Topological Kondo Insulators:Happy Birthday Gil! Magnetism meets Topology Piers Coleman & Onur Erten Center for Materials Theory, Rutgers U, USA Royal Holloway, University of London, UK.

Piers Coleman, Rutgers CMT. Topological Kondo Insulators: Magnetism meets Topology Piers Coleman & Onur Erten Center for Materials Theory, Rutgers U, USA Royal Holloway, University of London, UK. Topological Kondo Insulators: Magnetism meets Topology Piers Coleman & Onur Erten Center for Materials Theory, Rutgers U, USA Royal Holloway, University of London, UK.

Collaborators Maxim Dzero Kent State Kai Sun Michigan Maryland " 4"% ",'* Vic Onur Vic Alexandrov, IAS Onur Erten, Rutgers Joel Moore, Berkeley Phil Anderson, Princeton Suchitra Sebastian Cambridge Gil Lonzarich Cambridge Topological Kondo Insulators: Magnetism meets Topology Piers Coleman & Onur Erten Center for Materials Theory, Rutgers U, USA Royal Holloway, University of London, UK.

Collaborators Maxim Dzero Kent State Kai Sun Michigan Victor Galitski Maryland " 4"% ",'* Vic Onur Vic Alexandrov, IAS Onur Erten, Rutgers Joel Moore, Berkeley Phil Anderson, Princeton Suchitra Sebastian Cambridge Gil Lonzarich Cambridge

M. Dzero, K. Sun, V. Galitski, PC Phys. Rev. Lett. 104, 106408 (2010) T. Takimoto, J. Phys. Soc. Jpn. 80, 123710 (2011). V. Alexandrov, P. Coleman, O. Erten, Phys. Rev. Lett. 114:177202 (2015). Maxim Dzero, Jing Xia, Victor Galitski, Piers Coleman, Annual Reviews CMP (2016), ArXiv 1506.05635 Topological Kondo Insulators: Magnetism meets Topology Piers Coleman & Onur Erten Center for Materials Theory, Rutgers U, USA Royal Holloway, University of London, UK.

Outline

• SmB6 and the rise of topology • TKIs: a link with superfluid He-3 • Is SmB6 topological? • The Magnetic Connection. Kondo insulators: History &,!91!$*&$$A HH9HOK9GOLOB (($"&"+#A'%%&,+ GL9GKK9GOOHB

B6

2.7+ Sm SmB6

K Kondo insulators: History &,!91!$*&$$A HH9HOK9GOLOB (($"&"+#A'%%&,+ GL9GKK9GOOHB

"%($+, '&' /

B6

2.7+ Sm SmB6

K Kondo insulators: History Hybridization picture.

&,!91!$*&$$A HH9HOK9GOLOB Maple + Wohlleben, 1972 (($"&"+#A'%%&,+ GL9GKK9GOOHB Mott Phil Mag, 30,403,1974 Allen and Martin, 1979

"%($+, '&' /

H =(|kσVσα(k)α| +H.c)

E(k)

k Ef

L Kondo insulators: History Hybridization picture.

&,!91!$*&$$A HH9HOK9GOLOB Maple + Wohlleben, 1972 (($"&"+#A'%%&,+ GL9GKK9GOOHB Mott Phil Mag, 30,403,1974 Allen and Martin, 1979

"%($+, '&' /

H =(|kσVσα(k)α| +H.c)

E(k)

k Ef

“In SmB6 and high-pressure SmS a very small gap separates occupied from unoccupied states, this in our view being due to hybridization of 4f and 4d bands.” Mott 1974 L Kondo insulators: History Hybridization picture.

&,!91!$*&$$A HH9HOK9GOLOB Maple + Wohlleben, 1972 (($"&"+#A'%%&,+ GL9GKK9GOOHB Mott Phil Mag, 30,403,1974 Allen and Martin, 1979        

Cooley, Aronson, et al. 1995 H =(|kσVσα(k)α| +H.c)

E(k)

k Ef

In SmB6 and high-pressure SmS a very small gap separates occupied from unoccupied states, this in our view being due to hybridization of 4f and 4d bands. M !*"+''('$' 5;

N !*"+''('$' 5;

g=1 g=3 g=genus g=0 g=2 N !*"+''('$' 5;

