Papers

We apply tensor-network methods to a field-coupled XY model for long-only portfolio construction. Daily return statistics set asset-specific fields and correlation couplings. Correlation distance, four-point Gromov hyperbolicity, and Ward clustering are used to construct a sparse interaction path. A Fourier-Bessel expansion maps the continuous angular partition function to a finite-current tensor network; bottom-up and top-down contractions then give the one-site marginals and equilibrium cosine scores. A softmax map converts these scores into positive, fully invested portfolio weights. We study five equity markets over continuous ranges of inverse temperature beta and concentration gamma, and compare selected parameter pairs with long-only Markowitz frontiers and standard benchmarks. The same-sample comparisons demonstrate a tractable route from financial time series to network-adjusted allocations; predictive trading performance is outside their scope.

We suggest a model of a diffusive process inside a fractal sponge structure, which is a generalization of the diffusion processes on a comb and fractal mesh structure. The sponge model is considered as the direct product of Cantor sets. It is shown that the corresponding one-dimensional diffusion process is governed by a generalized Fokker-Planck equation with a power-law memory kernel and a position-dependent diffusion coefficient. That is, the fractal structure of the medium induces memory effects and heterogeneity in the transport system. The considered model may be of interest to describe anomalous heat transport in porous fractal media.

Clocks' performance, especially their precision, is fundamentally limited by thermodynamic laws as they are physical devices. Yet, it is known that the established thermodynamic uncertainty relation (TUR), which imposes an upper bound on the uncertainty product of the precision of oscillations and entropy production, is violated for underdamped systems such as stochastic pendulum clocks. Here, we show that for a general class of stochastic pendulum clocks described as a weakly nonlinear oscillator the uncertainty product of the phase of a pendulum clock and entropy production takes a universal and simple form that depends solely on the degree of nonlinearity. Its validity and limitations are examined using several representative models of pendulum clocks. Our framework reveals a universal thermodynamic principle governing stochastic pendulum clocks beyond the conventional TUR and provides a foundation for designing optimal pendulum clocks that operate efficiently in stochastic environments.

Naturally occurring bulk quasicrystals are rare, and magnetic instances are rarer still, with chemical constraints typically permitting the synthesis of approximants rather than true quasicrystalline magnets. Here, we present an artificial spin ice based on a recently discovered Einstein lattice, the Hat tiling, which is built from the first known shape - the hat - that tiles the plane only aperiodically. The Einstein artificial spin ice has long-range structural order with no translational symmetry, and low-connectivity vertices with well-defined local ground states and excitations. Together, these properties provide a model two-dimensional quasicrystalline magnet, with magnetic correlations that we probe with magnetic force microscopy and parallel-tempered Monte Carlo simulations. We identify a two-stage partial ordering process, driven by the competition between two possible positions for magnetic excitations. This competition is resolved in the ground-state manifold, where exactly one magnetic excitation is caged on each antihat, yet the manifold remains macroscopically degenerate. This yields an unusual type of medium-range order, where the underlying quasiperiodic long-range order is randomly modulated by a strictly constrained disorder. Our findings establish the Einstein artificial spin ice as a blueprint for understanding quasicrystalline magnetism, demonstrating how quasiperiodic monotile geometries can be exploited to engineer unconventional magnetic phases with no direct equivalent in periodic systems.

We obtain the exact full counting statistics of charge and flavor transport in the one-dimensional infinite-$U$ $SU(N)$ Hubbard model. The no-passing constraint freezes the ordered flavor sequence while the particle coordinates evolve as free spinless fermions. Consequently, charge transfer is governed by a free-fermion determinant, whereas flavor transfer is exactly subordinated to the number of particles crossing the observation cut. For an arbitrary traceless Cartan generator this yields an exact relation between the charge and flavor cumulants. The complete asymptotic flavor distribution is a universal non-Gaussian M-Wright law. Together, these results establish spin–charge subordination as a kinematic mechanism for anomalously slow internal-state transport in one-dimensional impenetrable quantum gases.

We discuss the possibility to train a thermodynamic system, whose micro-variables are fully determined through the Hamiltonian by the usual statistical mechanical rules, to perform tasks such as memorization and generalization. Training is achieved by the application of suitable external fields, playing the role of data. At variance with conventional machine learning, no other logical or algorithmic rules are introduced. The system is amenable, in principle, to exact analytical calculations. We specialize this general approach to a prototypical Ising system with annealed dichotomous couplings and study its learning ability. Results indicate excellent memorization and good generalization capacity already for small systems, and a tendency to improve with system size.

