Introduction

In quantum mechanics, a given initial state of a closed interacting quantum system will evolve unitarily according to its Hamiltonian H. A fundamental assumption in statistical physics is that a generic closed quantum many-body system thermalizes under its own dynamics1,2. This implies that, under unitary time evolution, the reduced density matrix of any generic initial state \({|{\Psi (0)}\rangle }\)tends to evolve towards the equilibrium Gibbs state within that subsystem. Since any initial state can be spanned by the eigenstates of the Hamiltonian, then the reduced matrix constructed with the eigenstates should also evolve toward equilibrium. This is the notion underlying the known eigenstate thermalization hypotesis (ETH)3,4,5. At a first glance, it seems that any out-of-equilibrium state evolves towards equilibrium. However, this is not always the case. It was shown that disorder can induce localization in a variety of interacting systems6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21, a phenomenon dubbed many-body-localization (MBL), as a generalization of the well known Anderson localization proposed by P. W. Anderson about fifty years ago22,23,24.

The intriguing phenomenon of ergodicity break in many-body localized systems has motivated great effort in the scientific community to understand the mechanisms that lead to MBL25,26,27,28. Indeed, the phenomena of localization in interacting quantum systems is a puzzling problem still under study29,30,31,32,33,34,35,36,37,38. From a theoretical point of view, it is well established that the breakdown of the ETH in disorder induced many-body localized systems can be captured by the level-spacing statistics2,5,30,39or entanglement properties of their eigenstates10,15,40. The ETH usually understood in two classes: strong sense, in which all states thermalize and weak sense, where almost all states thermalize. The main features of disorder-induced MBL phases have been observed in various experimental platforms41,42,43,44,45. In spite of the complexities in the MBL phenomena, theoretical and experimental results suggest that, at least in one dimension, strong disorder induces the emergence of nearly conserved local quantities, leading to integrability46,47,48,49,50.

Recently, Schulz et al. showed that localization can occur in interacting systems even in the absence of disorder51. The key ingredient here is the presence of a nearly uniform gradient potential in an interacting system and this is closely tied to the single-particle localization process known as Wannier-Stark localization52.This phenomenon was called Stark Many-Body Localization (SMBL), and it shares similarities with the traditional MBL, such as level statistics, but differs in aspects such as a strong dependence on the initial conditions, as shown in53for a spin 1/2 Heisenberg chain. The mechanism leading to ETH violation in SMBL was believed to be Hilbert space fragmentation53,54,55,56,57, in which the Hilbert space fragments into many disconnected subspaces, preventing themalization. However, this argument has been questioned58. Striking experiments in quantum simulators were performed to demonstrate Stark MBL59,60 and confirm that localization can arise in disorder-free systems.

In a recent study, one of us has extended the results of Ref53. by including exchange interaction up to second nearest neighbors61. It was shown that the SMBL phenomenon is robust for arbitrary ratio \(J_2/J_1\), a feature also found in spinless fermions with long-range interactions62. SMBL has been also investigated in the context of bosonic lattices63,64and it’s worth noting that other mechanisms can give rise to MBL, such as quasi-periodic potentials65,66and periodic driving67,68.

Nonetheless, it is remarkable that all existing studies on spin lattices have exclusively focused on SMBL in spin 1/2 systems, which does not allow interactions between the systems’ elementary excitations, kinks or spinors. Interactions arise naturally, in systems with spin \(S\ge 1\), for instance, magnon-magnon interactions are present in spin\(-1\) FM chains, giving rise to a nonlinear term \(D\sum _{j=1}^L \left( S_j^{z}\right) ^2\), where D represents the strength of uniaxial anisotropy, which is a fundamental phenomenon in magnetic materials, where the spins of electrons or ions exhibit a preference for certain orientations relative to the system’s axes. Anisotropy can arise from various sources, such as exchange interactions between spins, crystal symmetry, structural deformations, and confinement effects. Interestingly, weak ergodicity breaking in the form of quantum many-body scars (QMBS) appears in the spin\(-1\)XY model69. We should mention that investigations on magnon-bound states has been conducted examining the impact of a weak anisotropy (D much smaller than the exchange interaction J) on thermalization time scales of spin\(-1\)systems70,71. Notably, tunable single-ion anisotropy have been realized experimentally in trapped ions72, ultra-cold atoms73, and compounds such as \(\hbox {NiNb}_2\)\(\hbox {O}_6\)74 and [Ni(\(\hbox {HF}_2\))(\(3-\)Clpyradine)\(_4\)]\(\hbox {BF}_4\)75.

In this paper, we investigate localization of quantum states in a \(J_1-J_2\) model in the presence of a non-uniform magnetic field, and incorporating an additional term to address uniaxial single-ion anisotropy, which naturally arises in magnetic models with \(S \ge 1/2\)76,77,78. By employing numerically exact diagonalization, we calculate the temporal evolution of complementary quantifiers, namely imbalance, entanglement entropy, and participation entropy. These measures are commonly used to characterize the ergodic-to-MBL phase diagram in quantum many-body systems. Our results reveal that for some initial product states: (i) Similar to the case of \(S=1/2\) chain, MBL induced by finite \(h_0\) is also capable of localizing systems of spin \(S>1/2\); (ii) For \(S > 1/2\), uniaxial anisotropy alone localizes the system; (iii) If the system is localized for a particular value of \(h_0\), an anysotropy \(D \sim h_0\) suppresses localization and (iv) for \(D,h_0 \gg J\) the system localizes. These results suggest that the localization phenomena for both D and \(h_0\) are of the same nature. Indeed, they seem to result from a suppression of the spin dynamics by local energy constraint. Although our numerically exact method is limited by the system size, our results propose an alternative mechanism for inducing localization in quantum many-body systems. Additionally, our work provides insight into the nature of SMBL, which is currently a focus of extensive research, and lays the groundwork for future experimental exploration of these effects.

