Abstract The critical behavior of spin systems is fundamentally governed by dimensionality and connectivity. Moving beyond translationally invariant lattices, we explore a new paradigm where fractality itself becomes a tunable parameter, engineering magnetic order and critical behavior. By implementing the classical Ising model on a three-dimensional fractal lattice—a static realization of a spin cluster with Hausdorff dimension d H = 2.5 and boundary dimension d = 2—we demonstrate how fractal geometry dictates unique critical phenomena. Using the higher-order tensor renormalization group (HOTRG) method, we identify a finite-temperature phase transition at T c ≈ 2.65231 with exotic critical exponents ( β ≈ 0.059, δ ≈ 35) and a diverging specific heat consistent with a logarithmic singularity (i.e., α = 0) within the present numerical accuracy—a hallmark absent in lower-dimensional fractals. This work establishes that fractal geometry serves as a powerful and untapped degree of freedom for spintronics. It provides a blueprint for designing materials with programmable magnetic phase transitions, paving the way for next-generation, geometry-driven devices in magnonics and neuromorphic computing.

Uniform spin-orbit coupling in an open single-band Hubbard chain is an exactly removable SU(2) gauge field at the Hamiltonian level, but not at the level of laboratory-frame spin correlations. We study this separation using density matrix renormalization group calculations for the repulsive one-dimensional Rashba-Hubbard chain. For open boundary conditions, a site-dependent spin rotation maps the model with hopping t and Rashba spin-orbit strength λ onto the ordinary Hubbard chain with renormalized hopping t_λ=√(t^2+λ^2). Consequently, charge and energy diagnostics are affected only through the bandwidth renormalization, which is quadratic in weak λ/t. Spin correlations, however, respond already at linear order because the same transformation rotates the local spin basis by the wave vector k_ so=2arctan(λ/t). We use DMRG to verify this observable consequence across the filling diagram of finite open chains. The filling structure follows the gauge-equivalent Hubbard model, whereas the spin structure factor shows the predicted spin-orbit sidebands. A dominant Hubbard-chain magnetic wave vector k_0 is transformed into components at k_0± k_ so, folded into the open-chain Brillouin zone. At half filling, where k_0=π, the two sidebands fold onto a single, in-plane, spin spiral wave with k=π-k_ so<π. Away from half filling, the incommensurate Hubbard spin response splits into two distinct spin-orbit-shifted components, producing a real-space beating pattern. Our results provide a filling-resolved tensor-network benchmark for the exactly removable limit of one-dimensional spin-orbit coupling, and establish a controlled reference point for ladders, multiorbital chains, rings, proximitized wires, and higher-dimensional Hubbard systems where spin-orbit coupling can no longer be gauged away.

We propose a tensor-network-based algorithm to study the classical Ising model on an infinitely large hyperbolic lattice with a regular 3D tesselation of identical dodecahedra. We reformulate the corner transfer matrix renormalization group (CTMRG) algorithm from 2D to 3D to reproduce the known results on the cubic lattice. We subsequently generalize the CTMRG to a hyperbolic lattice with dodecahedral cells, which is an infinite-dimensional lattice. We analyze the spontaneous magnetization, von Neumann entropy, and correlation length to find a continuous noncritical phase transition on the dodecahedral lattice. We estimate the phase-transition temperature and find the magnetic critical exponents β=0.4999 and δ=3.007, which confirm the mean-field universality class, in accord with predictions from Monte Carlo and high-temperature series expansions. The algorithm can be applied to arbitrary multistate spin models.

We consider the magnetic field induced by the steady or the quasi-steady electric currents that flow along the straight wires, which are equidistantly arranged on the hyperboloid. The spatial distribution of the magnetic field and the force acting on each wire are calculated. The continuum limit, where the wires are densely aligned, is also considered. We discuss the application of the hyperbolic current configuration to the generation of high magnetic fields.

The Corner Transfer Matrix Renormalization Group (CTMRG) algorithm is modified to measure the magnetization at the boundary of the system, including the corners of the square-shaped lattice. Using automatic differentiation, we calculate the magnetization's first derivative, allowing us to determine the boundary critical exponent β accurately.

The critical behavior of the classical Ising model on a three-dimensional fractal lattice with Hausdorff dimension d_H = ln32 / ln4 = 2.5 is investigated using the higher-order tensor renormalization group (HOTRG) method. We determine the critical temperature T_c ≈ 2.65231 and the critical exponents for magnetization β≈ 0.059 and field response δ≈ 35. Unlike a previously studied 2D fractal with d_H ≈ 1.792, the specific heat for this 3D fractal exhibits a divergent singularity at T_c. The results are compared with those for regular lattices and other fractal structures to elucidate the role of dimensionality in critical phenomena.