In this study, we demonstrate the characteristics of the stochastic Loewner evolution (SLE) as a one-dimensional (1D) turbulence model. First, we show that the diffusion process $x(t)$ obtained by the time coordinate change in the SLE becomes the time dependent diffusion process that satisfies Richardson's law of turbulence.Subsequently, we derive the condition on the diffusivity parameter $\kappa$ such that the diffusion process $x(t)$ satisfies Kolmogorov's 4/5th law, which determines the relation between the mean energy dissipation and velocity in the particles of turbulence. Further mathematical analysis showed that this condition is seen as the condition on Loewner entropy $S_{\mathrm{Loew}}$, which includes the white Gaussian noise term $W_{s}$ in the Loewner driving function of the present model. In addition, the numerical simulations were performed to verify the scaling relations of this 1D turbulence model. These results suggest that the standard SLE becomes a beneficial model of 1D turbulence by introducing an appropriate time coordinate change and imposing the conditions on $\kappa$ and $S_{\mathrm{Loew}}$.

Mean-field theory is a widely used approximation for describing phase transitions, particularly effective above the upper critical dimension. Its origins can be traced back to the van der Waals theory of the liquid-gas transition and Weiss's molecular field theory of the paramagnetic-ferromagnetic transition. However, it was Lev D. Landau who provided a unifying and general framework applicable to a broad class of physical systems. The mean-field approach typically involves neglecting thermal fluctuations, which is a reasonable assumption in many classical contexts. However, its application to quantum phase transitions at zero temperature is less common. The aim of this short pedagogical paper is to explore and clarify the use of the mean-field approach in the less familiar domain of quantum phase transitions. We specifically consider the Ising model and the Blume-Capel model.

Higher-order networks exhibit rich critical phenomena that cannot be captured by traditional pairwise models. Here, we develop an analytical dimension-reduction framework that maps higher-order networked dynamics onto an effective low-dimensional system, allowing accurate prediction of tipping boundaries, bistability regions, and the nature of phase transitions. We demonstrate the power of this framework across a range of dynamical processes, revealing distinct effects of higher-order interactions on transition continuity and hysteresis. Furthermore, we find that system resilience exhibits a profound dependence on the alignment between pairwise and higher-order connectivity, with assortative mixing enhancing tipping toward active states. Our findings establish a general theory for understanding the critical transitions in higher-order networks, offering new insights for anticipating and managing systemic risk in complex systems.

The overdamped approximation is widely used to describe Brownian motion in complex environments, but its validity becomes nontrivial when the friction coefficient itself fluctuates in time. We investigate this problem by comparing underdamped and overdamped Langevin dynamics subject to the same fluctuating friction and satisfying the fluctuation–dissipation relation at a common temperature. We show that environmental averaging and inertial elimination generally lead to different long-time transport when environmental fluctuations are fast compared with velocity relaxation. For rapidly fluctuating friction, the underdamped dynamics is governed by the arithmetic mean friction and yields $D_{\rm eff}^{\rm under}=k_{\rm B}T/\langle\gamma\rangle$, whereas the overdamped dynamics gives $D_{\rm eff}^{\rm over}=k_{\rm B}T\langle\gamma^{-1}\rangle$. For a two-state Markov friction, we derive the finite-time effective diffusion coefficient exactly and identify the full crossover between these two regimes, controlled by the competition between velocity relaxation and environmental switching. We further establish the fast-fluctuation result for general stationary friction processes and verify it for a continuous log-Ornstein–Uhlenbeck environment, for which the characteristic crossover time can also be determined independently. Our results reveal a noncommutativity between environmental averaging and inertial elimination, establish a timescale-dependent criterion for the validity of overdamped dynamics in temporally heterogeneous environments, and show that rapid environmental fluctuations can enhance, rather than suppress, the consequences of inertia.

Von Neumann proposed two quantum entropies, one in 1927 and the other in 1929. He intended to define thermodynamic entropy for quantum states in both cases. The 1927 entropy is the well-known von Neumann entropy, which is zero for all pure states. The 1929 entropy, which is nonzero for almost all pure states, has largely been ignored. After reviewing the history and properties of both entropies, we argue that the 1927 entropy is not thermodynamic entropy and should be used simply as a measure of entanglement, whereas the 1929 entropy is the true thermodynamic entropy for quantum states.

