Sign-problem-free Quantum Monte Carlo of the onset of antiferromagnetism in metals

Transcription

1 Sign-problem-free Quantum Monte Carlo of the onset of antiferromagnetism in metals Subir Sachdev sachdev.physics.harvard.edu HARVARD

2 Max Metlitski Erez Berg HARVARD

3 Max Metlitski Erez Berg Sean Hartnoll Diego Hofman Matthias Punk HARVARD

4 Increasing SDW order s Nd 2 x Ce x CuO 4 T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner, M. Lambacher, A. Erb, J. Wosnitza, and R. Gross, Phys. Rev. Lett. 103, (2009).

5 Resistivity ρ 0 + AT n BaFe 2 (As 1 x P x ) 2 K. Hashimoto, K. Cho, T. Shibauchi, S. Kasahara, Y. Mizukami, R. Katsumata, Y. Tsuruhara, T. Terashima, H. Ikeda, M. A. Tanatar, H. Kitano, N. Salovich, R. W. Giannetta, P. Walmsley, A. Carrington, R. Prozorov, and Y. Matsuda, Science 336, 1554 (2012).

6 BaFe 2 (As 1 x P x ) 2 K. Hashimoto, K. Cho, T. Shibauchi, S. Kasahara, Y. Mizukami, R. Katsumata, Y. Tsuruhara, T. Terashima, H. Ikeda, M. A. Tanatar, H. Kitano, N. Salovich, R. W. Giannetta, P. Walmsley, A. Carrington, R. Prozorov, and Y. Matsuda, Science 336, 1554 (2012).

7 Lower Tc superconductivity in the heavy fermion compounds G. Knebel, D. Aoki, and J. Flouquet, arxiv: Tuson Park, F. Ronning, H. Q. Yuan, M. B. Salamon, R. Movshovich, J. L. Sarrao, and J. D. Thompson, Nature 440, 65 (2006)

8 Outline 1. Weak coupling theory 2. Universal critical theory 3. Quantum Monte Carlo without the sign problem 4. Features of strong coupling

9 Outline 1. Weak coupling theory 2. Universal critical theory 3. Quantum Monte Carlo without the sign problem 4. Features of strong coupling

10 The Hubbard Model H = i<j t ij c iα c jα + U i n i 1 n i µ i c iα c iα t ij hopping. U local repulsion, µ chemical potential Spin index α =, n iα = c iα c iα c iα c jβ + c jβ c iα = δ ijδ αβ c iα c jβ + c jβ c iα =0

11 Fermi surface+antiferromagnetism Metal with large Fermi surface + The electron spin polarization obeys S(r, τ) = ϕ (r, τ)e ik r where K is the ordering wavevector.

12 The Hubbard Model Decouple U term by a Hubbard-Stratanovich transformation S = d 2 rdτ [L c + L ϕ + L cϕ ] L c = c aε( i )c a ϕ 2 α 2 L ϕ = 1 2 ( ϕ α) 2 + r 2 ϕ2 α + u 4 L cϕ = λϕ α e ik r c a σ α ab c b. Yukawa coupling between fermions and antiferromagnetic order: λ 2 U, the Hubbard repulsion

13 The Hubbard Model Decouple U term by a Hubbard-Stratanovich transformation S = d 2 rdτ [L c + L ϕ + L cϕ ] L c = c aε( i )c a ϕ 2 α 2 L ϕ = 1 2 ( ϕ α) 2 + r 2 ϕ2 α + u 4 L cϕ = λϕ α e ik r c a σ α ab c b. Hertz, Moriya, Millis: integrate out fermions, and focus on effective theory of damped excitations of the order parameter ϕ α.methodfails in d = 2.

14 Fermi surface+antiferromagnetism Metal with large Fermi surface

15 Fermi surface+antiferromagnetism Fermi surfaces translated by K =(π, π).

16 Fermi surface+antiferromagnetism Hot spots

17 Fermi surface+antiferromagnetism Electron and hole pockets in antiferromagnetic phase with ϕ =0

18 Fermi surface+antiferromagnetism ncreasing SDW ϕ =0 ϕ =0 Metal with electron and hole pockets Metal with large Fermi surface Increasing interaction S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

19 Pairing glue from antiferromagnetic fluctuations V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983) D.J. Scalapino, E. Loh, and J.E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986) S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, (2010)

20 c kα c kβ = ε αβ (cos k x cos k y ) Unconventional pairing at and near hot spots

21 Fermi surface+antiferromagnetism T * Fluctuating Fermi pockets Quantum Strange Critical Metal Large Fermi surface ncreasing SDW Spin density wave (SDW) Underlying SDW ordering quantum critical point in metal at x = x m

22 Fermi surface+antiferromagnetism Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Quantum Strange Critical Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) QCP for the onset of SDW order is actually within a superconductor

