Topological order in insulators and metals

Transcription

1 HARVARD Topological order in insulators and metals 34th Jerusalem Winter School in Theoretical Physics New Horizons in Quantum Matter December 27, January 5, 2017 Subir Sachdev Talk online: sachdev.physics.harvard.edu

2 High temperature superconductors Cu O CuO 2 plane YBa 2 Cu 3 O 6+x

3 H = J S i S j ij Undoped insulating antiferromagnet

4 Insulating spin liquid =( "#i #"i) / p 2 Lattice realization of a topological quantum field theory (TQFT) L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949) P. W. Anderson, Materials Research Bulletin 8, 153 (1973) G. Baskaran and P. W. Anderson, Phys. Rev. B 37, 580(R) (1988)

5 Insulating spin liquid =( "#i #"i) / p 2 Lattice realization of a topological quantum field theory (TQFT) L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949) P. W. Anderson, Materials Research Bulletin 8, 153 (1973) G. Baskaran and P. W. Anderson, Phys. Rev. B 37, 580(R) (1988)

6 Insulating spin liquid =( "#i #"i) / p 2 Lattice realization of a topological quantum field theory (TQFT) L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949) P. W. Anderson, Materials Research Bulletin 8, 153 (1973) G. Baskaran and P. W. Anderson, Phys. Rev. B 37, 580(R) (1988)

7 Insulating spin liquid =( "#i #"i) / p 2 Lattice realization of a topological quantum field theory (TQFT) L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949) P. W. Anderson, Materials Research Bulletin 8, 153 (1973) G. Baskaran and P. W. Anderson, Phys. Rev. B 37, 580(R) (1988)

8 Insulating spin liquid =( "#i #"i) / p 2 Lattice realization of a topological quantum field theory (TQFT) L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949) P. W. Anderson, Materials Research Bulletin 8, 153 (1973) G. Baskaran and P. W. Anderson, Phys. Rev. B 37, 580(R) (1988)

9 Insulating spin liquid =( "#i #"i) / p 2 Lattice realization of a topological quantum field theory (TQFT) L. Pauling, Proceedings of the Royal Society London A 196, 343 (1949) P. W. Anderson, Materials Research Bulletin 8, 153 (1973) G. Baskaran and P. W. Anderson, Phys. Rev. B 37, 580(R) (1988)

10 Why is this a TQFT? Place insulator on a torus:

11 Why is this a TQFT? Place insulator on a torus: Number of dimers crossing branch-cut is conserved modulo 2

12 Why is this a TQFT? =( "#i #"i) / p 2 Place insulator on a torus: Number of dimers crossing branch-cut is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

13 Why is this a TQFT? =( "#i #"i) / p 2 Place insulator 4 on a torus: Number of dimers crossing branch-cut is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

14 Why is this a TQFT? =( "#i #"i) / p 2 Place insulator 4 on a torus: Number of dimers crossing branch-cut is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

15 Why is this a TQFT? =( "#i #"i) / p 2 Place insulator 4 on a torus: Number of dimers crossing branch-cut is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

16 Why is this a TQFT? =( "#i #"i) / p 2 Place insulator 2 on a torus: Number of dimers crossing branch-cut is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

17 Why is this a TQFT? =( "#i #"i) / p 2 Place insulator 2 on a torus: Number of dimers crossing branch-cut is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

18 Why is this a TQFT? =( "#i #"i) / p 2 Place insulator 0 on a torus: Number of dimers crossing branch-cut is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

19 Why is this a TQFT? =( "#i #"i) / p 2 Place insulator 2 on a torus: Number of dimers crossing branch-cut is conserved modulo 2 D.J. Thouless, PRB 36, 7187 (1987) S.A. Kivelson, D.S. Rokhsar and J.P. Sethna, Europhys. Lett. 6, 353 (1988)

20 Introduce a pseudo spin, z` on every link `. = 1i Odd dimer constraint: = +1i Y ` on site i z` = 1. Topologically conserved charge: Y W C = z` = ±1. ` cuts contour C

21 =( "#i #"i) / p 2 2 W C =1

22 Operator to change W C =( "#i #"i) / p 2 2 W C =1 V C 0 = Y ` on C 0 x` C 0

23 Operator to change W C =( "#i #"i) / p 2 1 W C = 1 V C 0 = Y ` on C 0 x` C 0

24 Why is this a TQFT? C 0 C W C V C 0 = V C 0W C V C W C 0 = W C 0V C

25 The TQFT The Z 2 spin liquid: Described by the simplest, non-trivial, topological field theory with time-reversal symmetry: L = 1 Z 4 K IJ d 3 xa I ^ da J where a I, I =1, 2 are U(1) gauge connections, and the K matrix is 0 2 K = 2 0 The Wilson loops V C =exp i Z C dx a 1, W C =exp i Z C dx a 2 obey W C V C 0 = V C 0W C when C and C 0 wrap separate cycles of the torus. N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991); X.-G. Wen, Phys. Rev. B 44, 2664 (1991); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000); M. Freedman, C. Nayak, K. Shtengel, K. Walker, and Z. Wang, Annals of Physics 310, 428 (2004) T. H. Hansson, Vadim Oganesyan, S. L. Sondhi, Annals of Physics 313, 497 (2004)

26 b r ψ w rr 1 X H w 1 b y r rr H:c: 0 w 2 w 2 u y r;r 0 2r X rr 0 r 00 X y X rr 0 r 0 r 00 H:c: u b n b r 2 hrr 0 i n rr 0 2 U X r N 2 r : r 2N r 2n b r X r 0 2r n rr 0: Average of one boson per site: hn r i =1 R. Jalabert and S. Sachdev Phys. Rev. B 44, 686 (1991); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000); O.I. Motrunich and T. Senthil, PRL 89, (2002).

