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.
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.
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.
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.
Using the transfer matrix method for Axial Next-Nearest Neighbor Ising model without an external field on a closed chain of spins of width 2 in the direction of interaction of only nearest neighbors and length L in the direction of interaction of nearest neighbors and next nearest neighbors, we found exact values of the partition function in a finite strip of length L , free energy, internal energy per node, heat capacity in a finite strip of length L and in the thermodynamic limit. The entire spectrum of the 16x16 transfer matrix and the structure of all transfer matrix eigenvectors are found. The problem of finding the spectrum of the transfer matrix is reduced to finding the spectrum of two matrixes of size 8x8. The characteristic polynomial of the first matrix is decomposed into two linear polynomials of multiplicity two and two square trinomials. The characteristic polynomial of the second matrix is decomposed into two polynomials of the fourth degree, the roots of which are found by the Ferrari method. The free energy in the thermodynamic limit has a very simple form and is expressed in terms of the logarithm of the root of the quadratic equation. Two theorems on the physical characteristics of the ANNNI model in a finite closed strip of length L and in the thermodynamic limit are formulated and proved. An example of calculation is given for some values of the model parameters in the thermodynamic limit, heat capacity graphs are plotted. A comparative analysis of the heat capacity for the double-stranded ANNNI model and the ANNNI model on the entire flat lattice is given. The closest connection and similarity of the physical characteristics of the two-chain model and the model on the entire flat lattice is shown.