Transfer-matrix spectra and finite-size scaling for two-dimensional spin systems with non-reciprocal couplings

The strip transfer-matrix method with sparse-matrix factorization recently delivered benchmark-level results for the spin-1 Blume-Capel model on the triangular lattice - the tricritical point $(\Delta_t,T_t)=[2.938568(1),0.9847240(2)]$, validation of the tricritical-Ising conformal-field-theory spectrum, and interfacial tensions from persistence-length scaling - for strip widths up to $L=19$. Its authors announced as the explicitly stated next step the extension to models with non-reciprocal interactions introduced through asymmetric coupling matrices, relevant to active-matter-inspired dynamics, where conventional simulation methods often struggle. This extension is open: how the transfer-matrix spectrum and the finite-size-scaling framework behave when the couplings are made asymmetric - so that the strip transfer matrix is genuinely non-symmetric (non-Hermitian), left and right eigenvectors differ, and subdominant eigenvalues may be complex - has not been established. The problem is to formulate the strip transfer-matrix framework for non-reciprocal spin systems (deciding between directed-bond equilibrium embeddings and genuinely kinetic, detailed-balance-violating constructions), to generalize the spectral diagnostics of the Hermitian framework (exponential degeneracy of leading eigenvalues as a first-order-transition criterion, correlation and persistence lengths, interfacial-tension extraction, conformal-field-theory-based finite-size scaling of the central charge and scaling dimensions) to this setting, and to validate the method on a non-reciprocal Ising or Blume-Capel model against the Monte-Carlo, mean-field-bifurcation, or renormalization-group results established since 2024.

Background

The Blume-Capel model is a spin-1 generalization of the Ising ferromagnet in which each site carries $s_i\in\{-1,0,+1\}$, the state $s_i=0$ representing a vacancy, with Hamiltonian $\mathcal{H}=-J\sum_{\langle i,j\rangle}s_is_j+\Delta\sum_i s_i^2$; the crystal-field coupling $\Delta$ controls the vacancy density. In two dimensions its $(\Delta,T)$ phase diagram contains a line of continuous Ising-class transitions meeting a first-order segment at a tricritical point, where the universality class is that of the tricritical Ising conformal field theory (CFT), with central charge $c=7/10$ and magnetic and thermal scaling dimensions $x_\sigma=3/40$ and $x_\epsilon=1/5$. (A scaling dimension $x$ fixes the power-law decay of an operator's correlation function with distance; the central charge measures the number of effective critical degrees of freedom of the CFT.) In the strip transfer-matrix method one computes the partition function of an $L\times M$ strip with periodic boundary conditions across the width as $Z=\mathrm{Tr}\,\mathbf{T}^M$, where $\mathbf{T}$ is a $3^L\times 3^L$ matrix acting on transverse row configurations; its leading eigenvalues determine the free energy, correlation lengths, and finite-size-scaling (FSS) data of the infinite strip. Because $3^L$ grows exponentially, full diagonalization is impossible beyond small $L$, so Mataragkas, Vasilopoulos, Fytas and Kim (2025) - building on a sparse-matrix factorization introduced by Blöte, Wu and Wu (1990) for the honeycomb-lattice Ising model and computing a few extremal eigenvalues with the thick-restart Lanczos algorithm (a Krylov-subspace eigensolver for large symmetric matrices) - constructed a symmetric factorized $\mathbf{T}$ for the triangular-lattice Blume-Capel model and reached strip widths up to $L=19$. From the leading eigenvalues $\lambda_i$ they formed free-energy levels $f_i=-(T/\sqrt{3}L)\ln\lambda_i$, located first-order transition points through the exponential closing of the spectral gap with $L$ (the finite-size signature of level degeneracy at first-order transitions), extracted the correlation length $\xi_L$ and persistence length $\tilde\xi_L$ from eigenvalue ratios (the persistence length measures how far the strip's statistical weight propagates along its axis and diverges with the interfacial tension at coexistence), obtained the interfacial tension $\sigma$ from the scaling $\tilde\xi^\ast_L\sim L\exp[\beta\sigma L]$ along the coexistence curve, and validated the tricritical point through the CFT finite-size forms $f(L)=f_\infty-\pi c/(6L^2)$, $2\pi x_\epsilon=L/(\sqrt{3}\,\tilde\xi_L)$, and $2\pi x_\sigma=L/(\sqrt{3}\,\xi_L)$. This machinery presumes a symmetric (Hermitian, real-spectrum) transfer matrix. Non-reciprocal interactions - couplings with $J_{ij}\neq J_{ji}$, as arise generically in active matter and in systems whose interactions are mediated by a driven, non-equilibrium environment - violate action-reaction symmetry and, in spatially extended spin models, break detailed balance (the balance of forward and backward transition rates that guarantees an equilibrium Boltzmann measure). Since 2024 such models have been studied intensively by Monte Carlo, mean-field bifurcation analysis, and dynamic renormalization group: Avni, Fruchart, Martin, Seara and Vitelli introduced the two-species nonreciprocal Ising model with disordered, statically ordered, and time-dependent swap phases (PRL 2025; PRE 2025); Rajeev and Anil Kumar (2024) simulated a two-dimensional Ising model whose couplings are modified along a non-reciprocity vector, and Garcés and Levis (2024) studied single-species vision-cone variants (in which each spin interacts only with neighbours inside a directional cone); Saha and Mohanty (2024) analyzed directed non-reciprocal Potts models including an equilibrium variant; Arjun R, Patra and Anil Kumar (Phys. Rev. E 2026) studied a two-species nonreciprocal Blume-Capel model with antisymmetric single-ion anisotropies; and Akritidis, Garcés, Vasilopoulos, Carosi, Levis and Fytas (2026) showed by Monte Carlo and dynamic renormalization group that two-dimensional Ising criticality is robust against vision-cone non-reciprocity. A structural caveat delimits the problem: for pure pair interactions the antisymmetric part of the coupling matrix cancels in the energy $\sum_{ij}J_{ij}s_is_j$, so genuine non-reciprocity requires either directed-bond or multi-spin equilibrium constructions or a genuinely kinetic (Glauber-type, detailed-balance-violating) model definition - and in the kinetic case no scalar partition function exists, so whether and how a transfer-matrix formulation applies is itself part of what must be established. For an entry-wise positive but non-symmetric transfer matrix the Perron-Frobenius theorem still guarantees a real, non-degenerate dominant eigenvalue, but subdominant eigenvalues can occur in complex-conjugate pairs and left and right eigenvectors are biorthogonal rather than transposes of one another - precisely the regime in which the spectral diagnostics and CFT-based finite-size-scaling forms above have not been justified.