27 b r ψ w rr 1 w 2 b w/u Superfluid Insulator with topological Fractionalized order Conventional Mott Insulator U/u b Average of one2. boson Schematic per site: phase hn diagram r i =1of the boson H R. Jalabert and S. Sachdev Phys. Rev. B 44, 686 (1991); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000); O.I. Motrunich and T. Senthil, PRL 89, (2002).

28 b r ψ w rr 1 w 2 b w/u Superfluid Insulator with topological Fractionalized order O(2) Conventional Mott Insulator U/u b Average of one boson per site: hn r i =1 2. Schematic phase diagram of the boson H R. Jalabert and S. Sachdev Phys. Rev. B 44, 686 (1991); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000); O.I. Motrunich and T. Senthil, PRL 89, (2002).

29 Topological phase transitions b r ψ w rr 1 w 2 b w/u Superfluid Insulator with topological Fractionalized order Conventional Mott Insulator U/u b Average of one boson per site: hn r i =1 2. Schematic phase diagram of the boson H R. Jalabert and S. Sachdev Phys. Rev. B 44, 686 (1991); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000); O.I. Motrunich and T. Senthil, PRL 89, (2002).

30 = b ; N r = 1 for all r Trivial insulator Average of one boson per site: hn r i =1

31 = b ; = 1 p 2! + or Trivial insulator Average of one boson per site: hn r i =1

32 = b ; = 1 p 2! + or Trivial insulator Average of one boson per site: hn r i =1

33 = b ; = 1 p 2! + or Average of one boson per site: hn r i =1

34 = b ; = 1 p 2! + or Average of one boson per site: hn r i =1

35 = b ; = 1 p 2! + or N r = 1 for all r Average of one boson per site: hn r i =1

36 In the limit of large U, we expect to prefer states in which N r = 1 at all r. By the usual procedure of perturbation theory in 1/U, we obtain an e ective Hamiltonian within the subspace of states with N r = 1. To order 1/U, this e ective Hamiltonian is e X H 0 eff H u b ;u J bond X y rr 0 2 b r b r H:c: 0 hrr 0 i X K ring y y H:c: ; 0 00 stands for the on-site repulsion terms a u X n 0 2 rr hrr 0 i X u b n b r 2 Bosons at unit density on the square lattice; N r = 1 for all r r

37 = b ; = 1 p 2! + or Bosons at unit density on the square lattice; N r = 1 for all r

38 = b ; = 1 p 2! + or Bosons at unit density on the square lattice; N r = 1 for all r

39 = b ; = 1 p 2! + or Insulator with topological order Bosons at unit density on the square lattice; N r = 1 for all r

40 = b ; = 1 p 2! + or Insulator with topological order Bosons at unit density on the square lattice; N r = 1 for all r

41 = b ; = 1 p 2! + or Insulator with topological order Bosons at unit density on the square lattice; N r = 1 for all r

42 = b ; = 1 p 2! + or Insulator with topological order Bosons at unit density on the square lattice; N r = 1 for all r

43 = b ; = 1 p 2! + or Insulator with topological order Bosons at unit density on the square lattice; N r = 1 for all r

44 e H 0 eff H u b ;u X J X bond y rr 2 b 0 r b r 0 H:c: hrr 0 i X K ring y y H:c: ; 0 00 stands for the on-site repulsion terms a u X n 0 2 rr hrr 0 i X u b n b r 2 This e ective Hamiltonian has exactly the form of a U(1) lattice gauge theory. We define b r e i" r r where " r = ±1 on the two sublattices, and rr 0 e i" ra r,wherer 0 = r +ê, = x, y. Then the above theory can be written on cubic spacetime lattice in a relativistic from with action r S = J X r cos( µ r 2a rµ ) K X cos( µ a ) The boson e i r has U(1) gauge charge 2. The Gauss law for this lattice gauge theory is equivalent to the constraint N r = 1 at all r.

45 = b ; = 1 p 2! + or Insulator with topological order Bosons at unit density on the square lattice; N r = 1 for all r

46 = b ; = 1 p 2! + or N r =1 Insulator with topological order Add a boson on site.

47 = b ; = 1 p 2! + or N r = 1 2 Insulator with topological order At large U, energy is lowered when the boson to splits into 2 half bosons.

