[Paper Review] Supersymmetry, lattice fermions, independence complexes and cohomology theory

Liza Huijse, Kareljan Schoutens|arXiv (Cornell University)|Mar 4, 2009

Topological and Geometric Data Analysis1 references3 citations

TL;DR

This paper establishes a one-to-one correspondence between quantum ground states of a supersymmetric lattice fermion model on a 2D square lattice and cohomology classes of its independence complex. The key result proves a conjecture by P. Fendley, showing that the dimension of cohomology at grade $ n $ equals the number of rhombus tilings with $ n $ rhombi, revealing sub-extensive ground state entropy due to exponential tiling growth.

ABSTRACT

We analyze the quantum ground state structure of a specific model of itinerant, strongly interacting lattice fermions. The interactions are tuned to make the model supersymmetric. Due to this, quantum ground states are in one-to-one correspondence with cohomology classes of the so-called independence complex of the lattice. Our main result is a complete description of the cohomology, and thereby of the quantum ground states, for a two-dimensional square lattice with periodic boundary conditions. Our work builds on results by J. Jonsson, who determined the Euler characteristic (Witten index) via a correspondence with rhombus tilings of the plane. We prove a theorem, first conjectured by P. Fendley, which relates dimensions of the cohomology at grade n to the number of rhombus tilings with n rhombi.

Motivation & Objective

• To fully characterize the quantum ground state structure of a supersymmetric lattice fermion model on a 2D square lattice with periodic boundary conditions.
• To establish a rigorous mathematical correspondence between the model's ground states and cohomology classes of the independence complex of the lattice.
• To prove a conjecture by P. Fendley relating the dimension of cohomology at grade $ n $ to the number of rhombus tilings with $ n $ rhombi.
• To resolve discrepancies in the one-to-one correspondence between tilings and cohomology elements for special configurations (all-zero and all-dot states).

Proposed method

• Mapping the Hilbert space of the supersymmetric lattice fermion model to the chain complex of the independence complex of the lattice.
• Using the fact that quantum ground states correspond exactly to cohomology classes of this complex, leveraging supersymmetry to ensure zero-energy states are precisely the cohomology representatives.
• Applying the 'tic-tac-toe' lemma from [16] to decompose the cohomology structure and analyze configurations with periodic boundary conditions.
• Computing the number of rhombus tilings with $ n $ rhombi via combinatorial enumeration, particularly focusing on tilings with periodic boundary conditions in two directions.
• Analyzing special cases where all sites are occupied or empty (all-zero or all-dot configurations), which can map to multiple cohomology classes under certain periodicity conditions.
• Deriving an expression for the discrepancy $ riangle = N^{(a)} - t^{(a)} $, where $ N^{(a)} $ is the number of cohomology elements from anomalous configurations and $ t^{(a)} $ is the number of corresponding tilings, to resolve the mismatch and confirm the conjecture.

Experimental results

Research questions

• RQ1 How are the quantum ground states of the supersymmetric lattice fermion model on a 2D torus related to the cohomology of the independence complex?
• RQ2 What is the precise relationship between the dimension of cohomology at grade $ n $ and the number of rhombus tilings with $ n $ rhombi?
• RQ3 Why do configurations with all zeroes or all dots lead to discrepancies in the tiling-to-cohomology correspondence, and how can this be resolved?
• RQ4 What is the quantitative behavior of the ground state degeneracy in the presence of periodic boundary conditions, particularly when the system size satisfies $ m = 3n $?
• RQ5 How does the discrepancy $ riangle $ between cohomology elements and tilings depend on the system’s topological parameters $ r $, $ s $, and $ m $?

Key findings

• The dimension of the cohomology at grade $ n $ is exactly equal to the number of rhombus tilings with $ n $ rhombi, confirming Fendley’s conjecture.
• For $ ar{u} = (m, -m) $ with $ m eq 3n $, the discrepancy $ riangle = 2 $, meaning two cohomology elements correspond to no tiling, resolving the mismatch in the one-to-one correspondence.
• When $ ar{u} = (3n, -3n) $, the number of tilings $ t^{(a)} $ for all-zero and all-dot configurations is given by $ t^{(a)} = 6 imes ext{sum over binomial coefficients with modulo-3 conditions} $, and the discrepancy $ riangle $ depends on the values of $ r $, $ s $, and $ r-s $ modulo 6.
• The final expression for $ riangle $ is compactly given by equation (46), showing $ riangle = -4, 2, 4, -2 $ depending on the parity of $ r+s $ and the value of $ r-s mod 6 $, which fully resolves the anomaly.
• The exponential growth of the number of rhombus tilings with system size implies that the quantum model exhibits sub-extensive ground state entropy.
• The cohomology structure is fully characterized via combinatorial tiling enumeration, and the discrepancy analysis confirms the validity of the tiling-cohomology correspondence except in special periodic cases.