From the pseudogap to the strange metal

Transcription

1 HARVARD From the pseudogap to the strange metal S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, PRB 94, (2016) S. Sachdev and S. Chatterjee, arxiv: APS March meeting March 13, 2017 Talk online: sachdev.physics.harvard.edu

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

3 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

4 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.

5 Begin with the spin-fermion model. Electrons c i on the square lattice with dispersion H c = X t c i, c i+v, + c i+v, c i, µ X c i, c i, + H int i, i are coupled to an antiferromagnetic order parameter ` = x, y, z X H int = `(i)c i i, ` c i, + V where i = ±1 on the two sublattices. i `(i), When `(i) =constant independent of i, we have long-range AFM, and a gap in the fermion spectrum at the anti-nodes.

6 Begin with the spin-fermion model. Electrons c i on the square lattice with dispersion H c = X t c i, c i+v, + c i+v, c i, µ X c i, c i, + H int i, i are coupled to an antiferromagnetic order parameter ` = x, y, z X H int = `(i)c i i, ` c i, + V where i = ±1 on the two sublattices. i `(i), When `(i) =constant independent of i, we have long-range AFM, and a gap in the fermion spectrum at the anti-nodes.

7 Begin with the spin-fermion model. Electrons c i on the square lattice with dispersion H c = X t c i, c i+v, + c i+v, c i, µ X c i, c i, + H int i, i are coupled to an antiferromagnetic order parameter ` = x, y, z X H int = `(i)c i i, ` c i, + V where i = ±1 on the two sublattices. i `(i), When `(i) =constant independent of i, we have long-range AFM, and a gap in the fermion spectrum at the anti-nodes. In

8 In LGW-Hertz criticality of antiferromagnetism (A) Antiferromagnetic metal (B) Fermi liquid with large Fermi surface h i6=0 h i =0

9 In LGW-Hertz criticality of antiferromagnetism (A) Antiferromagnetic metal Criticality in Fe-based and electron-doped-cuprate materials (B) Fermi liquid with large Fermi surface h i6=0 h i =0

10 Can we get a stable zero temperature state with fluctuating antiferromagnetism and a small Fermi surface (and so a gap near the anti-nodes)?

11 Can we get a stable zero temperature state with fluctuating antiferromagnetism and a small Fermi surface (and so a gap near the anti-nodes)? Yes, provided the metal has topological order T. Senthil, M. Vojta and S. Sachdev, PRB 69, (2004)

12 For fluctuating antiferromagnetism, we transform to a rotating reference frame using the SU(2) rotation R i ci" i,+ = R i, c i# in terms of fermionic chargons s and a Higgs field H a (i) i, ` `(i) =R i a H a (i) R i The Higgs field is the AFM order in the rotating reference frame.

13 For fluctuating antiferromagnetism, we transform to a rotating reference frame using the SU(2) rotation R i ci" i,+ = R i, c i# in terms of fermionic chargons s and a Higgs field H a (i) i, ` `(i) =R i a H a (i) R i The Higgs field is the AFM order in the rotating reference frame. Note that this representation is ambiguous up to a SU(2) gauge transformation, V i! V i i,+ i, i,+ R i! R i V i a H a (i)! V i b H b (i) V i. i,

14 Fluctuating antiferromagnetism The simplest e ective Hamiltonian for the fermionic chargons is the same as that for the electrons, with the AFM order replaced by the Higgs field. H = X i, t i,s i+v,s + i+v,s i,s µ X i i,s i,s + H int H int = X i H a (i) i,s a ss 0 i,s 0 + V H i

15 Fluctuating antiferromagnetism The simplest e ective Hamiltonian for the fermionic chargons is the same as that for the electrons, with the AFM order replaced by the Higgs field. H = X i, t i,s i+v,s + i+v,s i,s H int = X i i H a (i) i,s µ X i a ss 0 i,s 0 + V H i,s i,s + H int IF we can transform to a rotating reference frame in which H a (i) = a constant independent of i and time, THEN the fermions in the presence of fluctuating AFM will inherit the anti-nodal gap of the electrons in the presence of static AFM. In

16 Fluctuating antiferromagnetism We cannot always find a single-valued SU(2) rotation R i to make the Higgs field H a (i) a constant! A.V. Chubukov, T. Senthil and S. Sachdev, PRL 72, 2089 (1994); S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, PRB 94, (2016) Vortex in AFM order

17 Fluctuating antiferromagnetism We cannot always find a single-valued SU(2) rotation R i to make the Higgs field H a (i) a constant! A.V. Chubukov, T. Senthil and S. Sachdev, PRL 72, 2089 (1994); S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, PRB 94, (2016) R Vortex in AFM order R

18 Topological order We cannot always find a single-valued SU(2) rotation R i to make the Higgs field H a (i) a constant! A.V. Chubukov, T. Senthil and S. Sachdev, PRL 72, 2089 (1994); S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, PRB 94, (2016) R Vortex in AFM order R Vortices associated with 1 (SO(3)) = Z 2 must be suppressed: such a metal with fluctuating antiferromagnetism has Z 2 TOPOLOGICAL ORDER and fermions which inherit the Fermi surfaces of the antiferromagnetic metal i.e. a pseudogap.