48 = b ; = 1 p 2! + or N r = 1 2 Insulator with topological order The half charge bosons can then move freely through the lattice

49 = b ; = 1 p 2! + or N r = 1 2 Insulator with topological order The half charge bosons can then move freely through the lattice

50 = b ; = 1 p 2! + or N r = 1 2 Insulator with topological order The half charge bosons can then move freely through the lattice

51 = b ; = 1 p 2! + or N r = 1 2 Insulator with topological order The half charge bosons can then move freely through the lattice

52 = b ; = 1 p 2! + or N r = 1 2 Insulator with topological order The half charge bosons can then move freely through the lattice

53 We can move beyond the N r = 1 subspace, and account for these half-charged states, by introducing an operator e i r which has U(1) gauge charge 1, and global boson number charge " r /2. Then gaugeinvariant boson operators for the site and bond bosons now become b r = e i" r( r 2 r ), r = e i" r(a r r ) It is easy to verify that these new representations leave the J and K terms in the previous spacetime lattice action for N r =1independent of r. We can also write the half-charge hopping terms illustrated in the previous slides in this formulation. The final form of the action so obtained is simplest in terms of a new field r in the mapping r = r when " r =1, r = r + r when " r = 1 Note that e i r which has U(1) gauge charge 1, and global boson number charge 1/2 Bosons at unit density on the square lattice

54 Collecting these transformations, we obtain the complete action for the full phase diagram is S = t X r cos( µ r a rµ A rµ /2) J X r cos( µ r 2a rµ ) K X cos( µ a ) In this form, e i r has U(1) gauge charge 1, and boson number charge 1/2; e i r has U(1) gauge charge 2, and boson number charge 0. We have also included an external (fixed) gauge field A µ, which couples to the boson number. The gauge-invariant boson operator is and this only has A µ charge 1/2. b r e i2 r+i r Bosons at unit density on the square lattice

55 What we have achieved so far: We started with a model of bosons, b i on the square lattice, at a density of one boson per site, with short-range interactions. This model has a global U(1) symmetry under which b! be i. We showed that this model can be rewritten (under certain conditions) in terms of half-charged boson h e i and a Higgs field e i, so that b (h ) 2 This theory has the additional gauge invariance under which! e 2i#(x), h! he i#(x) Then the phases of the theory can be described by the Lagrangian L = (@ µ ia µ ia µ /2)h 2 + m 2 h h 2 + u h h 4 + (@ µ 2ia µ ) 2 + m u 4 K cos( µ a )... Bosons at unit density on the square lattice

56 The phases are: (1) Superfluid: h i6= 0, hhi 6= 0. The global U(1) symmetry is broken, and so there is Goldstone mode. The gauge fluctuations are completely Higgsed. (2) Trivial insulator: h i = 0, hhi = 0 and h i = 0, hhi 6= 0: Strong gauge fluctuations confine and h into the composite b h 2.Thebquanta are gapped excitations. Such a description is obtained in the confining phase of the compact U(1) gauge theory where there is no Higgs condensate. However, it is also obtained in the Higgs phase where there is a h condensate, and such a Higgs phase is smoothly connected to the confining phase. Bosons at unit density on the square lattice

57 (3) Insulator with topological order: h i6= 0, hhi = 0. Now the field is condensed and this gaps the U(1) gauge excitations, and h quanta are deconfined gapped excitations. However, there is another gapped excitation: the analog of the Abrikosov vortex in, which we denote by v. We perform a particle-vortex duality transform to obtain an e ective action for the gapped field v. Then we obtain the field theory with the continuum Lagrangian (we are being a little sloppy about the role of monopoles here) L = (@ µ ia µ )h 2 + m 2 h h 2 + u 2 h h 4 + (@ µ ib µ )v 2 + m 2 v v 2 + u 2 v v 4 + i µ a b + i 2 µ A b Bosons at unit density on the square lattice

58 (3) Insulator with topological order: h i6= 0, hhi = 0. Now the field is condensed and this gaps the U(1) gauge excitations, and h quanta are deconfined gapped excitations. However, there is another gapped excitation: the analog of the Abrikosov vortex in, which we denote by v. We perform a particle-vortex duality transform to obtain an e ective action for the gapped field v. Then we obtain the field theory with the continuum Lagrangian (we are being a little sloppy about the role of monopoles here) L = (@ µ ia µ )h 2 + m 2 h h 2 + u 2 h h 4 + (@ µ ib µ )v 2 + m 2 v v 2 + u 2 v v 4 + i µ a b + i 2 µ A b TQFT Bosons at unit density on the square lattice

59 b r ψ w rr 1 w 2 b w/u Superfluid Insulator with topological Fractionalized order Conventional Mott Insulator U/u b Average of one2. boson Schematic per site: phase hn diagram r i =1of the boson H R. Jalabert and S. Sachdev Phys. Rev. B 44, 686 (1991); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000); O.I. Motrunich and T. Senthil, PRL 89, (2002).

60 A.V. Chubukov, T. Senthil and S. Sachdev, PRL 72, 2089 (1994) O(2)* b r ψ w rr 1 w 2 b w/u Superfluid Insulator with topological Fractionalized order Conventional Mott Insulator U/u b Average of one2. boson Schematic per site: phase hn diagram r i =1of the boson H R. Jalabert and S. Sachdev Phys. Rev. B 44, 686 (1991); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000); O.I. Motrunich and T. Senthil, PRL 89, (2002).