23 Outline 1. Weak coupling theory 2. Universal critical theory 3. Quantum Monte Carlo without the sign problem 4. Features of strong coupling

24 Outline 1. Weak coupling theory 2. Universal critical theory 3. Quantum Monte Carlo without the sign problem 4. Features of strong coupling

25 Hot spots

26 Low energy theory for critical point near hot spots

27 Low energy theory for critical point near hot spots

28 Theory has fermions ψ 1,2 (with Fermi velocities v 1,2 ) and boson order parameter ϕ, interacting with coupling λ v 1 v 2 k y k x ψ 1 fermions occupied ψ 2 fermions occupied

29 Theory has fermions ψ 1,2 (with Fermi velocities v 1,2 ) and boson order parameter ϕ, interacting with coupling λ L = ψ 1α ( τ iv 1 r ) ψ 1α + ψ 2α ( τ iv 2 r ) ψ 2α ( r ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 ψ λϕ 1α σ αβψ 2β + ψ 2α σ αβψ 1β Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, (2004).

30 Theory has fermions ψ 1,2 (with Fermi velocities v 1,2 ) and boson order parameter ϕ, interacting with coupling λ L = ψ 1α ( τ iv 1 r ) ψ 1α + ψ 2α ( τ iv 2 r ) ψ 2α ( r ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 ψ λϕ 1α σ αβψ 2β + ψ 2α σ αβψ 1β M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010) S.A. Hartnoll, D.M. Hofman, M. A. Metlitski and S. Sachdev, Phys. Rev. B 84, (2011)

31 Theory has fermions ψ 1,2 (with Fermi velocities v 1,2 ) and boson order parameter ϕ, interacting with coupling λ L = ψ 1α ( τ iv 1 r ) ψ 1α + ψ 2α ( τ iv 2 r ) ψ 2α ( r ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 ψ λϕ 1α σ αβψ 2β + ψ 2α σ αβψ 1β Theory flows to strong-coupling in d = 2. M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010) S.A. Hartnoll, D.M. Hofman, M. A. Metlitski and S. Sachdev, Phys. Rev. B 84, (2011)

32 Outline 1. Weak coupling theory 2. Universal critical theory 3. Quantum Monte Carlo without the sign problem 4. Features of strong coupling

33 Outline 1. Weak coupling theory 2. Universal critical theory 3. Quantum Monte Carlo without the sign problem 4. Features of strong coupling

34 Theory has fermions ψ 1,2 (with Fermi velocities v 1,2 ) and boson order parameter ϕ, interacting with coupling λ v 1 v 2 k y k x ψ 1 fermions occupied ψ 2 fermions occupied

35 Theory has fermions ψ 1,2 (with Fermi velocities v 1,2 ) and boson order parameter ϕ, interacting with coupling λ v 1 v 2 k y k x To faithfully realize low energy theory in quantum Monte Carlo, we need a UV completion in which Fermi lines don t end and all weights are positive.

36 Low energy theory for critical point near hot spots

37 We have 4 copies of the hot spot theory...

38 and their Fermi lines are connected as shown:

39 Reconnect Fermi lines and eliminate the sign problem!

40 QMC for the onset of antiferromagnetism K Hot spots in a single band model

41 QMC for the onset of antiferromagnetism K E. Berg, M. Metlitski, and S. Sachdev, arxiv: Hot spots in a two band model

42 QMC for the onset of antiferromagnetism K E. Berg, M. Metlitski, and S. Sachdev, arxiv: Hot spots in a two band model

43 QMC for the onset of antiferromagnetism K E. Berg, M. Metlitski, and S. Sachdev, arxiv: No sign problem in fermion determinant Monte Carlo! Determinant is positive because of Kramer s degeneracy, and no additional symmetries are needed; holds for arbitrary band structure and band filling, provided K only Hot spots in a two band model connects hot spots in distinct bands

44 QMC for the onset of antiferromagnetism Electrons with dispersion ε k interacting with fluctuations of the antiferromagnetic order parameter ϕ. Z = S = + Dc α Dϕ exp ( S) dτ c kα τ ε k c kα k dτd 2 1 x 2 ( x ϕ ) 2 + r 2 ϕ λ dτ i ϕ i ( 1) x i c iα σ αβc iβ

45 QMC for the onset of antiferromagnetism Electrons with dispersions ε (x) and ε (y) k k interacting with fluctuations of the antiferromagnetic order parameter ϕ. Z = Dc (x) α Dc (y) α Dϕ exp ( S) S = dτ c (x) kα τ ε(x) c (x) k kα k + dτ c (y) kα τ ε(y) c (y) k kα k + dτd 2 1 x 2 ( x ϕ ) 2 + r 2 ϕ λ dτ ϕ i ( 1) x i c (x) iα σ αβc (y) iβ i + H.c. E. Berg, M. Metlitski, and S. Sachdev, arxiv:

