Topological order in the pseudogap metal

Transcription

1 HARVARD Topological order in the pseudogap metal High Temperature Superconductivity Unifying Themes in Diverse Materials 2018 Aspen Winter Conference Aspen Center for Physics Subir Sachdev January 16, 2018 Review: arxiv: Talk online: sachdev.physics.harvard.edu

2 Mathias Scheurer Wei Wu Shubhayu Chatterjee arxiv: arxiv: Michel Ferrero Antoine Georges

3 Topological materials Descendants of the integer quantum Hall effect Protected gapless edge states, while bulk excitations are trivial Descendants of the fractional quantum Hall effect Bulk topological excitations which cannot be created from the ground state by the action of a local operator. Can also appear in gapless metallic states.

4 Topological materials Descendants of the integer quantum Hall effect Protected gapless edge states, while bulk excitations are trivial Descendants of the fractional quantum Hall effect Bulk topological excitations which cannot be created from the ground state by the action of a local operator. Can also appear in gapless metallic states.

5 Classical XY model Z XY = Y i Z 2 0 d i 2 exp ( H/T ) H = J X hiji cos( i j ) i e i i Describes non-zero T phase transitions of superfluids, magnets with `easy-plane spins,..

6 Classical XY model in D=3 XY LRO h i i6=0 Symmetry breaking phase transition i j h i i =0 exp( r i r j / ) r i r j XY SRO No topological order J Jc

7 Classical XY model in D=2 Ordering, metastability and phase transitions in twodimensional systems J M Kosterlitz and D J Thouless J. Phys. C 1973 Journal of Physics C: Solid State Physics, Volume 6, Number 7 A new definition of order called topological order is proposed for two-dimensional systems in which no long-range order of the conventional type exists. The possibility of a phase transition characterized 1 by a change in the response of the system to an external i j r i r j exp( r i r j / ) i j r i r j 1/2 XY QLRO Topological order Topological phase transition: Kosterlitz Thouless XY SRO No topological order Vortices expelled Vortices proliferate TKT T

8 Classical XY model in D=3 Can we have a topological phase transition in D=3? XY LRO h i i6=0 Symmetry breaking phase transition i j h i i =0 exp( r i r j / ) r i r j XY SRO No topological order J Jc

9 ez XY = Y i Z 2 0 d i 2 exp eh/t eh = J X hiji cos( i j ) + X ijk` K ijk` cos( i + j k `)+... Add terms which suppress ±2 but not ±4 vortices.

10 ez XY = Y i Z 2 0 d i 2 exp eh/t eh = J X hiji cos( i j ) + X ijk` K ijk` cos( i + j k `)+... Add terms which suppress ±2 but not ±4 vortices. A convenient form is obtained using an auxiliary variable ij = ±1 on the links of the cubic lattice. X Y Z 2 d i ez XY = 2 exp eh/t { ij }=±1 eh = J X hiji i 0 ij cos [( i j )/2] K X Y (ij)2 ij

11 Attach Z 2 flux (vison) to the core of a ±2 vortex -1 ij = 1

12 Classical XY model in D=3 Can we have a topological phase transition in D=3? XY LRO h i i6=0 Symmetry breaking phase transition i j h i i =0 exp( r i r j / ) r i r j XY SRO No topological order J Jc

13 Classical XY model in D=3 XY SRO Z2 topological order Symmetry breaking and topological phase transition XY LRO i j Odd (±2, ±6...) vortices expelled Even (±4, ±8...) vortices proliferate h i i =0 exp( r i r j / ) Symmetry breaking phase transition r i r j 2 Topological phase transition h i i6=0 h i i =0 i j exp( r i r j / ) r i r j XY SRO No topological order All (±2, ±4...) vortices proliferate K J Jc

14 Square lattice Hubbard model at generic density ncreasing SDW ` 6=0 Symmetry breaking and topological phase transition SDW SRO Z2 or U(1) topological order. Z2 vortices or hedgehogs expelled. ` =0 ncreasing SDW Topological phase transition ` =0 g SDW LRO Symmetry breaking phase transition SDW SRO No topological order. U/t

15 We can (exactly) transform the Hubbard model to 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 SDW order parameter `(i), ` = x, y, z H int = X i i `(i)c i, ` c i, + V where i = ±1 on the two sublattices. (For suitable V, integrating out the ` yields back the Hubbard model). When `(i) = (non-zero constant) independent of i, we have longrange SDW order, which transforms the Fermi surfaces from large to small.

