Topology, quantum entanglement, and criticality in the high temperature superconductors

Transcription

1 HARVARD Topology, quantum entanglement, and criticality in the high temperature superconductors Exploring quantum phenomena and quantum matter in ultrahigh magnetic fields, National Science Foundation, Alexandria VA Subir Sachdev September 21, 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 Undoped insulating antiferromagnet

4 Antiferromagnet with p mobile holes per square

5 Filled Band

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

7 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

8 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

9 c B (T) 50 5 Tl2201 t/b (a.u.) 0 5 YBa 2 Cu 4 O /B (T 1 ) 2 Overdoped Figure 1 Quantum oscillations in Tl2201. a, Raw data on interlayer resistance (R H ) with B//c for a Tl2201 single crystal at T K. R H (B) rises rapidly above the irreversibility field B irr, passes through a small. Vignolle B, Carrington A, Cooper RA, French MMJ, Mackenzie AP, Jaudet C, Vignolles D, Proust C and Hussey NE Nature 455: plateau, then grows quasi-quadratically with field up to 60 T. Inset, magnified view of the high-field region of the down sweep. Small but welldefined oscillations are clearly resolved in the raw data, with an amplitude that grows with increasing field strength. The maximum amplitude of the oscillations is only 0.5 mv. b, Averaged magnetic torque data (from five sweeps at temperatures between 0.6 K and 0.8 K) with B close to the c axis for a different Tl2201 crystal. The temperatures of the torque sweeps are subject to an additional uncertainty of 150 mk due to the weak thermal link inside the dilution refrigerator 5 and the high currents needed to observe the oscillations. Below B 5 14 T, the torque shows hysteretic behaviour due to flux trapping and expulsion in the superconducting mixed state. Again, welldefined oscillations are clearly resolved in the expanded region shown in the inset. The value of T c for both crystals is 10 K (defined by their zero-resistive state), compared with the maximal TS. c insebastian Tl2201 of 92 K. and The torque C. Proust, crystal showed a very small kink in the zero-field r H data around 20 K, suggesting Annual Reviews thatof some Condensed fraction of the crystal, Matter presumably Physics, the surface 6 (2015) layer, had a higher T value. Note that the difference in B exhibited by both crystals is

10 c B (T) 50 5 Tl2201 t/b (a.u.) 0 5 YBa 2 Cu 4 O /B (T 1 ) 2 Underdoped Doiron-Leyraud N, Proust C, LeBoeuf D, Levallois J, Bonnemaison JB, Liang R, Bonn DA, Hardy WN, Taillefer L Nature 447: Overdoped Figure 1 Quantum oscillations in Tl2201. a, Raw data on interlayer resistance (R H ) with B//c for a Tl2201 single crystal at T K. R H (B) rises rapidly above the irreversibility field B irr, passes through a small. Vignolle B, Carrington A, Cooper RA, French MMJ, Mackenzie AP, Jaudet C, Vignolles D, Proust C and Hussey NE Nature 455: plateau, then grows quasi-quadratically with field up to 60 T. Inset, magnified view of the high-field region of the down sweep. Small but welldefined oscillations are clearly resolved in the raw data, with an amplitude that grows with increasing field strength. The maximum amplitude of the oscillations is only 0.5 mv. b, Averaged magnetic torque data (from five sweeps at temperatures between 0.6 K and 0.8 K) with B close to the c axis for a different Tl2201 crystal. The temperatures of the torque sweeps are subject to an additional uncertainty of 150 mk due to the weak thermal link inside the dilution refrigerator 5 and the high currents needed to observe the oscillations. Below B 5 14 T, the torque shows hysteretic behaviour due to flux trapping and expulsion in the superconducting mixed state. Again, welldefined oscillations are clearly resolved in the expanded region shown in the inset. The value of T c for both crystals is 10 K (defined by their zero-resistive state), compared with the maximal T c in Tl2201 of 92 K. The torque crystal S. Sebastian and C. Proust, Annual Reviews of Condensed Matter Physics, 6 (2015) showed a very small kink in the zero-field r H data around 20 K, suggesting that some fraction of the crystal, presumably the surface layer, had a higher T value. Note that the difference in B exhibited by both crystals is