We propose a vertex representation of the tensor network (TN) for classical spin systems on hyperbolic lattices. The tensors form a network of regular p-sided polygons (p>4) with the coordination number 4. The response to multistate spin systems on the hyperbolic TN is analyzed for their entire parameter space. We show that entanglement entropy is sensitive to distinguish various hyperbolic geometries, whereas other thermodynamic quantities are not. We test the numerical accuracy of vertex TNs in the phase transitions of the first, second, and infinite order at the point of maximal entanglement entropy. The hyperbolic structure of TNs induces noncritical properties in the bulk, although boundary conditions significantly affect the total free energy in the thermodynamic limit. Thus a developed vertex-type TN can be used for the lowest-energy quantum states on the hyperbolic lattices.

The two-dimensional q-state clock models exhibit the Berezinskii–Kosterlitz–Thouless (BKT) transition for q≥5 since they are a subset of the isotropic XY model. We examine the 6-state clock model with an anisotropic deformation. Selecting the 6-state Potts model as a source of the deformation, the model naturally violates the discrete rotational symmetry of the clock model. We introduce the anisotropic deformation parameter α in the clock model interpolating the clock (α=1) and the Potts (α=0) models. We employ the corner transfer matrix renormalization group method to analyze the phase transitions on the square lattice in the thermodynamic limit. Three different phases and phase transitions are identified. The phase diagram is constructed, and we determine a tricritical point at αc=0.21405(4) and Tc=0.834017(5). Analyzing the latent heat and the entanglement entropy in the vicinity of the Tc(αc), we observe a single discontinuous phase transition and two BKT phase transitions meeting in the tricritical point. The tricritical point exhibits a phase transition of the second order with the critical exponents β≈1/10 and δ≈14. We conjecture that an infinitesimal surrounding of the tricritical point consists of the three fundamental phase transitions, in which the first and the BKT orders gradually weaken into the second-order tricritical point.

The phase transition of the classical Ising model on the Sierpiński carpet, which has the fractal dimension log_{3}^{}8≈1.8927, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase transition is observed at the critical temperature T_{c}^{}≈1.478. Position dependence of local functions is studied through impurity tensors inserted at different locations on the fractal lattice. The critical exponent β associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas T_{c}^{} is not affected. Furthermore, we employ automatic differentiation to accurately and efficiently compute the average spontaneous magnetization per site as a first derivative of free energy with respect to the external field, yielding the global critical exponent of β≈0.135.

We study classical Ising spin-1/2 models on a two-dimensional (2D) square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution, which prevents application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely, the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field, we calculate with high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent β and the anomaly number c are shown to be constant along the critical line, equal to their values β=1/8 and c=1/2 for the 2D Ising model in a zero magnetic field. The 2D ferromagnets behave in analogy with their 1D counterparts defined on a chain of sites, namely, there exists a transient temperature which splits the temperature range into its high-temperature and low-temperature parts. The free energy and the magnetization are well defined in the high-temperature region. In the low-temperature region, the free energy exhibits singularities at the Yang-Lee zeros of the partition function and the magnetization is also ill-defined: It varies chaotically with the size of the system. The transient temperature is determined as a function of the imaginary magnetic field by using the fact that from the high-temperature side both the first derivative of the free energy with respect to the temperature and the magnetization diverge at this temperature.

We generalize a tensor-network algorithm to study the thermodynamic properties of self-similar spin lattices constructed on a square-lattice frame with two types of couplings, J_{1}^{} and J_{2}^{}, chosen to transform a regular square lattice (J_{1}^{}=J_{2}^{}) onto a fractal lattice if decreasing J_{2}^{} to zero (the fractal fully reconstructs when J_{2}^{}=0). We modified the higher-order tensor renormalization group (HOTRG) algorithm for this purpose. Single-site measurements are performed by means of so-called impurity tensors. So far, only a single local tensor and uniform extension-contraction relations have been considered in HOTRG. We introduce 10 independent local tensors, each being extended and contracted by 15 different recursion relations. We applied the Ising model to the J_{1}^{}-J_{2}^{} planar fractal whose Hausdorff dimension at J_{2}^{}=0 is d^{(H)}=ln12/ln4≈1.792. The generalized tensor-network algorithm is applicable to a wide range of fractal patterns and is suitable for models without translational invariance.

We generalize a tensor-network algorithm to study thermodynamic properties of self-similar spin lattices constructed on a square-lattice frame with two types of couplings, $J_{1}^{}$ and $J_{2}^{}$, chosen to transform a regular square lattice ($J_{1}^{} = J_{2}^{}$) onto a fractal lattice if decreasing $J_{2}^{}$ to zero (the fractal fully reconstructs when $J_{2}^{} = 0$). We modified the Higher-Order Tensor Renormalization Group (HOTRG) algorithm for this purpose. Single-site measurements are performed by means of so-called impurity tensors. So far, only a single local tensor and uniform extension-contraction relations have been considered in HOTRG. We introduce ten independent local tensors, each being extended and contracted by fifteen different recursion relations. We applied the Ising model to the $J_{1}^{}-J_{2}^{}$ planar fractal whose Hausdorff dimension at $J_{2}^{} = 0$ is $d^{(H)} = \ln 12 / \ln 4 \approx 1.792$. The generalized tensor-network algorithm is applicable to a wide range of fractal patterns and is suitable for models without translational invariance.

