Facets of Many-body Localization

Many body localization (MBL) has emerged as a powerful paradigm for understanding non-equilibrium quantum dynamics. Folklore based on perturbative arguments holds that MBL only arises in systems with short range interactions. Here we advance non-perturbative arguments indicating that MBL can arise in systems with long range (Coulomb) interactions. In particular, we show using bosonization that MBL can arise in one dimensional systems with ~ r interactions, a problem that exhibits charge confinement. We also argue that (through the Anderson-Higgs mechanism) MBL can arise in two dimensional systems with log r interactions, and speculate that our arguments may even extend to three dimensional systems with 1/r interactions. Our arguments are `asymptotic' (i.e. valid up to rare region corrections), yet they open the door to investigation of MBL physics in a wide array of long range interacting systems where such physics was previously believed not to arise.

Many-body localization (MBL) features are studied here for a large spin chain model with long range interactions. The model corresponds to cold atoms placed inside a cavity and driven by an external laser field with long range interactions coming from rescattering of cavity photons. Earlier studies were limited to small sizes amenable to exact diagonalization. It is shown that nonergodic features and MBL may exist in this model for random disorder as well as in the presence of tilted potential on experimental time scales also for experimentally relevant system sizes using tensor networks algorithms.

We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of random interactions ${V}_{ij}$ decaying as power-law ${V}_{ij}/{({r}_{ij})}^{\ensuremath{\alpha}}$ with distance ${r}_{ij}$. We demonstrate that MBL survives even for $\ensuremath{\alpha}<1$ and is preceded by a broad nonergodic subdiffusive phase. Starting from parameters at which the short-range interacting system shows an infinite temperature MBL phase, turning on random power-law interactions results in many-body mobility edges in the spectrum with a larger fraction of ergodic delocalized states for smaller values of $\ensuremath{\alpha}$. Hence, the critical disorder ${h}_{c}^{r}$, at which ergodic to nonergodic transition takes place, increases with the range of interactions. Time evolution of the density imbalance $I(t)$, which has power-law decay $I(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\gamma}}$ in the intermediate to large time regime, shows that the critical disorder ${h}_{c}^{I}$, above which the system becomes diffusionless (with $\ensuremath{\gamma}\ensuremath{\sim}0$) and transits into the MBL phase, is much larger than ${h}_{c}^{r}$. In between ${h}_{c}^{r}$ and ${h}_{c}^{I}$ there is a broad nonergodic subdiffusive phase, which is characterized by the Poissonian statistics for the level spacing ratio, multifractal eigenfunctions, and a nonzero dynamical exponent $\ensuremath{\gamma}\ensuremath{\ll}1/2$. The system continues to be subdiffusive even on the ergodic side ($h<{h}_{c}^{r}$) of the MBL transition, where the eigenstates near the mobility edges are multifractal. For $h<{h}_{0}<{h}_{c}^{r}$, the system is superdiffusive with $\ensuremath{\gamma}>1/2$. The rich phase diagram obtained here is unique to the random nature of long-range interactions. We explain this in terms of the enhanced correlations among local energies of the effective Anderson model induced by random power-law interactions.

Many-body localization in an $XY$ model with a long-range interaction is investigated. We show that in the regime of a high strength of disordering compared to the interaction an off-resonant flip-flop spin-spin interaction (hopping) generates the effective Ising interactions of spins in the third order of perturbation theory in a hopping. The combination of hopping and induced Ising interactions for the power law distance dependent hopping $V(R) \propto R^{-\alpha}$ always leads to the localization breakdown in a thermodynamic limit of an infinite system at $\alpha < 3d/2$ where $d$ is a system dimension. The delocalization takes place due to the induced Ising interactions $U(R) \propto R^{-2\alpha}$ of "extended" resonant pairs. This prediction is consistent with the numerical finite size scaling in one-dimensional systems. Many-body localization in $XY$ model is more stable with respect to the long-range interaction compared to a many-body problem with similar Ising and Heisenberg interactions requiring $\alpha \geq 2d$ which makes the practical implementations of this model more attractive for quantum information applications. The full summary of dimension constraints and localization threshold size dependencies for many-body localization in the case of combined Ising and hopping interactions is obtained using this and previous work and it is the subject for the future experimental verification using cold atomic systems.

