Thermodynamic limit of spin systems on random graphs

The paper is devoted to stochastic equations describing the evolution of classical and quantum unbounded spin systems on discrete lattices and on Euclidean spaces. Existence and asymptotic properties of the corresponding transition semigroups are studied in a unified way using the theory of dissipative operators on weighted Hilbert and Banach spaces. This paper is an enlarged and rewritten version of the paper [7].

Alternating metalloporphyrin-acceptor systems provide a ‘laboratory’ for study of quasi-one-dimensional magnetic phenomena. For example, the [octaethylporphyrinatomanganate(III)][hexacyanobutadiene], [MnOEP][HCBD], compound is a model alternating classical (spin 2) and quantum (spin 1/2) spin system. While the high temperature susceptibility is well fit to Seiden's model for alternating S = 2 and S = 1/2 chains with antiferromagnetic exchange J/kB = 172 K, and g for [MnOEP] = 1.92, the low temperature data reveal a transition to a two-dimensional antiferromagnet, followed by a transition to a three-dimensional canted antiferromagnet. The similarly structured [tetraphenyl-porphyrinatomanganate(III)][tetracyanoethenide], [MnTPP][TCNE], also has ferrimagnetic behavior within the chains, though it has ferromagnetic interactions between the chains. We associate the change in sign of interchain exchange (from antiferromagnetic to ferromagnetic) with an increase in interchain separation and a change in the interchain exchange pathways. Subtle disorder, the magnitude of which depends upon the solvent incorporated within the structure, causes cluster glass-like behavior at low temperatures. The roles of the various constituents of these compounds in determining the ferromagnetic or antiferromagnetic nature of the ground state are discussed.

We demonstrate that a quantum field theory (QFT) in general two-dimensional curved spacetimes can be realized by a system of quantum spins or qubits. We consider a spin-1/2 model on a one-dimensional ring with spatially and temporally varying exchange couplings and magnetic fields. This model reduces to a QFT of Majorana fermions in the continuum limit. From this correspondence, we establish a dictionary for translating between the spacetime-dependent parameters of the spin model and the general metric on which the QFT is defined. After addressing the general case, we consider the Friedmann-Lemaître-Robertson-Walker (FLRW) metric as a simple example. According to the dictionary, the QFT of Majorana fermions on the FLRW metric corresponds to the Ising model with a time-dependent transverse magnetic field. We demonstrate that the production of Majorana particles in the expanding universe can be simulated with the transverse-field Ising model by increasing the strength of the magnetic field. Furthermore, we examine the Unruh effect through the spin system by using our prescription and show the direct relation between the entanglement (or modular) Hamiltonian in the spin system and the Rindler Hamiltonian. This approach provides an experimentally viable system for probing various phenomena in QFT within curved spacetime, while also opening the door to uncovering nontrivial phenomena in spin systems inspired by curved spacetime physics. It offers fresh perspectives on both QFT in curved spacetimes and quantum many-body spin systems, revealing profound connections between these fields. Published by the American Physical Society 2025

Convolutional neural networks (CNNs) have been employed along with variational Monte Carlo methods for finding the ground state of quantum many-body spin systems with great success. However, it remains uncertain how CNNs, with a model complexity that scales at most linearly with the number of particles, solve the “curse of dimensionality” and efficiently represent wavefunctions in exponentially large Hilbert spaces. In this work, we use methodologies from information theory, group theory and machine learning, to elucidate how CNN captures relevant physics of quantum systems. We connect CNNs to a class of restricted maximum entropy (MaxEnt) and entangled plaquette correlator product state (EP-CPS) models that approximate symmetry constrained classical correlations between subsystems. For the final part of the puzzle, inspired by similar analyses for matrix product states and tensor networks, we show that the CNNs rely on the spectrum of each subsystem's entanglement Hamiltonians as captured by the size of the convolutional filter. All put together, these allow CNNs to simulate exponential quantum wave functions using a model that scales at most linear in system size as well as provide clues into when CNNs might fail to simulate Hamiltonians. We incorporate our insights into a new training algorithm and demonstrate its improved efficiency, accuracy, and robustness. Finally, we use regression analysis to show how the CNNs solutions can be used to identify salient physical features of the system that are the most relevant to an efficient approximation. Our integrated approach can be extended to similarly analyzing other neural network architectures and quantum spin systems. Published by the American Physical Society 2025