61 Transition from superfluid to insulator with Z 2 topological order The boson field e i is condensed in both phases. This gaps out the a µ field. So the quantum criticality is just the critical theory of the half-charged boson h e i L = (@ µ A µ /2)h 2 + m 2 h h 2 + u h 4 This is the O(2) Wilson-Fisher theory. The refers to the fact that the spectrum of the theory only contains operators which are invariant under h! h: it is not possible to create a single half-boson, and they always appear in pairs. Bosons at unit density on the square lattice The critical theory therefore involves a fractionalized field, and not the order parameter: this is an example of a deconfined critical point.

62 R. Jalabert and S. Sachdev Phys. Rev. B 44, 686 (1991); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000); O.I. Motrunich and T. Senthil, PRL 89, (2002). b r ψ w rr 1 w 2 b w/u Superfluid Insulator with topological Fractionalized order Ising* Conventional Mott Insulator U/u b 2. Schematic phase diagram of the boson H Bosons at unit density on the square lattice

63 Transition from trivial insulator to insulator with Z 2 topological order The half-charged boson h e i is gapped in both phases. So as a first attempt, we just write down the theory of the critical Higgs field e i L = (@ µ 2ia µ ) 2 + m u e 2 ( a ) 2 However, this turns out to be incorrect: we cannot ignore the monopoles is the compact U(1) gauge field, a µ. Bosons at unit density on the square lattice

64 Transition from trivial insulator to insulator with Z 2 topological order The correct theory can be obtained by performing a particle-vortex duality on both the half-charged boson h e i (to a double-vortex d) and the Higgs field e i (to the vortex v) L = (@ µ ic µ )d 2 + m 2 d d 2 + (@ µ ib µ )v 2 + m 2 v v 2 + i 2 µ b A + i 2 µ a (c 2b ) K X cos( µ a ) After integrating over a µ this becomes equivalent to the field theory L = (@ µ i2b µ )d 2 + m 2 d d 2 + (@ µ ib µ )v 2 + m 2 v v 2 + i 2 µ b A y m d v 2 +c.c., where y m is the monopole fugacity. Bosons at unit density on the square lattice

65 Transition from trivial insulator to insulator with Z 2 topological order The double vortex d is condensed in both phases: so we can replace d by a constant, and ignore the gapped b µ gauge field. Because of the monopole fugacity term, the symmetry v! ve i' is broken. Only the real part of v becomes critical at the transition. Denoting w v + v, the vison field, we have the critical theory L = (@ µ w) 2 + m 2 ww 2 + uw 4 This is the Ising Wilson-Fisher critical theory. Again, the refers to the fact that all observable operators must be invariant under w! w. Bosons at unit density on the square lattice

66 An alternative formulation in two dimensions preemptively accounts for the strong e ects of monopoles. We take the strong-coupling limit, in which the Higgs field is locally condensed, to a Z 2 gauge field where e i r = 1 and e ia rµ = ±1. Then we have the Hamiltonian of a Z 2 gauge field coupled to a half-charged boson e i r : X H = t r z cos( r A r /2) r, =x,y K X z z z z g X r, =x,y x r The Z 2 gauge field is dual to the Ising* theory encountered earlier. Bosons at unit density on the square lattice

67 R. Jalabert and S. Sachdev PRB 44, 686 (1991); A.V. Chubukov, T. Senthil and S. Sachdev, PRL 72, 2089 (1994); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000); O.I. Motrunich and T. Senthil, PRL 89, (2002). O(2)* b r ψ w rr 1 w 2 b w/u Superfluid Insulator with topological Fractionalized order O(2) Ising* Conventional Mott Insulator U/u b 2. Schematic phase diagram of the boson H Bosons at unit density on the square lattice

68 S. Sachdev and R. Jalabert, Modern Physics Letters B 4, 1043 (1990); R. Jalabert and S. Sachdev Phys. Rev. B 44, 686 (1991); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000). b r ψ w rr 1 w 2 b w/u Superfluid Insulator with topological Fractionalized order Conventional Mott Insulator Valence bond solid U/u 2. Schematic phase diagram of the boson H b Bosons at half-integer density on the square lattice

69 Topological phase transitions b r ψ w rr 1 w 2 b w/u Superfluid Insulator with topological Fractionalized order Conventional Mott Insulator Valence bond solid U/u 2. Schematic phase diagram of the boson H b Bosons at half-integer density on the square lattice

70 A.V. Chubukov, T. Senthil and S. Sachdev, PRL 72, 2089 (1994) O(2)* b r ψ w rr 1 w 2 b w/u Superfluid Insulator with topological Fractionalized order Conventional Mott Insulator Valence bond solid U/u 2. Schematic phase diagram of the boson H b Bosons at half-integer density on the square lattice

71 T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004) Deconfined criticality b r ψ w rr 1 w 2 b w/u Superfluid Insulator with topological Fractionalized order Conventional Mott Insulator Valence bond solid U/u 2. Schematic phase diagram of the boson H b Bosons at half-integer density on the square lattice

72 VBS states on the square lattice Columnar VBS =( "#i #"i) / p 2 Equivalently, this can be interpreted as a model of bosons at 1/2 filling, with the mapping "i ) 0i #i ) b 0i

73 VBS states on the square lattice Columnar VBS =( "#i #"i) / p 2 Equivalently, this can be interpreted as a model of bosons at 1/2 filling, with the mapping "i ) 0i #i ) b 0i

