Quenched Disorder and Ves0gial Nema0city

Transcription

1 Quenched Disorder and Ves0gial Nema0city Laimei Nie and Gilles Tarjus, Sam Lederer, Yoni Shatner, Erez Berg Eduardo Fradkin, Also Ian Fisher, Jiun- Haw Chu, S. Sachdev, L. Sierens, Roger Melko, Akash Maharaj, John Tranquada Rafael Fernandez, Dung- Hai Lee, Fa Wang, Vadim Oganesyan, Andy Mackenzie, Erica Carlson, Karin Dahmen,Yi Zhang, Sri Raghu, Eun- Ah Kim, Michael Lawler, Hong Yao, Jiaping Hu, John Robertson, Ted Geballe Paris 2015

2 Effec0ve Field Theory Approach Emphasis on broken symmetries, Sharp dis0nc0ons between phases of ma[er (mostly based on long- range order). Long- distance low energy emergent proper0es.

3 Ves?gial Order When there is a sequence of transi0ons separa0ng an ordered (broken symmetry) state from a disordered (symmetric) state, intermediate phases that restore some but not all of the symmetries broken in the fully ordered state can be said to have ves0gial order. Electron Nema?c Order Spontaneous breaking of rota0onal (point group) symmetry as an emergent property of the electron fluid (in a crystal). Role of Quenched Disorder When spontaneous breaking of spa0al symmetries occurs in solids, quenched disorder is always qualita0vely important.

4 Theory of Ves?gial Nema?c Order We will consider a concrete model problem (which may, however, be directly applicable to the charge- ordering phenomena found in the cuprate high temperature superconductors and spin and charge ordering in the Fe- based superconductors)

5 Incommensurate CDW Order (~r) = + x(~r)e i Q ~ x r + y (~r)e i Q ~ y r +c.c Stripe ordered phase : h x i 6= 0 or h y i 6= 0 so h x 2 y 2 i6=0 Checkerboard ordered phase : h x i = h y i 6= 0 with h x 2 y 2 i =0 Nematic phase : h x i = 0 and h y i =0 but h x 2 y 2 i N 6= 0 L. Nie, G. Tarjus, and SAK PNAS (2013)

6 Snapshot of a fluctua?ng stripe nema?c Stripes are one of those revolving trends, frequently fluctua0ng in and out of style. Well, stripes are officially in again, thanks to...

7 Fluctua?ng stripe nema?c in a MoJ insulator S. A. Kivelson, E. Fradkin and V. J. Emery, Nature 393, (11 June 1998)

8 Snapshot of a state with ves0gial nema0c order Red region has N=[ < ψ x 2 > - < ψ y 2 > ] > 0 Courtesy J.E. Hoffman, E. Hudson, C- L Song E.W.Carlson and co.

9 Incommensurate CDW Order (~r) = + x(~r)e i ~ Q x r + y (~r)e i ~ Q y r +c.c (~r )! (~r + ~ R) a (~r )! a (~r )e i a where a = ~ Q a ~R O(2) O(2) Z 2 symmetry L. Nie, G. Tarjus, and SAK PNAS (2013)

10 Incommensurate CDW Order (~r) = + x(~r)e i ~ Q x r + y (~r)e i ~ Q y r +c.c (~r )! (~r + ~ R) a (~r )! a (~r )e i a where a = ~ Q a ~R O(2) O(2) Z 2 symmetry S(~r )= h Commensurate SDW i x(~r )e i Q ~ x ~r + y(~r )e i Q ~ y ~r +H.C ~Q x =(, 0); ~ Qy =(0, ) SO(3) Z 2 Fang, Yao, Tsai, Hu, and Kivelson, PRB 77, (2008) Xu, Muller, and Sachdev, PRB 78, (2008).

11 Incommensurate CDW Order (~r) = + x(~r)e i ~ Q x r + y (~r)e i ~ Q y r +c.c (~r )! (~r + ~ R) a (~r )! a (~r )e i a where a = ~ Q a ~R O(2) O(2) Z 2 symmetry S(~r )= h Commensurate SDW i x(~r )e i Q ~ x ~r + y(~r )e i Q ~ y ~r +H.C ~Q x =(, 0); ~ Qy =(0, ) SO(3) Z 2 Incommensurate SDW SO(3) O(2) O(2) Z 2

12 Incommensurate CDW Order (~r) = + x(~r)e i ~ Q x r + y (~r)e i ~ Q y r +c.c H = apple h x x (j) 2 y y (j) 2i + apple? y x (j) 2 x y (j) 2i 2 2 m h x (j) 2 + x (j) 2i + u h x (j) 2 + x (j) 2i x (j) 2 y (j) 2 + J x? (j + 1) x (j)+ 2 y(j + 1) y (j)+c.c. Can promote a from a U(1) to O(N) field Exactly solvable in N!1limit Same solution provides good mean field description forn = 2 Stripes 00 for >0; checkerboards 00 for <0 L. Nie, G. Tarjus, and SAK PNAS (2013)