The rest of this paper is organized as follow: In Sect. 2 we introduce the model and numerical methods, in Sect. 3 we present our numerical results and discussions. Finally, our work is summarized in Sect. 4

Model and methods

To be more specific, we consider a chain of L interacting spins modeled by a \(J_1-J_2\) Hamiltonian, which also includes terms accounting for external magnetic field and single-ion interaction. Explicitly, the Hamiltonian can be written as,

$$\begin{aligned} H=J_1\sum _{j=1}^{L-1} \textbf{S}_j \cdot \textbf{S}_{j+1}+ J_2\sum _{j=1}^{L-2} \textbf{S}_j \cdot \textbf{S}_{j+2}+\sum _{j=1}^L \left( h_j+DS_j^{z}\right) S_j^{z}, \end{aligned}$$
(1)

where the first two terms of our model Hamiltonian represent the exchange interaction between neighboring spins, capturing the mutual influence of these adjacent entities. \(J_1\) and \(J_2\) define the coupling strength between nearest and next-nearest neighbors, respectively. In the last summation of Eq. (1), \(h_jS_j^{z}\) (with \(h_j=jh_0\)) introduces the effect of a non-uniform magnetic field along the z-direction, allowing the study of Stark-many-body localization. Lastly, D accounts for a single-ion anisotropy, which takes into account the directional preference of spins along the z-axis. This particular form of the Hamiltonian (1), provides a theoretical framework for investigating fundamental properties of the system for spin \(S=1/2\) and \(S\ge 1\) as well. We should mention that the effect of D can only be observed in the dynamics of system of spins \(S>1/2\). In the special case of \(S=1/2\), the effect of D is just to shift the spectrum of the Hamiltonian by an amount DL/4.

For convenience, following Burssil et al79. and our previous work61, we parameterize the exchange couplings \(J_1\) and \(J_2\) as \((J_1, J_2) = (J_0 \cos \theta , J_0 \sin \theta )\), with \(0 \le \theta < 2\pi\). As a result, the possible values for \(J_1\) and \(J_2\) form a circle in the \((J_1, J_2)\) plane, where \(J_0\) is the radius of this circle, representing the maximum magnitude of the exchange couplings. The different combinations of \(J_1\) and \(J_2\) depend on \(\theta\). As \(\theta\) varies, \((J_1, J_2)\) takes all possible values on the circle of radius \(J_0\), including both positive and negative values, allowing us to investigate the localization process of the state under different coupling strengths. The ground state properties for \(h_0 = \gamma = 0\) in Eq. (1) have received great attention over the past decades80,81. However, our main focus here is on thermalization processes, which potentially involve all eigenstates of the system consistent with the particular symmetry of the initial state. The absence of translational invariance due to a finite field gradient naturally leads us to adopt open boundary conditions. This choice allows us to explore the static and dynamic properties of the system in its entirety, at the cost of finite-size effects. Here, we aim to understand how the system thermalizes, i.e., how it evolves from a given initial out-of-equilibrium state towards a thermal equilibrium state. More specifically, we will cover initial states of the form \({|{\Psi _0}\rangle }=~\mid \cdots m,m,m,-m,-m,-m,\cdots \rangle\) in which \(m=S^z_j\). This particular type of states correspond to a state with a single domain wall, but states with multiple domain walls will also be considered. This type of initial state has already been analyzed in other studies where it was possible to unveil several interesting characteristics of Stark many-body localization phenomena53,61. Here we will explore mainly the case of \(S > 1/2\) by considering the effects of the single-ion anisotropy D and magnetic fields gradients \(h_0\) in order to gain a better understanding of both local and global effects over the collective behavior of the spin dynamics.

Time-evolution analysis

To study the phenomena of localization and thermalization in our closed quantum system, we adopt the Schrödinger representation and perform time evolution of a given initial state \({|{\Psi _0}\rangle }\equiv {|{\Psi (t=0)}\rangle }\) at \(t=0\) as \({|{\Psi (t)}\rangle }=U(t){|{\Psi _0}\rangle }\) where \(U(t)=\exp {\left( -iHt / \hbar \right) }\) is the time evolution operator. Having the time-evolved quantum state \({|{\Psi (t)}\rangle }\), we are able to calculate the relevant physical quantity that witnesses localization/thermalization phenomenon, such as imbalance, participation and entanglement entropies. Since they are in general individually inconclusive they may provide corroborative results.

Imbalance.  Imbalance, denoted as \({{\mathcal {I}}}(t)\), is a key measure in investigating the dynamics of quantum systems, particularly in the context of Heisenberg models51,53,59,82. It provides insights on whether the system retains or loses information about local magnetization as it evolves in time. We define the imbalance for a chain of L spins S as

$$\begin{aligned} {{\mathcal {I}}}(t)=\frac{1}{LS^2}\sum _{j=1}^L \langle S_j^z(t)\rangle \langle S_j^z(0)\rangle , \end{aligned}$$
(2)