74 VBS states on the square lattice Columnar VBS =( "#i #"i) / p 2 Equivalently, this can be interpreted as a model of bosons at 1/2 filling, with the mapping "i ) 0i #i ) b 0i

75 VBS states on the square lattice Columnar VBS =( "#i #"i) / p 2 Equivalently, this can be interpreted as a model of bosons at 1/2 filling, with the mapping "i ) 0i #i ) b 0i

76 VBS states on the square lattice Plaquette VBS Equivalently, this can be interpreted as a model of bosons at 1/2 filling, with the mapping "i ) 0i #i ) b 0i

77 VBS states on the square lattice Plaquette VBS Equivalently, this can be interpreted as a model of bosons at 1/2 filling, with the mapping "i ) 0i #i ) b 0i

78 VBS states on the square lattice Plaquette VBS Equivalently, this can be interpreted as a model of bosons at 1/2 filling, with the mapping "i ) 0i #i ) b 0i

79 VBS states on the square lattice Plaquette VBS Equivalently, this can be interpreted as a model of bosons at 1/2 filling, with the mapping "i ) 0i #i ) b 0i

80 S. Sachdev and R. Jalabert, Modern Physics Letters B 4, 1043 (1990); R. Jalabert and S. Sachdev Phys. Rev. B 44, 686 (1991); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000). b r ψ w rr 1 w 2 b w/u Superfluid Insulator with topological Fractionalized order Conventional Mott Insulator Valence bond solid U/u 2. Schematic phase diagram of the boson H b Bosons at half-integer density on the square lattice

81 S. Sachdev and R. Jalabert, Modern Physics Letters B 4, 1043 (1990); R. Jalabert and S. Sachdev Phys. Rev. B 44, 686 (1991); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000). At half-integer density, we obtain an additional Berry phase term (" r = ±1 on the two sub lattices) S = i X r " r a r t X r J X r K X cos( µ r a rµ A rµ /2) cos( µ r 2a rµ ) cos( µ a ) The Berry phases prohibit a trivial phase with no broken symmetry and no topological order: instead we obtain a phase with valence bond solid order and broken translational symmetry. Also, the Z 2 spin liquid is now a symmetry enriched topological (SET) state. Bosons at half-integer density on the square lattice

82 S. Sachdev and R. Jalabert, Modern Physics Letters B 4, 1043 (1990); R. Jalabert and S. Sachdev PRB 44, 686 (1991); A.V. Chubukov, T. Senthil and S. Sachdev, PRL 72, 2089 (1994); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000); T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004) O(2)* b r ψ w rr 1 w 2 b w/u CP 1 Superfluid Insulator with topological Fractionalized order Z8* Conventional Mott Insulator Valence bond solid U/u b 2. Schematic phase diagram of the boson H Bosons at half-integer density on the square lattice

83 Anomaly constraints on phase diagram Consider a general lattice Hamiltonian with 2 global symmetries: translation by a lattice spacing, ˆTx, and a global U(1) symmetry, U, associated in our case with conservation of boson number. The global U(1) symmetry is generated by U =exp i X! ˆn i i where is the rotation angle, and ˆn i is the boson number on site i. Theretwosymmetries clearly commute ˆT x U = U ˆT x Now place the system on a L x L y torus, let us consider a spatially-dependent rotation angle (i.e. we gauge the U(1) symmetry)! U G =exp i 2 X x iˆn i. L x It is now easy to show that ˆT x and U G do not commute, and ˆT x U G =exp i2 N U G ˆTx L x i where N is the total number of bosons. U(1) ˆT x global symmetry. This anomaly is an obstruction to gauging the

84 Anomaly constraints on phase diagram Flux quantum L y L x The operator U G is equivalent to the adiabatic insertion of one flux quantum of the external gauge field, A µ, which couples to the boson number. Specifically, we have A x = f(t)/l x,wheref(t) increases slowly from 0 to 2. However, in the action of the TQFT we have the term Z Z Z i i exp d 3 df r µ b A =exp dydt b y =exp i dy b y W y 2 2 dt Here b µ is the gauge field that couples to the vison, and W y is the branch-cut operator. So we have the basic result that in an insulator with Z 2 topological order, when acting on the low-energy sector U G = W y

85 Anomaly constraints on phase diagram Flux quantum W y L y L x The operator U G is equivalent to the adiabatic insertion of one flux quantum of the external gauge field, A µ, which couples to the boson number. Specifically, we have A x = f(t)/l x,wheref(t) increases slowly from 0 to 2. However, in the action of the TQFT we have the term Z Z Z i i exp d 3 df r µ b A =exp dydt b y =exp i dy b y W y 2 2 dt Here b µ is the gauge field that couples to the vison, and W y is the branch-cut operator. So we have the basic result that in an insulator with Z 2 topological order, when acting on the low-energy sector U G = W y

86 Anomaly constraints on phase diagram Combining our results we have for bosons at density ˆT x W y = exp(2 i L y ) W y T x ˆT y W x = exp(2 i L x ) W x T y As the Wilson loop operator transports a vison around the torus, in a translationally invariant system, we can write these relations as a constraint on the translation operator acting on single vison states ˆT x ˆTy =exp(2 i ) ˆT y ˆTx. So the vison sees each boson as a 2 flux quantum. At = 1/2, this implies that each vison state is at least doublydegenerate, and it transforms non-trivially under lattice translations.

