Schrieffer-Wolff transformation for quantum many-body systems

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.

• The paper introduces a rigorous formalization of the SW transformation to derive low-energy effective Hamiltonians in quantum many-body systems.
• It develops a systematic perturbative expansion with diagrammatic techniques to compute high-order corrections for accurate simulations.
• The work shows that the transformation preserves locality in quantum spin lattices, underpinning its significance in quantum computing and condensed matter research.

Overview of the Schrieffer-Wolff Transformation for Quantum Many-Body Systems

The paper entitled "Schrieffer-Wolff transformation for quantum many-body systems" by Sergey Bravyi, David P. DiVincenzo, and Daniel Loss provides a comprehensive analysis and generalization of the Schrieffer-Wolff (SW) transformation, a critical component in the degenerate perturbation theory. This transformation is particularly relevant for systems where a low-energy effective Hamiltonian is needed for an accurate description of the dynamics by eliminating the high-energy degrees of freedom. The authors focus on both the theoretical formalism and the practical applications, especially within quantum spin lattices.

Key Contributions

1. Formalization of SW Method: The paper begins by meticulously defining the Schrieffer-Wolff transformation in terms of a unitary operation that maps a Hilbert space's low-energy subspace to its high-energy counterpart. The authors provide a detailed proof of its existence under the condition that the subspaces are non-orthogonal, exhibiting a rigorous theoretical foundation for the method.
2. Perturbative Expansion: A systematic perturbative expansion of the SW transformation is developed, complete with a diagrammatic technique to calculate high-order corrections. This aspect of the work is crucial for practical computations where low-order approximations are insufficient.
3. Application to Spin Systems: The authors extend the application of the SW transformation to quantum spin lattices with short-range interactions. They demonstrate its utility by showing the transformation's ability to map a local Hamiltonian in the high-energy regime to a local effective Hamiltonian in the low-energy regime while maintaining a form of unitary equivalence.
4. Linked Cluster Theorem: An important result within the paper is the proof of the linked cluster theorem in the context of the SW transformation. This theorem asserts that the effective low-energy Hamiltonian only involves interactions among connected clusters of spins, thereby preserving the notion of locality crucial for the practical application of the theory in large systems.
5. Numerical and Computational Techniques: By leveraging both analytical and numerical techniques, the authors provide tools for deriving the SW series and effective Hamiltonians. These techniques can be readily implemented in computer programs to aid in simulations and practical calculations of quantum many-body systems.

Implications and Future Directions

The work presented has profound implications for the study and simulation of quantum many-body physics. In practice, the SW transformation enables the simplification of complex systems by reducing the complexity of interaction terms, which is significant in fields such as condensed matter physics and quantum computing.

Furthermore, the systematic approach and rigorous theoretical underpinning make the results broadly applicable, potentially impacting the development of simulation algorithms for quantum mechanics and aiding in the design of quantum devices where local interactions dominate system behavior.

Future research stemming from this work could involve exploring the limits of the SW transformation in more complex systems, such as those with long-range interactions, and investigating the interplay of this transformation with modern quantum computing techniques and algorithms.

In summary, the paper presents a robust formal treatment of the Schrieffer-Wolff transformation, extending its utility in studying quantum many-body systems, particularly in maintaining locality in effective Hamiltonian models—a critical requirement in the realistic simulation and understanding of quantum materials and devices.