High-temperature modulations, mesoscale interactions, and hyperscaling breakdown in Ising models with frustration: Some insights from thermodynamic geometry
In this work we revisit the axial third-nearest-neighbor Ising chain and examine in detail some aspects of its phase behavior ensuing from competing interactions and resulting frustration. We probe the phase behavior with two complimentary tools: a microscopic two-point correlation function C(n), which we carefully construct after appropriate spin transformations, and a macroscopic, thermodynamic curvature R which we obtain from the free entropy. We report observations of phenomena such as hyperscaling breakdown and intermediate-ranged modulations among others. The zero-field thermodynamic curvature R_{0} is shown to systematically subdivide the ground-state phases into regions of attractive or repulsive effective interactions of varying strength. Furthermore, R_{0} brings forth the significance of some third-order moments in describing the effects of frustration, including the multiphase lines. Combined use of both the tools in the short-ranged modulated order regime confirms and further clarifies the discussion in [S. Khatua et al., Phys. Rev. E 111, L022105 (2025)10.1103/PhysRevE.111.L022105] regarding the appropriate measure of high temperature correlation length in this regime.
In this work we revisit the axial third-nearest-neighbor Ising chain and examine in detail some aspects of its phase behavior ensuing from competing interactions and resulting frustration. We probe the phase behavior with two complimentary tools: a microscopic two-point correlation function C(n), which we carefully construct after appropriate spin transformations, and a macroscopic, thermodynamic curvature R which we obtain from the free entropy. We report observations of phenomena such as hyperscaling breakdown and intermediate-ranged modulations among others. The zero-field thermodynamic curvature R_{0} is shown to systematically subdivide the ground-state phases into regions of attractive or repulsive effective interactions of varying strength. Furthermore, R_{0} brings forth the significance of some third-order moments in describing the effects of frustration, including the multiphase lines. Combined use of both the tools in the short-ranged modulated order regime confirms and further clarifies the discussion in [S. Khatua et al., Phys. Rev. E 111, L022105 (2025)10.1103/PhysRevE.111.L022105] regarding the appropriate measure of high temperature correlation length in this regime.
This Letter examines the full scope of long-standing conjectures identifying the invariant thermodynamic curvature R as the correlation volume ξ^{d} and also as a measure of underlying statistical interactions. To this end, we set up a two-parameter axial next nearest neighbor Ising (ANNNI) chain featuring two next nearest neighbor (nnn) and a nearest neighbor (nn) interaction. Competing interactions and resulting frustrations engender a rich phase behavior including a crossover between two ferrimagnetic subphases. We show that R attests to all its conjectured attributes with valuable physical insights into the character of mesoscopic fluctuating substructures. In a remarkable demonstration of its relevance at a far-from-critical point, R is shown to resolve a hitherto unnoticed tricky issue involving ξ. A physically transparent expression for the zero field R helps bring into focus the pivotal role played by some third order fluctuation moments.
Microscopic models of realistic thermodynamic systems usually involve a number of parameters, not all of equal macroscopic relevance. We examine a decorated (1+3) Ising spin chain containing two microscopic parameters: a stiff parameter K mediating the long-range interactions, and a sloppy J operating within local spin groups. We show that K dominates the macroscopic behavior, with varying J having only a weak effect, except in regions where J brings about transitions between phases through its conditioning of the local spin groups with which K interacts. We calculate the heat capacity C(H), the magnetic susceptibility χ(T), and the thermodynamic curvature R. For large |J/K|, we identify four magnetic phases: ferromagnetic, antiferromagnetic, and two ferrimagnetic, according to the signs of K and J. We argue that for characterizing these phases, the strongest picture is offered by the thermodynamic geometric invariant R, proportional to the correlation length ξ. This picture has correspondences to other cases, such as fluids.
This work examines the full scope of long-standing conjectures identifying the invariant thermodynamic curvature R as the correlation volume ξd and also as a measure of underlying statistical interactions. To this end, we set up a two-parameter ANNNI (Axial Next Nearest Neighbour Ising) chain featuring two next nearest neighbour (nnn) and a nearest neighbour (nn) interaction. Competition between interactions and resulting frustration engender a rich phase behaviour including a cross-over between two ferrimagnetic sub-phases. We show that R attests to all its conjectured attributes with valuable insights into the character of mesoscopic fluctuating substructures. In a remarkable demonstration of its relevance at a far-from-critical point, R is shown to resolve a hitherto unnoticed tricky issue involving ξ. A physically transparent expression for the zero field R helps bring into focus the pivotal role played by some third order fluctuation moments.