where, \({\langle {S^z_j(t)}\rangle }=\langle \Psi (t) \mid S_j^z \mid \Psi (t) \rangle\) represents the expectation value of \(S^z\) of the j-th spin of the chain within the evolving quantum state \(\mid \Psi (t)\rangle\). As such, we clearly have \({{\mathcal {I}}}(0)=1\) if local spins are fully polarized, along either the positive or negative \(S^z\) projections. Moreover, in the regime of strong localization, \({{\mathcal {I}}}(t)\) remains close to its value at t=0, indicating that information about local magnetization is preserved during the evolution. On the other hand, in the thermal regime, \({{\mathcal {I}}}(t)\) tends to some lower value at long time scales. The precise asymptotic value of the imbalance in the thermal regime depends on the initial state. For instance, let us assume an initial product state \({|{\Psi _0}\rangle }\) as defined above for a chain of L spins, where \(N_{\uparrow }\) and \(N_{\downarrow }\) represent the number of spins with maximum and minimum \(S^z\), respectively. Thus, the total spin projection along the z-axis is given by \(S^z_{\textrm{tot}}=(N_{\uparrow }-N_{\downarrow })S\). Since in the thermal state, this quantity will be uniformly distributed along the chain, each site will exhibit a magnetization \(S^z_{\textrm{tot}}/L\), if the total \(S^z\) is conserved. Thus, based on Eq. (2), it is demonstrated that the thermal value of the imbalance is determined by

$$\begin{aligned} {{\mathcal {I}}}_{\textrm{thermal}}=\left( \frac{S^z_{\textrm{tot}}}{LS}\right) ^2. \end{aligned}$$
(3)

In particular, for an initial state in the sector \(S^z_{\textrm{tot}}=0\) we obtain \({{\mathcal {I}}}_{\textrm{thermal}}=0\). We should remark that, as defined in Eq. (2), the imbalance is not capable of detecting dynamics of spins that occur only on the xy-plane. In this case \({{\mathcal {I}}}(t)=0\), regardless the dynamics of the initial state.

Entanglement entropy.  Another important quantity to monitor the localization of quantum states is the well known entanglement entropy \(S_\ell (t)\) defined as

$$\begin{aligned} S_\ell (t)=-\frac{1}{\ln 2}\textrm{Tr}\left[ \rho ^{\ell }_A(t) \ln \rho ^{\ell }_A(t)\right] , \end{aligned}$$
(4)

where \(\rho ^{\ell }_A(t)=Tr_B[\rho ^\ell _B]\) is the reduced density matrix of a subsystem A. Here, given the 1D nature of a chain, \(\ell\) simply represents the site that defines the interface between two portions of the chain which we call subsystem A and B.

While the imbalance gives qualitative local data that is good for describing the dynamics of individual states, the use of entanglement entropy for many-body systems allows for the investigation of their collective and emergent properties, providing insights into the nature of quantum correlations and how information distributes throughout the entire system83.

Participation entropy. Participation entropy (PE) has gained attention in studies of dynamics of many-body systems. Here we follow Ref52. and defined this quantity as

$$\begin{aligned} S_2 = -\ln \left( \sum _{mn}|c_n |^2 |c_m |^2\delta _{E_n,Em}\right) , \end{aligned}$$
(5)

where \(c_n={\langle {n}|}{{\Psi _0}\rangle }\) is the projection of the initial state \({|{\Psi _0}\rangle }\) onto the eigenbasis \(\{{|{n}\rangle }\}\). For practical purpose, to calculate \(S_2\) we first define the quantity \(f(t)={\langle {n}|}\exp (-iHt/\hbar ){|{\Psi _0}\rangle }\), which is the projection of the time evolved initial state onto the eigenstate \({|{n}\rangle }\). Upon Fourier transforming f(t) to the energy domain, we obtain \(F(\omega )\). The inverse participation ratio (IPR) can be obtained as

$$\begin{aligned} IPR=\int F^2(\omega ) d\omega =\sum _{mn}|c_n |^2 |c_m |^2\delta _{E_n,Em}. \end{aligned}$$
(6)

With this, the participation entropy (5) is just \(S_2=-\ln (IPR)\). For the non degenerate case, this quantity coincides with the quantity \(S_q\) for \(q=2\)defined in Ref84.. Defined as such, \(S_2\) provides a measures of how the projections of the initial state are distributed among the eigenstates of the Hamiltonian. In the absence of symmetries, for a delocalized state, \(S_2 \sim \ln ({{\mathcal {N}}})\), where \({{\mathcal {N}}}\) is the dimension of the Hilbert space. This increases logarithmicaly with the system size. On the other hand, in a strongly localized state, only a limited subset of coefficients contributes significantly to the sum, resulting in a constant PE \(S_2\). In other words, the state is fully spanned on a restricted region of the Hilbert space, indicating the presence of localization.

The quantity denoted as \(S_q\)as defined in Ref84. represents an alternative for characterizing participation entropy. However, in this approach, it is not necessary to perform dynamics on the system; instead, one ultimately diagonalizes the Hamiltonian.

$$\begin{aligned} S_q = \frac{1}{1-q}\ln \left( \sum _{\alpha =1}^{\mathcal {N}}{|\psi _\alpha |}^{2q}\right) , \end{aligned}$$
(7)

where \(\psi _\alpha\) is the projection of a given state \({|{\Psi }\rangle }\) onto the spin basis \(\{{|{\alpha }\rangle }\}\) and \(\mathcal {N}\) is the dimension of the Hilbert space. The participation entropy \(S_q\), as defined above, provides information on how the coefficients of a given state are distributed among the different basis states. Here we consider \({|{\alpha }\rangle }\) as an eigenvector of the Hamiltonian. Thus, \(S_q(t)\) measures how the projections of the initial state are distributed among the eigenvectors of the Hamiltonian. Similar to \(S_2\), for the case of thermalization, we have \(S_q \sim \ln \mathcal {N}\), and in the localized case, it will be constant.

