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International Conference on Applied Statistics, Computational Mathematics, and Software Engineering (ASCMSE)

Perturbative Schwinger-Dyson-equations-of-motion for quantum spins 1/2: Ising model of two spins in tilted fields

Chang Cheng, Wen Xin Ding

Published Aug 23, 2023• 0 citations
Publisher

Abstract

The “noncanonicality” of the underlying operators is the fundamental difficulty in formulating theories for most strong interaction systems such as the t-J model, the Hubbard-type model under the large U limit, and quantum spin etc. But B.S. Shastry proposed Schwinger-Dyson equations of motion to study the t-J model, which can rigorously solve this noncanonicality. In this article, we use the perturbation Schwinger-Dyson equation of motion method to study two spin 1/2 transverse field Ising models perturbed by longitudinal fields, and discuss potential generalizations of the corresponding lattice models.

DOI
10.1117/12.2692892

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The “noncanonicality” of the underlying operators is the fundamental difficulty in formulating theories for most strong interaction systems such as the t-J model, the Hubbard-type model under the large U limit, and quantum spin etc. But B.S. Shastry proposed Schwinger-Dyson equations of motion to study the t-J model, which can rigorously solve this noncanonicality. In this article, we use the perturbation Schwinger-Dyson equation of motion method to study two spin 1/2 transverse field Ising models perturbed by longitudinal fields, and discuss potential generalizations of the corresponding lattice models.
Abstract The essential difficulty of a large portion of the strongly interacting systems can be traced down to the “noncanonicality” of the underlying operators which are used to formulate the theory, like t-J model, Hubbard-type models in the large-U limit, quantum spin, etc. We proposed a new Schwinger-Dyson equations-of-motion method to compute the dynamical correlation functions of quantum spins at zero temperature. This method can also compute all the static observables from the dynamical correlation functions. We propose that this method can formulate dynamical perturbation calculations for quantum spin-1/2 models.
Variational methods are of fundamental importance and widely used in theoretical physics, especially for strongly interacting systems. In this work, we present a set of variational equations of state (VES) for pure states of an interacting quantum Hamiltonian. The VES can be expressed in terms of the variation of the density operators or static correlation functions. We derive the algebraic relationship between a known pure state density matrix and its variation, and obtain the VES by applying this relation to the averaged Heisenberg-equations-of-motion for the exact density matrix. Additionally, we provide a direct expression of the VES in terms of correlation functions to make it computable. We present three nontrivial applications of the VES: a perturbation calculation of correlation functions of the transverse field Ising model in arbitrary spatial dimensions, a study of a longitudinal field perturbation to the one-dimensional transverse field Ising model at the critical point and variational calculation of magnetization and ground state energy of the two-dimensional spin-1/2 Heisenberg model on a square lattice. For the second one, our results not only recover the scaling limit, but also indicate the possibility of continuous tuning of the critical exponents by adjusting the longitudinal fields differently from the scaling limit. For the Heisenberg model, we obtained results numerically comparable to established results with simple calculations. The VES approach provides a powerful and versatile tool for studying interacting quantum systems.
We developed a systematic non-perturbative method base on Dyson-Schwinger theory and the derivable theory for Ising model at broken phase. Based on these methods, we obtain critical temperature and spin spin correlation beyond mean field theory. The spectrum of Green function obtained from our methods become gapless at critical point, so the susceptibility become divergent at T c . The critical temperature of Ising model obtained from this method is fairly good in comparison with other non-cluster methods. It is straightforward to extend this method to more complicate spin models for example with continue symmetry.
We investigate the dynamical spreading of spatial correlations after a quantum quench starting from a magnetically disordered state in the transverse-field Ising model at one (1D) and two spatial dimensions (2D). We analyze specifically the longitudinal and transverse spin-spin correlation functions at equal time with use of several methods. From the comparison of the results in 1D obtained by the linear spin-wave approximation (LSWA) and those obtained by the rigorous analytical approach, we show that the LSWA can asymptotically reproduce the exact group velocity in the limit of strong transverse fields while it fails to capture the detailed time dependence of the correlation functions. By applying the LSWA to the 2D case, in which the rigorous analytical approach is unavailable, we estimate the propagation velocity to be $Ja/(2\ensuremath{\hbar})$ at the strong-field limit, where JJJ is the Ising interaction and aaa is the lattice spacing. We also utilize the tensor-network method based on the projected-entangled pair states for 2D and quantitatively compute the time evolution of the correlation functions for a relatively short time. Our findings provide useful benchmarks for quantum simulation experiments of correlation spreading and theoretical refinement of the Lieb-Robinson bound in the future.
Abstract For a transverse-field Ising chain with weak long-range interactions we develop a perturbative scheme, based on quantum kinetic equations, around the integrable nearest-neighbour model. We introduce, discuss, and benchmark several truncations of the time evolution equations up to eighth order in the Jordan–Wigner fermionic operators. The resulting set of differential equations can be solved for lattices with O(102) sites and facilitates the computation of spin expectation values and correlation functions to high accuracy, at least for moderate timescales. We use this scheme to study the relaxation dynamics of the model, involving prethermalisation and thermalisation. The techniques developed here can be generalised to other spin models with weak integrability-breaking terms.
The quantum dynamics of many-qubit systems is an outstanding problem that has recently driven significant advances in both numerical methods and programmable quantum processing units. In this work, we employ a comprehensive toolbox of state-of-the-art numerical approaches to classically simulate the dynamics of the two-dimensional transverse field Ising model. Our methods include three different tensor network techniques -- matrix product states, tree-tensor networks, and two-dimensional tensor-networks under the belief propagation approximation -- as well as time-dependent variational Monte Carlo with Neural Quantum States. We focus on two paradigmatic dynamical protocols: (i) quantum annealing through a critical point and (ii) post-quench dynamics. Our extensive results show the quantitative predictions of various state-of-the-art numerical methods providing a benchmark for future numerical investigations and experimental studies with the aim to push the limitations on classical and QPUs. In particular, our work connects classical simulability to different regimes associated with quantum dynamics in Rydberg arrays - namely, quasi-adiabatic dynamics, the Kibble-Zurek mechanism, and quantum quenches.