bibkey: beccaria2012staggered authors: Matteo Beccaria; Christian Hagendorf year: 2012 title: “A staggered fermion chain with supersymmetry on open intervals” doi: 10.1088/1751-8113/45/36/365201 url: https://arxiv.org/abs/1206.4194v2 claim: “Section 3.2.3 conjectures the endpoint-density identity for staggering II on every open M1 chain with N = 3n.” strata_touched:

• D5/S3/Quantum/SpinChains/SupersymmetricFermion/JordanWigner
• D5/S3/Quantum/SpinChains/SupersymmetricFermion/HardCoreModel
• D5/S3/Quantum/SpinChains/StaggeredM1EndpointDensity license: citation-only triage: anchor

A staggered fermion chain with supersymmetry on open intervals

M. Beccaria and C. Hagendorf, J. Phys. A 45 (2012) 365201, arXiv:1206.4194v2. Section 2.1, printed page 2, states the canonical anticommutation relations $\{c_i,c_j\}=\{c_i^{†},c_j^{†}\}=0$ and $\{c_i,c_j^{†}\}={\delta }_{ij}$.

Section 2.1, printed page 3, specifies:

nearest-neighbour exclusion: at most one of two adjacent sites may be occupied.

We shall work with free boundary conditions what is equivalent to add two inaccessible but empty sites at j = 0 and j = N + 1.

The hard-core configurations use the existing admissible-word predicate AdmissibleCount.Adm; ForbiddenNeighbourDeterminant.adm_iff_no_adjacent_true identifies it with the stated nearest-neighbour exclusion.

The dressed annihilator is $d_j=P_{j-1}{c}_j{P}_{j+1}$, with $P_j=1-c_j^{†}{c}_j$. In the hard-core realization, HardCoreModel.annihilator_number identifies $c_j^{†}{c}_j$ with the occupation diagonal, including zero at inaccessible sites. The supercharge is $Q={\sum }_{j=1}^N{\lambda }_j{d}_j$ and the Hamiltonian is $H=Q{Q}^{†}+Q^{†}Q$. Equation (2) gives staggering II as ${\lambda }_{3p-2}=y$, ${\lambda }_{3p-1}=y$, ${\lambda }_{3p}=1$.

Section 3.2.3, printed page 13, states:

As similar pattern is found by probing if a particle is present on the last site. The data is consistent with ${\rho }_n^{(N)}(y)=y^{-2}{\rho }_n^{(1)}(y)$ for finite $n\le 8$. We conjecture this to hold for arbitrary system sizes.

Here $N=3n$, $n\ge 1$, and the density is the normalized expectation of site occupation in the zero-energy ground state. The formal claim quantifies over every nonzero zero-energy state for real $y\ne 0$. It therefore contains the stated ground-state relation without using existence or uniqueness. The source’s cohomology statement about the unique ground state cites its reference [14]; that statement is not a premise of the endpoint argument.

Verified locator

• DOI: https://doi.org/10.1088/1751-8113/45/36/365201
• Source: https://arxiv.org/abs/1206.4194v2