A variant of energy scale deformation is considered for the S = 1/2 antiferromagnetic Heisenberg model on polyhedra. The deformation is induced by the perturbations to the uniform Hamiltonian, whose coefficients are determined by the bond coordinates. On the tetrahedral, octahedral, and cubic clusters, the perturbative terms do not affect the ground state of the uniform Hamiltonian when they are sufficiently small. On the other hand, for the icosahedral and dodecahedral clusters, it is numerically confirmed that the ground state of the uniform Hamiltonian is almost insensitive to the perturbations unless they lead to a discontinuous change in the ground state. The obtained results suggest the existence of a generalization of sine-square deformation in higher dimensions.

We generalize a tensor-network algorithm to study the thermodynamic properties of self-similar spin lattices constructed on a square-lattice frame with two types of couplings, J_{1}^{} and J_{2}^{}, chosen to transform a regular square lattice (J_{1}^{}=J_{2}^{}) onto a fractal lattice if decreasing J_{2}^{} to zero (the fractal fully reconstructs when J_{2}^{}=0). We modified the higher-order tensor renormalization group (HOTRG) algorithm for this purpose. Single-site measurements are performed by means of so-called impurity tensors. So far, only a single local tensor and uniform extension-contraction relations have been considered in HOTRG. We introduce 10 independent local tensors, each being extended and contracted by 15 different recursion relations. We applied the Ising model to the J_{1}^{}-J_{2}^{} planar fractal whose Hausdorff dimension at J_{2}^{}=0 is d^{(H)}=ln12/ln4≈1.792. The generalized tensor-network algorithm is applicable to a wide range of fractal patterns and is suitable for models without translational invariance.

We investigate the Berezinskii-Kosterlitz-Thouless transitions for the square-lattice six-state clock model with the corner-transfer matrix renormalization group (CTMRG). Scaling analyses for effective correlation length, magnetization, and entanglement entropy with respect to the cutoff dimension m at the fixed point of the CTMRG provide transition temperatures consistent with a variety of recent numerical studies. We also reveal that the fixed-point spectrum of the corner-transfer matrix in the critical intermediate phase of the six-state clock model is characterized by the scaling dimension consistent with the c = 1 boundary conformal field theory associated with the effective Z(6) dual sine-Gordon model.

The combination of an exact and Corner Transfer Matrix Renormalization Group (CTMRG) methods is used to study an influence of external electric and magnetic fields on existence of intriguing reentrant magnetic transitions in a coupled spin-electron model on a decorated square lattice. The two-dimensional (2D) decorated square lattice with localized nodal spins and delocalized electrons is taken into account. It was found that the competition among all involved interactions (the electron hopping, spin-spin and spin-electron interaction, external electric and magnetic fields) in combination with thermal fluctuations can produce new type of reentrant magnetic transitions. Depending on the model parameters the non-zero fields can stabilize or destabilize magnetic reentrance. In addition, an alternative and more effective way, for modulating the magnetic reentrance is found. An origin of intriguing low-temperature round maximum in the specific heat was explained as a consequence of rapid changes in the sublattice magnetizations, which is induced through a competition of all presented interactions.

Entanglement entropy is a powerful tool to detect continuous, discontinuous and even topological phase transitions in quantum as well as classical systems. In this work, von Neumann and Renyi entanglement entropies are studied numerically for classical lattice models in a square geometry. A cut is made from the center of the square to the midpoint of one of its edges, say the right edge. The entanglement entropies measure the entanglement between the left and right halves of the system. As in the strip geometry, von Neumann and Renyi entanglement entropies diverge logarithmically at the transition point while they display a jump for first-order phase transitions. The analysis is extended to a classical model of non-overlapping finite hard rods deposited on a square lattice for which Monte Carlo simulations have shown that, when the hard rods span over 7 or more lattice sites, a nematic phase appears in the phase diagram between two disordered phases. A new Corner Transfer Matrix Renormalization Group algorithm (CTMRG) is introduced to study this model. No logarithmic divergence of entanglement entropies is observed at the phase transitions in the CTMRG calculation discussed here. We therefore infer that the transitions neither can belong to the Ising universality class, as previously assumed in the literature, nor be discontinuous.

Magnetic properties of the transverse-field Ising model on curved (hyperbolic) lattices are studied by a tensor product variational formulation that we have generalized for this purpose. First, we identify the quantum phase transition for each hyperbolic lattice by calculating the magnetization. We study the entanglement entropy at the phase transition in order to analyze the correlations of various subsystems located at the center with the rest of the lattice. We confirm that the entanglement entropy satisfies the area law at the phase transition for fixed coordination number, i.e., it scales linearly with the increasing size of the subsystems. On the other hand, the entanglement entropy decreases as power-law with respect to the increasing coordination number.

The entanglement entropy S is calculated on the system boundary of the square-lattice +/- J Ising model by the time-evolving block decimation (TEBD) method. The random average < S > is evaluated on the Nishimori line, through the successive multiplications of transfer matrices, whose width N is up to 300. It is confirmed that < S > shows critical singularity around the Nishimori point.