We present a self-contained numerical study of the one-dimensional transverse-field Ising model (TFIM), tracing its ground-state entanglement structure from exact diagonalization at small system size (L=8,14,20) through density-matrix renormalization group (DMRG) calculations up to L=100. A single, consistent methodology - identical field grid, identical convergence diagnostic (independent runs at bond dimensions chi=100 and chi=200 at every point) - is used across all seven system sizes, and we show explicitly where exact diagonalization and DMRG must agree exactly rather than merely approximately. Along the way we document and resolve a bond-dimension convergence artifact that produced a spurious discontinuity in an earlier, less systematic dataset, as a worked cautionary example for practitioners. Using the resolved dataset we extract the central charge of the transition via two complementary routes - the leading logarithmic finite-size scaling of the mid-chain entropy, and the full Calabrese-Cardy formula applied to every bond of every system size simultaneously - obtaining c_eff to 0.51-0.52 as short-distance lattice corrections are systematically excluded, consistent with the exact two-dimensional Ising value c=1/2. We review the quantum-classical (Suzuki-Trotter) correspondence that underlies this agreement and discuss how the Widom-Kadanoff scaling hypothesis transplants from the classical to the quantum problem. The manuscript is intended as both a physics result and a worked methodological example of careful finite-size numerics.
We present a self-contained numerical study of the one-dimensional transverse-field Ising model (TFIM), tracing its ground-state entanglement structure from exact diagonalization at small system size (L=8,14,20) through density-matrix renormalization group (DMRG) calculations up to L=100. A single, consistent methodology - identical field grid, identical convergence diagnostic (independent runs at bond dimensions chi=100 and chi=200 at every point) - is used across all seven system sizes, and we show explicitly where exact diagonalization and DMRG must agree exactly rather than merely approximately. Along the way we document and resolve a bond-dimension convergence artifact that produced a spurious discontinuity in an earlier, less systematic dataset, as a worked cautionary example for practitioners. Using the resolved dataset we extract the central charge of the transition via two complementary routes - the leading logarithmic finite-size scaling of the mid-chain entropy, and the full Calabrese-Cardy formula applied to every bond of every system size simultaneously - obtaining c_eff to 0.51-0.52 as short-distance lattice corrections are systematically excluded, consistent with the exact two-dimensional Ising value c=1/2. We review the quantum-classical (Suzuki-Trotter) correspondence that underlies this agreement and discuss how the Widom-Kadanoff scaling hypothesis transplants from the classical to the quantum problem. The manuscript is intended as both a physics result and a worked methodological example of careful finite-size numerics.
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.
The scaling behavior of the entanglement entropy in the two-dimensional random transverse field Ising model is numerically studied through the strong disordered renormalization group method. We find that the leading term of the entanglement entropy always linearly scales with the block size. However, besides this area law contribution, we find a subleading logarithmic correction at the quantum critical point. This correction is discussed from the point of view of an underlying percolation transition, both at finite and at zero temperature.
We numerically study the one-dimensional long-range transverse field Ising model (TFIM) in the antiferromagnetic (AFM) regime at zero temperature using generalized Hartree-Fock (GHF) theory. The spin-spin interaction extends to all spins in the lattice and decays as $1/{r}^{\ensuremath{\alpha}}$, where r denotes the distance between two spins and $\ensuremath{\alpha}$ is a tunable exponent. We map the spin operators to Majorana operators and approximate the ground state of the Hamiltonian with a fermionic Gaussian state (FGS). Using this approximation, we calculate the ground-state energy and the entanglement entropy, which allows us to map the phase diagram for different values of $\ensuremath{\alpha}$. In addition, we compute the scaling behavior of the entanglement entropy with the system size to determine the central charge at criticality for the case of $\ensuremath{\alpha}>1$. For $\ensuremath{\alpha}<1$ we find a logarithmic divergence of the entanglement entropy even far away from the critical point, a feature of systems with long-range interactions. We provide a detailed comparison of our results to outcomes of density matrix renormalization group (DMRG) and the linked cluster expansion (LCE) methods. In particular, we find excellent agreement of GHF with DMRG and LCE in the weak long-range regime $\ensuremath{\alpha}\ensuremath{\ge}1$, and qualitative agreement with DMRG in the strong-long range regime $\ensuremath{\alpha}\ensuremath{\le}1$. Our results highlight the power of the computationally efficient GHF method in simulating interacting quantum systems.
Abstract The physical properties of a quantum many-body system can, in principle, be determined by diagonalizing the respective Hamiltonian, but the dimensions of its matrix representation scale exponentially with the number of degrees of freedom. Hence, only small systems that are described through simple models can be tackled via exact diagonalization. To overcome this limitation, numerical methods based on the renormalization group paradigm that restrict the quantum many-body problem to a manageable subspace of the exponentially large full Hilbert space have been put forth. A striking example is the density-matrix renormalization group (DMRG), which has become the reference numerical method to obtain the low-energy properties of one-dimensional quantum systems with short-range interactions. Here, we provide a pedagogical introduction to DMRG, presenting both its original formulation and its modern tensor-network-based version. This colloquium sets itself apart from previous contributions in two ways. First, didactic code implementations are provided to bridge the gap between conceptual and practical understanding. Second, a concise and self-contained introduction to the tensor-network methods employed in the modern version of DMRG is given, thus allowing the reader to effortlessly cross the deep chasm between the two formulations of DMRG without having to explore the broad literature on tensor networks. We expect this pedagogical review to find wide readership among students and researchers who are taking their first steps in numerical simulations via DMRG. Graphic abstract
The square-lattice J 1 - J 2 transverse-field (TF) Ising model was investigated with the exact diagonalization (ED) method. In order to analyze the TF-driven phase transition, we applied the longitudinal-field fidelity susceptibility χ F ( h ) , which is readily evaluated via the ED scheme. Here, the longitudinal field couples with the absolute value of the magnetic moment | M | rather than the raw M so that the remedied fidelity susceptibility exhibits a peak around the critical point; note that the spontaneous magnetization does not appear for the finite-size systems. As a preliminary survey, the modified fidelity susceptibility χ F ( h ) is applied to the analysis of criticality for J 2 = 0 , where a number of preceding results are available. Thereby, properly scaling the distance from the multi-criticality, η = 0 . 5 − J 2 , the χ F ( h ) data were cast into the crossover-scaling formula, and the multi-critical exponent for χ F ( h ) is estimated. The result is cross-checked by the numerically evaluated β -function behavior.
We develop an efficient method to perform density matrix renormalization group simulations of the SU(N) Heisenberg chain with open boundary conditions taking full advantage of the SU(N) symmetry of the problem. This method is an extension of the method previously developed for exact diagonalizations and relies on a systematic use of the basis of standard Young tableaux. Concentrating on the model with the fundamental representation at each site (i.e., one particle per site in the fermionic formulation), we have benchmarked our results for the ground-state energy up to N=8 and up to 420 sites by comparing them with Bethe ansatz results on open chains, for which we have derived and solved the Bethe ansatz equations. The agreement for the ground-state energy is excellent for SU(3) (12 digits). It decreases with N, but it is still satisfactory for N=8 (six digits). Central charges c are also extracted from the entanglement entropy using the Calabrese-Cardy formula and agree with the theoretical values expected from the SU(N)1 Wess-Zumino-Witten conformal field theories.