Abstract

Exactly solvable models are essential in physics. For many-body spin-1/2systems, an important class of such models consists of those that can be mapped to free fermions hopping on a graph. We provide a complete characterization of models which can be solved this way. Specifically, we reduce the problem of recognizing such spin models to the graph-theoretic problem of recognizing line graphs, which has been solved optimally. A corollary of our result is a complete set of constant-sized commutation structures that constitute the obstructions to a free-fermion solution. We find that symmetries are tightly constrained in these models. Pauli symmetries correspond to either: (i) cycles on the fermion hopping graph, (ii) the fermion parity operator, or (iii) logically encoded qubits. Clifford symmetries within one of these symmetry sectors, with three exceptions, must be symmetries of the free-fermion model itself. We demonstrate how several exact free-fermion solutions from the literature fit into our formalism and give an explicit example of a new model previously unknown to be solvable by free fermions.

Highlights

  • Solvable models provide fundamental insight into physics without the need for difficult numerical methods or perturbation theory

  • This reduces the problem of solving the n-spin system over its full 2n-dimensional Hilbert space to one of solving a single-particle system hopping on a lattice of O(n) sites

  • These fermions are resolved as nonlocal Pauli operators in the spin picture, and the nonlocal nature of this mapping may suggest that finding generalizations to this mapping for more complicated spin systems is a daunting task

Read more

Summary

Introduction

Solvable models provide fundamental insight into physics without the need for difficult numerical methods or perturbation theory. In the particular setting of many-body spin-1/2 systems, a remarkable method for producing exact solutions involves finding an effective description of the system by noninteracting fermions This reduces the problem of solving the n-spin system over its full 2n-dimensional Hilbert space to one of solving a single-particle system hopping on a lattice of O(n) sites. They become universal for quantum computation with the introduction of non-matchgates such as the SWAP gate [17, 18], certain measurements and resource inputs [19, 20], and when acting on nontrivial circuit geometries [21] These circuits share an interesting connection to the problem of counting the number of perfect matchings in a graph, which is the context in which they were first developed [22,23,24,25]. We expect that further investigation of the graph structure of manybody Hamiltonians will continue to yield important insights into their physics

Summary of Main Results
Frustration Graphs
Majorana Fermions
Fundamental Theorem
Symmetries
Orientation and Full Solution
Small Systems
The Kitaev honeycomb model
Frustrated Hexagonal Gauge 3D Color Code
Sierpinski-Hanoi model
Proof of Theorem 1
Proof of Theorem 2
Discussion
Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.