Numerical results

To show our numerical results for the time evolution of the initial many-body state, we first set \(J_0=1\) as energy unit and assume \(\hbar =1\), so that time has units of inverse energy. Since the system is closed, exact numerical unitary evolution can be performed for small chains. To this end, here we use the python package Quspin85,86 that allows us to perform time evolution and calculation of all the physical quantities we need. Since the Hilbert space increase as \((2S+1)^L\), where L is the length of the chain, for a spin-1/2, for instance we can handle chains up to \(L=22\) spins or so, depending on the symmetry of the relevant sector of the Hilbert space.

Spin-1/2 system: effect of magnetic field

For the sake of completeness, we reproduce and elaborate on the main results obtained by one of us in Ref61. for spin-1/2 SMBL. We validate that the fundamental characteristics are qualitatively well captured at a scale involving 12 and 20 spins. In Fig. 1 we use imbalance and entanglement entropy to show how the system evolves from a thermalized to localized regime as \(h_0\) increases. We use an initial state within the Hilbert subspace with \(S_{\textrm{tot}}^z=0\) consisting of an island of spin up at the center of the chain. For a given value of \(h_0\), we perform time evolution up to \(J_0 t = 500\), just as in Ref53,61,87,88., where this time scale was found to be sufficient to capture the physics of the evolved system. For consistency, we maintain this time scale for the remaining simulations throughout the work. Since both \({{\mathcal {I}}}(t)\) and \(S_\ell (t)\) oscillate around a fixed value for large t, we take their respective time average, \(\bar{{\mathcal {I}}}\) and \({\bar{S}}_\ell\), for \(J_0t\in [400,500]\).

Figure 1
figure 1

Localization in a spin 1/2 chain, for different values of \(h_0\) and \(\theta\) with \(D = 0\), using the average of the imbalance \(\bar{{\mathcal {I}}}\) vs \(h_0\) and the average entanglement entropy \({\bar{S}}_\ell\) vs \(h_0\). The system is configured in the space \(S_z = 0\) with \(L = 12\) and \(L=20\) with a magnetization island of 1/2 in the middle of the chain and \(-1/2\) at the ends.

Full size image

In Fig. 1(a) and 1(b), \(\bar{{\mathcal {I}}}\) as a function of \(h_0\) is depicted for system sizes \(L=12\) and \(L=20\), respectively, for various values of \(\theta\). These different values of \(\theta\) covers distinct ratios \(J_2/J_1\), including their signs, which is important when discussing the effect of the anisotropy D. We observe that \(\bar{{\mathcal {I}}} \approx 0\) for \(h_0 = 0\), but increases with increasing \(h_0\) reaching almost the unity for \(h_0=2\) for all values of \(\theta\). The vanishing \(\bar{{\mathcal {I}}}\) for small \(h_0\) reveals that the local information of the initial state is lost at long time through the unitary dynamics, while large value of \(\bar{{\mathcal {I}}}\) indicates that the information of the initial state is kept local at long times. There is an interesting qualitative agreement between the results for \(L=12\) and \(L=20\), which shows that localization can be observed for relatively small chains. Similarly, the entanglement entropy shown in Fig. 1(c) and 1(d) (for \(L=12\) and \(L=20\), respectively) confirms localization in the system as \(h_0\) increases. Since the initial state is a product state, with entanglement entropy \(S_{L}(t)=0\), large values of \({\bar{S}}_L\) for \(h_0\rightarrow 0\) indicate that the system becomes entangled, signifying the spread of information throughout the entire system in the thermalized regime. Conversely, vanishing \({\bar{S}}_L\) in the localized regime for large \(h_0\) confirms that information is kept local at long times. It is noteworthy to mention a symmetry pertaining to the imbalances \(\theta\) and \(\theta +\pi\), as previously highlighted in Ref61..

In what follows we will turn our attention to systems with \(S>1/2\) on which D exerts non-trivial effects. The natural progression leads us to the case of \(S=1\). However, in this instance, flipping a given spin from \(S_z=-1\) to \(S_z=+1\) requires passing through \(S_z=0\), which remains inert to the magnetic field along the z-direction. To explore the interplay between \(h_0\) and D, the case of \(S=1\) does not encompass the most general scenario. Consequently, we shift our focus to the case of \(S=3/2\), which presents a richer set of dynamics. Further details regarding the \(S=1\) case can be found in the supplemental material.

Spin-3/2 system: separate effects of magnetic field and single-ion anisotropy

Effect of magnetic field alone.  The \(S=3/2\) case offers a much richer Hilbert space as compared to \(S=1/2\). In this case each spins exhibits \(S^z = (3/2,1/2,-1/2,-3/2)\) where the two local subspaces defined by \(|S_j^z|=1/2\) and \(|S_j^z|=3/2\) are both active to external magnetic field gradient \(h_0\) and anisotropy D. This leads to more complex dynamics of the system towards thermalization. We will first study the localization in the presence of \(h_0\) and contrast it with the case of \(S=1/2\) and then investigate the effect of D alone. In a third stage we will study the interplay between \(h_0\) and D in the dynamics of the system. Following the same strategy used before, we will analyze the time-evolution for an initial state where there is an island of L/2 spins with \(S_j^z=-3/2\) centered at L/2 while the rest of the system has \(S_j^z=3/2\). Similar to the initial state we used for \(S=1/2\), this is a non-equilibrium product state and therefore non-entangled.

