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Communications in Theoretical Physics

Constructing integrable spin-boson model with open boundary conditions

Hong Yao, Kun Hao

Published Nov 21, 2024• 0 citations
Publisher

Abstract

Abstract We investigate the integrability of the Rabi model, which is traditionally viewed as not Yang-Baxter integrable despite its solvability. Building on efforts by Bogoliubov, Amico, Batchelor, and Zhou, who explored special limiting cases of the model, we develop a spin-boson interaction Hamiltonian under more general boundary conditions, particularly focusing on open boundary conditions with off-diagonal terms. Our approach maintains the direction of the spin in the $z$ direction, and also preserves the boson particle number operator $a^\dagger a$, marking a progression beyond previous efforts that have primarily explored reduced forms of the Rabi model from Yang-Baxter algebra. We also address the presence of ``unwanted" quadratic boson terms $a^2$ and $a^\dagger{}^2$, which share coefficients with the boson particle number operator. Interestingly, these terms vanish when spectral parameter $u = \pm \theta_s$, simplifying the model to a limiting case of operator-valued twists, a scenario previously discussed by Batchelor and Zhou.

DOI
10.1088/1572-9494/ad9568

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Abstract We investigate the integrability of the Rabi model, which is traditionally viewed as not Yang-Baxter integrable despite its solvability. Building on efforts by Bogoliubov, Amico, Batchelor, and Zhou, who explored special limiting cases of the model, we develop a spin-boson interaction Hamiltonian under more general boundary conditions, particularly focusing on open boundary conditions with off-diagonal terms. Our approach maintains the direction of the spin in the zzz direction, and also preserves the boson particle number operator a†aa^\dagger aa†a, marking a progression beyond previous efforts that have primarily explored reduced forms of the Rabi model from Yang-Baxter algebra. We also address the presence of ``unwanted" quadratic boson terms a2a^2a2 and a†2a^\dagger{}^2a†2, which share coefficients with the boson particle number operator. Interestingly, these terms vanish when spectral parameter u=±θsu = \pm \theta_su=±θs​, simplifying the model to a limiting case of operator-valued twists, a scenario previously discussed by Batchelor and Zhou.
In this paper, we address the problem of Yang-Baxter integrability of doubled quantum circuit of qubits (spins 1/2) with open boundary conditions where the two circuit replicas are only coupled at the left or right boundary. We investigate the cases where the bulk is given by elementary six vertex unitary gates of either the free fermionic XX type or interacting XXZ type. By using the Sklyanin’s construction of reflection algebra, we obtain the most general solutions of the boundary Yang-Baxter equation for such a setup. We use this solution to build, from the transfer matrix formalism, integrable circuits with two step discrete time Floquet (aka brickwork) dynamics. We prove that, only if the bulk is a free-model, the boundary matrices are in general non-factorizable, and for particular choice of free parameters yield non-trivial unitary dynamics with boundary interaction between the two chains. Then, we consider the limit of continuous time evolution and we give the interpretation of a restricted set of the boundary terms in the Lindbladian setting. Specifically, for a particular choice of free parameters, the solutions correspond to an open quantum system dynamics with the source terms representing injecting or removing particles from the boundary of the spin chain.
The Rabi model describes the simplest interaction between light and matter via a two-level quantum system interacting with a bosonic field. We demonstrate that the fully quantized version of the Rabi model is integrable in the Yang-Baxter sense at two parameter values. The model is argued to be not Yang-Baxter integrable in general. This is in contrast to the claim that the quantum Rabi model is integrable based on a phenomenological criterion of quantum integrability not presupposing the existence of a set of commuting operators. Similar Yang-Baxter integrable points are identified for the generalized Rabi model and the fully quantized Dicke model. The integrable points have particular implications for the level statistics of the Dicke model.
This work is based on the author's PhD thesis. The main result of the thesis is the use of the boost operator to develop a systematic method to construct new integrable spin chains with nearest-neighbour interaction and characterized by an R-matrix of non-difference form. This method has the advantage of being more feasible than directly solving the Yang-Baxter equation. We applied this approach to various contexts, in particular, in the realm of open quantum systems, we achieved the first classification of integrable Lindbladians. These operators describe the dynamics of physical systems in contact with a Markovian environment. Within this classification, we discovered a novel deformation of the Hubbard model spanning three sites of the spin chain. Additionally, we applied our method to classify models with su(2)⊕su(2)\mathfrak{su}(2)\oplus \mathfrak{su}(2)su(2)⊕su(2) symmetry and we recovered the matrix part of the S-matrix of AdS5×S5AdS_5 \times S^5AdS5​×S5 derived by requiring centrally extended su(2∣2)\mathfrak{su}(2|2)su(2∣2) symmetry. Furthermore, we focus on spin 1/2 chain on models of 8-Vertex type and we showed that the models of this class satisfy the free fermion condition. This enables us to express the transfer matrix associated to some of the models in a diagonal form, simplifying the computation of the eigenvalues and eigenvectors. The thesis is based on the works: 2003.04332, 2010.11231, 2011.08217, 2101.08279, 2207.14193, 2301.01612, 2305.01922.
Abstract Yang–Baxter integrable vertex models with a generic Z 2 -staggering can be expressed in terms of composite R -matrices given in terms of the elementary R-matrices. Similarly, integrable open boundary conditions can be constructed through generalized reflection algebras based on these objects and their representations in terms of composite boundary matrices K ± . We show that only two types of staggering yield a local Hamiltonian with integrable open boundary conditions in this approach. The staggering in the underlying model allows for a second hierarchy of commuting integrals of motion (in addition to the one including the Hamiltonian obtained from the usual transfer matrix), starting with the so-called quasi momentum operator. In this paper, we show that this quasi momentum operator can be obtained together with the Hamiltonian for both periodic and open models in a unified way from enlarged Yang–Baxter or reflection algebras in the composite picture. For the special case of the staggered six-vertex model, this allows constructing an integrable spectral flow between the two local cases.
We consider integrable Matrix Product States (MPS) in integrable spin chains and show that they correspond to "operator valued" solutions of the so-called twisted Boundary Yang-Baxter (or reflection) equation. We argue that the integrability condition is equivalent to a new linear intertwiner relation, which we call the "square root relation", because it involves half of the steps of the reflection equation. It is then shown that the square root relation leads to the full Boundary Yang-Baxter equations. We provide explicit solutions in a number of cases characterized by special symmetries. These correspond to the "symmetric pairs" (SU(N),SO(N))(SU(N),SO(N))(SU(N),SO(N)) and (SO(N),SO(D)⊗SO(N−D))(SO(N),SO(D)\otimes SO(N-D))(SO(N),SO(D)⊗SO(N−D)), where in each pair the first and second elements are the symmetry groups of the spin chain and the integrable state, respectively. These solutions can be considered as explicit representations of the corresponding twisted Yangians, that are new in a number of cases. Examples include certain concrete MPS relevant for the computation of one-point functions in defect AdS/CFT.