We study the stability of collective neutrino oscillations using a distributional approach motivated by the statistical mechanics of Kuramoto synchronization. Treating the ensemble of neutrino flavor polarization vectors in the thermodynamic limit $N\to\infty$, we derive an exact nonlinear Fokker–Planck (continuity) equation for the one-body distribution $F(\vec{\mathbf{S}},\omega,t)$ on the flavor sphere. This equation admits a two-parameter family of azimuthally symmetric stationary solutions, whose stability we analyze by linearizing around them. The resulting eigenvalue condition determines the growth or decay rate of small perturbations from any initial distribution – not merely from a state close to full flavor coherence – thereby going significantly beyond the conventional linear stability analysis of collective modes. In special limits the condition reproduces known synchronization thresholds in the two-beam model, providing a non-trivial check of the framework. We present analytical results for the eigenvalue equation and explore stability phase diagrams for physically relevant frequency distributions.

Fukui and Kawakami showed that the trigonometric Haldane–Shastry (HS) spin chain can be deformed by introducing twisted boundary conditions. In this work we revisit the resulting twisted HS chains, which we call Fukui–Kawakami (FK) chains. We analyse their spectra in dependence of the twist parameter, utilising a direct connection with the (untwisted) HS chain. This allows us to explain why part of the spectrum can be described by Fukui and Kawakami's twisted Bethe equations, and can be constructed from Yangian highest weight states. These states can be labelled by combinatorial data called `motifs', as in the HS chain, which cover some of the (deformed) descendants. We furthermore show that there are other descendants which do not follow from these Bethe equations, but whose energy can be described as a sum of two single-particle energies, suggesting additional hidden symmetries. We then focus on the special case of antiperiodic boundary conditions, and show that this chain coincides with the `minimally polarised' long-range model recently introduced by Basu-Mallick, Finkel, and González-López. Remarkably, this connection implies that the antiperiodic chain has a hidden $\mathfrak{gl}(1|1)$-symmetry, which we use to relate the HS motifs to their `supersymmetric' $\mathfrak{gl}(1|1)$ counterparts.

We derive nonasymptotic upper bounds on the probability that a generalized current of a geometrically ergodic diffusion observed for any amount of time, or its sample mean over any arbitrary sample size, deviates from the stationary mean by more than any given amount. The concentration-of-measure behavior of generalized currents is universally governed by the relaxation time of the underlying dynamics, the locally observed dissipation rate, and the intrinsic local fluctuations of the observable. We uncover stark qualitative and quantitative differences in fluctuations in and out of thermodynamic equilibrium. We further obtain refined inverse thermodynamic uncertainty relations, bounding the variance of generalized currents from above. We construct nonasymptotic confidence intervals for controlling uncertainty in thermodynamic inference from small data, i.e., from short trajectories and small samples, and provide the first quantitative answer to when a trajectory is sufficiently long and a sample is sufficiently large. As an illustration, we apply our results to currents observed on a two-dimensional Ornstein-Uhlenbeck process in and out of equilibrium and show how they can be used to rigorously detect broken detailed balance.

Quantum thermalization explains how an isolated many-body system naturally evolves towards a thermal state, rendering information about the initial conditions inaccessible to local measurements. This is precisely the mechanism utilized in quantum error correction, where information is protected by design through a nonlocal encoding. In this work, we leverage this connection to port the rigorous framework of (approximate) quantum error correction to the study of quantum thermalization. Treating typical late-time states as codewords, we characterize the error-correcting properties of generic thermalizing dynamics. We numerically uncover a universal relationship between the encoding rate, distance, and thermal entropy density of the emergent code. At infinite temperature, this universal curve saturates the quantum Singleton bound, achieving the same optimal limit as Haar-random codes. At finite temperature, we introduce a code family based on the Scrooge ensemble, the natural thermal analogue of the Haar ensemble, and prove it saturates the entropic quantum Singleton bound, establishing this family as optimal within entropic constraints. Our extracted universal curve independently saturates this same bound, revealing that finite-temperature thermalization is itself optimal. Finally, we show how conserved quantities limit the error-correcting behavior of thermalization: codewords with differing energies, or other conserved charges, leak only classical information, and correctability persists until the difference reaches the scale of thermal fluctuations. Our results reveal a universal optimal coding structure in thermalizing dynamics, while introducing new optimal codes that achieve fundamental limits of approximate quantum error correction.

