outline 1. coupled dimer antiferromagnetsqpt.physics.harvard.edu/leshouches/lecture3.pdf1. coupled...
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1. Coupled dimer antiferromagnets Landau-Ginzburg quantum criticality
2. Spin liquids and valence bond solids (a) Schwinger-boson mean-field theory - square lattice (b) Gauge theories of perturbative fluctuations (c) Non-perturbative effects: Berry phases (d) Schwinger-boson mean-field theory - triangular lattice (e) Visons and the Kitaev model 3. Cuprate superconductivity (a) Review of experiments, old and new (b) Fermi surfaces and the spin density wave theory (c) Fermi pockets and the underdoped cuprates
Outline
1. Coupled dimer antiferromagnets Landau-Ginzburg quantum criticality
2. Spin liquids and valence bond solids (a) Schwinger-boson mean-field theory - square lattice (b) Gauge theories of perturbative fluctuations (c) Non-perturbative effects: Berry phases (d) Schwinger-boson mean-field theory - triangular lattice (e) Visons and the Kitaev model 3. Cuprate superconductivity (a) Review of experiments, old and new (b) Fermi surfaces and the spin density wave theory (c) Fermi pockets and the underdoped cuprates
Outline
Half-filled band Mott insulator with spin S = 1/2
Triangular lattice of [Pd(dmit)2]2 frustrated quantum spin system
X[Pd(dmit)2]2 Pd SC
X Pd(dmit)2
t’tt
Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007)
H =!
!ij"
Jij!Si · !Sj + . . .
H = J!
!ij"
!Si · !Sj ; !Si ! spin operator with S = 1/2
What is the ground state as a function of J !/J ?
H =!
!ij"
Jij!Si · !Sj + . . .
H = J!
!ij"
!Si · !Sj ; !Si ! spin operator with S = 1/2
Anisotropic triangular lattice antiferromagnet
Neel ground state for small J’/J
Broken spin rotation symmetry
Anisotropic triangular lattice antiferromagnet
Possible ground states as a function of J !/J
• Neel antiferromagnetic LRO
Magnetic CriticalityT N
(K)
Neel order
Me4P
Me4As
EtMe3As
Et2Me2As Me4Sb
Et2Me2P
EtMe3Sb
Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007)
X[Pd(dmit)2]2Et2Me2Sb (CO)
!J !/J
=(|!"# $ |"!#)%
2
Anisotropic triangular lattice antiferromagnet
Possible ground state for intermediate J’/JN. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)
=(|!"# $ |"!#)%
2
Anisotropic triangular lattice antiferromagnet
Possible ground state for intermediate J’/JValence bond solid (VBS)
Broken lattice space group symmetry
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)
=(|!"# $ |"!#)%
2
Anisotropic triangular lattice antiferromagnetBroken lattice space group symmetry
Possible ground state for intermediate J’/JValence bond solid (VBS)
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)
=(|!"# $ |"!#)%
2
Anisotropic triangular lattice antiferromagnetBroken lattice space group symmetry
Possible ground state for intermediate J’/JValence bond solid (VBS)
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)
=(|!"# $ |"!#)%
2
Anisotropic triangular lattice antiferromagnetBroken lattice space group symmetry
Possible ground state for intermediate J’/JValence bond solid (VBS)
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)
Anisotropic triangular lattice antiferromagnet
Possible ground states as a function of J !/J
• Neel antiferromagnetic LRO
• Valence bond solid
Magnetic CriticalityT N
(K)
Neel order
Me4P
Me4As
EtMe3As
Et2Me2As Me4Sb
Et2Me2P
EtMe3Sb
Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007)
X[Pd(dmit)2]2Et2Me2Sb (CO)
!J !/J
Magnetic CriticalityT N
(K)
Neel order
Me4P
Me4As
EtMe3As
Et2Me2As Me4Sb
Et2Me2P
EtMe3Sb
EtMe3P
Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007)
X[Pd(dmit)2]2Et2Me2Sb (CO)
!J !/J
Spingap
Magnetic CriticalityT N
(K)
Neel order
Me4P
Me4As
EtMe3As
Et2Me2As Me4Sb
Et2Me2P
EtMe3Sb
EtMe3P
Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007)
X[Pd(dmit)2]2Et2Me2Sb (CO)
!J !/J
VBS order
Spingap
M. Tamura, A. Nakao and R. Kato, J. Phys. Soc. Japan 75, 093701 (2006)Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, Phys. Rev. Lett. 99, 256403 (2007)
Observation of a valence bond solid (VBS) in ETMe3P[Pd(dmit)2]2
Spin gap ~ 40 K J ~ 250 K
X-ray scattering
Magnetic CriticalityT N
(K)
Neel order
Me4P
Me4As
EtMe3As
Et2Me2As Me4Sb
Et2Me2P
EtMe3Sb
EtMe3P
Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007)
X[Pd(dmit)2]2Et2Me2Sb (CO)
!J !/J
VBS order
Spingap
Schwinger boson mean field theory on the square lattice and perturbative fluctuations
S =!