The effect of \(h_0\) in the time evolution of this state is shown in Fig. 2(a)-2(d) for \(h_0=0\), \(h_0=2.0\), \(h_0=3.0\) and \(h_0=4.0\), while keeping \(D=0\). In Fig. 2(a), corresponding to the case \(h_0 = 0\), the Hamiltonian of the system is exclusively governed by the exchange terms. We observe rapid thermalization in the system, leading to the decay of \({{\mathcal {I}}}(t)\). When the magnetic field gradient is increased to \(h_0 = 2\), as depicted in Fig. 2(b), the system continues to undergo thermalization for all angles, albeit at a slower pace for every \(\theta\). Notably, \({{\mathcal {I}}}(t,\theta )={{\mathcal {I}}}(t,\theta +\pi )\), a symmetry previously noted for spins \(S=1/2\). Fig. 2(c) and 2(d) exhibit the results \(h_0=3\) and \(h_0=4\), respectively. In both situations the imbalances remain close to unity, indicating localized states. The observed persistence of these imbalances in systems with spin 3/2 strongly suggests the occurrence of SMBL in large-spin chains.

Figure 2
figure 2

Imbalance as a function of time for the stark many-body localization (for increasing \(h_0\) and \(D=0)\) [panels (a)-(d)] and for \(h_0=0\) and increasing D [panels (f)-(h)]. Different curves corresponds to different values of \(\theta\) (see legend). The system consists of \(L=12\) spins and the initial state corresponds to an island of of spins with \(S^z_j=3/2\) in the middle of the chain, while all other have \(S^z_j=-3/2\).

Full size image

Effect of single-ion anisotropy We now turn our attention to the impact of single-ion anisotropy on the system dynamics, utilizing the same initial states as in the previous analysis. In Fig. 2(e)-2(h) we set \(h_0=0\) and present the dynamics of \({{\mathcal {I}}}(t)\) for \(D>0\). Specifically, Fig. 2(e) displays the results for \(D=0.5\) across various values of \(\theta\). All the curves exhibit rapid thermalization, except in the specific case of \(\theta =0\), where the imbalance remains finite for an extended duration, up to \(J_0t=500\). This observation strongly indicates that anisotropy has the potential to induce localization, with a notable dependence on the angle \(\theta\). Indeed, for \(D=4\), as shown in Fig. 2(h), the largest D considered here, the system remains localized, as all imbalances remain close to unity. The localization observed for large D can be easily understood by observing that for \(D\gg J_0\), the initial state becomes almost an eigenstate of the Hamiltonian (1) (for \(D \gg J_0\)), apart from some very small perturbation introduced by \(J_1\) and \(J_2\). In this scenario, localization is trivially achieved. The intriguing situation arises in the competing regime where \(D\sim J_0\). Let us take a closer look at Fig. 2(f) (where \(D=2\)) in more detail. Observe that the curves for \(\theta =0\), \(\theta =\pi /3\) and \(\theta =5\pi /3\) remains close to unity, indicating strong localization. Conversely, for \(\theta =2\pi /3\), \(\theta =\pi\) and \(\theta =4\pi /3\) the system undergoes thermalization. Note that \(J_2=0\) for both \(\theta =0\) and \(\theta =\pi\), yet the dynamics is entirely distinct. This difference can be attributed solely to the sign of \(J_1\). In other words, thermalization is primarily governed by the nearest-neighbor spin-flip processes facilitated by \(J_1\). To comprehend the mechanisms leading to localization (thermalization) \(J_1>0\) (\(J_1<0\)) at large t, let us analyze the early stages of the demolishing of the domain wall at a given interface between spins \(S^z=+3/2\) and \(S^z=-3/2\). The very first process that destroys the interface is given by the term \(J_1S_i^-S_{i+1}^+\) acting on the interface \(\mid{\cdots}3/2, 3/2, -3/2, -3/2, {\cdots}\rangle\), resulting in an smoother interface \(\mid{\cdots}3/2, 1/2, -1/2, -3/2, {\cdots}\rangle\). Due to finite D, there is a cost associated with smoothing the domain wall, as the local energy in sites with \(S_j^z = \pm 1/2\) significantly differs from that in sites with \(S_j^z = \pm 3/2\). This energy difference acts as a barrier that can only be surpassed through virtual processes during the system’s time evolution.

Origin of localization promoted by single-ion anisotropy. To gain a deeper understanding of the physical origin of this localization, one can explore the limit of very large |D| or, equivalently, \(J_0\rightarrow 0\). For simplicity, let us assume \(D>0\). When \(J_0=0\), our initial state is an eigenstate of the Hamiltonian (1) of highest energy, since all spins have the maximum \(|S^{z}_j|\). In fact, any product state constructed from the \(S^z\) basis is an eigenstate of the Hamiltonian for \(J_0=0\). The spectrum of the Hamiltonian consists of a series of delta-peaks among which the initial state corresponds to one of the states of highest energy (for \(D>0\)). For the sake of clarity, in Fig. 3(a) and 3(b) we illustrate the energy cost associated with placing a spin with different \(S_j^z\) along the chain, considering a constant-gradient magnetic field \(h_0\) and an anisotropy D, respectively. Figure 3(c) represents the entire Hilbert space of the system, with the white region denoting the sector of the Hilbert space where the states have spins fully aligned along the z-direction. Consider the simple case of the initial state \(\mid{\cdots}-3/2, -3/2, 3/2, 3/2, {\cdots}\rangle\). This state exhibits a single domain wall, represented by the red line within the sector of the Hilbert space indicated by the white region. The point where the curve touches the border of the white region represents the domain wall. At \(t=0^+\), only the region represented by the red line is active. The purple line indicates a state with a double wall, featuring two active regions. In the vicinity of the white sector, there are states where the two spins around the wall have projections \(\pm 1/2\), such as the state \(\mid{\cdots}-3/2, -1/2, 1/2, 3/2, {\cdots}\rangle\). One can readily confirm that for \(J_0=0\), this state is trivially localized, but for finite \(J_0\), its temporal evolution becomes highly complex. Consider that at \(t=0\), the system is prepared in the initial state with a single domain wall. At \(t=0^+\) the only active region of the state is around the domain wall, since H acting on this state modifies only the spins in the vicinity of the domain wall. Now, upon time evolution, in order to reach a thermal equilibrium, the mechanism involves states with \(S_j^z\pm 1/2\) in the vicinity of the wall (represented by the dark gray region). It turns out that, for \(D>0\) theses intermediate states have of much different energies for \(D \gg J_0\). Therefore, for a given \(J_0\) fixed, by increasing D these sectors become isolated from the rest of the Hilbert space.