Open-system lattice gauge theory (LGT) has so far been developed predominantly in Abelian settings, leaving open how genuinely non-Abelian gauge structure reshapes dissipative real-time dynamics. Here, we study a $1+1$D SU(2) Yang–Mills LGT with dynamical matter coupled to a thermal scalar environment through a gauge-preserving Lindblad evolution, which we solve using tensor networks. Starting from a quark–antiquark pair connected by a chromoelectric flux string, we find that the thermal medium melts the string by delocalizing the color charges and screening the flux; on resonance, this dissipative melting competes with and delays coherent string breaking. The thermalization time is non-monotonic in the environment coupling, decreasing through environment-assisted transport at weak dissipation before increasing in a quantum-Zeno regime. In the strong-dephasing limit, a Schrieffer–Wolff expansion maps the dynamics to a classical exclusion process and yields the Liouvillian thermalization time analytically. Beyond these generic open-system effects, the non-Abelian matter structure produces a systematic mesonic bias in the steady state, while the thermalization time decreases with temperature, in contrast to the Abelian Schwinger model trend and in qualitative agreement with pNRQCD studies of the quark–gluon plasma. These results establish a gauge-preserving framework for thermalization and string dynamics in open non-Abelian lattice gauge theories.

We investigate the structures of effective Hamiltonians governing monitored dynamics of a one-dimensional Majorana chain through the Lyapunov spectral analysis. We focus on a gapless phase characterized by finite-size scalings different from those in conventional critical and/or frustration-free systems; the spectral gap closing faster than $1/L$ but slower than $1/L^2$ and the entanglement entropy growing as $[\ln(L)]^2$ with $L$ being the system size. We find that the corresponding effective Hamiltonians have random long-range power-law hoppings with nontrivial magnitude correlations, rather than being independently and identically distributed. To elucidate the role of these non-Gaussian correlations, we construct random power-law hopping models that capture the essential features of the effective Hamiltonians. The spectral gaps of the constructed models decay faster than $1/L$ but slower than $1/L^2$. We find that, in the absence of hopping correlations, the ground-state entanglement exhibits $\ln(L)$ scaling. In the presence of correlations, by contrast, the entanglement entropy is enhanced and its system-size dependence is consistent with $[\ln(L)]^2$ scaling over the system sizes studied. These results suggest that correlations among long-range hopping magnitudes are responsible for the entanglement scaling that seldom appears in ground states of conventional isolated quantum systems.

A discrete time crystal is an out-of-equilibrium phase of matter characterized by the spontaneous breaking of discrete time-translation symmetry. Using extensive numerical simulations based on matrix-product-state methods, we provide evidence for discrete time-crystalline behavior in strongly disordered spin chains with Heisenberg interactions, including the isotropic point, subject to periodic driving. Starting from a many-body localized regime, we observe that rotations induced by delta kicks produce a pronounced subharmonic response at half the drive frequency in spin observables. We investigate the stability of this response against rotation-angle errors through entanglement entropy, quantum Fisher information, short-range spin correlations, and restricted-control ergotropy. Increasing the rotation error reveals an intermediate dynamical regime separating the time-crystalline and Floquet-localized responses. In this regime, most observables exhibit signatures of weakly correlated dynamics reminiscent of Anderson localization.

The number of metastable states of a random corrugated landscape controls what dynamics, inference and learning can reach. In the Sherrington-Kirkpatrick model these states are marginal, and counting them requires breaking the BRST symmetry. So far only the annealed complexity was known, predicting states below equilibrium. We compute the quenched complexity function with a two-group Ansatz and one-step replica symmetry breaking. It branches continuously off the annealed curve at its instability point and vanishes near the Parisi free energy, while the soft mode turns toward orthogonality to the magnetization.

Spatial confinement and chemical screening fundamentally reshape the non-equilibrium phase behavior of autophoretic active particles. Here, we present a systematic study mapping the collective dynamics of self-propelled particles governed by chemo-attractive translational forces ($\mu_t < 0$) and chemo-repulsive rotational torques ($\mu_r > 0$) across varying screening parameters $\kappa$, torque magnitudes $\mu_r$, and boundary condition coefficients $\Lambda^c$. Beyond standard chemotactic macro-phase separation and dynamic clustering, we report the emergence of novel boundary- and screening-induced bubbly phases, classified into boiling and bursting bubbles. Using a metric triad of steady-state cluster fraction $\langle S \rangle$, temporal fluctuation magnitude $\sigma_S$, and coordination number $\langle Q \rangle$, we draw phase diagrams to demarcate phases for no-flux boundaries ($\Lambda^c = 1$) and chemically permeable interfaces ($\Lambda^c = 0$) . Increasing chemical screening ($\kappa$) systematically suppresses long-range attraction, driving sequential phase transitions from macro-scale collapse toward bubbly states, dynamic micro-clusters, and homogeneous gas phases, while simultaneously inducing aggregate shape anisotropy. These findings provide predictive design rules for controlling active assembly and transport in microfluidic environments.

