1. Thermoelectric Effects in Correlated Matter Sriram Shastry UCSC Santa Cruz Work supported by NSF DMR 0408247 Work supported by DOE, BES DE-FG02- 06ER46319 Work supported by NSF DMR 0408247 Aspen Center for Physics: 14 August, 2008

2. Intro High Thermoelectric power is very desirable for applications. High Thermoelectric power is very desirable for applications. Usually the domain of semiconductor industry, e.g. Bi2Te3. However, recently correlated matter has found its way into this domain. Usually the domain of semiconductor industry, e.g. Bi2Te3. However, recently correlated matter has found its way into this domain. Heavy Fermi systems (low T), Mott Insulator Junction sandwiches (Harold Hwang 2004) Heavy Fermi systems (low T), Mott Insulator Junction sandwiches (Harold Hwang 2004) Sodium Cobaltate NaxCoO2 at x ~.7 Terasaki, Ong, ….. Sodium Cobaltate NaxCoO2 at x ~.7 Terasaki, Ong, …..

3. What is the Seebeck Coefficient S? T h ermopower S ( ! ) = L 12 ( ! ) TL 11 ( ! ) L oren t z N um b er L ( ! ) = · zc ( ! ) T ¾ ( ! ) F i gureo f M er i t Z ( ! ) T = S 2 ( ! ) L ( ! ) : ( 1 )

4. Similarly thermal conductivity and resitivity are defined with appropriate current operators. The computation of these transport quantities is brutally difficult for correlated systems. Hence seek an escape route……….That is the rest of the story!

5. Triangular lattice Hall and Seebeck coeffs: (High frequency objects) Notice that these variables change sign thrice as a band fills from 0->2. Sign of Mott Hubbard correlations.

6. Considerable similarity between Hall constant and Seebeck coefficients. Both gives signs of carriers---(Do they actually ???) Zero crossings tell a tale. These objects are sensitive to half filling and hence measure Mott Hubbard hole densities. Brief story of Hall constant to motivate the rest. The Hall constant at finite frequencies: S Shraiman Singh- 1993 High T_c and triangular lattices---

7. < e R H ( 0 ) = R ¤ H ( ­ ) + 2 ¼ Z ­ 0 = m R H ( º ) º d º : Consider a novel dispersion relation (Shastry ArXiv.org 0806.4629) Here  is a cutoff frequency that determines the RH*. LHS is measurable, and the second term on RHS is beginning to be measured (recent data exists). The smaller the , closer is our RH* to the transport value. We can calculate RH* much more easily than the transport value. For the tJ model, it would be much closer to the DC than for Hubbard type models. This is obvious since cut off is max{t,J} rather than U!! ~ ! À fj t j ; U g max ~ ! À fj t j ; J g max :

8. New Formalism SS (2006) is based on a finite frequency calculation of thermoelectric coefficients. Motivation comes from Hall constant computation (Shastry Shraiman Singh 1993- Kumar Shastry 2003) Perhaps  dependence of R_H is weak compared to that of Hall conductivity. Very useful formula since Captures Lower Hubbard Band physics. This is achieved by using the Gutzwiller projected fermi operators in defining J’s Exact in the limit of simple dynamics ( e.g few frequencies involved), as in the Boltzmann eqn approach. Can compute in various ways for all temperatures ( exact diagonalization, high T expansion etc…..) We have successfully removed the dissipational aspect of Hall constant from this object, and retained the correlations aspect. Very good description of t-J model. This asymptotic formula usually requires  to be larger than J ½ xy ( ! ) = ¾ xy ( ! ) ¾ xx ( ! ) 2 ! BR ¤ H f or! ! 1 R ¤ H = R H ( 0 ) i n D ru d e t h eory ANALOGY between Hall Constant and Seebeck Coefficients