Figure 3
figure 3

Effect of a linearly varying magnetic field (a) and a constant anisotropy (b) on the local energy of a spin-\(S=3/2\) chain. (c) Representation of the Hilbert space and typical initial state within the Hilbert space for a spin-3/2 chain. White region encloses only states in which all spins have their spin pointing along the z-axis, i.e., their projections are \(S_j^z=\pm 3/2\). Darker region contains states in which some of their spins have projections \(S^z_j=\pm 1/2\). The red and violet curves represent an initial state with single and double domain walls, respectively. The dots marks the active regions upon time \(t=0^+\) in the time evolution operator \(\exp (-iH 0^+ / \hbar )\).

Full size image

In Fig. 4(a)-4(d), we present the density of states for a system of \(L=8\) spins \(S=3/2\) for various values of \(J_0\) and fixed \(D=0.5\) and \(\theta =0\). For \(J_0=0\), in Fig.  4(a), the initial state precisely corresponds to one of the eigenstates associated with the rightmost degenerate energy peak. As \(J_0\) increases, these peaks hybridize with other states, resulting in peak splitting. For instance, at \(J_0=0.1\), the peaks are split due the the hybridization with the rest of the band. Therefore, the effect of \(D>0\) is evident in pushing the initial state to higher energy, creating a ”gap” between this state and the rest. Consequently, the entire white region of the Hilbert space, as illustrated in Fig. 3(c), becomes isolated from the rest. In contrast to QMBS, where the Hilbert space fragmentation is driven by emergent conserved quantities, the observed suppression of thermalization in this system arises from the inhibition of spin flip processes due to energy constraints.

Figure 4
figure 4

Density of states as a function of energy for \(J_0=0\) (a), \(J_0=0.1\) (b), \(J_0=0.2\) (c) and \(J_0=1.0\) (d). We have fixed \(D=0.5\) and \(\theta =0\). For all panels, we use the subspace of \(S_z=0\), with the spin-up magnetization island in the middle for L = 8.

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The picture drown above is useful to understand why for a moderate value of D, such as \(D=2\) and \(J_0=1\) shown in Fig. 2(f), by changing \(\theta =0\) to \(\theta =\pi\) (or equivalently \(J_0\) to \(-J_0\)), the system goes from localized to thermalized. The reason is that the spectrum is not symmetric when \(J_1\rightarrow -J_1\). In fact, for \(\theta =0\) the spectrum extends more towards the negative side of the energy axis. The opposite occurs for \(\theta =\pi\). Therefore, for a D fixed, the energy “gap” between the sector of the initial state is larger for \(\theta =0\), thus localization is obtained more easily. The evidence that increasing Dsuppresses dynamics of spins around the domain walls, places the localization mechanism here in the same perspective of the SMBL. As discussed in Ref58., in SMBL, only for infinite \(h_0\) the spins are completely frozen by “local energy” constraints. Likewise, here, the spin dynamics will be fully frozen only at infinite D. Nevertheless, our results suggest localization already for \(D \sim 2J_0\) as in Fig. 2(b).

aSince the demolishing of domain walls of the initial states occurs because of the active regions around the interfaces, let us analyse how the number of walls modifies the localization/thermalization. Figure 5 shows the time evolution of the imbalance starting at initial states containing one, two, three, four and five domain walls (all states within the zero total magnetization). Again, we use a chain of \(L=12\) spins \(S=3/2\) and set \(h_0=0\). Panels 5(a) and Fig. 5(b) show the case of \(D=0\) and \(\theta =0\) and \(\theta =2\pi /3\). For this we observe thermalization for all initial states shown. Now, for \(D=4\) (lower panels), while for \(\theta =0\) the system is still localized, as shown in Fig. 5(c), for \(\theta =2\pi /3\) localization becomes poorer as the number of walls increases. For instance, for \(\theta =2\pi /3\), note that for four and five interfaces the system thermalizes, while it remains localized for one and two walls. This is because, the number of active regions increases with the number of domain walls as illustrated in Fig. 3(a), thus favoring thermalization. This shows that the larger is the number of interfaces, the larger is the value of D and \(h_0\)necessary to produce the same localization, similar to what was previously reported in Ref53. for SMBL.

Figure 5
figure 5

Time evolution of imbalance for different initial conditions (different curves). Upper and lower panels correspond to \(D=0\) and \(D=4\), while left and right panels correspond to \(\theta =0\) and \(\theta =2\pi /3\). In all panels, we use a chain of spin 3/2 and \(L = 12\) and \(h_0=0\). All initial states lie within the \(S_z=0\) subspace.