16 We can (exactly) transform the Hubbard model to 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 SDW order parameter `(i), ` = x, y, z H int = X i i `(i)c i, ` c i, + V where i = ±1 on the two sublattices. (For suitable V, integrating out the ` yields back the Hubbard model). When `(i) = (non-zero constant) independent of i, we have longrange SDW order, which transforms ncreasing SDW the Fermi surfaces from large to small.

17 For (fluctuating) SDW SRO, 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 SDW 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. S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, PRB 80, (2009) i,

18 For (fluctuating) SDW SRO, 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 SDW 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. S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, PRB 80, (2009) i,

19 Fluctuating SDW The simplest e ective Hamiltonian for the fermionic chargons is the same as that for the electrons, with the SDW 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) SDW SRO will inherit the small Fermi surfaces of the electrons in the presence of SDW LRO. S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, PRB 80, (2009)

20 Fluctuating SDW The simplest e ective Hamiltonian for the fermionic chargons is the same as that for the electrons, with the SDW 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) SDW SRO will inherit the small Fermi surfaces of the electrons in the presence of SDW LRO. S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, PRB 80, (2009)

21 Fluctuating SDW We cannot always find a single-valued SU(2) rotation R i to make the Higgs field H a (i) a constant! vortex in AFM order (assume easy-plane AFM for simplicity) S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, PRB 94, (2016)

22 Fluctuating SDW We cannot always find a single-valued SU(2) rotation R i to make the Higgs field H a (i) a constant! vortex in AFM order R (assume easy-plane AFM for simplicity) R S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, PRB 94, (2016)

23 Fluctuating SDW We cannot always find a single-valued SU(2) rotation R i to make the Higgs field H a (i) a constant! vortex in AFM order R (assume easy-plane AFM for simplicity) The HIGGS PHASE, withh a condensed, has fluctuating R and SDW SRO with odd vortices expelled (for easy-plane SDW). Such a metal has topological order and the fermions which inherit the small Fermi surfaces of the metal with SDW LRO. R S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, PRB 94, (2016)

24 ncreasing SDW SDW SRO Higgs phase Z2 or U(1) topological order. Z2 vortices or hedgehogs expelled. ` =0 Symmetry breaking and topological phase transition hh a i6=0 ` 6=0 hri =0 ` =0 hh a i6= 0, hri 6=0 hh a i = 0, hri 6=0 ncreasing SDW Topological phase transition g SDW LRO Symmetry breaking phase transition SDW SRO Confinement No topological order. U/t

25 Electron Green s function in Higgs phase of SU(2) gauge theory The e ective Hamiltonian of the chargons in a constant Higgs potential hh a i = H a 0 is (the hoppings have been renormalized by hr i R ji): H = X i, t i,s i+v,s + i+v,s i,s X ( 1) i x+i y H0 a i,s ss a 0 i,s 0 i µ X i i,s i,s The chargon Fermi surface reconstructs into small pockets, even though translational and spin rotation symmetries remain unbroken. The diagonal chargon Green s function is G (!, ~ k)= 1! " ~k (!, ~ k), (!, ~ k)= H 2 0! " ~k+ ~ Q, ~ Q =(, ). This has poles at the pocket Fermi surfaces, and zeros at " ~k+ ~ Q. The electron Green s function is computed via a convolution with the spinons (R), and then the zeros are smeared to approximate zeros.

26 Common features of many cluster- DMFT computations of pseudogap metal: Momentum-space di erentiation: electron self-energy is enhanced at low frequencies in the anti-nodal region, and vanishes in the nodal region. FERMI ARCS AND HIDDEN ZEROS OF THE GREEN r(k) (π,0) (π,0) n = 0.78 (π,π) (0,0) (0,π) (0,0) (π,π) (π,0) (π,0) n = 0.92 C A B (π,π) (0,π) (π,π) Max Gapped spectrum in the antinodal region Fermi arcs in the nodal region Apparent zero of Green s function on a Luttinger surface. A(k) (0,0) (0,π) (0,0) (0,π) FIG. 4. Color online Renormalized energy r k upper panels and spectral function A k lower panels for the 2D Hubbard model with U=8t and T=0. The color code for the upper panels is green/gray r 0, blue/dark gray line r=0, yellow/light gray r 0, red dashed line r. The frequency dependence of the spectral function for the points marked by A, B, and C is shown in T.D. Stanescu and G. Kotliar, PRB 74, (2006) 0

