Quantum Ising model

Garoual, Anass

This preprint studies the hierarchical quantum transverse-field Ising model (HQTFIM) with couplings defined by a 2-adic hierarchy, 𝐽 𝑖 𝑗 ∝ 2 − 𝛼 𝑣 2 ( 𝑗 − 𝑖 ) J ij ∝2 −αv 2 (j−i) . Building on the classical Dyson hierarchical Ising model, we construct an explicit block renormalization scheme for the dimensionless transverse field 𝑥 = ℎ / 𝐽 x=h/J and derive a closed RG map with a non-trivial fixed point 𝑥 \* ( 𝛼 ) x \* (α). The corresponding field ℎ \* ( 𝛼 ) = 𝑥 \* ( 𝛼 ) 𝐽 0 h \* (α)=x \* (α)J 0 is interpreted as an “RG self-similarity field” deep inside the ordered phase. On the analytical side, we prove two exact structural results for the ideal Néel cat state |psi_cat> = (|Néel> + |anti-Néel>)/sqrt(2) on chains of length N = 2^H. First, for any dyadic bipartition A:B, the von Neumann entanglement entropy of the reduced state on A is exactly one bit. Second, we construct an explicit tree tensor network (TTN) representation of |psi_cat> on a depth-H binary tree with bond dimension chi = 2, independent of system size. This makes precise in what sense the Néel cat is an extreme example of an area-law state with a simple tensor-network structure. On the numerical side, we use dense exact diagonalization (Python / NumPy / SciPy) to compare the HQTFIM to the standard one-dimensional nearest-neighbour transverse-field Ising model (TFIM). For alpha = 1.5 and h/J0 = x^*(1.5) ≈ 0.404, the HQTFIM ground state in the even-parity sector exhibits: (i) a fidelity F_N with the ideal Néel cat that increases with N; (ii) a half-chain entanglement entropy S_N pinned extremely close to one bit; and (iii) a staggered order parameter M_stagg^2 very close to one. By contrast, the standard TFIM at the same ratio h/J has essentially zero Néel-cat fidelity and negligible staggered order. We further compare the HQTFIM at h = h^*(alpha) with the standard TFIM at its exact critical point h_c = J. The standard TFIM shows the expected logarithmic growth of entanglement, S_N ~ (c/3) log N with an effective central charge c_eff ≈ 0.7 on the small system sizes we access, and vanishing staggered order. In sharp contrast, at h^*(alpha) the HQTFIM shows entanglement consistent with a strict area law, S_N → 1 bit, and strong Néel order. A numerical scan in h for fixed N identifies a “true” critical field h_c(alpha) for the HQTFIM from the rapid drop of M_stagg^2(h); we find h_c(alpha) >> h^*(alpha), with ratios between ~5 and ~28 over alpha in [1.2, 3.0]. Thus the RG self-similarity field h^*(alpha) is not a quantum critical point but a special ordered-phase point with an unusually clean Néel-cat ground state. Finally, we characterize finite-size corrections at h^*(alpha) via power-law fits 1 − F_N ~ N^{-beta(alpha)} and 1 − M_stagg^2 ~ N^{-gamma(alpha)}, with beta(alpha) ≈ 1 and gamma(alpha) ≈ 1.8–2.1 and a systematic but mild dependence on alpha. We clearly separate mathematically rigorous statements (block spectrum, exact entanglement, TTN construction) from numerical or conjectural claims (properties of the HQTFIM in the thermodynamic limit and the location/universality class of the true critical point). The work suggests a concrete set of open problems on rigorous phase diagrams, entanglement scaling, and excitation spectra for quantum hierarchical models.

Geoffrey Grimmett

Subir Sachdev A

Jelena Stajic

Magnetism The material CoNb2O6 contains loosely bound zigzag chains of magnetic cobalt ions and is considered to be a model system for the so-called one-dimensional transverse field Ising model (TFIM). In this iconic model of quantum criticality, a string of particles with spin ½ orders with all of the spins pointing in the same direction, this ferromagnetic state melts at a critical value of an external transverse magnetic field. Morris et al. re-examined the applicability of this model to CoNb2O6 using time–domain terahertz spectroscopy and found that some of their results could not be explained by TFIM. Instead, the data fit what the authors dubbed a twisted Kitaev chain model, in which the Ising direction alternates along the chain. Nat. Phys. 10.1038/s41567-021-01208-0 (2021).

Somenath Jalal, Rishabh Khare, Siddhartha Lal

The thermal dynamics of the two-dimensional Ising model and quantum dynamics of the one-dimensional transverse-field Ising model (TFIM) are mapped to one another through the transfer-matrix formalism. We show that the fermionised TFIM undergoes a Fermi-surface topology-changing Lifshitz transition at its critical point. We identify the degree of freedom which tracks the Lifshitz transition via changes in topological quantum numbers (e.g., Chern number, Berry phase etc.). An emergent $SU(2)$ symmetry at criticality is observed to lead to a topological quantum number different from that which characterises the ordered phase. The topological transition is also understood via a spectral flow thought-experiment in a Thouless charge pump, revealing the bulk-boundary correspondence across the transition. The duality property of the phases and their entanglement content are studied, revealing a holographic relation with the entanglement at criticality. The effects of a non-zero longitudinal field and interactions that scatter across the singular Fermi surface are treated within the renormalisation group (RG) formalism. The analysis reveals that the critical point of the 1D TFIM and the 1D spin-1/2 Heisenberg chain are connected via a line of $SU(2)$-symmetric theories. We extend our analysis to show that the classical to quantum correspondence links the critical theories of Ising models in various dimensions holographically through the universal effective Hamiltonian that describes the Lifshitz transition of the 1D TFIM. We obtain in this way a unified perspective of transitions in Ising models that lie beyond the traditional Ginzburg-Landau-Wilson paradigm. We discuss the consequences of our results for similar topological transitions observed in classical spin models, topological insulators, superconductors and lattice gauge-field theories which are related to the Ising universality class.

George Allen Baker Jr.