Competing Phases of the J1 and J1-J2 Spin ½ Quantum Heisenberg Anti-Ferromagnet on the Kagome Lattice (A Variational Monte Carlo Approach)

By

Yasir Iqbal

Phd Student under Dr. Didier Poilblanc

Strongly Correlated Systems GroupLaboratoire de Physique Theorique CNRS & Universite Paul Sabatier Toulouse, France

OUR GROUP

OUTLINE OF THE TALK:

What makes the Kagome geometry so special ?

Quantitative measure and manifestation of frustration

What do experimental probes point out to ?

Mean Field Theory for handling spin disordered systems

c Gutwiller Projection

Construction & properties of parent LR-RVB states for NN model

VBS ordering - Dimerization

J1-J2 model – Construction, Properties, and VBS ordering.

What protects the gapless nature of the excitations ?

Bravais Lattice vectors: ,

Reciprocal Lattice vectors: ,

A

BC

A

A

√3× √3

q=0

α= 0

β=120°

γ=240°

Spin – ½ HAF on the Kagome lattice: A Classical Treatment

By treating the spins Si at each site as ordinary classical vectors of length 1/2

PROVIDES A MEASURE OF FRUSTRATION

Features of Classical Ground state:

1.) The angle between neighboring spins is 120°.

2.) Infinite degeneracy of the classical GS. Strong Frustration

3.) High geometric frustration of the Kagome lattice leads to unsatisfied bonds leading to increase of classical GS energy, THIS INCREASE OF ENERGY SERVES AS A QUANTITATIVE MEASURE OF FRUSTRATION. For Kagome and Triangular lattices EGS / bond = -s2/2 = -0.125 as compared to non frustrated lattices such as square and honeycomb for which EGS / bond = -1s2 = - 0.25.

4.) Drastic reduction of order parameter for the Kagome lattice as compared to bipartite unfrustrated lattices such as square and honeycomb. Cont…

Lattice

Coordination No

mquan/mclass

Square 4 0.635

Hexagonal

3 0.558

Triangle

6 0.386

Kagome 4 0.000

Comparison of the order parameter ‘m’ of S=1/2 HAFM(Obtained by finite size extrapolation)

m = [(1/N2)∑|<Si Sj>|]1/2 Staggered magnetizationFor Bi-partite systems

All theoretical analysis points towards a Quantum Paramagnetic state T=0,i.e. a Quantum disordered system such as “Quantum spin liquid”.

Experiment: A structurally perfect Spin ½ QAF – Herbertsmithite

Conclusions of 17O NMR, ESR & μSR studies:

C1.) No sign of ordering – magnetic or otherwise down to at least 50 mK despite a sizeable NN Cu2+ AF interaction of the order of J ≈ 200 K,

2.) A vanishing spin gap Long Range RVB

LIQUID BEHAVIOR

3.) Frozen magnetic moments and spin glass behavior also not observed.

44.) Spatial isotropy of the in-plane magnetic interaction.

5.) Intersite mixing defect of Cu and Zn species, non-magnetic Zn2+ substitute

Cu in the kagome plane leading to dilution of the magnetic kagome lattice, and to preserve stoichiometry the same amount of magnetic Cu2+ then occupies interplane sites acting like a bridge between kagome layers, thereby destroying the essential 2-D nature of the lattice.

Heisenberg Model (Half-filling + Large on-site repulsion limit of Hubbard Model)

(J>0, AF coupling)Mean Field - Approx

Replace

Solve HMFto get GS

The are chosen such that the self-consistency condition is satisfied.

PROBLEM: This approach fails for spin disordered systems since SOLUTION: Go over into the Fermionic representation by introducing spin ½ charge neutral spinon operators , in terms of which the spin operators are expressed as bilinears in , as

In terms of the spinon operators the Hamiltonian ‘H’ reads as,

Some features of the Fermionic representation:

1.) Enlargement of the Hilbert space: In contrast to the standard mean field treatment, here not only the Hamiltonian but also the Hilbert space is changed (enlarged), from 2 states per site to 4 states per site. Valid only in the subspace

of one fermion per site

Need to impose the local constraint, &

to go back to the physically realistic Hilbert space.

Zeroth order Mean Field treatment: (All fluctuations neglected)

Replacing the constraint

by its Ground state averageand ignoring its fluctuations. Enforced by adding a site dependent and time independent Lagrangian multiplier term, in the Hamiltonian.

Replacing the operator by its GS average value i.e. , and again ignoring its fluctuations(both phase and amplitude).

We get the following zeroth order, quadratic, tight binding spinon Hamiltonian describing free spinons, mathematically represented by

Cont ….

Here, the in the above Hamiltonian must satisfy the self-consistency condition

Important:

The on site, site dependent chemical potential is chosen such that condition is satisfied by the Mean field ground state.

