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arXiv:2409.01643

Probing the mesoscopics of competing interactions with the thermodynamic curvature: the case of a two-parameter ANNNI chain

Khatua, Soumen, Anurag Sahay

Published Sep 3, 2024• 0 citations
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Abstract

This work 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 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.

DOI
10.48550/arxiv.2409.01643

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This work examines the full scope of long-standing conjectures identifying the invariant thermodynamic curvature RRR as the correlation volume ξdξ^dξd 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 RRR 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, RRR is shown to resolve a hitherto unnoticed tricky issue involving ξξξ. A physically transparent expression for the zero field RRR helps bring into focus the pivotal role played by some third order fluctuation moments.
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.
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.
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.
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=16L=16L=16 to L=128L=128L=128. Our results demonstrate that the ground state of the system comprises four phases: ferromagnetic, paramagnetic, floating, and ⟨2,2⟩\langle 2,2 \rangle⟨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 present a kinetic lattice system that evolves in general towards steady nonequilibrium states due to dynamical conflict between nearest- and next-nearest-neighbour interactions. Under two simple particular limits, the system would reach asymptotically the canonical equilibrium states for the ordinary Ising model and for the axial next-nearest-neighbour Ising (ANNNI) model, respectively. We find more generally that, independently of the lattice dimension, the steady state probability distribution for a given class of transition rates has a quasi-canonical structure with a short-range effective Hamiltonian. We solve exactly the one-dimensional version of this case, and compare its behaviour to the one for the ordinary ANNNI model. In particular, the system is shown to exhibit several spatially modulated phases and impure critical points. We also obtain some information on the phase diagram for the two-dimensional lattice.