outline 1. coupled dimer antiferromagnetsqpt.physics.harvard.edu/leshouches/lecture3.pdf1. coupled...

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


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


X[Pd(dmit)2]2Et2Me2Sb (CO)

!J !/J

=(|!"# $ |"!#)%

2


Possible ground state for intermediate J’/JN. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)

=(|!"# $ |"!#)%

2


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)



Possible ground states as a function of J !/J

• Neel antiferromagnetic LRO

• Valence bond solid


(K)

Neel order

Me4P

Me4As

EtMe3As

Et2Me2As Me4Sb

Et2Me2P

EtMe3Sb



!J !/J


(K)

Neel order

Me4P

Me4As

EtMe3As

Et2Me2As Me4Sb

Et2Me2P

EtMe3Sb

EtMe3P



!J !/J

Spingap


(K)

Neel order

Me4P

Me4As

EtMe3As

Et2Me2As Me4Sb

Et2Me2P

EtMe3Sb

EtMe3P



!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


(K)

Neel order

Me4P

Me4As

EtMe3As

Et2Me2As Me4Sb

Et2Me2P

EtMe3Sb

EtMe3P



!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

%


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


S =!

d2xd!

"|("! ! iA! )z"|2 + c2|("x ! iAx)z"|2 + s|z"|2

+ u#|z"|2

$2 +1

2e2(#µ#$"#A$)2

%


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






!ii! ! !ii!ei(!(i)+!(i!))


!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

Theory is stable to non-perturbative effects

Topological excitation of a Z2 spin liquid (vison)


An Abrikosov vortex in the ! condensate

!d2r(!" !A) = "

!! !e2i!




!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.




!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

-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.

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.


“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.



changes the sign of a given term in the superposition (a “vison”). Charges and vortices interact via topological Aharonov-Bohm

interactions.



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

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


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)


!

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


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




SDW order parameter is a vector, !",whose amplitude vanishes at the transition

to the Fermi liquid.

!!!!


Electron pockets

N. P. Armitage et al., Phys. Rev. Lett. 88, 257001 (2002).

Photoemission in NCCO


Spin density wave theory in hole-doped cuprates

!




!!




!!!


Electron pockets

Hole pockets



!!!


!

Hole pockets

SDW order parameter is a vector, !",whose amplitude vanishes at the transition

to the Fermi liquid.

outline 1. coupled dimer antiferromagnetsqpt.physics.harvard.edu/leshouches/lecture3.pdf1. coupled...

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