Theory of Network Contractor Dynamics for Exploring Thermodynamic Properties of Two-dimensional Quantum Lattice Models

I. INTRODUCTION

• NCD combines three proposed phase-transition indicators with an efficient, well-controlled algorithm for two-dimensional models representable as tensor networks.The introduction names the 2D Heisenberg and Ising models as examples.Source: §I · p. 1
• NCD results for the square Ising model and honeycomb spin-1/2 Heisenberg antiferromagnet compare well with exact results, other methods, and quantum Monte Carlo simulations.The comparison is reported as demonstrating the method’s efficiency and accuracy.Source: §I · p. 1
• NCD represents the finite-temperature density operator as a tensor product density operator and reduces infinite tensor-network contractions to local contractions of a tensor cluster and boundary contractor vectors.The cluster is treated as the only inequivalent tensor, and its nonlinear mapping’s fixed point supplies the contractor vectors; this strategy addresses bond-space growth and infinite contractions.Source: §I · p. 1
• The imaginary-time-sweep algorithm controls Trotter-Suzuki and truncation errors through the time slice and discarding weight, and controls deviation from globally optimal truncation through the loop character.When the loop character decays to zero, the effect of destroyed larger loops is negligible and truncation can be considered globally optimal; the scheme is free from the negative sign problem.Source: §I · pp. 1–2
• Quasi-entanglement entropy S, Lyapunov exponent I^lya, and loop character I^loop characterize tensor-network properties and describe nonlocality, enabling detection of possible transitions in classical and quantum systems.Their transition-detection capability is demonstrated for the square Ising model and spin-1/2 anisotropic honeycomb Heisenberg antiferromagnet.Source: §I · p. 2

TEMPERATURE DENSITY OPERATOR AND A TENSOR NETWORK

• Nearest-neighbor local evolution operators and the Trotter-Suzuki decomposition transform the density operator at inverse temperature β into a three-dimensional brick-wall tensor network.SVD introduces geometrical bonds, and translational invariance is applied to the honeycomb-lattice sublattices.Source: TEMPERATURE DENSITY OPERATOR AND A… · p. 2
• Repeated imaginary-time contractions of the TPDO tensors build ρ(β), while geometrical-bond dimensions grow exponentially and require truncation.The evolution begins from the TPDO representation at infinitesimal inverse temperature τ.Source: TEMPERATURE DENSITY OPERATOR AND A… · p. 2
• The partition function becomes a contraction of the resulting infinite two-dimensional tensor network, and thermal averages are calculated from the same density operator.The thermal average is defined as Tr[ρ(β) O]/Z(β).Source: TEMPERATURE DENSITY OPERATOR AND A… · p. 2

III. NETWORK CONTRACTOR DYNAMICS

• NCD defines unit-norm contractors as fixed points of mappings built from a local tensor, equivalent to a rank-1 decomposition of that tensor.Permutation invariance requires equal contractors on paired bonds.Source: §III · pp. 2–3
• NCD applies to tensor-network representations with one inequivalent tensor that is permutation-invariant within each paired set of indices.The theory can also be applied to other prototype tensors or networks meeting this requirement.Source: §III · pp. 3–4
• Replacing a minimal set of local tensors with rank-1 tensors breaks all loops of the remaining tensors, reducing the infinite partition-function contraction to a local contraction with contractors.The defective network is tree-like, and the fixed-point condition yields the local contraction.Source: §III · p. 3, Figure 2 · p. 4
• Thermal averages are computed by contracting the operator and its related tensors with contractors on the boundary of a local tensor cluster.The rest of the network is the same as for the partition function.Source: §III · p. 3
• A smaller Lyapunov exponent Ilya indicates faster convergence of the mapping toward its fixed point against perturbations.The initial contractor vectors may be arbitrary when the total mapping time is sufficiently large.Source: §III · p. 4
• Starting from a cluster that retains loops recovers missing terms and can improve accuracy, but contraction cost grows exponentially with the number of included loops.Outside the starting cluster, the defective network contains no loops of the local tensor.Source: §III · p. 4

