Truncated string state space approach and its application to nonintegrable spin-$\frac{1}{2}$ Heisenberg chain

One-dimensional quantum systems characterized by exact solutions and quantum integrability offer a fascinating arena to study many-body physics. Notable examples include the one-dimensional (1D) spin-$\frac{1}{2}$ XXZ model [1, 2, 3, 4], Gaudin-Yang model [5, 6, 7], Lieb-Liniger model [8, 9], and quantum Ising models [10, 11, 12, 13]. Although these models have paved the way for determining the eigenstates and eigenenergies of those systems, it has long been a challenge to calculate their form factors and thus dynamical response, which was partly tackled recently [14, 7, 15, 16, 17, 18, 19, 20]. Empowered by the theoretical development, the spin dynamics of celebrated many-body quasiparticles, such as spinons [21, 22, 23], strings [24, 25, 26], $E_{8}$ [27, 28, 29, 30, 31] and $D_{8}^{(1)}$ particles [32, 33, 34], have been extensively explored, providing crucial guidance for the experimental observations in quasi-1D materials [3, 35, 36, 37, 38, 39, 40, 30, 31]. This progress has led to the cooperative effort of both theorists and experimentalists to unveil the intricate nature of these exotic phenomena.

Bearing real materials in mind, it becomes crucial to ask how robust integrable physics is against nonintegrable perturbations that may partially or fully break the conservation laws of integrable systems. This has inspired extensive research focused on nonintegrable models such as spin-$\frac{1}{2}$ ladders [41, 42], chains with a staggered magnetic field [43, 44], frustrated spin chains [45], and dimerized spin chains [46, 47]. Most studies are performed by using effective field theory [43, 44] or numerical methods such as the exact diagonalization (ED) [45, 46], matrix product state [47], and quantum Monte Carlo methods [48]. However, on the one hand, numerical methods in general lack a clear understanding of the essential physical picture, on the other hand, the effective field theory can provide only limited insight within the low-energy and long-wavelength limit. Therefore, a method able to go beyond those limitations is always desired.

At first glance it may seem promising to apply Bethe states to study nonintegrable systems. However, a notorious open problem persists: finding the precise complex solutions of the Bethe ansatz equation (BAE) for string states [24, 49, 50, 51, 52, 53]. In the past few decades, many approaches have been explored, including a carefully-designed iterative method [54] and a rational $Q$-system method [55, 56, 57]. The former can easily access large system size but suffers from many unphysical solutions with repeated roots. Although the latter can solve the BAE for all exact solutions simultaneously, it is limited to a small system size. Those shortcomings impede the practical application of Bethe states to understand nonintegrable systems of reasonable size.

In this paper, we first outline a solving machine for the spin-$\frac{1}{2}$ Heisenberg chain, which can obtain exact solutions for Bethe string states. Based on these string states, we develop a truncated string-state-space approach (TS³A) to study nonintegrable Hamiltonian, specifically the spin-$\frac{1}{2}$ Heisenberg chain with a staggered field. The TS³A can determine the eigenstate and eigenenergy for the nonintegrable Hamiltonian in the truncated Hilbert space. Additionally, we evaluate its efficiency for small systems under various truncation schemes, involving different energy cutoffs and string lengths, by comparing its performance to that of ED calculations.

Following the TS³A, we analyze the nonintegrable spin dynamics in the spin-$\frac{1}{2}$ Heisenberg chain with staggered field characterized by wavevector $Q$. In addition to the $Q$-ordering of the system, a series of elastic peaks appear at $nQ\ (|n|=2,3,\ldots)$, indicating the ground state contains multi-$Q$ Bethe string states. Moreover, the staggered field plays the role of the confining field for the Bethe string states, constraining the motion of spins along the chain. The confinement of Bethe strings results in two separated continua in dynamical spectra at low magnetization. Notably, the above results were successfully applied to experimental observations of the quasi-1D antiferromagnet $\rm YbAlO_{3}$, aligning with the unified Bethe-string-based framework provided by the TS³A [58].

The rest of this paper is organized as follows. Section II introduces the Hamiltonian of the 1D Heisenberg model with a staggered field. Section III illustrates the Bethe string state and then presents a method for obtaining exact solutions from the BAE. The framework of the TS³A is developed, and its efficiency is investigated in Sec. IV. Then Sec. V discusses the spin dynamics of the nonintegrable Hamiltonian. Section VI contains the conclusion and outlook.

