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The Journal of Chemical Physics

The structure and symmetry of modular state space for complex quantum systems

Guohua Tao

Published Mar 18, 2025• 1 citation
Publisher

Abstract

Understanding the state space structure of complex quantum systems can help to effectively characterize the system properties and explore underlying mechanisms. The structure of the state space could be quite complicated for quantum many-body systems, and the systematic decomposition of the state space is normally involved. Recently, a modular tensor diagram approach was proposed to reorganize the state space hierarchically based on a modular basis. Here, we review the construction of spin eigenfunctions for multiple exciton systems and further develop modular tensor diagrams to exemplify the hierarchical symmetry of the state space. The newly constructed spin eigenfunctions for quadruple excitons, along with the results for triple excitons, are used to demonstrate the effective decomposition of the state space into hierarchical tensorial structures. A universal recursive relation is derived to determine the coefficients of spin eigenfunctions exhibiting transformation symmetry between different classes of elementary modules for an arbitrary number of exciton units. Interestingly, different coupling schemes mapped to quantum many-body interactions lead to different spin adapted basis states, which may correspond to different realistic systems upon the breakdown of spin degeneracy. This work highlights the hierarchical symmetry of the tensorial structure of quantum many-body systems, which may facilitate a better understanding of the structure property relationship toward the object-oriented materials design.

