Talk online: : Sachdev

Ground states of quantum antiferromagnets in two dimensions

Leon BalentsMatthew Fisher

Olexei MotrunichKwon Park

Subir SachdevT. Senthil

Ashvin Vishwanath

Mott insulator: square lattice antiferromagnet

Parent compound of the high temperature superconductors: 2 4La CuO

Ground state has long-range magnetic Néel order, or “collinear magnetic (CM) order”

0n

jiij

ij SSJH

Néel order parameter: 1 x yi i

i iS

n

Possible theory for fractionalization and topological order

Possible theory for fractionalization and topological order

Z2 gauge theory for fractionalization and topological order

N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)

J1

Experimental realization: CsCuCl3

g

J1 0 ; 0z

0 ; 0z 0 ; 0z

0 ; 0z

Topologically ordered phase described by Z2 gauge theory

?

Central questions:

Vary Jij smoothly until CM order is lost and a paramagnetic phase with is reached.

• What is the nature of this paramagnet ?

• What is the critical theory of (possible) second-order quantum phase transition(s) between the CM phase and the paramagnet ?

0n

Quantum fluctuations in U(1) theory for collinear antiferromagnets

Berry PhasesBerry Phases

iSe A

A

Area of triangle formed by , , and 1

2 an arbitrary reference A

0

n n

n

2 mod 4A

A

Berry PhasesBerry Phases

Area of triangle formed by , , and 1

2 an arbitrary reference A

0

n n

n

iSe A

A

2 mod 4A

A

Theory for quantum fluctuations of Neel state

S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990).

g0

2,

This is compact QED in 2+1 dimensions with

static charges 2 on two sub

1exp

lattices

cos

.

2a a a a aaa

Z dA A A i S Ae

S

At large g , we can integrate z out, and work with effective action for the Aa

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).

For S=1/2 and large e2 , low energy height configurations are in exact one-to-one correspondence with dimer coverings of the square lattice

2+1 dimensional height model is the path integral of the Quantum Dimer Model

There is no roughening transition for three dimensional interfaces, which are smooth for all couplings There is a definite average height of the interface Ground state has bond order.

D. Rokhsar and S.A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988); E. Fradkin and S. A. Kivelson, Mod. Phys. Lett. B 4, 225 (1990))

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).

Two possible bond-ordered paramagnets for S=1/2

There is a broken lattice symmetry, and the ground state is at least four-fold degenerate.

Distinct lines represent different values of on linksi jS S

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).

g

J1 0 ; 0z

0 ; 0z 0 ; 0z

0 ; 0z

Topologically ordered phase described by Z2 gauge theory

g0

S=1/2

A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002)

Bond order in a frustrated S=1/2 XY magnet

2 x x y yi j i j i j k l i j k l

ij ijkl

H J S S S S K S S S S S S S S

g=

First large scale numerical study of the destruction of Neel order in a S=1/2 antiferromagnet with full square lattice symmetry

Use a sequence of simpler models which can be analyzed by duality mappings

A. Non-compact QED with scalar matterB. Compact QED with scalar matter

C. N=1: Compact QED with scalar matter and Berry phases D. theory

E. Easy plane case for N=2N

Nature of quantum critical point

Use a sequence of simpler models which can be analyzed by duality mappings

A. Non-compact QED with scalar matterB. Compact QED with scalar matter

C. N=1: Compact QED with scalar matter and Berry phases D. theory

E. Easy plane case for N=2N

Nature of quantum critical point

A. N=1, non-compact U(1), no Berry phases

C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981).

Use a sequence of simpler models which can be analyzed by duality mappings

A. Non-compact QED with scalar matterB. Compact QED with scalar matter

C. N=1: Compact QED with scalar matter and Berry phases D. theory

E. Easy plane case for N=2N

Nature of quantum critical point

Use a sequence of simpler models which can be analyzed by duality mappings

A. Non-compact QED with scalar matterB. Compact QED with scalar matter

C. N=1: Compact QED with scalar matter and Berry phases D. theory

E. Easy plane case for N=2N

Nature of quantum critical point

B. N=1, compact U(1), no Berry phases

Use a sequence of simpler models which can be analyzed by duality mappings

A. Non-compact QED with scalar matterB. Compact QED with scalar matter

C. N=1: Compact QED with scalar matter and Berry phases D. theory

E. Easy plane case for N=2N

Nature of quantum critical point

Use a sequence of simpler models which can be analyzed by duality mappings

A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry

phases D. theoryE. Easy plane case for N=2

N

Nature of quantum critical point

C. N=1, compact U(1), Berry phases

C. N=1, compact U(1), Berry phases

C. N=1, compact U(1), Berry phases

Use a sequence of simpler models which can be analyzed by duality mappings

A. Non-compact QED with scalar matterB. Compact QED with scalar matter

C. N=1: Compact QED with scalar matter and Berry phases D. theory

E. Easy plane case for N=2N

Nature of quantum critical point

Identical critical theories for S=1/2 !

Use a sequence of simpler models which can be analyzed by duality mappings

A. Non-compact QED with scalar matterB. Compact QED with scalar matter

C. N=1: Compact QED with scalar matter and Berry phases D. theory

E. Easy plane case for N=2N

Nature of quantum critical point

Identical critical theories for S=1/2 !

D. , compact U(1), Berry phasesN

E. Easy plane case for N=2

g0

Critical theory is not expressed in terms of order parameter of either phase

S=1/2