Our parent Hamiltonian is the 1D Heisenberg spin-$\frac{1}{2}$ model with longitudinal field $h_{z}$,

with $N$ being the total number of sites, $J$ being antiferromagnetic coupling, and $\mathbf{S}_{n}$ being spin operators with components $S_{n}^{\mu}$ ($\mu=x,y,z$) at site $n$. With the introduction of a staggered field $\mathbf{h}_{Q}=h_{Q}\sum_{i}\cos(Qr_{i})\hat{z}$ which couples to the spin chain, the total Hamiltonian becomes nonintegrable,

where

$h_{Q}$ is the strength of the staggered field, and the ordering wave vector $Q=(1-m)\pi$, where the magnetization density $m=M_{z}/M_{s}$, which is the ratio of magnetization $M_{z}$ to its saturation value $M_{s}$. In practice, the staggered field can be effectively induced from three-dimensional (3D) magnetic ordering of quasi-1D materials, such as $\rm YbAlO_{3}$ [59, 60, 61, 58], $\rm SrCo_{2}V_{2}O_{8}$ [62] and $\rm BaCo_{2}V_{2}O_{8}$ [30]. We note that the staggered field can be both commensurate and incommensurate, depending on whether $2\pi/Q$ is a rational or irrational number, respectively.

In this section, we begin with an introduction to the coordinate Bethe ansatz and Bethe string states for the Hamiltonian $H_{0}$ [Eq. (1)]. Then an efficient method is presented for obtaining the exact solutions from the BAE.

In many physical problems, our focus is on the low-energy subspace rather than the entire Hilbert space [71, 72, 73]. In this study, we employ the TS³A method, as illustrated in [Fig. 1f], to gain insights into the low-energy physics of nonintegrable Heisenberg models Eq. (2). The detailed construction is as follows.

When considering a nonintegrable perturbation, such as $H^{\prime}$ [Eq. (3)], the Bethe string state is no longer the eigenstate. Therefore, it becomes necessary to find out the new ground state and low-energy excited states before conducting any calculations of physical quantities, such as correlation functions. The first step is to construct the matrix representation of the nonintegrable Hamiltonian $H$ [Eq. (2)] within a truncated low-energy subspace of Bethe string states, denoted as $H^{tr}_{ab}=\delta_{ab}E^{B}_{a}+\langle B_{a}|H^{\prime}|B_{b}\rangle$, with $E^{B}_{a}\leq E^{B}_{cut}$. The truncated dimension of $H^{tr}_{ab}$ is typically much less than $2^{N}$, which is determined by the energy cutoff $E^{B}_{cut}$ and types of Bethe string states $|B_{a}\rangle$. Following the diagonalization of $H^{tr}$, a new ground state $|GS\rangle$ and low-energy excited states are obtained, which are considered as approximate eigenstates of the Hamiltonian $H$.

However, an immediate question arises: How do we select the types of Bethe string states and determine the energy cutoff? To answer the first question, we investigate the form factor between Bethe string states. It is evident from Fig. 2(a) that the form factors for string states generally diminish rapidly as the difference in string length increases. For instance, the ground state of $H_{0}$ Eq. (1) is a 1-string state, and then we can safely truncate the string state space into relevant subspace in terms of string lengths. For the second issue, we study the asymptotic behavior of ground state energy $E_{GS}$ and the staggered magnetization $M^{z}_{Q}=\sum_{j}e^{-iQj}\langle GS|S_{j}^{z}|GS\rangle/\sqrt{N}$ as the energy cutoff of the truncated space increase. In Fig. 2(b,c), the calculation includes all allowed string states within a given $E_{cut}$. As $E_{cut}$ increases the results converge rapidly and approach the exact values obtained from the ED calculation. Notably, even if only 1- and 2-string states are considered, the obtained results are already very close to the exact values, while the impact of 3- and $2*2$-string states is marginal. This phenomenon confirms the suggestion that the relevant string states primarily arise from those with small length differences, as illustrated in Fig. 2(a).

To investigate nonintegrable spin dynamics of $H$ [Eq. (2)], we focus on the zero-temperature dynamical structure factor (DSF) for spin along longitudinal ($z$) direction ($\hbar=1$),

with $q$ being the transfer momentum and $\omega$ being the transfer energy between the ground state $|GS\rangle$ and excited states $|\mu\rangle$, whose energies are $E_{GS}$ and $E_{\mu}$, respectively. In the following calculation, the eigenstates $|GS\rangle$ and $|\mu\rangle$ are obtain by the TS³A developed in Sec. IV. The form factor $\langle GS|S_{q}^{z}|\mu\rangle$ is deeply related to the form factor of Bethe string states, which can be elegantly expressed in terms of determinant [25, 74, 75, 22, 23, 26], with a summary in Appendix B.

To begin with, we investigate the TS³A results under different truncation schemes for $N=16$, $m=12.5\%$, and $h_{Q}=0.4J$. In Fig. 3(a1-a4), the DSF is calculated with different selected string types and fixed energy cutoff $E_{cut}=5J$, showing that 1- and 2-strings are the dominant states in the spin dynamics. In Fig. 3(b1-b4), the DSF is calculated with different energy cutoff and fixed string types (including 1-, 2-, 3- and $2*2$-string), showing that the dynamical spectrum converges quickly as $E_{cut}$ increases. Furthermore, we compare the DSF results at $N=16$ obtained from TS³A to that of the ED method [76, 77], whose results reveal a remarkable agreement in Fig. 4. This excellent comparison suggests that the TS³A is a highly efficient method for studying the nonintegrable spin dynamics.