d2xd!
"|("! ! iA! )z"|2 + c2|("x ! iAx)z"|2 + s|z"|2
+ u#|z"|2
$2 +1
2e2(#µ#$"#A$)2
%
Schwinger boson mean field theory on the square lattice and perturbative fluctuations
bi! ! bi!ei"(i) bj! ! bj!e!i"(j)
Qij ! Qijei(!(i)!!(j))
! with Qij = |Qij |eiAij , we have Aij ! Aij + !(i)" !(j)or Ai,i+µ ! Ai,i+µ + "µ!
Origin of gauge invariance
Schwinger boson mean field theory on the square lattice and perturbative fluctuations
S =!
d2xd!
"|("! ! iA! )z"|2 + c2|("x ! iAx)z"|2 + s|z"|2
+ u#|z"|2
$2 +1
2e2(#µ#$"#A$)2
%
Schwinger boson mean field theory on the square lattice and perturbative fluctuations
S =!
d2xd!
"|("! ! iA! )z"|2 + c2|("x ! iAx)z"|2 + s|z"|2
+ u#|z"|2
$2 +1
2e2(#µ#$"#A$)2
%
Schwinger boson mean field theory on the square lattice and perturbative fluctuations
Nonperturbative e!ects lead to a gap in Aµ,confinement of z!,
and valence bond solid (VBS) order
S =!
d2xd!
"|("! ! iA! )z"|2 + c2|("x ! iAx)z"|2 + s|z"|2
+ u#|z"|2
$2 +1
2e2(#µ#$"#A$)2
%
Schwinger boson mean field theory: from the square to the triangular lattice
bi! ! bi!ei"(i) bj! ! bj!e!i"(j)
Qij ! Qijei(!(i)!!(j))
Schwinger boson mean field theory: from the square to the triangular lattice
bi! ! bi!ei"(i) bj! ! bj!e!i"(j)
Qij ! Qijei(!(i)!!(j))
!ii! ! !ii!ei(!(i)+!(i!))
Schwinger boson mean field theory: from the square to the triangular lattice
!ii! ! !ii!ei(!(i)+!(i!))
In the continuum limitz! ! zei",
Aµ ! Aµ + !µ",!! !e2i" .
Low energy theory for U(1) spin liquid
s > 0, z! gapped.
S =!
d2xd!
"|("! ! iA! )z"|2 + c2|("x ! iAx)z"|2 + s|z"|2
+ u#|z"|2
$2 +1
2e2(#µ#$"#A$)2
!
Low energy theory for Z2 spin liquid
s > 0, z! gapped.
S =!
d2xd!
"|("! ! iA! )z"|2 + c2|("x ! iAx)z"|2 + s|z"|2
+ u#|z"|2
$2 +1
2e2(#µ#$"#A$)2
+|(!! ! 2iA! )!|2 + !c2|(!x ! 2iAx)!|2 + !s|!|2 + !u|!|4"
!s < 0, ! condensed
Low energy theory for Z2 spin liquid
s > 0, z! gapped.
S =!
d2xd!