DOI
10.1063/5.0245447

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Understanding the state space structure of complex quantum systems can help to effectively characterize the system properties and explore underlying mechanisms. The structure of the state space could be quite complicated for quantum many-body systems, and the systematic decomposition of the state space is normally involved. Recently, a modular tensor diagram approach was proposed to reorganize the state space hierarchically based on a modular basis. Here, we review the construction of spin eigenfunctions for multiple exciton systems and further develop modular tensor diagrams to exemplify the hierarchical symmetry of the state space. The newly constructed spin eigenfunctions for quadruple excitons, along with the results for triple excitons, are used to demonstrate the effective decomposition of the state space into hierarchical tensorial structures. A universal recursive relation is derived to determine the coefficients of spin eigenfunctions exhibiting transformation symmetry between different classes of elementary modules for an arbitrary number of exciton units. Interestingly, different coupling schemes mapped to quantum many-body interactions lead to different spin adapted basis states, which may correspond to different realistic systems upon the breakdown of spin degeneracy. This work highlights the hierarchical symmetry of the tensorial structure of quantum many-body systems, which may facilitate a better understanding of the structure property relationship toward the object-oriented materials design.
The construction of spin eigenfunctions for multi-spin systems can become highly nontrivial as the system size increases. Recently, a modular tensor diagram approach was proposed to hierarchically decompose the state space in terms of tensorial modules based on spin pairs, resulting in an effectively organized state space and efficiently constructed symmetry-adapted basis states. Here, it is generalized to treat fermion systems with odd numbers of spins. Elementary modules made of primitive spin-pair modules augmented by an odd-spin tag module are classified into various module classes and further mapped to geometric building blocks of various sizes and shapes to illustrate the hierarchical structure and symmetry of the state space. Spin eigenfunctions are generated from linear combinations of single-spin-tagged or triple-spin-tagged elementary modules using only symmetry and orthogonality conditions, and universal recursive relations for systems with arbitrary odd numbers of spins can be obtained. This work explores the structure and symmetry of the state space of fermion systems that complement previous studies, which may provide new insights into general quantum many-body systems and spin dynamics.
We identify a hierarchical symmetry structure underlying the Hilbert space of quantum systems made of identical particles. By studying the linear map between single-body and many-body spectra, we show that the resulting spectral organization admits a natural partition into sectors induced by a symmetry governed by cyclotomic fields and their Galois groups. This approach offers a new perspective on the structure of quantum many-body spectra, defining a controlled coarse-graining of the Hilbert space in which degeneracies and spectral features can be systematically organized through a set of new quantum numbers. The resulting structure induces a hierarchy of spectral resolutions following a renormalization flow, which enables information loss control. We demonstrate that this hierarchical decomposition provides an efficient route to approximate MBDoS calculations for any identical particle system, while preserving physically relevant spectral properties. This approach provides a general symmetry-based framework for organizing and approximating many-body spectra, with potential applications to quantum thermalization, spectral statistics, and large-scale quantum simulations.
Abstract We study the representation and visualization of finite-dimensional, coupled quantum systems. To establish a generalized Wigner representation, multi-spin operators are decomposed into a symmetry-adapted tensor basis and are mapped to multiple spherical plots that are each assembled from linear combinations of spherical harmonics. We explicitly determine the corresponding symmetry-adapted tensor basis for up to six coupled spins 1/2 (qubits) using a first step that relies on a Clebsch–Gordan decomposition and a second step which is implemented with two different approaches based on explicit projection operators and coefficients of fractional parentage. The approach based on explicit projection operators is currently only applicable for up to four spins 1/2. The resulting generalized Wigner representation is illustrated with various examples for the cases of four to six coupled spins 1/2. We also treat the case of two coupled spins with arbitrary spin numbers (qudits) not necessarily equal to 1/2 and highlight a quantum system of a spin 1/2 coupled to a spin 1 (qutrit). Our work offers a much more detailed understanding of the symmetries appearing in coupled quantum systems.
A novel reduced-scaling, general-order coupled-cluster approach is formulated by exploiting hierarchical representations of many-body tensors, combined with the recently suggested formalism of scale-adaptive tensor algebra. Inspired by the hierarchical techniques from the renormalisation group approach, H/H2-matrix algebra and fast multipole method, the computational scaling reduction in our formalism is achieved via coarsening of quantum many-body interactions at larger interaction scales, thus imposing a hierarchical structure on many-body tensors of coupled-cluster theory. In our approach, the interaction scale can be defined on any appropriate Euclidean domain (spatial domain, momentum-space domain, energy domain, etc.). We show that the hierarchically resolved many-body tensors can reduce the storage requirements to O(N), where N is the number of simulated quantum particles. Subsequently, we prove that any connected many-body diagram consisting of a finite number of arbitrary-order tensors, e.g. an arbitrary coupled-cluster diagram, can be evaluated in O(NlogN) floating-point operations. On top of that, we suggest an additional approximation to further reduce the computational complexity of higher order coupled-cluster equations, i.e. equations involving higher than double excitations, which otherwise would introduce a large prefactor into formal O(NlogN) scaling.
The classical simulation of many-body quantum systems is an essential tool in understanding many fundamental aspects of condensed matter physics. But a major obstacle arises from the number of degrees of freedom involved in describing such systems, which is exponential in the system size. Recently, however, a class of numerical techniques based on structures called "tensor networks" has emerged, which allows many "typical" quantum states (such as the ground states of gapped, local Hamiltonians) to be represented much more efficiently. In this work we extend and apply these techniques to consider several central topics in quantum many-body physics. After reviewing the relevant background material from the field of tensor networks and tensor network states, we demonstrate a method for computing high order moments and cumulants of operators with respect to such states, including the so-called "Binder cumulant," a powerful tool for detecting phase transitions. Next, we employ tensor network algorithms to characterize the ground state phase diagram of a quantum spin model, including both symmetry-breaking phases and symmetry protected topological order, and find a signicant variety of phases and phase transitions. Finally, we consider the entanglement properties of quantum states exhibiting many-body localization, using a combination of exact diagonalization and tensor network techniques. | 231 pages
We focus on symmetries related to matrices and vectors appearing in the simulation of quantum many-body systems. Spin Hamiltonians have special matrix-symmetry properties such as persymmetry. Furthermore, the systems may exhibit physical symmetries translating into symmetry properties of the eigenvectors of interest. Both types of symmetry can be exploited in sparse representation formats such as Matrix Product States (MPS) for the desired eigenvectors. This article summarizes symmetries of Hamiltonians for typical physical systems such as the Ising model and lists resulting properties of the related eigenvectors. Based on an overview of MPS (Tensor Trains or Tensor Chains) and their canonical normal forms, we show how symmetry properties of the vector translate into relations between the MPS matrices and, in turn, which symmetry properties result from relations within the MPS matrices. In this context, we analyse different kinds of symmetries and derive appropriate normal forms for MPS representing these symmetries. Exploiting such symmetries by using these normal forms will lead to a reduction in the number of degrees of freedom in the MPS matrices. This article provides a uniform platform for both well-known and new results which are presented from the (multi-)linear algebra point of view.