13 L. Nie, G. Tarjus, and SAK PNAS (2013) J? /J

14 Electron Nema?c Phases Traceless symmetric order parameter Rela0on to spa0al symmetries makes it more like gravity than like internal symmetries Nema0c as a (ves0gial) composite order parameter e.g. N ab / 1 2 [ ab + ba ] ab D (Quadrapolar order) = X a aa Ising nema0c in a tetragonal crystal: N = [ xx yy ] [ xx + yy ]

15 Electron Nema?c Phases Traceless symmetric order parameter Rela0on to spa0al symmetries makes it more like gravity than like internal symmetries Nema0c as a (ves0gial) composite order parameter Examples of interes?ng systems for which compelling evidence of electron nema?c phases has been discovered recently: Nema0c Quantum Hall Metal and Nema0c Quan0zed Hall Fluid Field induced nema0c phase in Sr 3 Ru 2 O 7 Nema0c phases in Fe- based superconductors, e.g. BaFe 2- x Co x As 2 and various cuprate high temperature superconductors. Quadrapolar Ordered phase of YbRu 2 Ge 2 Colossal magneto- resis0ve Managanites e.g. La 1- x Ca x MnO 3...

16 Electron Nema?c Phases Traceless symmetric order parameter Rela0on to spa0al symmetries makes it more like gravity than like internal symmetries Nema0c as a (ves0gial) composite order parameter Examples of interes?ng systems for which compelling evidence of electron nema?c phases has been discovered recently: Nema0c Quantum Hall Metal and Nema0c Quan0zed Hall Fluid Field induced nema0c phase in Sr 3 Ru 2 O 7 Nema?c phases in Fe- based superconductors, e.g. BaFe 2- x Co x As 2 and various cuprate high temperature superconductors. Quadrapolar Ordered phase of YbRu 2 Ge 2 Colossal magneto- resis0ve Managanites e.g. La 1- x Ca x MnO 3...

17 Let s take a quick look at the Fe-based superconductors: BaFe 2-x Co x As 2 studies by I.R. Fisher group The AF striped phase breaks the C 4 symmetry of the host crystal The orthorhombic transi0on some0mes occurs simultaneously and some0mes slightly before the magne0c transi0on.

18 It is possible to measure an electron nema0c order parameter Resistive anisotropy in crystals of Fe-pnictide superconductor J-H. Chu et al, Science 329, 824 (2010)

19 η = (ρ xx ρ yy )/(ρ xx +ρ yy ) d /d" = nematic susceptibility " = strain = u xx u yy J.W. Chu et al, Science 337, 710 (2012) One of several new probes of electron nematicity R abcd = d ab du cd Resistoelastic tensor BaFe 2- x Co x As 2

20 Currie Weiss fit : d d" = 0 T T? J.W. Chu et al, Science 337, 710 (2012) BaFe 2- x Co x As 2

21 How generic is this behavior? Currie Weiss fit : d d" = 0 T T? T* is only ~ 10% below T s for small x Op0mal superconduc0ng T c occurs at roughly same x at which T* vanishes. - (2m - 2m 66-2m o 66 )- 1 (x10) (a) BaFe 2 As (b) 2 60 Co 2.5% Underdoped T s K- 280K K- 260K T s (c) Co 3.5% T s Underdoped K- 260K Co 6.8% (d) Optimal K- 250K Co 8% (e) Overdoped K- 240K (2m 66-2m o 66 )(T- T*) (10-2 K) T H- H. Kuo, J- H. Chu, S.A. K., I.R. Fisher, arxiv:

22 How generic is this behavior? Electron doped Isovalently doped Hole doped Iron chalcogenide -2m Ni 4.5% Optimal P 30%* Optimal K 42%* Optimal FeTe 0.6 Se 0.4 * Optimal -2m T (K) Co 6.8% Optimal T (K) T (K) T (K) T (K) All data taken above T c but fits to a Currie- Weiss law with T* which vanishes as a func0on of x approximately at op0mal doping. H- H. Kuo, J- H. Chu, S.A. K., I.R. Fisher, arxiv:

23 Curie- Weiss fits χ A/T (- )2m P 30% (a) Optimal Ni 4.5% (b) Optimal A 90 T c (- )(2m 66-2m o 66 )- 1 (x10) K- 250K K- 250K Temperat H- H. Kuo, J- H. Chu, S.A. K., I.R. Fisher, arxiv:

24 Rela?on between nema?c QCP and op?mal T c in Fe- based SCs Nema0c suscep0bility is universally large in neighborhood of op0mal doping The extrapolated nema0c cri0cal temperature, T*=T nem, is universally well below T c for op0mally doped materials. This is highly sugges0ve of a causal rela0on.

