Quantum models with the Yang-Lee phase transition

Technical Summary: Quantum Models with the Yang-Lee Phase Transition

Problem Statement
The Yang-Lee (YL) phase transition, originally identified through the distribution of partition function zeros in the complex magnetic field plane of classical Ising ferromagnets, is described by a non-unitary Conformal Field Theory (CFT). In two dimensions, this corresponds to the minimal model $M(2, 5)$. While the Euclidean $i\phi^3$ field theory provides a field-theoretic description of this criticality, standard Monte Carlo methods used for unitary models like the Ising model are inapplicable due to the non-unitary nature of the YL model. Consequently, numerical studies have largely been restricted to the canonical ferromagnetic quantum Ising model in a transverse field perturbed by an imaginary longitudinal magnetic field. The central problem addressed in this work is to investigate the universality of the YL CFT by demonstrating its realization in a broader class of 1+1D quantum microscopic models that preserve $\mathcal{PT}$ symmetry but differ in their underlying lattice structures and symmetries.

Methodology
The authors employ a combination of analytical derivations and numerical simulations to establish the presence of YL criticality in four distinct quantum models:

1. Analytical Framework:

   • Polyakov-Hubbard (PH) Transformation: Used to map classical lattice models (Ising, Blume-Capel, Clock) to effective Euclidean Landau-Ginzburg-Wilson (LGW) actions. This allows the authors to derive the continuum field theories and identify the conditions (specifically imaginary magnetic field deformations) under which the action reduces to the massless $i\phi^3$ theory.
   • Bosonization: Applied to the massive Schwinger model to map the fermionic theory to a bosonic field theory, facilitating the identification of the relevant deformation terms that preserve $\mathcal{PT}$ symmetry while breaking the $Z_2$ symmetry.
   • Mean-Field Analysis: Utilized to estimate critical values for the imaginary fields and to determine the phase diagrams of the models.

2. Numerical Approach:

   • Hamiltonian Formulation: The authors construct lattice Hamiltonians for the antiferromagnetic Ising model, the Schwinger model, the Blume-Capel model, and the three-state quantum clock model, each deformed by specific imaginary fields (longitudinal, staggered, or pseudo-scalar).
   • Density Matrix Renormalization Group (DMRG): Numerical spectra of the non-Hermitian but $\mathcal{PT}$-symmetric Hamiltonians are computed using the DMRG method (via the iTensor library) for finite system sizes ($N$).
   • Finite-Size Scaling (FSS) and State-Operator Correspondence: The scaling dimensions of operators are extracted from the energy gaps of the Hamiltonian spectrum using the relation $E_n - E_0 \propto \frac{2\pi v}{N} \Delta_n$. The pseudo-critical points are identified using criticality criteria (e.g., Eq. 6.5) and extrapolated to the thermodynamic limit ($N \to \infty$) using the Bulirsch-Stoer (BST) algorithm.
   • Correlation Functions: Two-point functions of the primary operator $\phi$ are computed numerically to verify their growth with distance, a signature of the negative scaling dimension in the YL CFT.

Key Contributions and Results
The paper presents four distinct quantum models that realize the Yang-Lee critical point under $\mathcal{PT}$-symmetric deformations:

1. Antiferromagnetic Quantum Ising Model:

   • The authors show that a ferromagnetic Ising model in an imaginary longitudinal field yields YL criticality, whereas the antiferromagnetic version in a uniform imaginary field does not (it remains in the Ising universality class).
   • However, by introducing an imaginary staggered longitudinal magnetic field, the antiferromagnetic model is shown to possess a surface of Yang-Lee critical points.
   • Numerical results for scaling dimensions ($\Delta_\phi \approx -0.4003$) and the effective central charge ($c_{\text{eff}} \approx 0.3996$) agree with the exact $M(2, 5)$ values ($\Delta_\phi = -2/5$, $c_{\text{eff}} = 2/5$) within $10^{-4}$.

2. Massive Schwinger Model:

   • The Schwinger model at $\theta = \pi$ (Ising critical point) is deformed by an imaginary pseudo-scalar mass term.
   • Bosonization reveals that this deformation breaks the charge conjugation ($Z_2$) symmetry but preserves $\mathcal{PT}$ symmetry, leading to the $i\phi^3$ theory.
   • Numerical extraction of the critical mass $m_{5,c}$ and scaling dimensions confirms the YL universality class, with results matching theoretical predictions to within $10^{-3}$.

3. Spin-1 Blume-Capel Model:

   • The model is deformed by an imaginary longitudinal magnetic field.
   • The study confirms the existence of "wing" surfaces of YL critical points attached to the Ising critical line in the $(\alpha, h_x)$ plane.
   • The authors verify the existence of YL edge lines emanating from the tricritical Ising point, which were previously conjectured to be described by the $M(2, 7)$ minimal model.

4. Three-State Quantum Clock (Potts) Model:

   • This model is deformed by a term breaking $Z_3$ symmetry but preserving a specific $\mathcal{PT}$ combination.
   • The LGW analysis predicts that for certain deformation parameters, the massless YL field decouples from a massive field.
   • Numerical spectra confirm the presence of both massless YL states and massive states. The scaling dimensions of the massless sector align with the YL CFT, while the massive states exhibit energy gaps that grow linearly with system size.
   • Additionally, the paper explores a deformation of the clock model that preserves $Z_2$ symmetry, successfully identifying an Ising critical point coexisting with massive states, distinct from the YL transition.

Verification of CFT Predictions
Using the ferromagnetic quantum Ising model as a baseline, the authors numerically compute the structure constant $C_{\phi\phi\phi}$ and the two-point correlation function $\langle I | \phi_0 \phi_x | I \rangle$.

• The structure constant is calculated as $C_{\phi\phi\phi} \approx 1.91i$, matching the exact value derived from the minimal model.
• The two-point function is shown to grow with distance $x$, consistent with the negative scaling dimension $\Delta_\phi = -2/5$, a unique feature of non-unitary CFTs.

Significance and Claims
The paper claims to demonstrate the universality of the Yang-Lee critical point across diverse 1+1D quantum systems. By deriving the effective $i\phi^3$ field theory for each model via PH transformations or bosonization, the authors establish that the YL phase transition is a robust phenomenon occurring in systems with $\mathcal{PT}$ symmetry, provided the deformation breaks the relevant discrete symmetry (e.g., $Z_2$ or $Z_3$) without breaking $\mathcal{PT}$.

The work validates the use of the Hamiltonian approach combined with DMRG for studying non-unitary criticality, overcoming the limitations of Monte Carlo methods. Furthermore, it provides a detailed numerical characterization of the YL edge lines in the Blume-Capel model and elucidates the coexistence of massless YL and massive sectors in the quantum clock model. The results serve as a comprehensive confirmation that the $M(2, 5)$ minimal model describes the critical behavior of these deformed quantum systems, with numerical data showing agreement with exact theoretical results at the level of $10^{-3}$ to $10^{-4}$.