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arXiv:2208.03637

Algebraic-Dynamical Theory for Quantum Spin-1/2: the Two-spin limit and Implications for Lattice Models

Wenxin Ding, Chuanru Dai, Zhengfei Hu

Published Aug 7, 2022• 0 citations
arXivHTMLPDF

Abstract

Recently, an {\it algebraic-dynamical theory} (ADT) for strongly interacting many-body quantum Hamiltonians in W. Ding, arXiv: 2202.12082 (2022). By introducing the complete operator basis set, ADT proposes a generic framework for systematically constructing dynamical theories for interacting quantum Hamiltonians, using quantum entanglement as the organizing principle. In this work, we study exact ADT solutions of interacting two-spin problems which can be used as "free theories" for perturbation study on relevant solvable limits. Then we perform ADT perturbation calculations and obtain the correct ground state under the perturbation, which shows that the ADT framework is capable of constructing correct dynamical perturbation theories for strongly interacting quantum spin models. We also discuss the implication for relevant lattice models.

DOI
10.48550/arxiv.2208.03637

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Recently, an {\it algebraic-dynamical theory} (ADT) for strongly interacting many-body quantum Hamiltonians in W. Ding, arXiv: 2202.12082 (2022). By introducing the complete operator basis set, ADT proposes a generic framework for systematically constructing dynamical theories for interacting quantum Hamiltonians, using quantum entanglement as the organizing principle. In this work, we study exact ADT solutions of interacting two-spin problems which can be used as "free theories" for perturbation study on relevant solvable limits. Then we perform ADT perturbation calculations and obtain the correct ground state under the perturbation, which shows that the ADT framework is capable of constructing correct dynamical perturbation theories for strongly interacting quantum spin models. We also discuss the implication for relevant lattice models.
Perturbation theories provide valuable insights on quantum many-body systems. Systems of interacting particles, like electrons, are often treated perturbatively around exactly solvable Gaussian points. Systems of interacting qubits have gained increasing prominence as another class of models for quantum systems thanks to the recent advances in experimentally realizing mesoscopic quantum devices. Stabilizer states, innately defined on systems of qudits, have correspondingly emerged as another class of classically simulatable starting point for the study of quantum error-correcting codes and topological phases of matter in such devices. As a step towards analyzing more general quantum many-body problems on these platforms, we develop a systematic stabilizer perturbation theory in qubit systems. Our approach relies on the local Schrieffer-Wolff transformation, which we show can be efficiently performed through the binary encoding the Pauli algebra. As demonstrations, we first benchmark the stabilizer perturbation theory on the transverse field Ising chain in one dimension. The method is then further applied to Z2\mathbb{Z}_2Z2​ toric code on square lattice and kagome lattice to probe the tendency toward confinement for anyonic excitations.
A central problem in modern mathematical physics is to study interacting systems beyond the reach of perturbation theory. This thesis studies interacting quantum systems using the mathematical theory of operator algebras. In the first part of this thesis, we consider how to construct quantum many-body systems from one-particle systems. This problem has a well-established solution - namely, the second quantization (or the Gaussian functor). However, it seems difficult for the second quantization to go beyond generalized free field constructions without the help of perturbation theory. The second quantization takes real Hilbert spaces as inputs. Using only information from one-particle Hilbert spaces, it is mathematically impossible for the second quantization to produce connected higher-point correlation functions. Moreover, from the perspectives of mathematical quantization and operator algebra theory, it is more natural to have a quantization procedure that takes an algebra along with a functional as the input. Such a pair of mathematical objects contains more information about the underlying one-particle system and includes the Hilbert space of states (via the GNS construction) as part of the data. Inspired by quantum probability theory, we introduce an alternative approach to construct quantum many-body systems. This approach is based on a noncommutative generalization of the classical Poisson random measure. We call this construction Poissonization. Mathematically, Poissonization is a functor from the category of von Neumann algebras with normal semi nite faithful weights to the category of von Neumann algebras with normal faithful states. It is a natural adaptation of the second quantization to the context of von Neumann algebras. This thesis discusses several properties of Poissonization, and uses Poissonization as a tool to construct various toy models of algebraic quantum field theories relevant to high energy physics. This collection of examples aims at demonstrating the versatility of Poissonization. In the second part of this thesis, we study the noisy interactions between an open quantum system and its environment. One of the challenges in quantum information science is to control open quantum systems with a large number of qubits. An important aspect of many-body systems is the emergence of collective phenomena. One collective noise model is an open atomic system in an electromagnetic environment. This model was considered by Dicke in the 50's. In this thesis, we study the entropic decay in Dicke's model and other related collective noise models. Specifically, we develop a general framework to estimate the spectral gap and modified logarithmic Sobolev constant of these collective noise models. In addition, we study the necessary mixing conditions a general Dicke's model must satisfy in order to have a unique equilibrium state.
Many-body entangled systems, in particular topologically ordered spin systems proposed as resources for quantum information processing tasks, often involve highly nonlocal interaction terms. While one may approximate such systems through two-body interactions perturbatively, these approaches have a number of drawbacks in practice. In this Letter, we propose a scheme to simulate many-body spin Hamiltonians with two-body Hamiltonians nonperturbatively. Unlike previous approaches, our Hamiltonians are not only exactly solvable with exact ground state degeneracy, but also support completely localized quasiparticle excitations, which are ideal for quantum information processing tasks. Our construction is limited to simulating the toric code and quantum double models, but generalizations to other nonlocal spin Hamiltonians may be possible.
Exactly solvable quantum systems represent a fascinating and critically important class of models in theoretical physics, offering exact solutions to non-trivial problems that often shed light on the behavior of more complex, intractable systems. The remarkable feature of these systems lies not only in their solvability but also in the underlying mathematical structures that facilitate such solutions. This paper explores the concept of a universal algebraic structure that underpins exactly solvable quantum systems. We delve into how diverse models, ranging from one-dimensional spin chains to many-body systems and field theories, exhibit common algebraic symmetries, such as Lie algebras, affine Lie algebras, and quantum groups. These algebraic frameworks provide a unified language for understanding integrability, the existence of an infinite number of conserved quantities, and the ability to construct exact eigenstates. We review the historical development of these ideas, from the Bethe ansatz to the development of the inverse scattering method and quantum groups. Our discussion highlights how these mathematical structures are not merely tools for calculation but fundamental descriptors of the physical reality of these systems, offering profound insights into non-perturbative quantum phenomena. The identification of a universal algebraic architecture suggests a deeper coherence in the landscape of quantum mechanics, paving the way for new discoveries and classifications of quantum models.
Despite nearly a century of study of the S=1/2S=1/2S=1/2 Heisenberg model on the square lattice, there is still disagreement on the nature of its high-energy excitations. By tuning toward the Heisenberg model from the exactly soluble Ising limit, we find that the strongly attractive magnon interactions of the latter naturally account for a number of spectral features of the Heisenberg model. This claim is backed up both numerically and analytically. Using the density matrix renormalization group method, we obtain the dynamical structure factor for a cylindrical geometry, allowing us to continuously connect both limits. Remarkably, a semiquantitative description of certain observed features arises already at the lowest nontrivial order in perturbation theory around the Ising limit. Moreover, our analysis uncovers that high-energy magnons are localized on a single sublattice, which is related to the entanglement properties of the ground state.