"*&-$'%,*5'('$' 5

g=1 g=3 g=genus g=0 g=2 N !*"+''('$' 5;

<1++?'&&,!'*%=1++A1&(1$"+!B9'&&,AGNJNB

"*&-$'%,*5'('$' 5 1 Ω κdA = =(1− g) 4π 4π

g=1 g=3 g=genus g=0 g=2 N !*"+''('$' 5;

"*&-$'%,*5  ',!321&-'&'**

O Berry v. Klitzing Laughlin Thouless Haldane !*"+''('$' 5;

"*&-$'%,*5  ',!321&-'&'**

Integer Quantum Hall

von Klitzing, Dorda & Pepper (1980) O Berry v. Klitzing Laughlin Thouless Haldane !*"+''('$' 5;

"*&-$'%,*5  ',!321&-'&'**

Integer Quantum Hall

   ak = i un,k|∇k|un,k   ,$9AGONHB n=1,2N

von Klitzing, Dorda & Pepper (1980) O  Berry v. Klitzing Laughlin Thouless Haldane   ak = i un,k|∇k|un,k   ,$9AGONHB n=1,2N !*"+''('$' 5;

Kane Mele Zhang Molencamp Hasan Balents Moore Roy Fu

"*&-$'%,*5  ',!321&-'&'**

Z2 Topological Insulators Integer Quantum Hall

"&!& 92; '!5+AHFGFB; von Klitzing, Dorda & Pepper (1980) O ?

P

11%

GF '&2&-'&$& "&+1$,'*

?

P

11%

GF Conventional band insulator: adiabatic continuation of the vacuum.

'&2&-'&$& "&+1$,'*

?



!

P

11%

GF

'('$' "$<"&+1$,'*= A**5'&&-'&<,3"+,=B

?



!

P

11%

GG Topological insulator : adiabatically disconnected from the vacuum.

Gap must close at interface between '('$' "$<"&+1$,'*= two different vacua A**5'&&-'&<,3"+,=B

?



!

P

11%

Metallic surfaces.GG 1D Topological Insulator

Igor William Tamm Shockley

Tamm 1932, Shockley, Phys Rev, 56, 317 (1939): 1D TI 1D Topological Insulator

Igor William Tamm Shockley

P=-1 tp p + + - V - -t P=+1 + +

Tamm 1932, Shockley, Phys Rev, 56, 317 (1939): 1D TI 1D Topological Insulator

Igor William Tamm Shockley

P=-1 tp p + + - V - -t P=+1 + +

Tamm 1932, Shockley, Phys Rev, 56, 317 (1939): 1D TI Open BCs- broken translation symmetry odd/even states mix to form edge states. Band Crossing of odd and even parity states Yields a Z2 Topological Insulator (Fu, Kane, Mele, 2007) Band Crossing of odd and even parity states Yields a Z2 Topological Insulator (Fu, Kane, Mele, 2007) Band Crossing of odd and even parity states Yields a Z2 Topological Insulator (Fu, Kane, Mele, 2007)

Topological Texture of Berry Connection Band Crossing of odd and even parity states Yields a Z2 Topological Insulator (Fu, Kane, Mele, 2007)

Z2= 1 Z2= -1

Topological Texture of Berry Connection Band Crossing of odd and even parity states Yields a Z2 Topological Insulator (Fu, Kane, Mele, 2007)

 Z2 = δ(Γi) i

1& & AHFFNB:*",5)1-'&;

Z2= 1 Z2= -1

Topological Texture of Berry Connection FIG. 27 ARPES data for Bi2Se3 thin films of thickness (a) 1QL (b) 2QL (c) 3QL (d) 5QL (e) 6QL, measured at room temperature (QL stands for quintuple layer). From Zhang et al., 2009.

H >1&,1%$$+ @ FIG. 25 ARPES data for the dispersion of the surface states of ¯ ¯ ¯ ¯ Bi2Se3, along directions (a) Γ − M and (b) Γ − K in the surface ¯ ¯ A '&" ,$9FM>NB Brillioun zone. Spin-resolved ARPES data is shown along Γ−M for a fixed energy in (d), from which the spin polarization in momentum space (c) can be extracted. From Xia et al., 2009 and Hsieh et al., @I Bi2Se3 ,Bi2Te 3, Sb2Te 3 2009. (Zhang et al ,Xia et al ,Chen et al, Hsieh et al (09))

GK FIG. 27 ARPES data for Bi2Se3 thin films of thickness (a) 1QL (b) 2QL (c) 3QL (d) 5QL (e) 6QL, measured at room temperature (QL stands for quintuple layer). From Zhang et al., 2009.