We have developed a method for evaluating the specific heat of lattice spin systems. It is based on the knowledge of high-temperature series expansions, the total entropy of the system, and the low-temperature expected behavior of the specific heat as well as the ground-state energy. By the choice of an appropriate variable (entropy as a function of energy), a stable interpolation scheme between low and high temperature is performed. Contrary to previous methods, the constraint that the total entropy is $\mathrm{log}(2S+1)$ for a spin S on each site is automatically satisfied. We present some applications to quantum spin models on one- and two-dimensional lattices. Remarkably, in most cases, a good accuracy is obtained down to zero temperature.

When studying interacting systems, computing their statistical properties is a fundamental problem in various fields such as physics, applied mathematics, and machine learning. However, this task can be quite challenging due to the exponential growth of the state space as the system size increases. Many standard methods have significant weaknesses. For instance, message-passing algorithms can be inaccurate and even fail to converge due to short loops, while tensor network methods can have exponential computational complexity in large graphs due to long loops. In this Letter, we propose a new method called "tensor network message passing." This approach allows us to compute local observables like marginal probabilities and correlations by combining the strengths of tensor networks in contracting small subgraphs with many short loops and the strengths of message-passing methods in globally sparse graphs, thus addressing the crucial weaknesses of both approaches. Our algorithm is exact for systems that are globally treelike and locally dense-connected when the dense local graphs have a limited tree width. We have conducted numerical experiments on synthetic and real-world graphs to compute magnetizations of Ising models and spin glasses, and have demonstrated the superiority of our approach over standard belief propagation and the recently proposed loopy message-passing algorithm. In addition, we discuss the potential applications of our method in inference problems in networks, combinatorial optimization problems, and decoding problems in quantum error correction.

The ground-state properties of a generalized quantum $s=\frac{1}{2}$ $\mathrm{XY}$ model (including the ferromagnetic and antiferromagnetic cases) in a $Z$ field on a triangular lattice are examined. The exact values of critical fields above which the gap opens are obtained. A real-space renormalization-group method is used to estimate the ground-state energy and to analyze the critical behavior. There is some evidence that certain features of frustration in classical spin systems still persist in quantum systems.

Dynamical non-equilibrium effects after a quantum quench in a quantum spin system, which permits exact analytical solution in both closed and open cases, are studied. The exact analytic results are obtained for the bi-axial paramagnet, both for the “easy axis”- and the “easy-plane”-like situations, and for the field directed along both principal axes of the system. Quantum quench of the external magnetic field produces a nonlinear response. For the closed system the average magnetic moment oscillates with time and with the final value of the external field. Such oscillations exist also for the open system, connected to the bath, in the dynamical regime. For the steady-state regime in the open case the oscillations are damped. Non-equilibrium effects yield specific hysteresis phenomena in the considered single spin system.

For inhomogeneous mean-field quantum spin systems with random sites, we prove that the weak*-limit points of the sequence of n-particle Gibbs states, on a suitable C*-algebra, are superpositions of inhomogeneous product states, which are given by solutions of a generalized Curie–Weiss equation. Under certain conditions there is only one weak*-limit point. As applications, the BCS model and a spin system with cosine interaction are studied.

The effect of the modulation of a high-frequency ac magnetic field parallel to the dc one (longitudinal pumping) by the lower-frequency ac field acting on quantum spin systems is considered for closed and open systems. For most magnetically ordered spin systems, the modulation causes the oscillation of the threshold of the parametric instability of spin waves as a function of the amplitude and the frequency of the modulation. On the other hand, our results for quantum spin systems with fermionic eigenstates imply that the modulating field causes the combined resonance and the modulation of the magnitude of time-dependent oscillations of the magnetization and the absorbed by the system power of the ac field for the closed system and for the open system in the dynamical and the steady-state regimes.