Problem Statement

Extend the strip transfer-matrix framework - the symmetric sparse factorization with Krylov eigenvalue computation and its finite-size-scaling diagnostics, as validated by Mataragkas, Vasilopoulos, Fytas and Kim for the triangular-lattice Blume-Capel model - to two-dimensional spin systems with non-reciprocal interactions introduced through asymmetric coupling matrices, the explicitly announced next step of that work. A complete answer should: (i) specify the model class and the strip construction such that the couplings are genuinely asymmetric and the resulting transfer matrix is well-defined but non-symmetric - a solution may treat either an equilibrium embedding (for example directed-bond or multi-spin weights of the kind used in directed Potts models) or a kinetic, detailed-balance-violating transfer formalism, provided it states clearly which setting it solves; the kinetic branch, in which no scalar partition function exists, may be excluded by an explicit argument; (ii) establish how the spectral diagnostics carry over: the behavior of the free-energy levels $f_i$ and of the exponential spectral-gap criterion for first-order transitions, the definition and scaling of the correlation and persistence lengths, and the extraction of interfacial tensions, when subdominant eigenvalues may be complex and left and right eigenvectors are biorthogonal; (iii) derive the corresponding finite-size-scaling forms - the analogue of $f(L)=f_\infty-\pi c/(6L^2)$ and of the eigenvalue-ratio relations for the central charge and the scaling dimensions, including whether a conformal-field-theory interpretation survives; and (iv) validate the resulting method on at least one concrete non-reciprocal Ising or Blume-Capel model against established results obtained by Monte Carlo, mean-field bifurcation analysis, or renormalization group (for example, the Ising-class location of the static-order transition in the 2D nonreciprocal Blume-Capel model, or the dependence of critical points and exponents on the non-reciprocity strength). These components jointly serve one objective - a validated transfer-matrix spectral and finite-size-scaling methodology for non-reciprocal spin systems - and any proposed method must reduce to the published triangular-lattice Blume-Capel transfer-matrix results in the symmetric-coupling limit.

