Schrieffer-Wolff transformation for quantum many-body systems

By Sergey Bravyi, David P. DiVincenzo, Daniel Loss. 2011

577 citations recorded by the source corpus.

Abstract The Schrieffer–Wolff (SW) method is a version of degenerate perturbation theory in which the low-energy effective Hamiltonian H eff is obtained from the exact Hamiltonian by a unitary transformation decoupling the low-energy and high-energy subspaces. We give a self-contained summary of the SW method with a focus on rigorous results. We begin with an exact definition of the SW transformation in terms of the so-called direct rotation between linear subspaces. From this we obtain elementary proofs of several important properties of H eff such as the linked cluster theorem. We then study the perturbative version of the SW transformation obtained from a Taylor series representation of the direct rotation. Our perturbative approach provides a systematic diagram technique for computing high-order corrections to H eff . We then specialize the SW method to quantum spin lattices with short-range interactions. We establish unitary equivalence between effective low-energy Hamiltonians obtained using two different versions of the SW method studied in the literature. Finally, we derive an upper bound on the precision up to which the ground state energy of the n th-order effective Hamiltonian approximates the exact ground state energy.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/11b3877a67e288ec480f4980e8ed2e432eeb073f

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In Atomic and Molecular Physics, and Optics

In Condensed Matter Physics

Citations in the corpus

Optomechanical creation of magnetic fields for photons on a lattice (2015, methodology): “[48] Sergey Bravyi, David P.”
Quantum memories at finite temperature (2014, methodology): “A similar model whose ground-state degeneracy depends on microscopic details (Bravyi et al., 2011b) is introduced by Chamon in Ref.”
Circuit quantum electrodynamics (2020, methodology): “A general approach to perturbation theory which we follow here is known as a SchriefferWolff transformation (Bravyi et al., 2011; Schrieffer and Wolff, 1966).”
QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems part I: spin chains (2016, background): “Such multi-spin operators are often times generated by the nested commutators typically appearing in higher-order terms of perturbative expansions, such as the Schrieffer-Wolff transformation [19, 20, 21] and the inverse-frequency expansion [22, 23].”
Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations (2018, methodology): “In the limit of strong interactions (small ), we derive the effective Hamiltonian via the Schrieffer-Wolff transformation [44,45].”
Tunable Coupling Scheme for Implementing High-Fidelity Two-Qubit Gates (2018, background): “j=1,2 gj ∆j (σ j σ − c − σ− j σ c ) ] [25].”