27 Electron Green s function in Higgs phase of SU(2) gauge theory Red line indicates the locus of G(k,! = 0) = 0 Red line indicates the locus of Re G(k,! = 0) = 0 Full Brillouin zone spectra of chargons ( ) and electrons (c)

28 Electron Green s function in Higgs phase of SU(2) gauge theory T = t/30, U =7t, p =0.05 t 0 takes di erent negative values Anti-nodal spectra compared to cluster DMFT

29 Lifshitz transition compared to cluster DMFT ~k = ~k +Re ~k (! = 0) = Re G c (! =0, ~ 1 k) The p-t 0 dependence of the interacting Lifshitz transition, defined by the sign change of the renormalized quasiparticle energy (,0) at! peak > 0, is shown as solid blue lines calculated from the SU(2) gauge theory, part (a), and DCA, part (b). The black dashed lines show the location of the same transition for noninteracting electrons. The red lines indicate where the particle-hole asymmetry of the self-energy changes, i.e., where the peak position! peak of the anti-nodal Im(self-energy) changes sign.

30 Electron Green s function in Higgs phase of SU(2) gauge theory The imaginary part of the self-energy at the lowest Matsubara frequency! 0 = T determined from DQMC on the Hubbard model (U =7t, t 0 = 0.1t, T = 0.25t, p =0.042) and from the SU(2) gauge theory is shown in (a) and (b), respectively. To avoid too much broadening, we have applied a slightly smaller temperature of T =0.15t for the gauge theory. The inset in (b) shows the gauge theory prediction at zero frequency and low temperature (as before T = t/30). The black dashed line corresponds to the position of the Luttinger surface of the chargons.

31 New classes of quantum states with topological order Can be understood as: (a) defect suppression in states with fluctuating order associated with broken symmetries (b) Higgs phases of emergent gauge fields A metal with bulk topological order (i.e. long-range quantum entanglement) can explain existing experiments in cuprates, and agrees well with cluster-dmft arxiv: arxiv:

32 New classes of quantum states with topological order Can be understood as: (a) defect suppression in states with fluctuating order associated with broken symmetries (b) Higgs phases of emergent gauge fields A metal with bulk topological order (i.e. long-range quantum entanglement) can explain existing experiments in cuprates, and agrees well with cluster-dmft arxiv: arxiv:

33 New classes of quantum states with topological order Can be understood as: (a) defect suppression in states with fluctuating order associated with broken symmetries (b) Higgs phases of emergent gauge fields A metal with bulk topological order (i.e. long-range quantum entanglement) can explain existing experiments in cuprates, and agrees well with cluster-dmft arxiv: arxiv:

34 Square lattice Hubbard model at generic density ncreasing SDW Review: arxiv: ` 6=0 Symmetry breaking and topological phase transition SDW SRO Z2 or U(1) topological order. Z2 vortices or hedgehogs expelled. ` =0 ncreasing SDW Topological phase transition ` =0 g SDW LRO Symmetry breaking phase transition SDW SRO No topological order. U/t

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

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

37 Square lattice Hubbard model at generic density ncreasing SDW Review: arxiv: ` 6=0 Symmetry breaking and topological phase transition SDW SRO Z2 or U(1) topological order. Z2 vortices or hedgehogs expelled. ` =0 ncreasing SDW Topological phase transition ` =0 g SDW LRO Symmetry breaking phase transition SDW SRO No topological order. U/t

38 Square lattice Hubbard model at generic density ` 6=0 ncreasing SDW Symmetry breaking and topological phase transition SDW SRO Z2 or U(1) topological order. Z2 vortices or hedgehogs expelled. ` =0 ncreasing SDW Review: arxiv: Topological phase transition Optimal doping criticality. Fits doping dependence of Hall co-efficient (S. Chatterjee et al. PRB 96, (2017)) ` =0 g SDW LRO Symmetry breaking phase transition SDW SRO No topological order. U/t