As disorder strength increases in quantum many-body systems a new phase of matter, the so-called anybody localization, emerges across the whole spectrum. This transition is energy dependent, a phenomenon known as mobility edge, such that the mid-spectrum eigenstates tend to localize at larger values of disorder in comparison to eigenstates near the edges of the spectrum. Many-body localization becomes more sophisticated in long-range interacting systems. Here, by focusing on several quantities, we draw the phase diagram as a function of disorder strength and energy spectrum, for a various range of interactions. Regardless of the underlying transition type, either second-order or Kosterlitz-Thouless, our analysis consistently determines the mobility edge, i.e. the phase boundary across the spectrum. We show that long-range interaction enhances the localization effect and shifts the phase boundary towards smaller values of disorder. In addition, we establish a hierarchy among the studied quantities concerning their corresponding transition boundary and critical exponents. Interestingly, we show that deliberately discarding some information of the system can mitigate finite-size effects and provide results in line with the analytical predictions at the thermodynamic limit.

We investigate the many-body localization (MBL) transition in the quantum Ising model with long-range interactions. Unlike spin chains with short-range interactions, where the MBL transition point remains elusive due to strong finite-size effects and local fluctuations, long-range interactions suppress such fluctuations and enable clearer signatures of critical behavior. Using exact results from a related Bethe lattice localization problem, we estimate the MBL threshold within logarithmic accuracy and find consistency with exact diagonalization. Although the critical disorder diverges in the thermodynamic limit, our results demonstrate that the critical regime can still be probed and highlight the relevance of this model to systems with dipole-dipole, elastic, or indirect exchange interactions.

When a system thermalizes it loses all local memory of its initial conditions. This is a general feature of open systems and is well described by equilibrium statistical mechanics. Even within a closed (or reversible) quantum system, where unitary time evolution retains all information about its initial state, subsystems can still thermalize using the rest of the system as an effective heat bath. Exceptions to quantum thermalization have been predicted and observed, but typically require inherent symmetries or noninteracting particles in the presence of static disorder. The prediction of many-body localization (MBL), in which disordered quantum systems can fail to thermalize in spite of strong interactions and high excitation energy, was therefore surprising and has attracted considerable theoretical attention. Here we experimentally generate MBL states by applying an Ising Hamiltonian with long-range interactions and programmably random disorder to ten spins initialized far from equilibrium. We observe the essential signatures of MBL: memory retention of the initial state, a Poissonian distribution of energy level spacings, and entanglement growth in the system at long times. Our platform can be scaled to higher numbers of spins, where detailed modeling of MBL becomes impossible due to the complexity of representing such entangled quantum states. Moreover, the high degree of control in our experiment may guide the use of MBL states as potential quantum memories in naturally disordered quantum systems.

We numerically investigate the dynamics of entanglement in a chain of spinless fermions with nonrandom but long-range hopping and interactions, and with random on-site energies. For moderate disorder in the absence of interactions, the chain hosts delocalized states at the top of the band which undergo a delocalization-localization transition with increasing disorder. We find an interesting regime in this noninteracting disordered chain where the long-time entanglement entropy scales as $S(t) \sim \ln t$ and the saturated entanglement entropy scales with system size $L$ as $S(L,t \to {\infty}) \sim \ln L$. We further study the interplay of long-range hopping and interactions on the growth of entanglement and the many-body localization (MBL) transition in this system. We develop an analogy to higher-dimensional short-range systems to compare and contrast such behavior with the physics of MBL in a higher dimension.