The is called the ansatz of the spin liquid state

The and do not change under a spin – rotation transformation, and so also the mean field ground state.

The predictions of zeroth order mean field theory can only be trusted forstable mean field states(in which the fluctuations are weak) and the interactionsinduced by the fluctuations vanish at low energies, i.e. the interactions are irrelevant perturbations.

Gutzwiller Projection – Enforcing single occupancy of sites

Ground State Not a valid wave state, since it has empty and doubly occupied sites

Enforcing one-fermionper site constraint

Via fluctuations of

(Valid spin wave function)

Projection of MF state to get WF:

--- is state with no f fermions i.e.

Connects the mean field ansatz(+its fluctuations)to physical spin WF.

--- Two mean field ansatz and related by the following gauge transformation,

give rise to the same physical spin wave function.

CONSTRUCTION OF A PARENT ‘LR-RVB’ STATE1.) -Flux state: Algebraic U(1) Linear Dirac Spin Liquid.

Ansatz:

, The magnitude of hoppings is spatially uniformover all links.

The only freedom is to then choose the sign of the hoppings, i.e. the fluxes through the plaquettes.Flux/plaquette =

For convenience we set ,the Hamiltonian is,

Electron hoppingproblem

--- on a site cluster

This state has the lowest energy amongtranslationally-invariant spin liquids

Time Reversal; -

Physical Properties of the -flux state:

1.) Spin – rotation symmetry: As the mean field ansatz does not depend on spin orientation, in the MF GS, every negative energy level is occupied by a spin up spin down fermion is also spin rotation invariant, and consequently also the physical spin wave function obtained by projection.

2.) Translational Invariance: The mean field ansatz is not invariant under translation in the x – direction by one Bravais lattice vector breaks translational invariance, but the physical spin wave function does not break translational symmetry since the translated ansatz is related to the original ansatz by a gauge transformation.

3.) Time Reversal Invariance: States with flux patterns of 0 and are time reversal or in other words spin reversal invariant, both the spin Hamiltonian and the mean field Hamiltonian are invariant under time reversal, the mean field ansatz flips its sign under time reversal.

BAND STRUCTURE CALCULATION

Writing the Hamiltonian in momentum representation:

Effecting the following transformation , we get as,

, with given by

where and for convenience we define,

& the Hamiltonian matrix now reads as,

Diagonalizing the above matrix and plotting the eigenvalues for the axis,we get the six-band structure.

Cont …

BAND STRUCTURE FOR THE - FLUX STATE

Low Energy excitations about these two points1.) The Fermi surfaces at half-filling are these two points, about which we have gapless spin excitations, which correspond to particle-hole excitations across these two Fermi points.2.) As the -flux state is a marginal mean field state, the striking results(spin ½ quasiparticle excitations) of zeroth order mean field theory cannot be trusted, the first order MF theory contains gapless Dirac fermions coupled to a U(1) gauge field.

2.) PROJECTED FERMI SEA STATE:

Ansatz:

,The magnitude and the sign of hoppings is spatially uniform over all links. No fluxthrough the system.This state will be henceforward called [0,0] state.

Setting , with sij=+1

Energy/site = -0.41200 J --- on a

site cluster

Time Reversal; -[π,0] [0,0]

This state has a lower energy than any ofthe chiral spin liquids(break time reversal)but its energy is higher than the flux state.

COMPARISON OF ENERGIES OF SPIN LIQUID STATES

Energies for all competing projected spin liquid Ground states.Energies are given per site in units of J.As is clearly seen the U(1) Dirac state - has the lowest energy,which is comparable to and lower than some other numerical estimates.

PARENTSTATE

INSTABILITY TOWARDS “VBS” ORDERING:(BY GIVING MASS TO FERMION FIELD BY PERTURBATIVE FLUCTUATIONS)

Physical Idea behind this proposal:

Given massless fermions, the system must ultimately try to break some symmetry by giving mass to the fermions.

Introducing Chirality Spin Solid – Via Dimerization of

The chiral mass term Mc is produced

by continuously deforming the state,to produce [ ] state, with and being flux through & hexagon resp.

,

[ , ](Chiral Spin liquid)Parent RVB

In a naïve unprojected MF, the chiral SL lowersthe energy as compared to parent state.---After projection, the fluctuations destroy the CSL.

Increases

Maximize the number of perfecthexagons, and increase the magnitude of on perfect hexagons while decreasing it on the defect triangles.

12 – SITE SUPERCELL DIMERIZATION:

Opening an energy gap:(Most efficient perturbation)

At the Dirac point, the mass perturbation to Dirac eqn is,

Energy cost to pay = J

To maximize the gap for given J , i.e.

