 0), connecting local to global Néel order, Adjoint scalar N(r i )=ψ i σ ψ i transforming as (3, 1, 0), measuring local Néel order.

53 Phase diagram

54 Phase diagram ε k Semi-metal k

55 Phase diagram ε k t Insulator Semi-metal with Neel k order

56 Phase diagram ε k k Semi-metal = 4 mass- SU(2) QCD with N f less Dirac quarks t Insulator with Neel L = 1 g 2 F 2 µν + ψγ µ ( µ ia µ )ψ order Could describe a CFT3 like SYM, but z could =0 also be, unstable N to =0 confinement.

57 Phase diagram ε k t Insulator Semi-metal with Neel k order Spin liquid: CFT of SU(2) QCD with Nf =4 massless Dirac quarks

58 Phase diagram ε k SU(2) is Higgsed down to U(1). k Semi-metal All matter fields are gapped. U(1) monopoles drive Polyakov z confinement =0, N =0 z Spectral =0 flow, in filled N fermion =0 bands leads to Berry phases of monopoles, endowing them with crystal Spin liquid: momentum. CFT of SU(2) QCD Confining state has valence bond with solid Nf =4 (VBS) massless order Dirac quarks t Insulator with Neel order

59 Phase diagram ε k t Insulator Semi-metal with Neel k order t Spin liquid: CFT of SU(2) QCD with Nf =4 massless Dirac quarks Confining insulator with VBS (kekule) order

60 Phase diagram ε k t Insulator Semi-metal with Neel k order M Spin liquid: CFT of SU(2) QCD with Nf =4 massless Dirac quarks Confining insulator with VBS (kekule) order t

61 Phase diagram ε k t Insulator Semi-metal with Neel k order M Spin liquid: CFT of SU(2) QCD with Nf =4 massless Dirac quarks Confining insulator with VBS (kekule) order t

62 Phase diagram ε k t k Semi-metal Spin liquid: CFT of SU(2) QCD with Nf =4 massless Multicritical point M: SU(2) QCD with N f = 4 massless Dirac quarks, 4 real fundamental scalars, and 3 real adjoint scalars. Dirac quarks M Insulator with Neel order L = 1 g 2 F 2 µν + ψγ µ ( µ ia µ )ψ + ( µ ia µ )z 2 +(( µ ia µ )φ) 2 Could describe a CFT3 like SYM, but could also be unstable to confinement. Confining insulator with VBS (kekule) order t

63 Phase diagram ε k t Insulator Semi-metal with Neel k order M Spin liquid: CFT of SU(2) QCD with Nf =4 massless Dirac quarks Confining insulator with VBS (kekule) order t