46 QMC for the onset of antiferromagnetism Electrons with dispersions ε (x) and ε (y) k k interacting with fluctuations of the antiferromagnetic order parameter ϕ. Z = Dc (x) α Dc (y) α Dϕ exp ( S) S = dτ c (x) kα τ ε(x) c (x) k kα k + dτ c (y) kα τ ε(y) c (y) k kα k + dτd 2 1 x 2 ( x ϕ ) 2 + r 2 ϕ λ dτ ϕ i ( 1) x i c (x) iα σ αβc (y) iβ i No sign problem! + H.c. E. Berg, M. Metlitski, and S. Sachdev, arxiv:

47 QMC for the onset of antiferromagnetism K E. Berg, M. Metlitski, and S. Sachdev, arxiv: Hot spots in a two band model

48 QMC for the onset of antiferromagnetism 1 a) k y /π K E. Berg, M. Metlitski, and S. Sachdev, arxiv: Center Brillouin zone at (π,π,)

49 QMC for the onset of antiferromagnetism 1 a) b) 0.5 E. Berg, M. Metlitski, and S. Sachdev, arxiv: k y /! 0 K k y /!!0.5!1!1 0 1 Move one of the Fermi surface by (π,π,)

50 QMC for the onset of antiferromagnetism 1 a) 0.5 E. Berg, M. Metlitski, and S. Sachdev, arxiv: k y /π Now hot spots are at Fermi surface intersections

51 QMC for the onset of antiferromagnetism 1 b) 0.5 E. Berg, M. Metlitski, and S. Sachdev, arxiv: k y /! 0!0.5!1!1 0 1 Expected Fermi surfaces in the AFM ordered phase

52 QMC for the onset of antiferromagnetism 1 r =!0.5 1 r = 0 1 r = k y /! !0.5!0.5! !1!1 0 1 k x /!!1!1 0 1 k x /!!1!1 0 1 k x /! Electron occupation number n k as a function of the tuning parameter r E. Berg, M. Metlitski, and S. Sachdev, arxiv:

53 QMC for the onset of antiferromagnetism! " /(L 2 #) a) b) L=8 L=10 L=12 L=14! r Binder cumulant ! r AF susceptibility, χ ϕ, and Binder cumulant as a function of the tuning parameter r E. Berg, M. Metlitski, and S. Sachdev, arxiv:

54 QMC for the onset of antiferromagnetism P ± (x max )! 10 x 10! _ P _ + P _ r c L = 14 L = 10 L = 12 0!2!2! r s/d pairing amplitudes P + /P as a function of the tuning parameter r E. Berg, M. Metlitski, and S. Sachdev, arxiv:

55 Outline 1. Weak coupling theory 2. Universal critical theory 3. Quantum Monte Carlo without the sign problem 4. Features of strong coupling

56 Outline 1. Weak coupling theory 2. Universal critical theory 3. Quantum Monte Carlo without the sign problem 4. Features of strong coupling

57 Features of strong coupling Shift in QCP due to superconductivity: backbending of SDW order. Pairing instability is (log) 2 with universal co-efficient. Leading subdominant instability is bond-modulated charge order. Intermediate fractionalized Fermi liquid (FL*) state with hole pockets and no broken symmetry.

58 Features of strong coupling Shift in QCP due to superconductivity: backbending of SDW order. Pairing instability is (log) 2 with universal co-efficient. Leading subdominant instability is bond-modulated charge order. Intermediate fractionalized Fermi liquid (FL*) state with hole pockets and no broken symmetry.

59 Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Quantum Strange Critical Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) QCP for the onset of SDW order is actually within a superconductor

60 Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Quantum Strange Critical Metal Large Fermi surface d-wave superconductor x s Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

61 E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). S. Sachdev, arxiv: T T * Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface E. G. Moon and S. Sachdev, Phy. Rev. B 80, (2009) T sdw d-wave SC SC+ SDW SC SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

62 QMC for the onset of antiferromagnetism P ± (x max )! 10 x 10! _ P _ + P _ r c L = 14 L = 10 L = 12 0!2!2! r s/d pairing amplitudes P + /P as a function of the tuning parameter r E. Berg, M. Metlitski, and S. Sachdev, arxiv:

63 QMC for the onset of antiferromagnetism P ± (x max )! 10 x 10! _ P _ + P _ r c L = 14 L = 10 L = 12 0!2!2! r Notice shift between the position of the QCP in the superconductor, and the position of maximum pairing. This is found in numerous experiments. E. Berg, M. Metlitski, and S. Sachdev, arxiv:

64 Features of strong coupling Shift in QCP due to superconductivity: backbending of SDW order. Pairing instability is (log) 2 with universal co-efficient. Leading subdominant instability is bond-modulated charge order. Intermediate fractionalized Fermi liquid (FL*) state with hole pockets and no broken symmetry.

