Technical Summary: Numerical Investigations of Phase Transitions in Lattice Field Theories

This thesis presents two complementary numerical investigations aimed at resolving critical phenomena and ensuring the reliability of simulations in lattice field theories. The work addresses the persistent demand for high-precision tools capable of distinguishing subtle critical behaviors and validating the thermodynamic consistency of Monte Carlo sampling. The study is divided into two distinct parts: a tensor network analysis of the two-dimensional generalized XY (gXY) model and the development of a configurational temperature estimator for compact U(1) lattice gauge theories.

Part I: Tensor Renormalization Study of the Generalized XY Model

Problem and Context
The two-dimensional generalized XY (gXY) model, proposed by Korshunov, Lee, and Grinstein, interpolates between distinct phases via a deformation parameter $\Delta$. The model features a competition between integer vortices (ferromagnetic phase) and half-integer vortices (nematic phase), separated by a disordered paramagnetic phase. The phase structure involves Berezinskii–Kosterlitz–Thouless (BKT) transitions for both integer and half-integer vortices, as well as an intervening Ising-like transition. Previous studies using Monte Carlo and matrix-product-state (MPS) methods have established the existence of these phases but have faced challenges in precisely mapping the phase boundaries and the nature of the multicritical region where these transition lines meet.

Methodology
The author employs the Higher-Order Tensor Renormalization Group (HOTRG) method, implemented with GPU acceleration to handle large bond dimensions. The approach involves:

1. Character Expansion: The Boltzmann weight of the gXY model is expanded using dual variables (character expansion), converting the partition function into a contraction of a tensor network.
2. HOTRG Coarse-Graining: The network is iteratively coarse-grained using Higher-Order Singular Value Decomposition (HOSVD). This allows for the truncation of bond dimensions while preserving dominant entanglement contributions across multiple directions simultaneously, offering higher accuracy than standard TRG.
3. Impurity Tensor Techniques: To extract thermodynamic observables such as magnetization and susceptibility, impurity tensors encoding external symmetry-breaking fields ($h$ and $h_1$) are inserted into the network. This allows for the calculation of derivatives of the free energy without relying on finite-difference approximations that can introduce numerical noise.
4. Extrapolation: Critical temperatures are determined by extrapolating susceptibility peaks to the zero-field limit, a necessary step given the Mermin–Wagner theorem which forbids spontaneous symmetry breaking in 2D without an external field.

Key Results

• Phase Diagram Mapping: The study provides a precise mapping of the phase diagram in the $(\Delta, T)$ plane. It identifies three distinct phases: ferromagnetic (integer vortices), nematic (half-integer vortices), and disordered.
• Transition Lines: The boundaries are characterized by two BKT transitions (integer and half-integer) and an Ising transition.
• Multicritical Point: The numerical results suggest a sequential merging scenario for the transition lines. The half-BKT and Ising lines appear to merge first around $\Delta \approx 0.34$, followed by a merger with the BKT line at $\Delta \approx 0.36(2)$. The multicritical point is located at $\Delta = 0.36(2)$ and $T_c = 0.716(3)$.
• Estimator Comparison: The study demonstrates that for BKT-type transitions, the peak of the specific heat systematically overestimates the critical temperature. In contrast, the zero-field extrapolation of magnetic susceptibility provides a more reliable estimator.
• Performance: The GPU-accelerated HOTRG implementation successfully computed results with a bond dimension of $D=91$ and a Fourier cutoff of $m \in [-50, 50]$, achieving high precision in the thermodynamic limit.

Significance
The work demonstrates that carefully controlled, GPU-accelerated HOTRG offers a highly accurate and efficient framework for resolving complex phase structures, improving upon earlier Monte Carlo and MPS studies. It provides a definitive numerical characterization of the multicritical region in the gXY model.