H >1&,1%$$+ @ FIG. 25 ARPES data for the dispersion of the surface states of ¯ ¯ ¯ ¯ Bi2Se3, along directions (a) Γ − M and (b) Γ − K in the surface ¯ ¯ A '&" ,$9FM>NB Brillioun zone. Spin-resolved ARPES data is shown along Γ−M for a fixed energy in (d), from which the spin polarization in momentum space (c) can be extracted. From Xia et al., 2009 and Hsieh et al., @I Bi2Se3 ,Bi2Te 3, Sb2Te 3 2009. (Zhang et al ,Xia et al ,Chen et al, Hsieh et al (09))

*,!*+,*'& $5'**$,'1&,*(*,+8 $,*'&+8 ,*'& $5"&,*-& 9"& &",("&'*",9'(*",5; GK Are Kondo insulators topological?

R. Martin & J. Allen, J. Applied Physics, 50,7561 (1979) Are Kondo insulators topological?

R. Martin & J. Allen, J. Applied Physics, 50,7561 (1979) Are Kondo insulators topological?

R. Martin & J. Allen, J. Applied Physics, 50,7561 (1979)

Dzero, Sun, Galitski, PC Phys. Rev. Lett. 104, 106408 (2010) Are Kondo insulators topological?

R. Martin & J. Allen, J. Applied Physics, 50,7561 (1979)

Dzero, Sun, Galitski, PC Phys. Rev. Lett. 104, 106408 (2010)

Many Body Localized Are Kondo insulators topological?

R. Martin & J. Allen, J. Applied Physics, 50,7561 (1979)

Dzero, Sun, Galitski, PC Phys. Rev. Lett. 104, 106408 (2010)

Many Body Many Body Localized Delocalization    Are Kondo insulators topological?      R. Martin & J. Allen, J. Applied Physics, 50,7561 (1979)

Dzero, Sun, Galitski, PC Phys. Rev. Lett. 104, 106408 (2010)

Many Body Many Body Localized Delocalization    Are Kondo insulators topological?      R. Martin & J. Allen, J. Applied Physics, 50,7561 (1979)

Dzero, Sun, Galitski, PC Phys. Rev. Lett. 104, 106408 (2010)

Many Body Many Body Localized Delocalization

Band Theory SmB6: T. Takimoto, J. Phys. Soc. Jpn. 80, 123710 (2011).

Maxim Dzero, Kai Sun, Piers Coleman and Victor Galitski, Phys. Rev. B 85 , 045130-045140 (2012). Gutzwiller + Band Theory F. Lu, J. Zhao, H. Weng, Z. Fang and X. Dai, Phys. Rev. Lett. 110, 096401 (2013).

Victor Alexandrov, Maxim Dzero and Piers Coleman PRL (2013). ,1*+',!&3%'$ Dzero et al, Annual Reviews of (2016), arXiv 1506.05635

GM ,1*+',!&3%'$ Dzero et al, Annual Reviews of Condensed Matter Physics (2016), arXiv 1506.05635

GM ,1*+',!&3%'$ Dzero et al, Annual Reviews of Condensed Matter Physics (2016), arXiv 1506.05635

Three crossings: THREE DIRAC CONES ON SURFACE. GM ,1*+',!&3%'$ Dzero et al, Annual Reviews of Condensed Matter Physics (2016), arXiv 1506.05635

Three crossings: THREE DIRAC CONES ON SURFACE. GN ,1*+',!&3%'$ Dzero et al, Annual Reviews of Condensed Matter Physics (2016), arXiv 1506.05635

sˆdˆ((kk)) == kˆ

Vαβ(k) = Vsk · σ αβ X

ˆ sk = (sin kx, sin ky, sin kz) ∼ k Three crossings: THREE DIRAC CONES ON SURFACE. GN ,1*+',!&3%'$ Dzero et al, Annual Reviews of Condensed Matter Physics (2016), arXiv 1506.05635

sˆdˆ((kk)) == kˆ

Vαβ(k) = Vsk · σ αβ X

ˆ sk = (sin kx, sin ky, sin kz) ∼ k Three crossings: THREE DIRAC CONES Hybridization of f (P=+) and d (P=-) vanishes at X point. ON SURFACE. GN ,1*+',!&3%'$ Dzero et al, Annual Reviews of Condensed Matter Physics (2016), arXiv 1506.05635