In the following, we present the DSF results obtained from the TS³A at $N=48$. Due to the staggered field perturbation term $H^{\prime}$ [Eq. (3)], where $\sum_{n}\cos(Qr_{n})S^{z}_{n}\propto(S^{z}_{Q}+S^{z}_{-Q})$, the non-vanishing matrix element of $H^{\prime}$ only appears between Bethe states with momentum difference $\Delta q=\pm Q$. As a result, the ground state consists of Bethe states with momenta $nQ\ (|n|=0,1,2,\ldots)$. For static structure factor $D^{zz}(q,\omega=0)$, a series of staggered-field-induced peaks appear at $nQ$ ($|n|=1,2,\ldots$), manifesting the presence of multi-$Q$ Bethe states in the ground state. For instance, in Fig. 5(a-d), there are satellite peaks at $q=2Q,-2Q+2\pi$ in addition to the predominant peaks at $q=Q,-Q+2\pi$, with $Q=(1-m)\pi$.

In Fig. 6, when $h_{Q}=0$, the dynamical spectra exhibit gapless excitations at $q=(1\pm m)\pi$, and the 2-string states are barely separated from the broad continuum of 1-string states. When $h_{Q}>0$, an energy gap emerges both near the elastic line and between the continua of 1- and 2-string states. This phenomenon arises because the staggered field acts as a confining field for the Heisenberg spin chains, effectively restricting the motion of spins [78, 79, 80, 43, 44]. These induced gaps reflect the energy cost for the excitation of Bethe strings, which is known as the confinement of Bethe strings. At small magnetization [Fig. 6(a1-a3)], 2-string states are effectively confined and become separated from the 1-string continuum. However, as magnetization $m$ increases, the 2-string continuum gradually dissipates into higher energy ranges.

Notably, the capability for larger size calculation of TS³A not only reduces the finite size effect but also renders the characteristics of spectra more transparent and discernible. And, the TS³A offers two key advantages compared to the ED method: First, it has higher efficiency, facilitating much larger systems ($N\gtrsim 50$); second, it naturally provides a unified Bethe-string-based physical picture for understanding the underlying physics.

We conclude this section by emphasizing that our model and findings offer a direct application in comprehending the low-energy spin dynamics observed in the quasi-1D antiferromagnet $\rm YbAlO_{3}$ [59, 61, 58]. In this material, the low-energy effective Hamiltonian is described by the 1D Heisenberg models Eq. (1), and Eq. (2) if the material is 3D ordered. In Fig. 5(e), the quasi-elastic signals obtained from neutron scattering align with the theoretical predictions, providing compelling evidence for the coexistence of multi-$Q$ Bethe states in the ordered phase of $\rm YbAlO_{3}$. Moreover, the staggered field, arising from the 3D ordering, plays the role of a confining field coupled with the spin chains within the material. As a result, the distinctive features characterizing the confined string states are observed through the inelastic neutron scattering spectra of $\rm YbAlO_{3}$ [58].

We exploited an efficient routine to find exact solutions for the Bethe string states from the BAE of the spin-$\frac{1}{2}$ Heisenberg spin chain. Based on the exact solutions we further developed the TS³A which enabled us to determine eigenstates and eigenenergies of nonintegrable spin-$\frac{1}{2}$ Heisenberg systems with U(1) symmetry preserved. The method was then applied to systematically study the spin dynamics of the spin-$\frac{1}{2}$ Heisenberg spin chain under staggered field. In the dynamical spectra, we revealed a series of elastic peaks located at the integer multiples of the ordering wavevector $Q$, signifying the existence of multi-$Q$ Bethe string states within the ground state. Moreover, the staggered field serves as a confining field for Bethe string states, inducing confinement gaps between the continua of 1- and 2-string states.

Our TS³A machine offers a Bethe-string-based scenario, contributing to a more comprehensive understanding of Heisenberg spin systems. We have demonstrated the efficiency and validity of this framework by interpreting experimental observations of the quasi-1D antiferromagnet $\rm YbAlO_{3}$. This intriguing consistency between theoretical predictions based on the TS³A and experimental results motivates its extended application to ladder and two-dimensional Heisenberg systems. This broadening of scope not only enhances the versatility of the Bethe string picture but also transcends its conventional one-dimensional limitations.

We thank Yunfeng Jiang for helpful discussion. This work is supported by National Natural Science Foundation of China No. 12274288 and the Innovation Program for Quantum Science and Technology Grant No. 2021ZD0301900, and the Natural Science Foundation of Shanghai with grant No. 20ZR1428400.