19 In LGW-Hertz criticality of antiferromagnetism (A) Antiferromagnetic metal Criticality in Fe-based and electron-doped-cuprate materials (B) Fermi liquid with large Fermi surface h i6=0 h i =0

20 In LGW-Hertz criticality of antiferromagnetism (A) Antiferromagnetic metal hri 6=0, hh a i6=0 Criticality in Fe-based and electron-doped-cuprate materials (B) Fermi liquid with large Fermi surface hri 6=0, hh a i =0

21 Global phase diagram In LGW-Hertz criticality of antiferromagnetism (A) Antiferromagnetic metal hri 6=0, hh a i6=0 (B) Fermi liquid with large Fermi surface hri 6=0, hh a i =0 In (C) Metal with Z2 topological order hri =0, hh a i6=0 Higgs criticality: Deconfined SU(2) gauge theory with large Fermi surface (D) SU(2) ACL eventually unstable to pairing and confinement hri =0, hh a i =0

22 Global phase diagram In LGW-Hertz criticality of antiferromagnetism (A) Antiferromagnetic metal hri 6=0, hh a i6=0 (B) Fermi liquid with large Fermi surface hri 6=0, hh a i =0 In (C) Metal with Z2 topological order hri =0, hh a i6=0 Higgs criticality: Deconfined SU(2) gauge theory with large Fermi surface (D) SU(2) ACL eventually unstable to pairing and confinement hri =0, hh a i =0 Proposal for optimal doping criticality in holedoped cuprates

23 Topological order More generally, the e ective Hamiltonian for the fermionic chargons can also have non-trivial SU(2) gauge connections U (i) along with the Higgs field H a (i). H = X i, t i,s i+v,s + H.c. 0 µ X i i,s i,s + H int H int = X i H a (i) i,s a ss 0 i,s 0 + V H i

24 Topological order S. Sachdev and S. Chatterjee, arxiv: More generally, the e ective Hamiltonian for the fermionic chargons can also have non-trivial SU(2) gauge connections U (i) along with the Higgs field H a (i). H = X i, t i,s U ss (i) 0 i+v,s + H.c. 0 µ X i i,s i,s + H int H int = X i H a (i) i,s a ss 0 i,s 0 + V H i

25 Topological order S. Sachdev and S. Chatterjee, arxiv: More generally, the e ective Hamiltonian for the fermionic chargons can also have non-trivial SU(2) gauge connections U (i) along with the Higgs field H a (i). H = X i, t i,s U ss (i) 0 i+v,s + H.c. 0 µ X i i,s i,s + H int H int = X i H a (i) i,s a ss 0 i,s 0 + V H i Such a gauge-connection can induce various gauge-invariant fluxes which can break one or more of time-reversal, inversion, and lattice rotation symmetries.

26 Topological order i j k Gauge-invariant combinations of Higgs fields and gauge connections which are proportional to the electrical current on links l m n O mj = i Tr ( a U mj U jk U km ) H a (m) i Tr ( a U jm U mn U nj ) H a (j) + i Tr ( a U mj U ji U im ) H a (m) i Tr ( a U jm U ml U lj ) H a (j) S. Sachdev and S. Chatterjee, arxiv: O mk = i Tr ( a U mj U jk U km ) H a (m) i Tr ( a U kj U jm U mk ) H a (k) + i Tr ( a U mn U nk U km ) H a (m) i Tr ( a U kn U nm U mk ) H a (k) Such a gauge-connection can induce various gauge-invariant fluxes which can break one or more of time-reversal, inversion, and lattice rotation symmetries.

27 Topological order S. Sachdev and S. Chatterjee, arxiv: I I I p 2I I p 2I (a) (b) States with topological order can have these patterns of spontaneous currents, while preserving translational symmetry. Both patterns are consistent with present neutron and light scattering experiments. Both patterns have Ising-nematic order: the Ising-nematic order of (a) is similar to that observed in the cuprates.

28 Global phase diagram In LGW-Hertz criticality of antiferromagnetism (A) Antiferromagnetic metal hri 6=0, hh a i6=0 (B) Fermi liquid with large Fermi surface hri 6=0, hh a i =0 In (C) Metal with Z2 topological order hri =0, hh a i6=0 Higgs criticality: Deconfined SU(2) gauge theory with large Fermi surface (D) SU(2) ACL eventually unstable to pairing and confinement hri =0, hh a i =0

29 Global phase diagram In LGW-Hertz criticality of antiferromagnetism (A) Antiferromagnetic metal hri 6=0, hh a i6=0 (B) Fermi liquid with large Fermi surface hri 6=0, hh a i =0 In (C) Metal with Z2 topological order and discrete symmetry breaking hri =0, hh a i6=0 Higgs criticality: Deconfined SU(2) gauge theory with large Fermi surface (D) SU(2) ACL eventually unstable to pairing and confinement hri =0, hh a i =0

30 S. Sachdev and S. Chatterjee, arxiv: Pseudogap metal SM FL at low p Lattice gauge theory for a metal with topological order co-existing with broken time-reversal and inversion symmetries, and Ising-nematic order

31 S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, PRB 80, (2009); D. Chowdhury and S. Sachdev, PRB 91, (2015); S. Sachdev and D. Chowdhury, arxiv: Gauge theory SM for a topological phase FL transition, and for the strange metal (SM)