=3   V s · σ H(k) = k k V sk · σ f k

sˆdˆ((kk)) == kˆ

Vαβ(k) = Vsk · σ αβ X

ˆ sk = (sin kx, sin ky, sin kz) ∼ k Three crossings: THREE DIRAC CONES Hybridization of f (P=+) and d (P=-) vanishes at X point. ON SURFACE. GN ,1*+',!&3%'$ Dzero et al, Annual Reviews of Condensed Matter Physics (2016), arXiv 1506.05635

=3Like He-3B: an adaptive insulator.  V s · σ H(k) = k k V sk · σ f k

sˆdˆ((kk)) == kˆ

Vαβ(k) = Vsk · σ αβ X

ˆ sk = (sin kx, sin ky, sin kz) ∼ k Three crossings: THREE DIRAC CONES Hybridization of f (P=+) and d (P=-) vanishes at X point. ON SURFACE. GN ,"$$1$-'&

(a)

Gutzwiller + DFT F. Lu, et al., Phys. Rev. Lett. 110:096401 (2013) Three crossings: THREE DIRAC CONES ON SURFACE. GO Is SmB6 a topological Kondo insulator? SmB6 Surface Conductivity SmB6 Surface Conductivity

RVert

Hall constant derives from the Surface. RLat u

+"+,"2

D. J. Kim et al, Scientific Reports 3, 3150 (2013) Wolgast et al, Phys Rev B, 88, 180405 (2013) SmB6 Surface Conductivity

Large Vertical Resistance indicates conductivity is from the surface.

RVert

Hall constant derives from the Surface. RLat u

+"+,"2

D. J. Kim et al, Scientific Reports 3, 3150 (2013) Wolgast et al, Phys Rev B, 88, 180405 (2013) SmB6 Surface Conductivity

Bulk Insulator

Surface Conductivity

Robustness/Sensitivity to potential/magnetic scattering. Large Vertical Resistance indicates conductivity is from the surface.

RVert

Hall constant derives from the Surface. RLat u

+"+,"2

D. J. Kim et al, Scientific Reports 3, 3150 (2013) Wolgast et al, Phys Rev B, 88, 180405 (2013) SmB6 TKI Check List.

           SmB6 TKI Check List.

4f j=5/2, 7/2 Multiplets

           SmB6 TKI Check List.

4f j=5/2, 7/2 Multiplets

5d-band

           SmB6 TKI Check List.

4f j=5/2, 7/2 Multiplets

          

5d-band

             SmB6 TKI Check List.

d-band crossing at X points

Odd number (3) of Surface FS (ARPES, dHVA, STM). 4f j=5/2, 7/2 Multiplets

          

5d-band

             Spin Resolved ARPES Nan Xu , X. Shi , P. Biswas , C. Matt , R. Dhaka , Y. Huang , N. Plumb , M. Radovic , J. Dil , E. Pomjakushina , K. Conder , A. Amato , Z. Salman , D. Paul , J. Mesot , Hong Ding , Ming Shi Nature Communications Volume: 5, 4566 (2014)

F High

Low Spin Resolved ARPES Nan Xu , X. Shi , P. Biswas , C. Matt , R. Dhaka , Y. Huang , N. Plumb , M. Radovic , J. Dil , E. Pomjakushina , K. Conder , A. Amato , Z. Salman , D. Paul , J. Mesot , Hong Ding , Ming Shi Nature Communications Volume: 5, 4566 (2014)

F High

Low

ky

_ M

_ _  Γ kx X

 Bulk f states Spin Resolved ARPES Nan Xu , X. Shi , P. Biswas , C. Matt , R. Dhaka , Y. Huang , N. Plumb , M. Radovic , J. Dil , E. Pomjakushina , K. Conder , A. Amato , Z. Salman , D. Paul , J. Mesot , Hong Ding , Ming Shi Nature Communications Volume: 5, 4566 (2014)

F High

Low

ky But no consensus yet! _ Phys. Rev. Lett. 111, 216402 (2013) M

_ _  Γ kx X arxiv/1502.01542  Bulk f states !% &-'&&-'&;

I. Kondo Breakdown II. Pressure:AFM III. Neutrons: Exciton IV. Field: dHvA and Quantum Criticality.

HJ V. Alexandrov, P. Coleman, O. Erten, I. Kondo Breakdown Phys. Rev. Lett. 114:177202 2015.