25 Electron Nema?c Phases Traceless symmetric order parameter Rela0on to spa0al symmetries makes it more like gravity than like internal symmetries Nema0c as a (ves0gial) composite order parameter Examples of interes?ng systems for which compelling evidence of electron nema?c phases has been discovered recently: Nema0c Quantum Hall Metal and Nema0c Quan0zed Hall Fluid Field induced nema0c phase in Sr 3 Ru 2 O 7 Nema?c phases in Fe- based superconductors, e.g. BaFe 2- x Co x As 2 and various cuprate high temperature superconductors. Hidden Order phase of URu 2 Si 2 Colossal magneto- resis0ve Managanites e.g. La 1- x Ca x MnO 3...

26 Effect of Quenched Randomness The random field problem is relevant to problems involving the breaking of pure spa0al symmetries.

27 Stripes, i.e. incommensurate charge density wave (CDW) order was discovered in one family of cuprate HTC in 1994 by Tranquada et al. (also bidirec?onal ) Short- range CDW order is now widely observed in hole- doped cuprates

28 Incommensurate CDW Order (~r) = + x(~r)e i Q ~ x r + y (~r)e i Q ~ y r +c.c Stripe ordered phase : h x i 6= 0 or h y i 6= 0 so h x 2 y 2 i6=0 Checkerboard ordered phase : h x i = h y i 6= 0 with h x 2 y 2 i =0 Nematic phase : h x i = 0 and h y i =0 but h x 2 y 2 i N 6= 0 L. Nie, G. Tarjus, and SAK (2013)

29 Effect of Quenched Randomness Random Field problem in Sta0s0cal Mechanics H[S] =H 0 [S]+ X j h j S j h j =0 h j h i = 2 ij D=2 is lower cri0cal dimension for Ising (Z 2 ) model (and presumably other models with discrete broken symmetries) D=4 is the lower cri0cal dimension for Heisenberg ( SO(3) ) model and most models with con0nuous broken symmetries. (There is the possible subtlety of a Bragg glass phase for D=3 and XY ( SO(2) ) symmetry.)

30 T Incommensurate Stripe Order T nem T str Stripe Nematic Homogeneous and Isotropic A 0 Assume a stripe- ordered ground- state and V z << J Σ 2 L. Nie, G. Tarjus, and SAK (2013)

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32 T Ves0gial Nema0c Order T nem T str Stripe Nematic Homogeneous and Isotropic A 0 Assume a stripe- ordered ground- state and V z << J Σ 2 L. Nie, G. Tarjus, and SAK (2013)

33 T Ves0gial Nema0c Order T nem T str Stripe In d > 2, Ising (discrete) symmetry breaking survives in the presence of weak enough random fields! Nematic Homogeneous and Isotropic A In d 4, no con0nuous symmetry breaking possible in the presence of random fields! Consider the effects of quenched disorder 0 Σ 2 L. Nie, G. Tarjus, and SAK (2013)

34 T Ves0gial Nema0c Order T nem T str Stripe Nematic Homogeneous and Isotropic 0 Σ 2 L. Nie, G. Tarjus, and SAK (2013)

35 Checkerboard T For Checkerboard Order For any form of incommensurate CDW, the transla0on symmetry breaking is short- range correlated, and looks rather similar whether we are dealing with stripes or checkerboards. The only fundamental dis0nc0ons involve Q=0 order (or, more weakly, commensurate order) i.e. 0me- reversal and/or point- group symmetry breaking. S0ll remain interes0ng sharp crossover near T cdw for σ small. σ 2 L. Nie, G. Tarjus, and SAK (2013)

36 Visualizing ves0gial nema0city when a unidirec0onal (stripe) ordered CDW is disrupted by quenched randomness Courtesy Jenny Hoffman

37 Bi x 650 A Z maps Fourier filtered data near Q**=(3/4,0) and (0,3/4) Pa[erns correspond to 2d surface of 3d RFIM near σc Courtesy J.E. Hoffman, E. Hudson, C- L Song E.W.Carlson and co.

38 A few things that are par0cularly interes0ng about electron nema0c order Nema0c QCP is possible even in the presence of quenched disorder Nema?c fluctua?ons enhance superconduc?vity in any channel. Quantum cri0cal nema0c fluctua0ons destroy FL on en0re FS. and, absent crystal field effects, in the en0re nema0c phase S. Lederer, Y.Scha[ner, E. Berg, and SAK, arxiv: T. A. Maier and D. J. Scalapino, arxiv: V. Oganesyan, SAK, and E. Fradkin, PRB 64, (2001) H. Watanabe and A. Vishwanath, PNAS accepted.