Metallic nanoring networks can serve as promising flexible transparent electrodes. These materials are crucial components in a wide range of applications, including solar cells, touchscreens and displays. In this work, a computational investigation considers in detail (i) the electrical conductance and optical performance of nanoring networks and (ii) the breakdown of these networks due to electrical damage. The electrical resistance of both the nanorings and the contacts between the rings (junctions) is taken into account. In part (i), the effects of 5 parameters on the electrical sheet resistance and optical transparency are presented. It is shown that several parameter combinations achieve better performance in comparison to indium tin oxide, currently the most widely used transparent electrode. In part (ii), due to electrical damage, the nanoring systems display the formation of a crack, running parallel to the vertical terminals, where a voltage difference is applied. The network degradation is measured by its sheet resistance, and a universal effect is observed: networks with varying filling factors exhibit the same degradation profile.

We present a theory for the optimal control of stochastic systems in structured environments, represented by penalty terms in the cost functional. We show that such control problems generically feature sharp finite-time transitions associated with a qualitative change in the control strategy at a critical time. Starting from an overdamped Langevin equation and a quadratic cost functional, we show that all resulting problems fall into three canonical equivalence classes (parabolic, hyperbolic, and elliptic), distinguished by the sign of the determinant of the control Hamiltonian. For each class, we obtain the optimal protocol, the cost function, and the critical time in closed form, and show that the transition exhibits features of a continuous phase transition at mean-field level. We then establish a mapping between the optimal control cost and the large-deviation rate function governing non-equilibrium relaxation after a potential quench. The mapping covers the parabolic and hyperbolic classes, while the elliptic class has no simple relaxation counterpart. This correspondence implies that recently discovered finite-time dynamical phase transitions, which are exponentially costly to sample directly, are accessible through ordinary averages over optimally controlled trajectories. To validate our theoretical findings, we report three experiments with optically trapped colloidal particles: a control transition for the mean stochastic work, and the finite-time dynamical phase transitions in free diffusion and in harmonic relaxation.

The Kibble-Zurek mechanism (KZM) and finite-time scaling (FTS) provide a foundational framework for driven critical dynamics, yet their predictive power has been largely confined to local observables. Here, we establish a universal finite-time scaling theory for the nonequilibrium dynamics of quantum entanglement. Using unbiased quantum Monte Carlo simulations, we investigate the corner entanglement entropy of (2+1)-dimensional interacting Dirac fermions driven from ordered phases toward a quantum critical point. We find that the corner entanglement accurately obeys a universal driven scaling governed by the driving rate and system size, persisting whether the initial ordered state is fully gapped or hosts gapless Goldstone modes. Crucially, this dynamical entanglement exhibits a logarithmic dependence on the driving rate, from which the universal corner coefficient of the underlying conformal field theory can be robustly extracted far from equilibrium. These results generalize the KZM from local observables to the intrinsic nonlocal quantum information measures, offering a practical blueprint for characterizing quantum criticality and entanglement on programmable quantum simulators.

For a simple model fluid, the Yukawa fluid, the condition for dynamic crossover, known as the Frenkel line, is defined. The condition is related to the fact that the roton minimum in the dispersion relation of longitudinal acoustic-like excitations exists only when the collective vibrational dynamics of particles dominates in the liquid. Based on the self-consistent relaxation theory for the Yukawa fluid, thermodynamic states are determined in which the roton minimum disappears. The obtained values of the state parameters for the Frenkel line are consistent with the results of studies in which the position of this line on the phase diagram of the Yukawa fluid was determined using molecular dynamics simulations. It is shown that the Frenkel line in this system can be determined directly from the structural characteristic – the static structure factor. A physical interpretation of the roton minimum frequency for the simple liquids near the Frenkel line is proposed.