IV. CELL TENSOR SIZE, QUASI-ENTANGLEMENT ENTROPY AND LOOP CHARACTER

• Larger cell tensors recover loops absent at smaller sizes and can improve NCD accuracy, but their dimensions grow exponentially with cell tensor size, requiring truncation.The truncations follow the same principle as those used to bound dimensions during imaginary-time contraction.Source: §IV · p. 4
• The quasi-entanglement entropy S is the entanglement entropy of the MPS representing the dominant eigenstate of the partition-function transfer matrix.The contractor is written as an MPS, and its normalized singular-value spectrum is used to obtain S.Source: §IV · pp. 4–5
• The loop character Iloop(γ) measures the difference between contractors for cell tensors of sizes γ and γ + 2, monitoring the effects of loops.When it converges to zero as γ increases, larger loops are negligible and it indicates error from the defective-network, rank-1 approximation.Source: §IV · p. 5

V. IMPLEMENTATION OF NCD SCHEME

• The enlarged-bond environment matrix is formed by contracting all shared bonds except that bond; its SVD truncation keeps the χ largest singular values and globally minimizes the partition-function error.NCD simplifies calculating the environment matrix using a defective tensor network.Source: §V · p. 5
• ITSA initializes the TPDO to the target temperature, then sweeps from the initial TPDO, renews truncations using the target-temperature partition-function network, and repeats until convergence.The network is constructed from ρ(β) and ρ(β̃−β); the converged TPDO enables calculation of the partition function and observables.Source: §V · p. 6
• ITSA sweeps obtain truncations by considering the full partition-function tensor network at the target temperature, avoiding error accumulation during imaginary-time evolution.The Trotter-Suzuki error is controlled by τ, truncation error by ε, and defective-network approximation by Iloop.Source: §V · pp. 5–6

• The NCD free-energy error approaches machine error away from the square-lattice Ising critical point and peaks at 1.26×10^-8 for χ = 32; its accuracy exceeds HOSRG away from criticality and is comparable near it.The figure compares NCD, TRG, SRG, and HOSRG; the cited prose also states that NCD accuracy exceeds TRG, SRG, and HOTRG particularly away from criticality.Source: Figure 6 · p. 6, §VI · p. 6
• For the square-lattice Ising model, S, Iloop, and Ilya locate the critical temperature with relative error about 10^-3.The reported estimates are T^S = 2.271, T^loop = 2.270, and T^lya = 2.271, versus exact Tc ≃ 2.2692.Source: §VI · p. 6, Figure 7 · p. 7
• NCD energies for the spin-1/2 honeycomb-lattice Heisenberg antiferromagnet agree well with QMC at Δ = 0.5 and 1, with ΔE around 10^-3 at Δ = 1 for χ = 16.The Δ = 1 crossover is identified as the most challenging parameter range; the figure shows energies and energy differences for the two anisotropies.Source: §VI · p. 6, Figure 8 · p. 7
• At Δ = 0.5, specific heat indicates a thermodynamic phase transition at TC = 0.345 in both NCD and QMC, while QEE peaks at TS = 0.3610 and vanishes away from criticality.The vanishing QEE is about 10^-3 to 10^-4, indicating that the dominant eigenstate of Z becomes more separable away from the critical vicinity.Source: §VI · p. 7, Figure 8 · p. 7, Figure 9 · p. 7
• For the honeycomb-lattice model at Δ = 0.5, Ilya and Iloop peak at Tlya = 0.3636 and Tloop = 0.3597, close to the specific-heat and QEE critical temperatures.QEE, Ilya, and Iloop peaks near criticality are attributed to state nonlocality: loop dependence requires larger cell tensors there, while larger Ilya corresponds to more mapping steps and contraction of a larger defective tensor network.Source: §VI · p. 7, Figure 9 · p. 7

VII. SUMMARY

• NCD agrees with exact results for the square Ising model and with quantum Monte Carlo simulations for the spin-1/2 honeycomb-lattice Heisenberg antiferromagnet.The square Ising model benchmarks NCD accuracy; the honeycomb-lattice calculation demonstrates its application to a quantum model.Source: §VII · p. 8
• The proposed ITSA provides a numerical realization of NCD that is free from the negative sign problem.The algorithm is proposed for implementing the NCD scheme.Source: §VII · p. 8
• Quasi-entanglement entropy, Lyapunov exponent, and loop character describe thermal-state properties and can detect possible thermodynamic phase transitions in classical and quantum systems.The summary introduces these three characteristics as new quantities within the NCD framework.Source: §VII · p. 8
• The NCD method can be straightforwardly extended to other classical or quantum lattice systems.Source: §VII · p. 8