87 S. Sachdev and R. Jalabert, Modern Physics Letters B 4, 1043 (1990); R. Jalabert and S. Sachdev PRB 44, 686 (1991); A.V. Chubukov, T. Senthil and S. Sachdev, PRL 72, 2089 (1994); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000); T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004) O(2)* b r ψ w rr 1 w 2 b w/u CP 1 Superfluid Insulator with topological Fractionalized order Z8* Conventional Mott Insulator Valence bond solid U/u b 2. Schematic phase diagram of the boson H Bosons at half-integer density on the square lattice

88 Transition from VBS insulator to insulator with Z 2 topological order Our previous incorrect attempt for this transition at = 1 turns out to correct at =1/2 (up to irrelevant terms at the critical point)! The half-charged boson h e i is gapped in both phases. So we just write down the theory of the critical Higgs field e i L = (@ µ 2ia µ ) 2 + m u e 2 ( a ) 2 The multiple vison species suppress monopoles in the compact U(1) gauge field, a µ. This theory is dual at an O(2) Wilson-Fisher theory, and that is the critical theory of the Z 8 transition. Bosons at half-integer density on the square lattice

89 S. Sachdev and R. Jalabert, Modern Physics Letters B 4, 1043 (1990); R. Jalabert and S. Sachdev PRB 44, 686 (1991); A.V. Chubukov, T. Senthil and S. Sachdev, PRL 72, 2089 (1994); S. Sachdev and M. Vojta, Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000); T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004) O(2)* b r ψ w rr 1 w 2 b w/u CP 1 Superfluid Insulator with topological Fractionalized order Z8* Conventional Mott Insulator Valence bond solid U/u b 2. Schematic phase diagram of the boson H Bosons at half-integer density on the square lattice

90 SM FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis

91 SM FL YBa 2 Cu 3 O 6+x d-wave superconductor Figure: K. Fujita and J. C. Seamus Davis

92 SM Insulating Antiferromagnet FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis

93 Undoped insulating antiferromagnet

94 Antiferromagnet with p mobile holes per square

95 Filled Band

96 Antiferromagnet with p mobile holes per square But relative to the band insulator, there are 1+ p holes per square

97 Antiferromagnet In a conventional metal (a Fermi liquid), with no broken symmetry, the area with p mobile holes per square enclosed by the Fermi surface must be 1+p But relative to the band insulator, there are 1+ p holes per square

98 The Hubbard Model H = X i<j t ij c i c j + U X i 1 1 n i" n i# 2 2 µ X i c i c i t ij! hopping. U! local repulsion, µ! chemical potential Spin index =", # n i = c i c i c i c j + c j c i = ij c i c j + c j c i =0 Will study on the square lattice

99 Fermi surfaces in electron- and hole-doped cuprates Hole states occupied Electron states occupied E ective Hamiltonian for quasiparticles: H 0 = X i<j t ij c i c j X k " k c k c k with t ij non-zero for first, second and third neighbor, leads to satisfactory agreement with experiments. The area of the occupied electron states, A e, from Luttinger s theory is A e = 2 2 (1 x) for hole-doping x 2 2 (1 + p) for electron-doping p The area of the occupied hole states, A h, which form a closed Fermi surface and so appear in quantum oscillation experiments is A h =4 2 A e.

100 Anomaly constraints on the Fermi surface size M. Oshikawa, PRL 84, 3370 (2000) A. Paramekanti and A. Vishwanath, PRB 70, (2004) Flux quantum L y L x The anomaly argument can be restated as a computation of the change in crystal momentum, P, of the many-body state due to piercing by a flux quantum P xf P xi = 2 N (mod 2 ) =2 L y (mod 2 ) L x where = N/(L x L y ) is the density of fermions.

101 Anomaly constraints on the Fermi surface size P x =2 L y (mod 2 ), P y =2 L x (mod 2 ) Now we compute the momentum balance assuming that the only low energy excitations are quasiparticles near the Fermi surface, and these react like free particles to a su ciently slow flux insertion. So each quasiparticle picks up a momentum ~ p (2 /Lx, 0), and then we can write (with n p the quasiparticle density excited by the flux insertion) P x = X p n p p x. Now n p = ±1 on a shell of thickness ~ p d S ~ p on the Fermi surface (where S ~ p is an area element on the Fermi surface). So we can write the above as a surface integral I Lx L y P x = p x ~ p d Sp ~ FS 4 2 Z = ( ~ Lx L y p ˆx) 4 2 dv by the divergence theorem. So ~ p FV

102 Anomaly constraints on the Fermi X surface size P x =2 L y (mod 2 ), P y =2 L x (mod 2 ) Now n p = ±1 on a shell of thickness ~ p d S ~ p on the Fermi surface (where S ~ p is an area element on the Fermi surface). So we can write the above as a surface integral I Lx L y P x = p x ~ p d Sp ~ FS 4 2 Z = ( ~ Lx L y p ˆx) 4 2 dv by the divergence theorem. So FV ~ p

103 Anomaly constraints on the Fermi surface size P x =2 L y (mod 2 ), P y =2 L x (mod 2 ) P x = 2 L x Lx L y 4 2 V FS, P y = 2 L y Lx L y 4 2 V FS where V FS is the volume of the Fermi surface. So, although the quasiparticles are only defined near the Fermi surface, by using Gauss s Law on the momentum acquired by quasiparticles near the Fermi surface, we have converted the answer to an integral over the volume enclosed by the Fermi surface. Now we equate these values to those obtained above, and obtain N L x L y V FS 4 2 = L xm x, N L x L y V FS 4 2 = L ym y for some integers m x, m y. Now choose L x, L y mutually prime integers; then m x L x = m y L y implies that m x L x = m y L y = pl x L y for some integer p. Then we obtain = N = V FS L x L y p. This is Luttinger s theorem.