The performance of bipartite entanglement indicators obtained directly from tomograms is compared with that of standard entanglement measures, in time-evolving hybrid quantum systems and spin systems. Bipartite and tripartite models are judiciously chosen so as to exhibit interesting effects such as entanglement sudden death and entanglement collapse to a constant nonzero value over significant time intervals during evolution. For specific initial states, the reliability of the tomographic indicators is investigated numerically. The IBM Q platform has been invoked to examine how well the equivalent circuits of specific model systems fare, to understand how experimental losses affect the entanglement. Using data obtained from a set of NMR experiments, we have examined the viability of tomographic indicators in monitoring bipartite entanglement, and find that they perform as satisfactorily as standard measures and monotones. As a trailer to the next chapter on the dynamics of quantum observables, we show that in a tripartite model examined in this chapter, an intensity-dependent bifurcation cascade appears at short times, and is captured both by the indicators and by the mean photon number.

The magnetization process is a very important probe to study magnetic materials, particularly in search of spin-liquid states in quantum spin systems. Regrettably, however, progress of the theoretical analysis has been unsatisfactory, mostly because it is hard to obtain sufficient numerical data to support the theory. Here we propose a machine-learning algorithm that produces the magnetization curve and the spin gap well out of poor numerical data. The plateau magnetization, its critical field and the critical exponent are estimated accurately. One of the hyperparameters identifies by its score whether the spin gap in the thermodynamic limit is zero or finite. After checking the validity for exactly solvable one-dimensional models we apply our algorithm to the kagome antiferromagnet. The magnetization curve that we obtain from the exact-diagonalization data with 36 spins is consistent with the DMRG results with 132 spins. We estimate the spin gap in the thermodynamic limit at a very small but finite value.

We study a quantum spin system on a honeycomb lattice, when it includes frustration and distortion in antiferromagnetic (AF) exchange interactions. We transform the spin system onto a nonlinear $\ensuremath{\sigma}$ model (NLSM), preserving the original spin degrees of freedom. Assisted by a renormalization-group argument, the NLSM provides a ground-state phase diagram consisting of an ordered AF phase and a disordered spin-gap phase. The spin-gap phase extends from a strong frustration regime to a strong distortion regime, showing that the disordered ground states are essentially the same in both regimes. In the spin-1/2 case, the spin-gap phase for spin system on a honeycomb lattice is larger than that for the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model on a square lattice.

We study the macroscopic entanglement properties of a low-dimensional quantum spin system by investigating its magnetic properties at low temperatures and high magnetic fields. The spin system chosen for this is copper nitrate (Cu(NO3)2 × 2.5H2O), which is a spin chain that exhibits dimerization. The temperature and magnetic field dependence of entanglement from the susceptibility and magnetization data are given, by comparing the experimental results with the theoretical estimates. Extraction of entanglement has been made possible through the macroscopic witness operator, magnetic susceptibility. An explicit comparison of the experimental extraction of entanglement with theoretical estimates is provided. It was found that theory and experiments match over a wide range of temperatures and fields. The spin system studied exhibits quantum phase transition (QPT) at low temperatures when the magnetic field is swept through a critical value. We show explicitly for the first time, using tools used in quantum information processing, that QPT can be captured experimentally using quantum complementary observables, which clearly delineate entangled states from separable ones across the QPT. We have also estimated the partial information sharing in this system from our magnetization and susceptibility data. The complementarity relation has been experimentally verified to hold in this system.

The authors study the singular behaviour occurring in delta alpha (q)/ delta q of the multifractal spectrum for the current distribution in random resistor networks. The singular behaviour may be understood as the divergent behaviour of specific heat in a thermodynamic phase transition. delta alpha (q)/ delta q shows a peak with the width q0(L)<q<qc(L), and the height of which diverges as the system size L increases. However the nature of the singular behaviour in delta alpha (q)/ delta q is very unusual compared with that of the ordinary phase transition in the following ways. First, the width of the peak becomes broader, rather than narrower, with the increasing system size L, because q0(L) to - infinity and qc(L) to 0 as L to infinity . Second, the singularities at q=q0 and q=qc are of different types. The singularity at q=q0 is of power-law type such as approximately Ld(2-m)m/ log L, and the singularity at q=qc is of logarithmic type such as approximately (log L)2m/, where d is spatial dimension and m is measured to be approximately 1.633+or-0.006.