  

HK V. Alexandrov, P. Coleman, O. Erten, I. Kondo Breakdown Phys. Rev. Lett. 114:177202 2015.

  

ARPES: vs~ 220-300 meVA

HK V. Alexandrov, P. Coleman, O. Erten, I. Kondo Breakdown Phys. Rev. Lett. 114:177202 2015.

  

ARPES: vs~ 220-300 meVA

Theory: vs~ 30-50 meVA 10x too small!

HK V. Alexandrov, P. Coleman, O. Erten, I. Kondo Breakdown Phys. Rev. Lett. 114:177202 2015. Lower co-ordination at surface dramatically suppresses the Kondo temperature 

    

ARPES: vs~ 220-300 meVA

Theory: vs~ 30-50 meVA 10x too small!

HK V. Alexandrov, P. Coleman, O. Erten, I. Kondo Breakdown Phys. Rev. Lett. 114:177202 2015. Lower co-ordination at surface dramatically suppresses the Kondo temperature  

  

    

ARPES: vs~ 220-300 meVA

Theory: vs~ 30-50 meVA 10x too small!

HK V. Alexandrov, P. Coleman, O. Erten, I. Kondo Breakdown Phys. Rev. Lett. 114:177202 2015. Lower co-ordination at surface dramatically Kondo Breakdown at surface. suppresses the Kondo temperature  

  

    

ARPES: vs~ 220-300 meVA

Theory: vs~ 30-50 meVA 10x too small!

HK V. Alexandrov, P. Coleman, O. Erten, I. Kondo Breakdown Phys. Rev. Lett. 114:177202 2015. Lower co-ordination at surface dramatically Kondo Breakdown at surface. suppresses the Kondo temperature  

  

    

ARPES: vs~ 220-300 meVA Theory: vs~ 30-50 meVA A / π 2 n 10x too small! FS (2 ) =Δ f

HK V. Alexandrov, P. Coleman, O. Erten, I. Kondo Breakdown Phys. Rev. Lett. 114:177202 2015. Lower co-ordination at surface dramatically Kondo Breakdown at surface. suppresses the Kondo temperature  

  

    

ARPES: vs~ 220-300 meVA Theory: vs~ 30-50 meVA A / π 2 n 10x too small! FS (2 ) =Δ f

Breakdown of at surface causes surface Dirac cones to dope, submerging the Dirac point and considerably enhancing the Fermi velocity.

HK V. Alexandrov, P. Coleman, O. Erten, I. Kondo Breakdown Phys. Rev. Lett. 114:177202 2015.

Kondo Breakdown at surface.

  

  

2 AFS/(2π) =Δnf

HL V. Alexandrov, P. Coleman, O. Erten, I. Kondo Breakdown Phys. Rev. Lett. 114:177202 2015.

Kondo Breakdown at surface.

  

  

2 AFS/(2π) =Δnf

Local moments on the surface form a 2D Kondo lattice with spin-orbit locked conduction bands.

HL V. Alexandrov, P. Coleman, O. Erten, I. Kondo Breakdown Phys. Rev. Lett. 114:177202 2015.

Kondo Breakdown at surface.

  

  

2 AFS/(2π) =Δnf

Local moments on the surface form a 2D Kondo lattice with spin-orbit locked conduction bands.

Surface Kondo physics? Magnetism, QCP even . HL II. The effect of Pressure: AFM

A. Barla et al, PRL 94, 2005

T

AFM? 50K Gap opens P 6GPa QCP? SmB6 0.3-4K Sm plateau

B6 III. Magnetic Fluctuations Fuhrman et al, PRL 114, 036401 (2015)

T

FM, AFM? AFM Fluctuations 50K Gap opens P 6GPa QCP?

