THE ISING PHASE IN THE J1-J2 MODEL

Valeria Lante and Alberto Parola

OUTLINE:

introduction to the model

our aim

analytical approach

numerical approach

Conclusions

the future

the model motivation phase diagram{

non linear sigma model

Lanczos exact diagonalizations

What is the J1-J2 model ?

J1

J2

INTRODUCTION:

Why the J1-J2 model ?

? Can quantum fluctuations stabilize a disordered phase in spin systems at T=0 ?

relevance of low dimensionalityrelevance of small spin

spin systems: symmetry breaking (magnetization) (no 1D for Mermin-Wagner theorem)T = 0

Simple quantum spin model: Heisenberg model (J2=0)< order parameter > ≠ 0 at T=0 in 2D

Frustration may enhance quantum fluctuations J1-J2 model

~0.4 ~0.6J2/J1

Néel phase Collinear phaseParamagnetic phase

T=0

Connection with high temperature superconductivity

AF

0.04 0.05

M ? SCT = 0

Holes moving in a spin disordered background (?)

It is worth studying models with spin liquid phases J1-J2 model

Vanadate compounds

VOMoO4

Li2VOSiO4

Li2VOGeO4

2D Heisenberg model at T=0 (J2=0)

Some definitions

SSi

i i { i

= 1 i +

-1 i -Néel state

GS {= classical (S→)

≠quantum

{m /Ss ~0.6quantum (S=1/2)

=1classical

Stot Sz

PHASE DIAGRAM OF THE J1-J2

MODEL

Classical (S) ground state (GS) at T=0

classical energy minimized by if J(q) is minimum

S = e cos(q·r) + e sin(q·r) r 21

0.5J2/J1

J2/J1< 0.5: J(q) minimum at q=()J2/J1> 0.5: two independent AF sublattices *J2/J1= 0.5: J(q) minimum at q=(qx) and q=(qy)

* thermal or quantum fluctuations select a collinear phase (CP) with q=(0) or q=(0)

Quantum ground state at T=0

~0.4 ~0.6J2/J1

O(3) X Z2O(3)

broken symmetries

m ≠ 0s

m ≠ 0s+

m ≠ 0s-

n + L / S+ +

n + L / S- --n + L / S+ +

-n + L / S- -

= n · n ≠ 0+ -

==VBC: valence bond

crystal RVB SL: resonating

valence bond spin liquid

| RVB > = A(C )|C >

Ci

ii

C = dimer coveringi

no long-ranged orderno SU(2) symmetry breakingno long-ranged spin-spin

correlations

dimer = 1/ √2 ( |> -|

VBC = regular pattern of singlets at nearest neighbours: dimers or plaquettes

long-ranged dimer-dimer or plaquette-plaquette orderno SU(2) symmetry breakingno long-ranged spin-spin

correlations

OUR AIM:

~0.6J2/J1

< > ≠ 0< n > ≠ 0

collinear

< > = 0< n > = 0

“disorder”

~0.6J2/J1

< > ≠ 0< n > ≠ 0

collinear

< > ≠ 0< n > = 0

“Ising”

< > = 0< n > = 0

“disorder”

?

ANALITYCAL APPROACH :

2D Quantum model at T=0

2+1 D Classical model at Teff ≠0

Haldane mapping

Non Linear Sigma Model method for =J2/J1 > 1/2

I. The partition function Z is written in a path integral representation on a coherent states basis.

II. For each sublattice every spin state is written as the sum of a “Néel” field and the respective fluctuation.

III. In the continuum limit, to second order in space and time derivatives and to lowest order in 1/S, Z results:

= n · n + -

checks:

classical limit (S → ∞ )

saddle point approximation for large : n = n + n+- +- -+0

static and homogeneous

same results of spin wave theory

Collinear long range ordered phase

NUMERICAL APPROACH:

Lanczos diagonalizations:

On the basis of the symmetries of the effective model, an intermediate phase with <n > = 0 and finite Ising order

parameter <> ≠ 0 may exist.+-

It can be either a:VB nematic phase, where bonds display

orientational orderingVBC ( translational symmetry breaking)

Analysis of the phase diagram for values of around 0.6

for a 4X4 and a 6X6 cluster

Lowest energy states referenced to the GSordered phases and

respective degenerate states

collinear(0,0)s S=0(0,0)d S=0(0,) S=1(,0) S=1

{columnar

VBC

(0,0)s S=0(0,0)d S=0(0, ) S=0(,0) S=0

{plaquette

VBC

(0,0)s S=0(0) S=0(0, ) S=0(, ) S=0

{conclusions:

0.60 :(0,0)s and (0,0)d singlets quasi degenerate → Z2 breaking

0.62 :(0, ) S=0 higher than (0, ) S=1 →no columnar VBC

() S=0 higher than the others →no plaquette VBC0.62 triplet states are gapped

4X4

6X6

Order Parameter

0.6 <<0.7: |s> and |d> quasi-degenerate s> + |d>)/√2 breaks Z2 Ôr = Ŝr · Ŝr+y - Ŝr · Ŝr+x

lim < Ôr > ≠ 0 and |s> and |d> degenerate (N → ∞) Z2 symmetry breaking

< Ôr >

Px

Py

conclusions:

0.60 : Py compatible with a disordered configuration0.60 : Px Px for Heisenberg chainsAsgrows Py 0 : vertical tripletscollinear phase

< Ôr > ≠ 0

Structure factor

S(k) = Fourier transform of the spin-spin correlation function

Blue (cyan) triangles: S(k) on the lowest s-wave (d-wave)

singlet for a 4x4 cluster. Red (green) dots: The same for a 6x6

cluster.

= (0,0) M= (0,) X= (,)

conclusions:

S(k) on |s > S(k) on |d > same physics

0.70 : S(,0) grows with size collinear order

0.600.62 : S(k) flat + no size dependence

0.62<towards transition to collinear phase

numerical data fitted by a SW function except at single points

From the symmetries of the non linear sigma model:

● At T=0 possibility of :

CONCLUSIONS:

Isingdisorder collinear

ISING PHASE = VB nematic phase

Isingdisorder collinear

? 0.62

< > ≠ 0< n > = 0

< > ≠ 0< n > ≠ 0

< > = 0< n > = 0

The Lanczos diagonalizations at T = 0

● Ising phase for ? < < 0.62

● ~ 0.60: collection of spin chains weakly coupled in the transverse

direction.

THE FUTURE:

About the J1-J2 model on square lattice

Monte Carlo simulation of the NLSM action

Numerical analysis (LD) of the phase:

looking for a chiral phase: Ŝr · (Ŝr+y Ŝr+x)

About the J1-J2 model on a two chain ladder

Numerical analysis (LD) of the phase diagram

“novel” phase diagram proposed by Starykh and Balents PRL (2004)