Describing and understanding the consequences of competing interactions remains profoundly challenging in both classical and quantum systems, as it is difficult to identify suitable order parameters, thereby hindering the characterization of certain phases such as the floating phase found in the one-dimensional axial next-nearest-neighbor Ising (ANNNI) model in the presence of the frustration interactions between the nearest and next-nearest neighbor sites. In this work, we employ the pattern picture to explore the frustration physics in such a model. This picture has been comprehensively detailed in our previous article [Yang and Luo, Phys. Rev. E \textbf{112}, 044102 (2025)]. Here, we apply it to the ANNNI model with periodic boundary conditions, considering system sizes ranging from L=16 to L=128. Our results demonstrate that the ground state of the system comprises four phases: ferromagnetic, paramagnetic, floating, and ⟨2,2⟩ antiphase. The transition from the ferromagnetic to the paramagnetic phase is continuous, analogous to that in the transverse-field Ising model, while the transition from the floating phase to the antiphase is first order. Furthermore, the floating phase exhibits particularly intriguing characteristics: states with distinct domain structures emerge successively as the frustration parameter increases. As the system size grows, this succession becomes progressively denser, leading to the reasonable inference that it eventually approaches a continuous variation in the thermodynamic limit. To validate the effectiveness of our picture, we computed the second derivative of the ground-state energy, which exhibits multiple dips within the floating phase−consistent with the pattern language.
We study the equilibrium and dynamical properties of the axial next-nearest-neighbor Ising chain at the multiphase point. An interesting property of the system is the macroscopic degeneracy of the ground state leading to finite zero-temperature entropy. In our equilibrium study we consider the effect of softening the spins. We show that the degeneracy of the ground state is lifted and there is a qualitative change in the low-temperature behavior of the system with a well-defined low-temperature peak of the specific heat that carries the thermodynamic "weight" of the ground state entropy. In our study of the dynamical properties, the stochastic Kawasaki dynamics is considered. The Fokker-Planck operator for the process corresponds to a quantum spin Hamiltonian similar to the Heisenberg ferromagnet but with constraints on allowed states. This leads to a number of differences in its properties, which are obtained through exact numerical diagonalization, simulations, and by obtaining various analytic bounds.
In spin systems, geometrical frustration describes the impossibility of minimizing simultaneously all the interactions in a Hamiltonian, often giving rise to macroscopic ground-state degeneracies and emergent low-temperature physics. In this thesis, combining tensor network (TN) methods to Monte Carlo (MC) methods and ground-state energy lower bound approaches, we study two-dimensional frustrated classical Ising models. In particular, we focus on the determination of the residual entropy in the presence of farther-neighbor interactions in kagome lattice Ising antiferromagnets (KIAFM). In general, using MC to determine the residual entropy is a significant challenge requiring ad-hoc updates, a precise evaluation of the energy at all temperatures to allow for thermodynamic integration, and a good control of the finite-size scaling behavior. As an alternative, we turn to TNs; however, we argue that, in the presence of frustration and macroscopic ground-state degeneracy, standard algorithms fail to converge at low temperatures on the usual TN formulation of partition functions. Inspired by methods for constructing ground-state energy lower bounds, we propose a systematic way to find the ground-state local rule using linear programming. Characterizing the rules as tiles that can be tessellated to form ground states of the model gives rise to a natural contractible TN formulation of the partition function. This method provides a direct access to the ground-state properties of frustrated models and, in particular, allows an extremely precise determination of their residual entropy. We then study two models inspired by artificial spin systems on the kagome lattice with out-of-plane (OOP) anisotropy. The first model is motivated by experiments on an array of chirally coupled nanomagnets. We argue that the farther-neighbor to nearest-neighbor couplings ratios in this system are much smaller than in the dipolar case, J2/J1 being of the order of 2%. A comparison of the experimental correlations with the results of extensive TN and MC simulations shows that (1) the experimental second- and third-neighbor correlations are inverted as compared to those of a pure nearest-neighbor model at equilibrium (even with a magnetic field), and (2) second-neighbor couplings as small as 1% of the nearest-neighbor couplings will affect the spin-spin correlations even at fairly high temperatures. Motivated by dipolar coupled artificial spin systems, we turn to the progressive lifting of the ground-state degeneracy of the KIAFM. We provide a detailed study of the ground-state phases of this model with up to third neighbor interactions, for arbitrary J2, J3 such that J1 >> J2, J3, obtaining exact results for the ground-state energies. When all couplings are antiferromagnetic, we exhibit three macroscopically degenerate ground-state phases and establish their residual entropy using our TN approach. Furthermore, in the phase corresponding to the dipolar KIAFM truncated to third neighbors, we use the ground-state tiles to establish the existence of a mapping to the ground-state manifold of the triangular Ising antiferromagnet.