Many-body localisation is studied in a disordered quantum spin-1/2 chain with long-ranged power-law interactions, and distinct power-law exponents for interactions between longitudinal and transverse spin components. Using a self-consistent mean-field theory centring on the local propagator in Fock space and its associated self-energy, a localisation phase diagram is obtained as a function of the power-law exponents and the disorder strength of the random fields acting on longitudinal spin-components. Analytical results are corroborated using the well-studied and complementary numerical diagnostics of level statistics, entanglement entropy, and participation entropy, obtained via exact diagonalisation. We find that increasing the range of interactions between transverse spin components hinders localisation and enhances the critical disorder strength. In marked contrast, increasing the interaction range between longitudinal spin components is found to enhance localisation and lower the critical disorder.

We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of long-range interactions or long-range hopping. Based on perturbative arguments there is a common belief that MBL can exist only in systems with short-range interactions and short-range hopping. We analyze effects of power-law interactions and power-law hopping, separately, on a system which has all the single particle states localized in the absence of interactions. Since delocalization is driven by proliferation of resonances in the Fock space, we mapped this model to an effective Anderson model on a complex graph in the Fock space, and calculated the probability distribution of the number of resonances up to third order. Though the most-probable value of the number of resonances diverge for the system with long-range hopping ($t(r) \sim t_0/r^\alpha$ with $\alpha < 2$), there is no enhancement of the number of resonances as the range of power-law interactions increases. This indicates that the long-range hopping delocalizes the many-body localized system but in contrast to this, there is no signature of delocalization in the presence of long-range interactions. We further provide support in favor of this analysis based on dynamics of the system after a quench starting from a charge density wave ordered state, level spacing statistics, return probability, participation ratio and Shannon entropy in the Fock space. We demonstrate that MBL persists in the presence of long-range interactions though long-range hopping with $1<\alpha <2$ delocalizes the system partially, with all the states extended for $\alpha <1$. Even in a system which has single-particle mobility edges in the non-interacting limit, turning on long-range interactions does not cause delocalization.

We study many-body localization (MBL) for interacting one-dimensional lattice fermions in random (Anderson) and quasiperiodic (Aubry-Andre) models, focusing on the role of interaction range. We obtain the MBL quantum phase diagrams by calculating the experimentally relevant inverse participation ratio (IPR) at half-filling using exact diagonalization methods and extrapolating to the infinite system size. For short-range interactions, our results produce in the phase diagram a qualitative symmetry between weak and strong interaction limits. For long-range interactions, no such symmetry exists as the strongly interacting system is always many-body localized, independent of the effective disorder strength, and the system is analogous to a pinned Wigner crystal. We obtain various scaling exponents for the IPR, suggesting conditions for different MBL regimes arising from interaction effects.

We provide evidence that a clean kicked Bose-Hubbard model exhibits a many-body dynamically localized phase. This phase shows ergodicity breaking up to the largest sizes we were able to consider. We argue that this property persists in the limit of large size. The Floquet states violate eigenstate thermalization and then the asymptotic value of local observables depends on the initial state and is not thermal. This implies that the system does not generically heat up to infinite temperature, for almost all the initial states. Differently from many-body localization here the entanglement entropy linearly increases in time. This increase corresponds to space-delocalized Floquet states which are nevertheless localized across specific subsectors of the Hilbert space: In this way the system is prevented from randomly exploring all the Hilbert space and does not thermalize.