"|("! ! iA! )z"|2 + c2|("x ! iAx)z"|2 + s|z"|2
+ u#|z"|2
$2 +1
2e2(#µ#$"#A$)2
+|(!! ! 2iA! )!|2 + !c2|(!x ! 2iAx)!|2 + !s|!|2 + !u|!|4"
!s < 0, ! condensed
Low energy theory for Z2 spin liquid
s > 0, z! gapped.
S =!
d2xd!
"|("! ! iA! )z"|2 + c2|("x ! iAx)z"|2 + s|z"|2
+ u#|z"|2
$2 +1
2e2(#µ#$"#A$)2
+|(!! ! 2iA! )!|2 + !c2|(!x ! 2iAx)!|2 + !s|!|2 + !u|!|4"
!s < 0, ! condensed
Theory is stable to non-perturbative effects
Topological excitation of a Z2 spin liquid (vison)
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
An Abrikosov vortex in the ! condensate
!d2r(!" !A) = "
!! !e2i!
Topological excitation of a Z2 spin liquid (vison)
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
An Abrikosov vortex in the ! condensate
!d2r(!" !A) = "
!! !e2i!
Visons are are the dark matter of spin liquids:they likely carry most of the energy, but are veryhard to detect because they do not carry chargeor spin.
Topological excitation of a Z2 spin liquid (vison)
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
An Abrikosov vortex in the ! condensate
!d2r(!" !A) = "
!! !e2i!
Parallel transport of a gapped spinon z!
around a vison yields z! ! "z!.Spinons and visons are mutual semions.
=
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
=
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
=
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
=
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
=
-1
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
=
-1
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
Properties of the Ground State
Ground state: all Ai=1, Fp=1 Pictorial representation: color each link with an up-spin. Ai=1 : closed loops. Fp=1 : every plaquette is an equal-amplitude superposition of
inverse images.
The GS wavefunction takes the same value on configurationsconnected by these operations. It does not depend on the geometry of the configurations, only on their topology.
Properties of Excitations
“Electric” particle, or Ai = –1 – endpoint of a line “Magnetic particle”, or vortex: Fp= –1 – a “flip” of this plaquette
changes the sign of a given term in the superposition. Charges and vortices interact via topological Aharonov-Bohm
interactions.
Properties of Excitations
“Electric” particle, or Ai = –1 – endpoint of a line (a “spinon”) “Magnetic particle”, or vortex: Fp= –1 – a “flip” of this plaquette
changes the sign of a given term in the superposition. Charges and vortices interact via topological Aharonov-Bohm
interactions.
Properties of Excitations
“Electric” particle, or Ai = –1 – endpoint of a line (a “spinon”) “Magnetic particle”, or vortex: Fp= –1 – a “flip” of this plaquette
changes the sign of a given term in the superposition (a “vison”). Charges and vortices interact via topological Aharonov-Bohm
interactions.
Properties of Excitations
“Electric” particle, or Ai = –1 – endpoint of a line (a “spinon”) “Magnetic particle”, or vortex: Fp= –1 – a “flip” of this plaquette
changes the sign of a given term in the superposition (a “vison”). Charges and vortices interact via topological Aharonov-Bohm
interactions.
Spinons and visons are mutual semions
Beyond Z2 spin liquids
String nets - M. A. Levin and X.-G. Wen, Phys. Rev. B 71, 045110 (2005).
Doubled SU(2)k Chern-Simons theory - M. Freedman, C. Nayak, K. Shtengel, K. Walker, and Z. Wang, Annals of Physics 310, 428 (2004).
Spin model on the honeycomb lattice - A. Y. Kitaev, Annals of Physics 321, 2 (2006).