104 Fermi surface+antiferromagnetism Hole states occupied Electron states occupied + The electron spin polarization obeys S(r, ) = (r, )e ik r where K is the ordering wavevector.

105 Fermi surface+antiferromagnetism We use the operator equation (valid on each site i): 1 1 U n " n # = 2U S ~ 2 + U 4 (1) Then we decouple the interaction via exp 2U X Z! Z d S 3 ~ i 2 = D J ~ i ( )exp i X i Z d apple 3 8U ~ J 2 i ~J i ~ Si! (2) We now integrate out the fermions, and look for the saddle point of the resulting e ective action for ~ J i. At the saddle-point we find that the lowest energy is achieved when the vector has opposite orientations on the A and B sublattices. Anticipating this, we look for a continuum limit in terms of afield~' i where ~J i = ~' i e ik r i (3)

106 Fermi surface+antiferromagnetism In this manner, we obtain the spin-fermion model Z Z = Dc D~' exp ( S) Z X S = d k Z c k d X " k c c i ~' i ~ c i e ik r i + Z d d 2 r apple 1 2 (r r ~' ) (@ ~' ) 2 + s 2 ~' 2 + u 4 ~' 4

107 Fermi surface+antiferromagnetism In the Hamiltonian form (ignoring, for now, the time dependence of ~' ), the coupling between ~' and the electrons takes the form X H sdw = ~' q c k+q, ~ c k+k, k,q,, where ~ are the Pauli matrices, the boson momentum q is small, while the fermion momenum k extends over the entire Brillouin zone. In the antiferromagnetically ordered state, we may take ~' / (0, 0, 1), and the electron dispersions obtained by diagonalizing H 0 + H sdw are E k± = " k + " k+k 2 ± s "k " k+k ~' 2 This leads to the Fermi surfaces shown in the following slides as a function of increasing ~'.

108 Square lattice Hubbard model at p=0 Increasing SDW order Hole pockets Electron pockets " k =0 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).

109 Square lattice Hubbard model at p=0 Increasing SDW order Hole pockets Electron pockets " k = 0 or " k+k =0 " k =0 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).

110 Square lattice Hubbard model at p=0 Increasing SDW order Hole pockets Electron pockets Hot spots where " k = " k+k 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).

111 Square lattice Hubbard model at p=0 Increasing SDW order Hole pockets Hole pockets Electron pockets Hot spots where " k = " k+k Fermi surface breaks up at hot spots into electron and hole pockets 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).

112 Square lattice Hubbard model at p=0 Increasing SDW order Insulator Hot spots where " k = " k+k Fermi surface breaks up at hot spots into electron and hole pockets 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).

113 Square lattice Hubbard model at p=0 h~' i6=0 and large h~' i6=0 and small h~' i =0 In ncreasing SDW Insulator Metal with electron and hole pockets Metal with large Fermi surface s

114 Spin density wave order, topological order, and Fermi surface reconstruction

115 Quantum phase transition with Fermi surface reconstruction ncreasing SDW h~' i6=0 h~' i =0 Metal with electron and hole pockets Metal with large Fermi surface

116 Separating onset of SDW order and Fermi surface reconstruction ncreasing SDW h~' i6=0 h~' i =0 Metal with electron and hole pockets Metal with large Fermi surface

117 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 h~' i6=0 h~' i =0 h~' i =0 Metal with electron and hole pockets Metal with large Fermi surface T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, (2003)

118 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 h~' i6=0 Metal with electron and hole pockets h~' i =0 Algebraic Charge liquid (ACL) or Fractionalized Fermi liquid (FL*) phase with no symmetry breaking and pocket Fermi surfaces h~' i =0 Metal with large Fermi surface T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, (2003)

119 Spin density wave order, topological order, and Fermi surface reconstruction Subir Sachdev, 1, 2 Erez Berg, 3 Shubhayu Chatterjee, 1 and Yoni Schattner 3 arxiv:

120 Hertz theory for XY SDW order The Hertz theory for the onset of SDW order can be described by the following Hamiltonian H sdw = H c + H + H Y, (1.1) where H c describes electrons (of density (1 H c = X i,j p)) hopping on the sites of a square lattice (t ij + µ ij ) c i c j (1.2) with c i the electron annihilation operator on site i with spin =", #. We represent the SDW order by a lattice XY rotor model, described by an angle i, and its canonically conjugate number operator N i, obeying X H = J ij cos( i j )+4 Ni 2 ; [ i,n j ]=i ij, (1.3) X i<j where J ij positive exchange constants, and is proportional to the bare spin-wave gap (the 4 is for future convenience). A term linear in N i is also allowed in H, but we ignore it for simplicity; such a linear term will not be allowed when we consider models with SU(2) symmetry in Section IV. X h i i