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Interplay between anisotropy and magnetic field

So far we have analysed separately the SMBL, induced by the magnetic field gradient, and the localization induced by anisotropy. Let us now address the interplay between these two terms of the Hamiltonian. In Fig. 6(a)-6(d) we repeat the same calculation of Fig. 2, but now for both \(h_0\) and D finite. Figure 6(a) shows the imbalance for \(h_0=D=2\). We observe that while localization were observed for some values of \(\theta\) for \(D=2\) and \(h_0=0\) shown in Fig.2(f), now localization is no longer observed for any value of \(\theta\). Interestingly, if we make \(h_0=2D=4\), as shown in Fig. 6(b), localization is again recovered. However, for \(D=2h_0=4\) [Fig. 6(c)], localization is observed only for some values of \(\theta\), in fact for those closer to \(\theta =0\) and \(\theta =2\pi\). Finally, for \(D=h_0=4\), shown in Fig. 6(d), all the curves approach to unity at large t. All these behaviours are in agreement with the corresponding curves for entanglement entropy shown in panels 6(e)-6(h). Despite the complexity of the behavior of the various curves, results indicate that for moderate values of \(D\approx h_0\), they compete against each other and fail to localize the system. However, if both are much larger than \(J_0\) localization can be recovered again. The competition between D and \(h_0\) observed here can be understood as follows: the energy price that strongly suppresses spin dynamics around the domain wall for D finite (and \(h_0=0\)) is now, to some extent, compensated by finite \(h_0\). Likewise, the suppression of the dynamics for finite \(h_0\) (and \(D=0\)) is now energetically relaxed if D is comparable to \(h_0\).

Figure 6
figure 6

Time evolution of imbalance (a)-(d) and entropy (e)-(h) for different values of \(h_0\) and D. The system and initial condition here are the same as the ones used in Fig. 2. The curves correspond to different values of \(\theta\) (see legend).

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In Fig. 7(a) we show the participation entropy as function of \({{\mathcal {N}}}\) (the dimension of the Hilbert space) for \((D,h_0)=(0,0)\) (red circles), \((D,h_0)=(8,0)\) (black squares), and \((D,h_0)=(0,8)\) (cyan triangles). Here we use an initial state consisting of chains of even number of sites from \(L=4\) to \(L=14\). The initial states are of the type \(\mid{\cdots}-3/2, 3/2, 3/2, -3/2, {\cdots}\rangle\). For all system sizes, the initial state contains an island of only two spin \(S_j^z=3/2\). For \((D,h_0)=(0,0)\) we note the \(S_2\) increases linearly with \(\ln {{\mathcal {N}}}\), as expected for an ergodic regime. In contrast, for both cases \((D,h_0)=(0,8)\) and \((D,h_0)=(8,0)\), \(S_2\) remains nearly constant. This indicates that for the ergodic regime, the number of eigenstates that participates on the dynamics increases with the dimension of the Hilbert space. In the localized regime, on the other hand, only a limited portion of the eigenstates contributes to the dynamics.

Figure 7
figure 7

(a) Participation entropy vs \(\ln {{\mathcal {N}}}\) for \((D,h_0)=(0,0)\) (red circles), \((D,h_0)=(8,0)\) (black squares) and \((D,h_0)=(0,8)\) (cyan triangles). Dashed lines corresponds to a linear fitting of the curves and serves to guide the eyes. To change the size of the Hilbert space, L includes all even numbers from 4 to 14. (b) and (c) show, respectively, latter time average of imbalance and entropy as function of D (for \(h_0=0\)) (red squares) and of \(h_0\) (for \(D_0=0\)) (blue triangles) for \(L=12\). For all panels, we set \(\theta =0\) and use an initial state of type \(\mid{\cdots}-3/2, 3/2, 3/2, -3/2, {\cdots}\rangle\), which corresponds to an island of spins \(S_j^z=3/2\) in the middle of a chain of spins \(S_j^z=-3/2\).

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Figure 7(b) shows \(\bar{{\mathcal {I}}}\) vs D for \(h_0=0\) (green squares) and \(\bar{{\mathcal {I}}}\) vs \(h_0\) for \(D=0\). We use a chain of \(L=12\) and the same initial state as in Fig. 7(a) and set \(\theta =0\). Again, \(\bar{{\mathcal {I}}}\) is the average of \({{\mathcal {I}}}(t)\) for \(t \in [400,500]\). We note that both curves exhibit very similarly evolution from a thermalized (\(\bar{{\mathcal {I}}}\approx 0.5\)) to a localized (\(\bar{{\mathcal {I}}}\approx 1\)) regime as D and \(h_0\) increases from zero to 8. Interestingly, the crossover occurs for similar values of D and \(h_0\). This is corroborated by the entropy \(\bar{S}_L\) depicted in Fig 7(c), where it is evident that \({{\bar{S}}}_L\) vanishes if either \(h_0\) and D increase, indicating the localization of the states.

To further elucidate the numerical results, we employ the Holstein-Primakoff (HP) transformation; mapping spin operators in a lattice system with spin\(-S\) to bosonic operators: \(S_{j}^{z}=(S-n_{j})\), \(S_{j}^{+}=\sqrt{2S-n_{j}}a_{j}\), and \(S_{j}^{-}=a_{j}^{\dagger }\sqrt{2S-n_{j}}a_{j}\), where \(a_{j}^{\dagger }(a_{j})\) is a bosonic creation (annihilation) and the number \(n_{j}=a_{j}^{\dagger }a_{j}\) operator at site j. Subsequently, a 1/S expansion enables us to derive an effective description of the quantum dynamics, expressed as \(H_{HP}=H_{G}+H_{1}+H_{2}+\mathcal {O(}1/S)\). The term \(H_{G}\) characterizes the classical ground-state energy, while \(H_{1}\) represents the magnon dispersion, both contingent on the parameters \(\left\{ J_{1},J\,_{2},D,h_{0}\right\}\), yet not influencing localization. The leading contribution for comprehending the interplay between \(h_{0}\) and D emerges from \(H_{2},\) since this term and higher-order ones, describe magnon-mangnon interactions. To simplify, we consider only nearest-neighbor interaction, i.e., \(J_{1}=J>0,J_{2}=0\) and \(D>0\). The HP representation leads to the effective Hamiltonian