We study the many-body localization aspects of single-particle mobility edges\nin fermionic systems. We investigate incommensurate lattices and random\ndisorder Anderson models. Many-body localization and quantum nonergodic\nproperties are studied by comparing entanglement and thermal entropy, and by\ncalculating the scaling of subsystem particle number fluctuations,\nrespectively. We establish a nonergodic extended phase as a generic\nintermediate phase (between purely ergodic extended and nonergodic localized\nphases) for the many-body localization transition of non-interacting fermions\nwhere the entanglement entropy manifests a volume law (`extended'), but there\nare large fluctuations in the subsystem particle numbers (`nonergodic'). We\nargue such an intermediate phase scenario may continue holding even for the\nmany-body localization in the presence of interactions as well. We find for\nmany-body states in non-interacting 1d Aubry-Andre and 3d Anderson models that\nthe entanglement entropy density and the normalized particle-number fluctuation\nhave discontinuous jumps at the localization transition where the entanglement\nentropy is sub-thermal but obeys the "volume law". In the vicinity of the\nlocalization transition we find that both the entanglement entropy and the\nparticle number fluctuations obey a single parameter scaling. We argue using\nnumerical and theoretical results that such a critical scaling behavior should\npersist for the interacting many-body localization problem with important\nconsequences. Our work provides persuasive evidence in favor of there being two\ntransitions in many-body systems with single-particle mobility edges, the first\none indicating a transition from the purely localized nonergodic many-body\nlocalized phase to a nonergodic extended many-body metallic phase, and the\nsecond one being a transition eventually to the usual ergodic many-body\nextended phase.\n

The accurate prediction of proton-induced neutron \((p, n)\) reaction cross sections is critical for applications in nuclear engineering, medical isotope production, and astrophysics. This study introduces a machine learning framework that combines high-predictive accuracy with interpretability and uncertainty quantification. We integrate the Hiking Optimization Algorithm (&lt;span class="sf"&gt;HOA&lt;/span&gt;) with the eXtreme Gradient Boosting (&lt;span class="sf"&gt;XGBoost&lt;/span&gt;) model and employ Monte Carlo (MC) Dropout for uncertainty estimation. The framework is interpreted using SHapley Additive exPlanations (&lt;span class="sf"&gt;SHAP&lt;/span&gt;). First, &lt;span class="sf"&gt;HOA&lt;/span&gt; is utilized to navigate the high-dimensional hyperparameter space, optimizing the &lt;span class="sf"&gt;XGBoost&lt;/span&gt; model for performance. Subsequently, the trained model’s predictions are benchmarked against experimental data from the EXFOR database and theoretical calculations from &lt;span class="sf"&gt;TALYS 2.0&lt;/span&gt;. Our &lt;span class="sf"&gt;HOA–XGBoost&lt;/span&gt; model demonstrates better predictive accuracy over other machine learning models and provides predictions closer to experimental values than &lt;span class="sf"&gt;TALYS 2.0&lt;/span&gt;. The inclusion of MC Dropout provides uncertainty bounds for the model’s predictions. A detailed &lt;span class="sf"&gt;SHAP&lt;/span&gt; analysis reveals the underlying physical drivers of the model’s decisions: the incident proton energy (\(EN\)) is identified as the most influential feature, with its strong interaction with the reaction \(Q\)-value and product proton number (\(Z_2\)) highlighting the model’s ability to learn fundamental concepts such as reaction thresholds and the Coulomb barrier. The product nuclide’s neutron (\(N_2\)) and proton (\(Z_2\)) numbers also show influence related to nuclear stability, while the product mass number (\(A_2\)) has a lesser impact. This work presents a complementary methodology for nuclear data evaluation, paving the way for more reliable predictions and targeted experimental design. Abstract Published by the Jagiellonian University 2025 authors

Many-body localization for a system of bosons trapped in a one-dimensional lattice is discussed. Two models that may be realized for cold atoms in optical lattices are considered. The model with a random on-site potential is compared with previously introduced random interactions model. While the origin and character of the disorder in both systems is different they show interesting similar properties. In particular, many-body localization appears for a sufficiently large disorder as verified by a time evolution of initial density wave states as well as using statistical properties of energy levels for small system sizes. Starting with different initial states, we observe that the localization properties are energy-dependent which reveals an inverted many-body localization edge in both systems (that finding is also verified by statistical analysis of energy spectrum). Moreover, we consider computationally challenging regime of transition between many body localized and extended phases where we observe a characteristic algebraic decay of density correlations which may be attributed to subdiffusion (and Griffiths-like regions) in the studied systems. Ergodicity breaking in the disordered Bose–Hubbard models is compared with the slowing-down of the time evolution of the clean system at large interactions.