121 X X Hertz theory for XY SDW order Finally, there is a Yukawa coupling between the XY order parameter, e i, and the fermions X i H Y = i he i i c i" c i# + e i i c i# c i", (1.4) where i i ( 1) x i+y i (1.5) is the staggering factor representing the opposite spin orientations on the two sublattices. Note that the Yukawa coupling, and the remaining Hamiltonian, commute with the total spin along the z direction S z = X i N i c i" c i" 1 2 c i# c i#. (1.6)

122 Quantum phase transition with Fermi surface reconstruction ncreasing SDW h~' i6=0 h~' i =0 Metal with electron and hole pockets Metal with large Fermi surface

123 Z2 gauge theory for fractionalized XY SDW order coupled to electrons H 1 = H c + H,Z2 + H Y H c = X i,j (t ij + µ ij ) c i c j H Y = X i i he i i c i" c i# + e i i c i# c i" H,Z2 = i X i<j J ij µ z ij cos (( i j )/2) + 4 X N 2 i g X hiji µ x ij K X " Y µ z ij #, i Z

124 Consider the phase with Z 2 topological order. In this state it is useful to perform a rotation about the z axis in spin space by introducing the fermion operators + = e i /2 c ", = e i /2 c #. Then the Yukawa coupling, H Y, takes a simple form independent of the orientation of the XY order: X h i H Y = i i+ i + i i+. i In other words, the ± fermions move in the presence of a spacetimeindependent XY order, even though the actual orientation of the XY order rotates from point to point. Moreover, from the electron hopping term in H c, we can obtain an e ective hopping Z ij t ij ( i+ j+ + i j )where Z ij = he ±i( i j )/2 i is a renormalization factor of order unity. So it appears we can realize a situation in which the ± fermions are approximately free, and their observation of constant XY order implies that they will form small pocket Fermi surfaces (or be fully gapped at p = 0).

125 In K (C) ACL/FL* Z 2 topological order Proliferation of double vortices e i =0 e i 6=0 In e i =0 Proliferation of single vortices (B) SDW metal (A) Fermi liquid

126 In K (C) ACL/FL* Z 2 topological order Proliferation of double vortices e i =0 e i 6=0 In Fe based superconductors e i =0 Proliferation of single vortices (B) SDW metal (A) Fermi liquid

127 In K (C) ACL/FL* Z 2 topological order Proliferation of double vortices Cuprate superconductors e i =0 e i 6=0 In e i =0 Proliferation of single vortices (B) SDW metal (A) Fermi liquid

128 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 h~' i6=0 Metal with electron and hole pockets h~' i =0 Algebraic Charge liquid (ACL) or Fractionalized Fermi liquid (FL*) phase with no symmetry breaking and pocket Fermi surfaces h~' i =0 Metal with large Fermi surface T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, (2003)

129 SM FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis

130 SM Insulating Antiferromagnet FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis

131 M. Platé, J. D. F. Mottershead, I. S. Elfimov, D. C. Peets, Ruixing Liang, D. A. Bonn, W. N. Hardy, S. Chiuzbaian, M. Falub, M. Shi, L. Patthey, and A. Damascelli, Phys. Rev. Lett. 95, (2005) SM FL A conventional metal: the Fermi liquid with Fermi surface of size 1+p

132 S. Badoux, W. Tabis, F. Laliberté, G. Grissonnanche, B. Vignolle, D. Vignolles, J. Béard, D.A. Bonn, W.N. Hardy, R. Liang, N. Doiron-Leyraud, L. Taillefer, and C. Proust, Nature 531, 210 (2016). Pseudogap metal SM FL at low p Many indications that this metal behaves like a Fermi liquid, but with Fermi surface size p and not 1+p.

133 T. Senthil, M. Vojta and S. Sachdev, PRB 69, (2004) Pseudogap metal SM FL at low p Many indications that this metal behaves like a Fermi liquid, but with Fermi surface size p and not 1+p. If present at T=0, a metal with a size p Fermi surface (and translational symmetry preserved) must have topological order

134 Hall effect measurements in YBCO b p* 1.5 SDW CDW FL n H = V / e R H p Fermi liquid (FL) with carrier density 1+p p p Badoux, Proust, Taillefer et al., Nature 531, 210 (2016)

135 Hall effect measurements in YBCO b p* n H = V / e R H SDW CDW FL p 1 + p Spin density wave (SDW) breaks translational invariance, and the Fermi liquid then has carrier density p p Badoux, Proust, Taillefer et al., Nature 531, 210 (2016)

136 Hall effect measurements in YBCO b p* 1.5 SDW CDW FL n H = V / e R H p 1 + p Charge density wave (CDW) leads to complex Fermi surface reconstruction and negative Hall resistance p Badoux, Proust, Taillefer et al., Nature 531, 210 (2016)

137 Hall effect measurements in YBCO b p* 1.5 SDW CDW FL Evidence for n H = V / e R H p a metal with topological order: Fermi surface of p size p! p Badoux, Proust, Taillefer et al., Nature 531, 210 (2016)