65 Two loop results: Non-Fermi liquid spectrum at hot spots v 1 v 2 G fermion 1 iω v.k k y k x A. J. Millis, Phys. Rev. B 45, (1992) Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, (2004)

66 Two loop results: Quasiparticle weight vanishes upon approaching hot spots k k y k k x G fermion = Z(k ) ω v F (k )k, Z(k ) v F (k ) k M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010)

67 Pairing by SDW fluctuation exchange Weak-coupling theory Fermi EF energy 1+λ 2 ρ(e F ) log ω Density of states at Fermi energy

68 Pairing by SDW fluctuation exchange Antiferromagnetic critical point Fermi 1+ sin θ 2π log2 EF ω energy θ is the angle between Fermi lines. Independent of interaction strength U in 2 dimensions. (see also Ar. Abanov, A. V. Chubukov, and A. M. Finkel'stein, Europhys. Lett. 54, 488 (2001)) M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010)

69 θ

70 Pairing by SDW fluctuation exchange Antiferromagnetic critical point 1+ sin θ 2π log2 EF ω Universal log 2 singularity arises from Fermi lines; singularity at hot spots is weaker. Interference between BCS and quantum-critical logs. Momentum dependence of self-energy is crucial. Not suppressed by 1/N factor in 1/N expansion. M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010)

71 Features of strong coupling Shift in QCP due to superconductivity: backbending of SDW order. Pairing instability is (log) 2 with universal co-efficient. Leading subdominant instability is bond-modulated charge order. Intermediate fractionalized Fermi liquid (FL*) state with hole pockets and no broken symmetry.

72 c kα c kβ = ε αβ (cos k x cos k y ) Log-squared instability to d-wave pairing

73 c k Q/2,α c k+q/2,α = Φ(cos k x cos k y ) Q is 2k F wavevector Φ Φ Similar log-squared instability in particle-hole channel to bond-modulated charge order

74 Bond-modulated charge order M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010) Φ Φ c k Q/2,α c k+q/2,α = Φ(cos k x cos k y )

75 Bond-modulated charge order M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010) Φ Φ Q c k Q/2,α c k+q/2,α = Φ(cos k x cos k y )

76 Bond-modulated charge order M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010) Φ Φ Q Q c k Q/2,α c k+q/2,α = Φ(cos k x cos k y )

77 Bond-modulated charge order M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, (2010) 1 Bond density measures amplitude for electrons to be in spin-singlet valence bond. 1 No modulations on sites, c rαc sα is modulated only for r = s. c k Q/2,α c k+q/2,α = Φ(cos k x cos k y )

78 Features of strong coupling Shift in QCP due to superconductivity: backbending of SDW order. Pairing instability is (log) 2 with universal co-efficient. Leading subdominant instability is bond-modulated charge order. Intermediate fractionalized Fermi liquid (FL*) state with hole pockets and no broken symmetry.

79 Holedoped Electrondoped?

80 Quantum phase transition with Fermi surface reconstruction ncreasing SDW ϕ =0 ϕ =0 Metal with electron and hole pockets Metal with large Fermi surface

81 Separating onset of SDW order and Fermi surface reconstruction ncreasing SDW ϕ =0 ϕ =0 Metal with electron and hole pockets Metal with large Fermi surface

82 Separating onset of SDW order and Fermi surface reconstruction ncreasing SDW Electron and/or hole Fermi pockets form in local SDW order, but quantum fluctuations destroy long-range SDW order ϕ =0 ϕ =0 ϕ =0 Metal with electron and hole pockets Fractionalized Fermi liquid (FL*) phase with no symmetry breaking and small Fermi surface Metal with large Fermi surface T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, (2003)

83 k_y 1.5 k_y k_x Hole pocket of a Z 2 -FL* phase in a single-band t-j model M. Punk and S. Sachdev, Phys. Rev. B 85, (2012)

84 Characteristics of FL* phase Fermi surface volume does not count all electrons. Such a phase must have neutral S =1/2 excitations ( spinons ), and collective spinless gauge excitations ( topological order). These topological excitations are needed to account for the deficit in the Fermi surface volume, in M. Oshikawa s proof of the Luttinger theorem. T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, (2003)

85 Outline 1. Weak coupling theory 2. Universal critical theory 3. Quantum Monte Carlo without the sign problem 4. Features of strong coupling

86 QMC for the onset of antiferromagnetism K Hot spots in a single band model

87 QMC for the onset of antiferromagnetism Faithful realization of low energy theory. Sign problem is absent as long as K connects hotspots in distinct K E. Berg, M. Metlitski, and S. Sachdev, arxiv: Particle-hole or point-group symmetries or commensurate densities not required! bands Hot spots in a two band model