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Part II: Configurational Temperature Estimator in Lattice Gauge Theories

Problem and Context
In canonical Monte Carlo simulations, particularly Hybrid Monte Carlo (HMC), there is a lack of straightforward conservation laws to validate algorithmic correctness. While conserved quantities like energy are used in molecular dynamics, canonical ensembles rely on comparing observables to known thermodynamic properties, which is problematic for novel systems or state points without reference data. Furthermore, discretization effects and algorithmic artifacts in lattice gauge theories can distort thermodynamic observables and critical temperatures.

Methodology
The thesis develops and benchmarks a configurational temperature estimator based on Rugh's geometric formulation of statistical mechanics.

1. Theoretical Basis: The estimator derives temperature from the gradient and Hessian of the Euclidean action ($S$) with respect to the field configurations, independent of momentum variables. The inverse configurational temperature $\beta_{config}$ is defined as:
   $$\beta_{config} = \left\langle \nabla_\phi \cdot \left( \frac{\nabla_\phi S}{|\nabla_\phi S|^2} \right) \right\rangle$$
   where $\nabla_\phi$ is the gradient with respect to all lattice field variables.
2. Gauge Invariance: The formulation is adapted to be gauge-invariant by utilizing covariant derivatives, making it applicable to compact gauge theories.
3. Diagnostic Application: In a correctly sampled ensemble where the probability distribution is $P[\phi] \propto e^{-S[\phi]}$, the estimator should yield $\beta_{config} = 1$ (in natural units). Deviations from unity indicate sampling inconsistencies, slow thermalization, or numerical errors.
4. Benchmarking: The estimator is tested in compact U(1) lattice gauge theories in one, two, and four Euclidean dimensions.
   • 1D and 2D: These cases serve as controlled testbeds with known analytical results (no phase transition in 1D/2D U(1)).
   • 4D: This case features a weak first-order confining–Coulomb phase transition around $\beta \approx 1.0$, providing a nontrivial benchmark.
   • Approximation: For large volumes (specifically in 4D), off-diagonal Hessian terms are shown to be negligible, allowing for a simplified estimator using only diagonal terms to reduce computational cost.

Key Results

• Validation: In 1D and 2D U(1) theories, the measured configurational temperature $\beta_M$ matches the input coupling $\beta$ (or $\beta_{config} \approx 1$) across a wide range of parameters, validating the implementation and the estimator's stability.
• Error Detection: The estimator successfully detects algorithmic errors. In a test where the Metropolis accept-reject step was intentionally flawed (using an incorrect random number range), $\beta_M$ deviated significantly from the input value, whereas standard observables might not have immediately revealed the issue.
• Thermalization Monitoring: The estimator tracks thermalization behavior, providing an early indicator of slow equilibration or metastability.
• Phase Transition Insensitivity: Unlike order parameters (e.g., plaquette susceptibility), the configurational temperature does not exhibit singular behavior at phase transitions; it remains smooth across the critical region. This confirms its role as a consistency check rather than an order parameter.

Significance
The configurational temperature estimator provides a gauge-invariant, momentum-free diagnostic tool for validating thermal sampling in lattice field theories. It offers an independent means to detect sampling pathologies, numerical errors, and slow thermalization before large-scale production runs. The study confirms that for large lattice volumes, the estimator can be implemented efficiently by neglecting off-diagonal Hessian terms without significant loss of accuracy.

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Overall Conclusions

The thesis concludes that these two approaches address complementary challenges in the numerical investigation of critical phenomena:

1. Tensor Networks (HOTRG): Provide a deterministic, high-precision route to mapping complex phase diagrams and critical behavior in systems with competing orders, effectively handling the thermodynamic limit.
2. Configurational Temperature: Provides an independent, low-overhead diagnostic to validate the reliability of stochastic sampling in Monte Carlo simulations.

The work suggests future directions, including applying HOTRG to generalized XY models with higher fractionalization ($q \ge 3$) or coupled gauge degrees of freedom, and integrating the configurational temperature monitor into large-scale Hybrid Monte Carlo simulations of non-Abelian gauge theories and finite-temperature QCD. The author notes that these methods are not limited to lattice field theories and are expected to be valuable in condensed matter systems where precise numerical control and reliable diagnostics are essential.