$$\begin{aligned} H_{\textrm{HP}} & \approx \sum _{j=1}^{L}\left( 2DS+2JS-h_{j}\right) n_{j} -JS\sum _{j=1}^{L-1}(a_{j}^{\dagger }a_{j+1}+H.c.)\nonumber \\ & +\frac{J}{8} \sum _{j}\left[ \left( a_{j}^{\dagger }\right) ^{2}a_{j}a_{j+1}+\left( a_{j+1}^{\dagger }\right) ^{2}a_{j+1}a_{j}+H.c.\right] \nonumber \\ & +D\sum _{j}^{L}n_{j}(n_{j}-1)+\frac{J}{2}\sum _{j=1}n_{j}n_{j+1}+H_{G} . \end{aligned}$$
(8)

The form of \(H_{\textrm{HP}}\) is similar to the generalized Bose-Hubbard model in a tilted optical lattice, with on-site chemical potential \(\mu _{j}=2DS+2JS-h_{j}\)(see Ref63.). The single-ion anisotropy D is mapped into a nonlinear repulsive on-site interaction U, symbolizing the additional energy needed to accommodate more than one boson at each location. In the context of a tilted lattice, where \(\mu _{j}=jA\), in which Arepresents the strength of the tilt, the standard Bose-Hubbard model exhibits SMBL63,64, closely resembling the behavior observed in our numerical results. Specifically, at large values of \(\ U\), high-energy states exhibit localized features at all values of A. This behavior is inherently connected to the segmentation of the Hilbert space into subbands, a characteristic that becomes more pronounced in regions with strong interactions. It leads to the freezing of dynamics on long time scales for initial states above a certain energetic threshold89. Another parallel emerges between U and \(\mu _{j}\), where an increase in U corresponds to an increase in the critical Stark field \(A_{c}\) necessary for the manifestation of the localization phenomenon. Additionally, the symmetry noted in our results, encompassing both positive and negative single-ion anisotropy D, is reflected in the bosonic scenario as an exact symmetry between two Hamiltonians with the sign of U changed and the energy ordering of eigenvalues and eigenvectors reversed.

Conclusions

In summary, we investigated the dynamics of a closed spin chain modeled by a \(J_1-J_2\) Hamiltonian in the presence of a finite-gradient magnetic field \(h_0\), which induces SMBL. An additional term accounting for single-ion uniaxial anisotropy D was included. Using exact numerical calculations, we computed various quantities such as imbalance and entanglement entropy, commonly used to probe localization phenomena. Our results indicate that the known SMBL observed in \(S=1/2\) chains also occurs for \(S>1/2\). Similarly, in the absence of a magnetic field, anisotropy alone can induce localization in the system for \(S>1/2\), particularly for initial product states where spins have maximum \(S_j^z\) projection.

Our findings suggest that the localization induced by \(D\) arises from the suppression of spin dynamics across the well-defined domain walls of the initial states. This suppression acts as an energy barrier imposed by anisotropy, which must be overcome to scramble spins from an initial product state where all spins have maximum or minimum \(S^z_j\) projections. While the sectors of the Hilbert space containing these families of product states do not completely decouple from the rest during time evolution, they become progressively isolated as \(D\)increases. This phenomenon bears similarities to what Zisling et al. discussed in Ref58. in the context of many-body localization (MBL). Moreover, we found that for \(D \approx h_0 \gtrsim J_0\), the magnetic field and anisotropy compete to localize the system, thereby favoring delocalization. We interpret this as an interplay between the energy costs incurred to melt the domain walls. In the competing regime, the energetic cost imposed by \(D\) is somewhat compensated by the cost associated with \(h_0\).

In our \(J_1\)-\(J_2\) model, the angle \(\theta\) is used to parameterize the relative strength of the couplings \(J_1/J_2\), including their sign. As shown it plays an important role in determining the localization and thermalization dynamics: stronger localization is observed for \(\theta =0\), \(\theta =\pi /3\), and \(\theta =5\pi /3\), while thermalization occurs for \(\theta =2\pi /3\), \(\theta =\pi\), and \(\theta =4\pi /3\). This difference can be attributed to the sign of \(J_1\), with thermalization primarily governed by nearest-neighbor spin-flip processes facilitated by \(J_1\).

Although our exact numerical calculations currently apply to few-site systems, we assert that these initial findings are relevant for understanding the dynamics of closed many-body quantum systems more broadly. Specifically, they contribute to our understanding of the connections between Hilbert space fragmentation and disordered free localized systems. Importantly, our results do not rule out the possibility of a gradual disappearance of localized regions within the Hilbert space in larger systems or in the long-time limit. Addressing these questions requires further exploration, both experimentally, using probes commonly employed in conventional MBL studies, and theoretically, through investigations that extend, for example, the theory of dynamical l-bits90 to spin lattices with \(S \ge 1\). It is well known that localization observed in disorder-free systems typically encounters significant finite-size effects91,92, where perturbative arguments often fail. To provide a preliminary qualitative assessment of these effects in our system, we present in the supplemental material the localization quantifiers, imbalance and entanglement entropy, for the system sizes feasible with our computational resources. Finally, it is worth noting that the models examined in this study can be experimentally realized using current quantum simulators, including cold atoms, trapped ions, and superconducting qubits.