Set of all matrices ‘M’ which anti-commute with

Open a gap =

Making the magnitudes of non-uniform.

Generated by

Dimerization of the system, hence changinga LR-RVB to a SR-RVB.

18 SITE SUPERCELL DIMERIZATION:

Physical Idea behind the proposal:

To maximize the number of perfecthexagons – i.e. hexagons with exactly 3 dimers.

RESONANCE

2nd order process in perturbation theory

LOWERING OF ENERGY

Proposal based on 1/N correctionto large N, mean field solution.

For every 18 sites there can be atmost one perfect hexagon.

COMPARISON OF ENERGIES OF PARENT AND VBS ORDEREDSTATES (for 12 and 18 site supercell ordering patterns)

Energies given are per site in units of J, calculated on a 12x12x3=432 site cluster with mixed periodic – anti periodic boundary conditions.

OBSERVATIONS:

1.) The system is extremely stable/robust against perturbative fluctuations which try to give the fermionic field any chiral/non-chiral mass term.

QUESTION:

What is the hidden symmetry that protects these gapless excitations?

18 site Dimer order

Uniform[0,0]12 site supercell

18 site Dimer order

Only hope left!, 36 SITE SUPERCELL

Some features:

It has two perfect hexagons – marked in pink.

It has one star - marked in green and yellow.

The effective pinwheel resonance which permutes the 12 spins on the star, vanishesprecisely for the NN Heisenberg model.

There are 3 centers of symmetry in this supercell.

About each center of symmetry, all the hoppingsconnected by rotations, and reflections have the same magnitude. (In all 6 classes of hoppings)The dashed lines on some of the bonds denoted negative hopping.

Energy per site in units of J, on a 36x4x4=576 site cluster, with mixed P-AP BC.

U(1) Dirac Spin Liquid (all ) = -0.42866(1) J

VBS Ordering (5% amplitude modulation of ) = -0.42860 J

Honey-comb Arrangemenent of Perfect Hexagons.

J1 – J2 Spin ½ QHAF on Kagome Lattice

Motivation:To add an effective pinwheel resonance term on the star Lowering of energy

Permutation of 12 spins on star (This term vanishes only for NN but is ≠ for NNN )

AFAF 2

U(1) Dirac SL Flux state (Parent RVB State)-- of original NN model.

(sets the unit of hopping)

(Sets the EnergyScale)

To be determined:

Flux pattern through the new plaquettes, formed by 2nd NN bonds i.e. sign pattern of .2

Optimized magnitude of for each J2 . 2

U(1) Dirac SL(parent RVB for NN) + PARENT LR-RVB STATE FOR J1 – J2 AF to be dimerized.

VANISHING OF PINWHEEL PROCESSES SHOWN FROM QUANTUM DIMER MODEL

The amplitude of the kinetic and potential pin wheel processes exactly vanishesat all orders for the NN Spin ½ QHAF on Kagome.Including a finite pinwheel term lifts a very special degeneracy of GS manifold.

NATURE OF DEGENERACY: (There is a degeneracy of the even and odd resonating pinwheels of the motif. )

FLUX PATTERN: (J1 – J2 model)

0

Ansatz for flux and signconvention for J1 – J2

---- Negative hopping Positive hopping

GENERIC BAND STRUCTURE FOR J1 – J2 MODEL

The matrix reads as: = J2 2

J2 2=0.3, =0.3518

Observations The double degenerate flat nature of two bands at E=2 is lost.

The presence of Dirac cones and the relativistic spectrum.

The position of Dirac cones(momentum values) remains fixed.

=0

BOND CLASSES+FLUXES: (J1 – J2 model)

U(1) DSL – BOND CLASSES NNN –BOND CLASSES

COMPARISON OF ENERGY OF REFERENCE SPIN LIQUID AND 36 SITE VBS ORDERED STATES FOR J1 – J2

Energy/Site in units of J, on a 36x4x4=576 site cluster with mixed P-AP B.C.

What protects gapless excitations in Spin Liquids ?

Quantum Spin Liquids are examples of Quantum Ordered states, i.e. which cannot becharacterized using broken symmetries and local order parameter.e.g. Free Fermion system Topology of its Fermi Surface is its Quantum Order.

For gapless mean field spin liquids Symmetry of the mean field ansatz.

Projective Symmetry Group(PSG)

If MF spin liquid is stable or marginal PSG is a universal property.

All physical properties Robust against perturbative fluctuations

The PSG of the Dirac U(1) Spin Liquid prevents the fermionic field from acquiringmass (i.e. Dimerization) via perturbative fluctuations.

The QUANTUM ORDER and the correspondingPSG protects gapless spinon excitations, even when they interact at all energy scales.