11 B. Vignolle, A. Carrington, R.A. Cooper, M.M.J. French, 0.15 A.P. Mackenzie, C. Jaudet, D. Vignolles, C. Proust and N.E. Hussey, Nature 455, 952 (2008) c B (T) Tl2201 SM t/b (a.u.) 0 5 YBa 2 Cu 4 O /B (T 1 ) FL A conventional Figure 1 Quantum oscillations in Tl2201. a, Raw resistance (R H ) with metal: B//c for a Tl2201 single crysta R H (B) rises rapidly above the irreversibility field B ir plateau, then the grows Fermi quasi-quadratically liquid with field u magnified view of the high-field region of the down defined oscillations are clearly resolved in the raw d that grows with with increasingfermi field strength. The maxi oscillations is only 0.5 mv. b, Averaged magnetic to sweeps at temperatures surface between of 0.6size K and 0.8 K) wit a different Tl2201 crystal. The temperatures of the to to an additional uncertainty 1+p of 150 mk due to the w the dilution refrigerator 5 and the high currents nee oscillations. Below B 5 14 T, the torque shows hyst flux trapping and expulsion in the superconducting defined oscillations are clearly resolved in the expan

12 N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois, J.B. Bonnemaison, R. Liang, D.A. Bonn, W.N. Hardy, L. Taillefer, Nature 447, 565 (2007) a 4 1 / (530 T) 2 SM R xy (mω) / B (T 1 ) FL Quantum oscillations of a Fermi surface of size close to p

13 Harrison N, Sebastian SE Phys. Rev. Lett. 106: Great deal of evidence that underdoped quantum oscillations are due to an electron pocket which is created by field-induced long-range charge density wave order

14 Allais A, Chowdhury D, Sachdev S Nature Commun. 5:5771 p al Q 1 0 Q 2 -p -p 0 p- Reconstructing the large Fermi surface by CDW order inevitably leads to many more Fermi surfaces, apart from the electron pocket

15 Specific heat measurements in field-induced normal state of under doped YBCO Zero field gamma 50 YBCO Riggs SC, Vafek O, Kemper JB, Betts JB, Migliori A, Balakirev FF, Hardy WN, Liang R, Bonn DA, Boebinger GS Nature Phys. 7: C/T (mj mol K ) γ= mj/mol K 0T Temperature (K ) Lowest values for LSCO values range from 2 2. YBCO ~2 mj/ mol K 0.4 to 4 mj/ mol K Value is only consistent with a single electron pocket (Wright et al, Handbook of High Tc) Wen et al Physical Review B 70, (2004)

16 Sawtooth waveform characteristic of an ideal 2D Fermi gas 2D electrons in GaAs/AlGaAs heterostructures Y. T. Hsu, M. Hartstein, J. Porras, T. Loew et al. (unpublished) A. Potts et al. J. Phys.: Condens. Matter 8, (1996) Consistent with a single electron pocket

17 Single reconstructed Fermi surface pocket in an underdoped single layer cuprate superconductor M. K. Chan, 1, 2, N. Harrison, 1, R. D. McDonald, 1 B. J. Ramshaw, 1 K. A. Modic, 1 N. Barišić, 3, 2 and M. Greven 2 c f/f (%) B (T) FIG. 2. Observation of quantum oscillations in Hg1201 with contactless resistivity. (a) Evolution of the PDO circuit frequency coupled to Hg1201 UD71 with applied magnetic field B along the c axis of the sample at T =1.8 K.Thesampleundergoesatransitionfromsuperconducting (SC, only black shaded a single region) electron to normal pocket (blue region) at B 35 T. (b) Derivative of the raw data with respect us to make an up Using m 2.7 m e we obtain a c-axis h revealing it to be nearest neighbor h CuO 2 planes. Our than the bare valu (t = 10 mev [45 normalization. Our ability to se in Hg1201 contrast timates of the c-ax from the effects of in Y123 range from the observed beat effects of bilayer-sp neling [8] or Fermi Fermi surface The simple crystal ideal system for rel Fermi surface pock ray scattering [18] the unreconstructe Nature Communications 7, (2016) And shape of quantum oscillations also support vides the magnitu Following Allais et reconstructed hole

18 The parent pseudogap metal, without CDW order, cannot have a large Fermi surface, and should be gapped in the anti-nodal region Such a Fermi surface violates the Luttinger theorem of Fermi-liquid theory However, such a Fermi surface can appear in a metal with bulk topological order, which has long-range, many-body quantum entanglement

19 The parent pseudogap metal, without CDW order, cannot have a large Fermi surface, and should be gapped in the anti-nodal region Such a Fermi surface violates the Luttinger theorem of Fermi-liquid theory However, such a Fermi surface can appear in a metal with bulk topological order, which has long-range, many-body quantum entanglement

20 The parent pseudogap metal, without CDW order, cannot have a large Fermi surface, and should be gapped in the anti-nodal region Such a Fermi surface violates the Luttinger theorem of Fermi-liquid theory However, such a Fermi surface can appear in a metal with bulk topological order, which has long-range, many-body quantum entanglement T. Senthil, M. Vojta and S. Sachdev, PRB 69, (2004) A. Paramekanti and A. Vishwanath, PRB 70, (2004).