We investigate the current response of a boundary-driven symmetric exclusion process with quenched site disorder. Hard-core particles hop symmetrically on a one-dimensional lattice with site-dependent rates and are injected and removed at the boundaries by two reservoirs of different densities. We approximate the steady-state density profile using a Galerkin projection at linear order and a mean-field closure at higher orders, and thereby obtain the current as a nonlinear function of the reservoir density difference. At linear order in the reservoir density difference, the current-response coefficient depends on the mean reservoir density $\rho$, in contrast to the homogeneous case. Through the linear-response relation, this dependence leads to an equilibrium current-fluctuation coefficient that is asymmetric under $\rho\rightarrow 1-\rho$. Beyond linear response, nonzero even-order current contributions break the antisymmetry of the current under reversal of the reservoir density difference, producing rectification-like behavior in individual disorder realizations. We further show that spatial-reflection symmetry of the equilibrium density profile rules out such behavior, so broken spatial-reflection symmetry of the profile is a necessary condition for rectification-like behavior. Within the present approximation, we further find that, for continuously distributed site disorder, rectification-like behavior occurs arbitrarily close to equilibrium for almost every disorder realization. At the ensemble level, however, the disorder-averaged current remains antisymmetric because the disorder ensemble is invariant under spatial reflection. These results provide a mechanism for rectification-like transport arising from sample-specific spatial heterogeneity rather than from an explicitly imposed directional asymmetry.

The multi-state Bennett acceptance ratio (MBAR) equations combine the data collected under different thermodynamic conditions in a statistically optimal way. Due to their practical importance, several solution strategies have been devised to optimize the convergence of the resulting set of coupled equations. However, even with graphics processing unit (GPU) acceleration, the convergence of these methods can still be either unstable or slow. We propose a new approach where the equations are interpreted as the limit of infinitesimal bin width of the respective weighted histogram analysis method (WHAM). In the proposed binned-to-unbinned MBAR (BU-MBAR) method, the bin width is adapted dynamically to ensure a stable and efficient convergence to the asymptotic solution.

How does particle shape affect driven transport through obstacle fields? We simulate a dissipative rod falling under gravity through randomly placed fixed obstacles. For sufficiently slender rods, the mean descent speed before trapping decreases, increases, and then decreases again as obstacle density rises, opening a window of densities in which rods fall faster in denser fields. Scaling arguments based on collision rates and rod geometry explain the three regimes and their crossovers. Particle shape can thus reverse the expectation that crowding slows driven transport.

The critical behavior of a system near a continuous phase transition is commonly characterized by power laws and a set of universal critical exponents. Yet universality at phase transitions has even more fundamental manifestations. It dictates that all observables follow universal scaling functions, which may extend beyond simple power laws. Furthermore, microscopic details are captured entirely by only two system-specific scale factors, that rescale any measured quantity onto its universal form. In this work, we report the experimental validation of this prediction in ultracold lattice Bose gases undergoing the superfluid transition. We extract the scale factors by collapsing the order-parameter cumulants measured at varying entropies. We show that these factors specify scaling functions governing both the order-parameter distribution and two-point correlations. By measuring these scaling functions, we also determine three critical exponents $\beta$, $\gamma$ and $\nu$. Our results provide an experimental demonstration that universal scaling is simultaneously obeyed by multiple observables in a quantum many-body system.

While spontaneous breaking of a discrete symmetry in one-dimensional classical systems with short-range interactions is absent, it is expected that a line defect embedded in a bulk criticality exhibits a stable discrete symmetry spontaneous breaking. Here, we investigate the behavior of a pinning-field line defect immersed in a 3D bulk that remains tuned to the $O(N)$ Wilson-Fisher critical point. Employing the fuzzy sphere technique, we provide convincing evidence of the existence of stable defect conformal fixed points, and we demonstrate their renormalization group stability by showing no relevant operator and less effective degrees of freedom than that at bulk fixed point via $g$-function. Moreover, we investigate the defect domain wall operator for various $N$, and we identify that it becomes irrelevance for $N\gtrsim 3$ but it is relevant for $N<3$.These evidence indicate that a one-dimensional defect coupled to a critical bulk cannot support a stable symmetry spontaneously broken defect fixed point due to domain wall proliferation for $N<3$ Wilson-Fisher universality, while in the case of $N \gtrsim 3$ a symmetry broken defect is possible.

We introduce a diagnostic framework based on the full probability distribution of quantum coherence to probe single-particle and many-body phases at finite temperature. We first apply this approach to the localization transition in the non-interacting Aubry-André model and then extend the analysis to its interacting counterpart, whose phase diagram is substantially richer. We show that the moments of the coherence distribution provide clear signatures of quantum phase transitions and capture the emergence of many-body phases. More importantly, we find that the entropy of the coherence distribution emerges as an even more powerful indicator, retaining clear signatures of the underlying quantum phases while remaining remarkably robust against increasing thermal fluctuations.