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)