9. Need similar high frequency formulas for S and thermal conductivity. Requirement::: L ij (  ) Did not exist, so had lots of fun with Luttinger’s formalism of a gravitational field, now made time dependent. K t o t = X K ( r )( 1 + Ã ( r ; t )) r ( Ã ( r ; T )) » rT ( r ; t )= T

10. i = 1 i = 2 C h arge E nergy I i ^ J x ( q x ) ^ J Q x ( q x ) U i ½ ( ¡ q x ) K ( ¡ q x ) Y i E x q = i q x Á q i q x à q : ( 1 ) L ij ( ! ) = i ­ ! c " h T ij i ¡ X n ; m p m ¡ p n " n ¡ " m + ! c ( I i ) nm ( I j ) mn # ; ( 1 ) h T ij i = ¡ l i m q x ! 0 h[ I i ; U j ]i 1 q x : ( 2 )

11. We thus see that a knowledge of the three operators gives us a interesting starting point for correlated matter:

12. ¿ xx = q 2 e ~ X ´ 2 x t ( ~ ´ ) c y ~ r + ~ ´ ; ¾ c ~ r ; ¾ or = q 2 e ~ X ~ k d 2 " ~ k dk 2 x c y ~ k ; ¾ c ~ k ; ¾

13. h © xx i = q e c k B T P m ; ¾ ; ~ k G ¾ ( k ; i ! m ) h d dk x ( v x k ( " k ¡ ¹ )) + d 2 " k dk 2 x § ¾ ( k ; i ! m ) i Unpublished- For Hubbard model using “Ward type identity” can show a simpler result for \Phi.

14. Hydrodynamics of energy and charge transport in a band model: This involves the fundamental operators in a crucial way: ½ @ @ t + 1 ¿ c ¾ ± J ( r ) = 1 ­ h ¿ xx i · 1 q 2 e @ ¹ @ n ( ¡ r ½ ) ¡ r Á ( r ) ¸ + 1 ­ h © xx i · 1 C ( T ) ( ¡ rK ( r )) ¡ rª ¸ ½ @ @ t + 1 ¿ E ¾ ± J Q ( r ) = 1 ­ h © xx i · 1 q 2 e @ ¹ @ n ( ¡ r ½ ) ¡ r Á ( r ) ¸ + 1 ­ h £ xx i · 1 C ( T ) ( ¡ rK ( r )) ¡ rª ¸ Einstein diffusion term of charge Energy diffusion term Continuity @ ½ @ t + rJ ( r ) = 0 @ K ( r ) @ t + rJ Q ( r ) = p ex t ( r ) Input power density These eqns contain energy and charge diffusion, as well as thermoelectric effects. Potentially correct starting point for many new nano heating expts with lasers.

15. And now for some results: Triangular lattice t-J exact diagonalization (full spectrum) Collaboration and hard work by:- J Haerter, M. Peterson, S. Mukerjee (UC Berkeley)

16. How good is the S* formula compared to exact Kubo formula? A numerical benchmark: Max deviation 3% anywhere !! As good as exact!

17. Results from this formalism: Prediction for t>0 material Comparision with data on absolute scale! T linear Hall constant for triangular lattice predicted in 1993 by Shastry Shraiman Singh! Quantitative agreement hard to get with scale of “t”

18. S* and the Heikes Mott formula (red) for Na_xCo O2. Close to each other for t>o i.e. electron doped cases

19. Predicted result for t<0 i.e. fiducary hole doped CoO_2 planes. Notice much larger scale of S* arising from transport part (not Mott Heikes part!!). Enhancement due to triangular lattice structure of closed loops!! Similar to Hall constant linear T origin.

20. Predicted result for t<0 i.e. fiducary hole doped CoO_2 planes. Different J higher S.

21. Predictions of S* and the Heikes Mott formula (red) for fiducary hole doped CoO2. Notice that S* predicts an important enhancement unlike Heikes Mott formula Heikes Mott misses the lattice topology effects.

22. Z*T computed from S* and Lorentz number. Electronic contribution only, no phonons. Clearly large x is better!! Quite encouraging.