Yao-Lee Model

§1. Yao-Lee model
§1.1. Hamiltonian
§1.1.1. Good quantum number:onsite total spin
§1.2. Subspaces
§1.3. as Pauli matrices
§1.4. Perturbative Treatments
§1.5. Majorana representation
§1.5.1. Kitaev's Majorana representation

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1 Yao-Lee model

ref: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.107.087205

Yao-Lee model is an exactly-solvable model that exhibits spin liquid in 2D. It is somewhat an extension to Kitaev honeycomb model.

It is defined on a decorated-honeycomb lattice

Figure 1: the decorated-honeycomb lattice, where labels the triangle (regardless of the orientation, however nearest neighbor triangulars always has different orientation), labels the sites in triangles, and the type of inter-triangle bonds are labeled with

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1.1 Hamiltonian

where we define

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1.1.1 Good quantum number:onsite total spin

We would prove that

Proof. First

Next the crossing terms, lets ignore the site index for simplicity where ¹

Sum this with and

⁠ 

Proof.

we know summing over one index of gives 0, so the and its original term vanishes

epsilon terms vanishes… so

It goes similarly with

The chirality term would takes more effort, let's write the full expasion first.

I'm tired and I think the summation of epsilon will lead us to the right way… ⁠ 

That tells us, the total spin of each site is a good quantum number.

Remark

Remember in AKLT, the total spin of each site is also conserved, thus we can project the local site into different spin sectors.

Remark

Parton - the right way to solve frustration

Frustation is namely the case where two constraints can't be satisfied at the same time. If we divide the single site into seperate degrees of freedom, we can solve the frustration by satisfying each parton degree of freedom.

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1.2 Subspaces

With the conserved onsite total spin quantum number, we can divide the onsite Hilbert space into different subspaces.

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1.3 as Pauli matrices

obeys algebra and Clifford algebra.

I check this by CAS using

QuantumAlgebra.jl

QuantumAlgebra.jl

See algebra.jlalgebra.jl

This implies are indeed Pauli matrices within the subspace.

The four states can be labelled as where , and

Remark

What is the orbital meaning of ?

In the remaining spin sector, , doesn't obey Pauli algebra.

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1.4 Perturbative Treatments

Let's assume from now, then all sites should be in the subspace.

In this subspace,

could be exactly solvable

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1.5 Majorana representation

This is somewhat more difficult to understand, need to delve into Kitaev's famous paper a little bit…

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1.5.1 Kitaev's Majorana representation

ref:https://arxiv.org/abs/cond-mat/0506438

Definition of Majorana Fermions could deviate in a minus sign, so we define here,

We represent two Fermion modes in 4 Majorana modes, the 4 Majorana Fermions are called

A simple observation is if we treat Majorana Fermions as independent degrees of freedom, the whole Hilbert space dimension is , larger than the original 4 Fermion modes . To deal with this, we need to impose a constraint on the Majorana modes.

Fermion Parity:

where is the number of Fermions in the system, or equivalently, the number of operators.

Any Hamiltonian should commute with , , since could never have odd number of Fermions.

The Majorana representation shall always satisfy the Fermion Parity.

Compute

This is the Fermion Parity operator under Majorana representation.

The whole Parity operator is

This can be regarded as a gauge transformation for the group .

We follow the notation in Kitaev's paper, denote the extended Hilbert space as , the physical Hilbert space is . The action is a cohomology class in .

The Pauli operators can then be written in space,

  1. why not begin with this formula…