Duality in spin systems via the $SU(4)$ algebra

Abstract

We provide several examples and an intuitive diagrammatic representation demonstrating the use of two-qubit unitary transformations for mapping coupled spin Hamiltonians to simpler ones and vice versa. The corresponding dualities may be exploited to identify phase transition points or to aid the diagonalization of such Hamiltonians. For example, our method shows that a suitable one-parameter family of coupled Hamiltonians whose ground states transform from an initially factorizing state to a final cluster state on a lattice of arbitrary dimension is dual to a family of trivial decoupled Hamiltonians containing local on-site terms only. As a consequence, the minimum enery gap (which determines the adiabatic run-time) does not scale with system size, which facilitates an efficient and simple adiabatic preparation of e.g. the two-dimensional cluster state used for measurement-based quantum computation.

13 pages, 7 figures, to appear in Journal of Statistical Mechanics: Theory and Experiment

References in corpus (7)

• Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension
• Identifying phases of quantum many-body systems that are universal for quantum computation
• Adiabatic Preparation of Topological Order
• Fate of the cluster state on the square lattice in a magnetic field
• Realizing all $so(N{)}_1$ quantum criticalities in symmetry protected cluster models
• Adiabatic preparation without Quantum Phase Transitions
• Adiabatic Rotation, Quantum Search and Preparation of Superposition States