0.3-4K plateau III. Magnetic Fluctuations Fuhrman et al, PRL 114, 036401 (2015)

T

FM, AFM? AFM Fluctuations 50K Gap opens P RPA 6GPa Theory QCP?

0.3-4K plateau IV. Field effect

T

FM, AFM? 50K Gap opens P 6GPa QCP?

0.3-4K plateau H IV. Field effect

Cooley et al, 1999

T

FM, AFM? 50K Gap opens P 6GPa QCP?

0.3-4K plateau H IV. Field effect

Cooley et al, 1999

T

FM, AFM?

50K ? Gap opens P 6GPa QCP?

0.3-4K plateau H IV. Field effect

Cooley et al, 1999

T

FM, AFM?

50K ? Gap opens P 6GPa6GPa QCP?QC

0.3-4K plateau Hc H >160T? IV. Field effect

Cooley et al, 1999

T

FM, AFM?

50K ? Gap opens P 6GPa6GPa QCP?QC

0.3-4K L=5, S=5/2, J=5/2, g~0.2-0.3 plateau Hc H Field effect is ORBITAL >160T? IV. Field effect

2D fluctuations: G. Li et al, Science 346, 1208 (2014).

  T

FM, AFM? 50K Gap opens P 6GPa6GPa QCP?QC

0.3-4K plateau Hc H >160T? IV. Field effect

Tan et al, Science (2015). 2D surface states: G. Li et al, Science 346, 1208 (2014).

  IV. Field effect

Tan et al, Science (2015). 2D surface states: G. Li et al, Science 346, 1208 (2014).

  IV. Field effect

3D orbits! Tan et al, Science (2015). 2D surface states: G. Li et al, Science 346, 1208 (2014).

  IV. Field effect

3D orbits! Tan et al, Science (2015). 2D surface states: G. Li et al, Science 346, 1208 (2014).

  IV. Field effect

3D orbits! 6x Rise in N(0)*: Q Criticality? Tan et al, Science (2015). 2D surface states: G. Li et al, Science 346, 1208 (2014).

  IV. Field effect

3D orbits! 6x Rise in N(0)*: Q Criticality? Tan et al, Science (2015). IV. Field effect

3D orbits! 6x Rise in N(0)*: Q Criticality? • 3D FS in a bulk insulator ! Tan et al, Science (2015). IV. Field effect

3D orbits! 6x Rise in N(0)*: Q Criticality? • 3D FS in a bulk insulator ! Tan et al, Science (2015).

• ωc ≳ V ? (Knolle &Cooper arXiv 1507.00885) IV. Field effect

3D orbits! 6x Rise in N(0)*: Q Criticality? • 3D FS in a bulk insulator ! Tan et al, Science (2015).

• ωc ≳ V ? (Knolle &Cooper arXiv 1507.00885)

• Majorana FS? Are KI gapless? ( Baskaran arXiv 1507 to appear; Miranda, PC, Tsvelik, Physica B, 186-188, 362, 1993)

• Quantum Criitcal phase separation? Congratulations Gil!

Piers Coleman, Rutgers CMT. 1%%*5C1+-'&+

@"+''('$' 5$+,'&3(",1*' '&' &+1$,'*+9 $'+*"&+("*",,' ?I,!&"$"'&

@!*'1+,&++',!1*,,+9#$'$"6-'&9 +1*'&1,&&9!+1 +,,!"+"+ ,'('$' "$ '&'"&+1$,'*;

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IL 1%%*5C1+-'&+

@"+''('$' 5$+,'&3(",1*' '&' &+1$,'*+9 $'+*"&+("*",,' ?I,!&"$"'&

@!*'1+,&++',!1*,,+9#$'$"6-'&9 +1*'&1,&&9!+1 +,,!"+"+ ,'('$' "$ '&'"&+1$,'*;

@!!" !1$#*+"+-2",5%#+,!"+&4$$&,*+*! %,*"$'*,'('$' "$+,1"+;

@,*'&    '&&-'&;

@ '&'*#'3&: +,!+1*"*&,9&&",1&* '(!+,*&+"-'&+8

@!1$#"++,*& $5"*&,&(*'$5 ($++&)1&,1%*"-$;

@ +,'('$' 5"%('*,&,'*',!*+,*'& $5'**$,+5+,%+?%,$+9+1(*'&1,'*+8

IL