References

1. D. Mataragkas, A. Vasilopoulos, N. G. Fytas, and D.-H. Kim, "Transfer-matrix approach to the Blume-Capel model on the triangular lattice", Physical Review Research 7, 033240 (2025), doi:10.1103/jfl3-f4kd, https://arxiv.org/abs/2506.16483
2. D. Mataragkas, A. Vasilopoulos, N. G. Fytas, and D.-H. Kim, "Tricriticality and finite-size scaling in the triangular Blume-Capel ferromagnet", Physical Review Research 7, 013214 (2025), doi:10.1103/PhysRevResearch.7.013214, arXiv:2411.11689, https://arxiv.org/abs/2411.11689
3. D. Mataragkas, A. Vasilopoulos, D.-H. Kim, and N. G. Fytas, "Crossover and universality breaking in the dilute Baxter-Wu model", arXiv:2605.13238 (2026), https://arxiv.org/abs/2605.13238
4. D. Mataragkas, A. Vasilopoulos, Z. D. Vatansever, Y. Kim, D.-H. Kim, and N. G. Fytas, "First-order phase transitions in the three-dimensional Blume-Capel ferromagnet", Physica A: Statistical Mechanics and its Applications 699, 131922 (2026), doi:10.1016/j.physa.2026.131922, arXiv:2606.17691, https://arxiv.org/abs/2606.17691
5. M. Akritidis, A. Garcés, A. Vasilopoulos, M. Carosi, D. Levis, and N. G. Fytas, "Fate of the Ising universality class under nonreciprocal interactions", arXiv:2606.06981 (2026), https://arxiv.org/abs/2606.06981
6. Arjun R, P. P. Patra, and A. V. Anil Kumar, "Nonreciprocal Blume-Capel Model with Antisymmetric Single-Ion Anisotropies", Physical Review E 113, 064140 (2026), doi:10.1103/y4b1-q6l7, arXiv:2512.18917, https://arxiv.org/abs/2512.18917
7. Y. Avni, M. Fruchart, D. Martin, D. Seara, and V. Vitelli, "Nonreciprocal Ising model", Physical Review Letters 134, 117103 (2025), doi:10.1103/PhysRevLett.134.117103, arXiv:2311.05471, https://arxiv.org/abs/2311.05471
8. Y. Avni, M. Fruchart, D. Martin, D. Seara, and V. Vitelli, "Dynamical phase transitions in the nonreciprocal Ising model", Physical Review E 111, 034124 (2025), doi:10.1103/PhysRevE.111.034124, arXiv:2409.07481, https://arxiv.org/abs/2409.07481
9. A. Garcés and D. Levis, "Phase Transitions in single species Ising Models with Non-Reciprocal couplings", Journal of Statistical Mechanics: Theory and Experiment 2025, 043205 (2025), doi:10.1088/1742-5468/adc896, arXiv:2411.03544, https://arxiv.org/abs/2411.03544
10. S. K. Saha and P. K. Mohanty, "Non-reciprocal interactions preserve the universality class of Potts model", arXiv:2412.19664 (2024), https://arxiv.org/abs/2412.19664
11. A. K. Rajeev and A. V. Anil Kumar, "Ising model with non-reciprocal interactions", arXiv:2403.06875 (2024), https://arxiv.org/abs/2403.06875
12. G. A. Weiderpass, M. Sharma, and S. Sethi, "Solving the Kinetic Ising Model with Non-Reciprocity", Physical Review E 111, 024107 (2025), doi:10.1103/PhysRevE.111.024107, arXiv:2410.23615, https://arxiv.org/abs/2410.23615
13. K. Blom, U. Thiele, and A. Godec, "Local Order Controls the Onset of Oscillations in the Nonreciprocal Ising Model", Physical Review E 111, 024207 (2025), doi:10.1103/PhysRevE.111.024207, arXiv:2407.20312, https://arxiv.org/abs/2407.20312
14. N. Grodzinski, R. L. Jack, and S. A. M. Loos, "Nonreciprocal Disorder Prevents Zero-Temperature Freezing in a Ferromagnet", arXiv:2606.21582 (2026), https://arxiv.org/abs/2606.21582
15. M. Scandolo, J. Pausch, M. E. Cates, and L. Di Carlo, "Non-equilibrium coupling to a diffusing density breaks Ising universality", arXiv:2607.02661 (2026), https://arxiv.org/abs/2607.02661
16. M. Jung and D.-H. Kim, "First-order transitions and thermodynamic properties in the 2D Blume-Capel model: the transfer-matrix method revisited", The European Physical Journal B 90, 245 (2017), doi:10.1140/epjb/e2017-80471-2, https://doi.org/10.1140/epjb/e2017-80471-2
17. H. W. J. Blöte, F. Y. Wu, and X. N. Wu, "Critical point of the honeycomb antiferromagnetic Ising model in a nonzero magnetic field: finite-size analysis", International Journal of Modern Physics B 4, 619 (1990), doi:10.1142/S0217979290000322, https://doi.org/10.1142/S0217979290000322
18. L. Moueddene, N. G. Fytas, and B. Berche, "Phase transition properties via partition function zeros: the Blume-Capel ferromagnet revisited", Journal of Statistical Mechanics: Theory and Experiment 2025, 104001 (2025), doi:10.1088/1742-5468/ae09a4, https://doi.org/10.1088/1742-5468/ae09a4
19. T. Akkurt, R. Erdem, and C. Ekiz, "Geometric characterization of the phase transitions in a Blume-Capel model via the Ruppeiner metric", The European Physical Journal Plus 141, 148 (2026), doi:10.1140/epjp/s13360-026-07330-3, https://doi.org/10.1140/epjp/s13360-026-07330-3
20. I. Tikader and M. Acharyya, "Multiple crossover in the decay of metastable volume fraction of a Blume-Capel ferromagnetic needle", The European Physical Journal B 98, 261 (2025), doi:10.1140/epjb/s10051-025-01115-w, https://doi.org/10.1140/epjb/s10051-025-01115-w
21. P. F. Dias, F. M. Zimmer, N. G. Fytas, and M. Schmidt, "Competing ferromagnetic and antiferromagnetic phases on the frustrated Ising honeycomb lattice", Physica A: Statistical Mechanics and its Applications 686, 131321 (2026), doi:10.1016/j.physa.2026.131321, https://doi.org/10.1016/j.physa.2026.131321