Quantum spin liquids with anyonic excitations
1. Coupled dimer antiferromagnets Landau-Ginzburg quantum criticality
2. Spin liquids and valence bond solids (a) Schwinger-boson mean-field theory - square lattice (b) Gauge theories of perturbative fluctuations (c) Non-perturbative effects: Berry phases (d) Schwinger-boson mean-field theory - triangular lattice (e) Visons and the Kitaev model 3. Cuprate superconductivity (a) Review of experiments, old and new (b) Fermi surfaces and the spin density wave theory (c) Fermi pockets and the underdoped cuprates
Outline
1. Coupled dimer antiferromagnets Landau-Ginzburg quantum criticality
2. Spin liquids and valence bond solids (a) Schwinger-boson mean-field theory - square lattice (b) Gauge theories of perturbative fluctuations (c) Non-perturbative effects: Berry phases (d) Schwinger-boson mean-field theory - triangular lattice (e) Visons and the Kitaev model 3. Cuprate superconductivity (a) Review of experiments, old and new (b) Fermi surfaces and the spin density wave theory (c) Fermi pockets and the underdoped cuprates
Outline
The cuprate superconductors
Ground state has long-range spin density wave (SDW) order
Square lattice antiferromagnet
H =!
!ij"
Jij!Si · !Sj
Order parameter is a single vector field !" = #i!Si
#i = ±1 on two sublattices!!"" #= 0 in SDW state.
The cuprate superconductors
! !
Phase diagram of electron-doped superconductorsNd2!xCexCuO4!y and Pr2!xCexCuO4!y
(a)
ΓQ
K1
K 2K3
K4
Q
2Q
4
Q 3 1N
SC
N+SC
x
(b)T
QCP
Neel order (N)
N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois, J.-B. Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy, and L. Taillefer, Nature 447, 565 (2007)
Quantum oscillations andthe Fermi surface in anunderdoped high Tc su-perconductor (ortho-II or-dered YBa2Cu3O6.5). Theperiod corresponds to acarrier density ! 0.076.
Quantum oscillations
Quantum oscillations
Onsager-Lifshitz relation:Frequency of oscillations in 1/H
=hc
2e
(Area of Fermi surface)2!2
Luttinger relation:Area of Fermi surface = 4!3(density of fermions)
Nature 450, 533 (2007)
Quantum oscillations
!
Fermi surfaces in electron- and hole-doped cupratesHole states
occupied
Electron states
occupied
!E!ective Hamiltonian for quasiparticles:
H0 = !!
i<j
tijc†i!ci! "
!
k
!kc†k!ck!
with tij non-zero for first, second and third neighbor, leads to satisfactory agree-ment with experiments. The area of the occupied electron states, Ae, fromLuttinger’s theory is
Ae ="
2"2(1! p) for hole-doping p2"2(1 + x) for electron-doping x
The area of the occupied hole states, Ah, which form a closed Fermi surface andso appear in quantum oscillation experiments is Ah = 4"2 !Ae.
Spin density wave theory
In the presence of spin density wave order, !" at wavevector K =(#, #), we have an additional term which mixes electron states withmomentum separated by K
Hsdw = !" ·!
k,!,"
ck,!!$!"ck+K,"
where !$ are the Pauli matrices. The electron dispersions obtainedby diagonalizing H0 + Hsdw for !" ! (0, 0, 1) are
Ek± =%k + %k+K
2±
"#%k " %k+K
2
$+ "2
This leads to the Fermi surfaces shown in the following slides forelectron and hole doping.
Increasing SDW order
Spin density wave theory in electron-doped cuprates
Increasing SDW order
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
!
Increasing SDW order
Spin density wave theory in electron-doped cuprates
Increasing SDW order
!!
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Increasing SDW order
Spin density wave theory in electron-doped cuprates
Increasing SDW order
!!!
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Hole pockets
Electron pockets
Increasing SDW order
Spin density wave theory in electron-doped cuprates
Increasing SDW order
SDW order parameter is a vector, !",whose amplitude vanishes at the transition
to the Fermi liquid.
!!!!
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Electron pockets
N. P. Armitage et al., Phys. Rev. Lett. 88, 257001 (2002).
Photoemission in NCCO
Increasing SDW order
Spin density wave theory in hole-doped cuprates
!
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Increasing SDW order
Spin density wave theory in hole-doped cuprates
!!
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Increasing SDW order
Spin density wave theory in hole-doped cuprates
!!!
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
Electron pockets
Hole pockets
Increasing SDW order
Spin density wave theory in hole-doped cuprates
!!!
S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).
!
Hole pockets
SDW order parameter is a vector, !",whose amplitude vanishes at the transition
to the Fermi liquid.