21 A simple model: Fluctuating antiferromagnetism leads to small Fermi surfaces and bulk topological order with long-range quantum entanglement

22 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.

23 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.

24 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

25 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.

26 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

27 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

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

29 Fluctuating antiferromagnetism We cannot always find a single-valued SU(2) rotation R i to make the Higgs field H a (i) a constant! n-fold vortex in AFM order R ( 1) n R A.V. Chubukov, T. Senthil and S. Sachdev, PRL 72, 2089 (1994); S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, PRB 94, (2016)

30 Topological order We cannot always find a single-valued SU(2) rotation R i to make the Higgs field H a (i) a constant! n-fold vortex in AFM order R ( 1) n R Vortices with n odd must be suppressed: such a metal with fluctuating antiferromagnetism has BULK Z 2 TOPOLOGICAL ORDER and fermions which inherit the pocket Fermi surfaces of the antiferromagnetic metal i.e. a pseudogap. Odd vortices in antiferromagnetism are stable bulk quasiparticles ( visons ) which could be observed in STM.

31 Pseudogap metal SM with bulk topological FL order?

32 Gauge theory for a SM topological FL Higgs-toconfinement phase transition

33 Gauge theory for a SM topological FL Higgs-toconfinement phase transition

34 Phase diagram in a high magnetic field Quantum oscillations and DW order B. J. Ramshaw, S. E. Sebastian, R. D. McDonald, James Day, B. S. Tan, Z. Zhu, J. B. Betts, Ruixing Liang, D. A. Bonn, W. N. Hardy, N. Harrison, Science 348, 6232 (2105).

35 Phase diagram in a high magnetic field b YBCO T NMR H = 50 T Badoux, Proust, Taillefer et al., Nature 531, 210 (2016) T ( K ) 40 T c 20 SDW CDW p* p Quantum oscillations and DW order

36 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)

37 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)

38 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)

39 Hall effect measurements in YBCO b p* 1.5 SDW CDW FL n H = V / e R H p 1 + p Rapid increase in carrier density due to a quantum phase transition at p=0.19? p Badoux, Proust, Taillefer et al., Nature 531, 210 (2016)

40 Scale-invariant magnetoresistance in a cuprate superconductor P. Giraldo-Gallo, 1 J. A. Galvis, 1 Z. Stegen, 1 K. A. Modic, 2 F. F Balakirev, 3 J. B. Betts, 3 X. Lian, 1 C. Moir, 1 S. C. Riggs, 1 J. Wu, 4 A. T. Bollinger, 4 X. He, 4, 5 I. Božović, 4, 5 B. J. Ramshaw, low temperatures 6, 3 R. D. McDonald, is poorly 3 G. understood, S. Boebinger, especially 1, 7 and A. near Shekhter critical 1, doping, x = Here we report a high-field magnetoresistance study of thin films of La 2 x Sr x CuO 4 cuprates in close vicinity to critical doping, apple x apple We find that the metallic state exposed by suppressing superconductivity is characterized by a magnetoresistance that is linear in magnetic field up to the highest measured fields of 80T. The slope of the linear-in-field resistivity is temperature-independent at very high fields. It mirrors the magnitude and doping evolution of the linear-in-temperature resistivity that has been ascribed to Planckian dissipation near a quantum critical point. This establishes true scale-invariant conductivity T arxiv: x crit x B

41 High field studies of cuprates have been crucial to unraveling their phase diagram Plethora of evidence for an exotic metal underlying the underdoped regime A metal with bulk topological order (i.e. long-range quantum entanglement) can explain existing experiments Novel quantum criticality likely associated with a deconfined-confined transition in a

42 Novel quantum criticality likely associated with a deconfined-confined transition in a gauge theory Higher field studies, with more experimental probes (STM ), promise to answer many open questions, and could lead to direct detection of topological order. An understanding of these issues is crucial to understanding the origin of the high critical temperature for superconductivity