Transcript Document

THE ISING PHASE IN THE J1-J2 MODEL
Valeria Lante and Alberto Parola
OUTLINE:
introduction to the model
our aim
{
the model
motivation
phase diagram
analytical approach
non linear sigma model
numerical approach
Lanczos exact diagonalizations
Conclusions
the future
INTRODUCTION:
What is the J1-J2 model ?
J1
Why the J1-J2 model ?
T= 0
?
J2
spin systems: symmetry breaking (magnetization)
(no 1D for Mermin-Wagner theorem)
Can quantum fluctuations stabilize
a disordered phase in spin systems at T=0 ?
relevance of low dimensionality
relevance of small spin
Simple quantum spin model: Heisenberg model (J2=0)
< order parameter > ≠ 0 at T=0 in 2D
Frustration may enhance quantum fluctuations
J1-J2 model
T=0
~0.4
~0.6
Néel phase
J2/J1
Collinear phase
Paramagnetic phase
Connection with high temperature superconductivity
T=0
AF
0.04
M?
SC
0.05

Holes moving in a spin disordered background (?)
It is worth studying models with spin liquid phases 
model
J 1 -J 2
Vanadate compounds
Li2VOGeO4
Li2VOSiO4
VOMoO4
Some definitions
Néel state

SS
i
i
i
 =
i
{
1 i +
-1 i  -
Stot S z
2D Heisenberg model at T=0 (J2=0)
GS
m s /S
{
{
=  classical (S→)
≠  quantum
= 1 classical
~ 0.6 quantum (S=1/2)
PHASE DIAGRAM
OF THE J1-J2
MODEL
Classical (S) ground state (GS) at T=0
classical energy minimized by
if J(q) is minimum
0.5
Sr= e cos(q·r) + e sin(q·r)
1
2
J2/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=()
Quantum ground state at T=0
broken symmetries
O(3)
O(3) X Z 2
+
ms ≠ 0
 = n+ · n- ≠ 0
ms ≠ 0
ms ≠ 0
~0.4
~0.6
J2/J1
-n - + L - / S
n + + L + / S
-n+  + L+ / S
n -  + L- / S
=
VBC: valence bond
crystal
RVB SL: resonating
valence bond spin liquid
VBC = regular pattern of singlets at
nearest neighbours: dimers or
plaquettes
dimer = 1/ √2 ( |> -|
| RVB > =  A(C )|C >
C
i
i
i
Ci = dimer covering
long-ranged dimer-dimer or
plaquette-plaquette order
no SU(2) symmetry breaking
no long-ranged spin-spin
correlations
no long-ranged order
no SU(2) symmetry breaking
no long-ranged spin-spin
correlations
OUR AIM:
<>=0
<n>=0
<>≠0
<n>≠0
“disorder”
collinear
J2/J1
~0.6
<>=0
<n>=0
<>≠0
<n>=0
<>≠0
<n>≠0
“disorder”
“Ising”
collinear
?
~0.6
J2/J1
ANALITYCAL APPROACH :
Non Linear Sigma Model method for  =
2D Quantum
model at T=0
Haldane mapping
J2/J1 > 1/2
2+1 D Classical
model at Teff ≠0
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 → ∞ )
static and
homogeneous
saddle point approximation for large :
n+ = n 0+ + n +
-
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 ordering
VBC ( 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 GS
ordered phases and
respective degenerate states
4X4
{
{
{
collinear
columnar
VBC
6X6
plaquette
VBC
conclusions:
(0,0)s
(0,0)d
(0,)
(,0)
S=0
S=0
S=1
S=1
(0,0)s
(0,0)d
(0, )
(,0)
S=0
S=0
S=0
S=0
(0,0)s
(0)
(0, )
(, )
S=0
S=0
S=0
S=0
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 VBC
0.62 triplet states are gapped
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 configuration
0.60 : Px Px for Heisenberg chains
As grows Py  0 : vertical triplets collinear
phase
< Ôr > ≠ 0
Structure factor
S(k) = Fourier transform of the spin-spin correlation function
= (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
0.600.62 : S(k) flat + no size
dependence
0.62< towards transition to
collinear phase
numerical data fitted by a SW
function except at single points
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.
CONCLUSIONS:
From the symmetries of the non linear sigma model:
disorder
Ising
collinear
● At T=0 possibility of :
The Lanczos diagonalizations at T = 0
<>=0
<n>=0
●
Ising phase for ? < < 0.62
●
<>≠0
<n>=0
disorder
Ising
?
<>≠0
<n>≠0
collinear
0.62
~ 0.60: collection of spin chains weakly coupled in the transverse direction.
ISING PHASE = VB nematic phase
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)