Quantum models with the Yang-Lee phase transition

The gist In this article, four different 1+1D quantum models are presented that realize the Yang-Lee (YL) phase transition under a deformation that preserves PT symmetry, and it is shown that in all these models, the YL critical point is described by a massless bosonic field with an iϕ3 interaction

Model Realizations and Theoretical Framework

The paper presents four different 1+1D quantum models realizing the Yang-Lee (YL) phase transition under a deformation that preserves PT symmetry, specifically the antiferromagnetic Ising spin chain in transverse and longitudinal magnetic fields, the massive Schwinger model, the Blume-Capel model, and the three-state quantum clock model Using bosonization for the Schwinger model and the Polyakov-Hubbard transformation for the other models, it is shown that in all of these models, the YL criticality is described by a massless Euclidean iϕ3 field theory In the quantum clock model, this critical field interacts with a massive bosonic field, and the massless and massive states in the Hamiltonian spectrum are identified

Critical Point Identification and Scaling Dimensions

The YL critical point is identified using the state-operator correspondence, which provides access to the scaling dimensions of the lowest operators in each model For example, in two dimensions, the non-unitary minimal model M(2, 5) provides an exact description of the Yang-Lee CFT with primary operators I and ϕ having scaling dimensions ∆I = 0 and ∆ϕ = −2/5 The numerical results for the structure constant Cϕϕϕ agree with the theoretical value in (4.2) within an error of order 10−3

Numerical Computations and CFT Agreement

The paper numerically computes the two-point function of ϕ at the Yang-Lee critical point and shows that it grows with distance, in agreement with theoretical expectations since the operator ϕ has negative scaling dimension ∆ϕ = −2/5 The structure constant Cϕϕϕ is computed numerically and agrees with the theoretical value in (4.2) within an error of order 10−3 These results for different models are summarized in Tables 4 through 8

Universality and Criticality Analysis

The analysis confirms that the critical point found in the antiferromagnetic quantum Ising model belongs to the 2D YL universality class, as evidenced by the effective central charge ceff being within 10−4 of the exact value ceff = 2/5 in 2D YL CFT For the three-state quantum clock model, massive states are observed in the spectrum in accordance with the LGW action

Concluding Remarks

In conclusion, all four models exhibit YL criticality described by the PT-symmetric massless iϕ3 theory The numerical results for the ferromagnetic quantum Ising spin chain at the Yang-Lee critical point show complete agreement with CFT predictions The paper confirms the existence of the YL edge line in the spin-1 Blume-Capel model, which is conjectured to be described by the M(2, 7) minimal model

How it works

The Yang-Lee criticality is realized through a deformation that preserves PT symmetry, where P: ϕ → −ϕ and T: i → −i This deformation breaks the Z2 symmetry of the Ising model but preserves PT symmetry The YL phase transition can be regarded as a transition associated with spontaneous breaking of the PT symmetry

Model Specific Deformations

The four models considered are the antiferromagnetic Ising model in transverse and longitudinal magnetic fields, the massive Schwinger model, the spin-1 Blume-Capel (BC) model, and the three-state quantum clock model (CM) For instance, in the Schwinger model, a deformation by an imaginary pseudo-scalar mass term satisfies P-odd and T-odd conditions to preserve PT symmetry

Lattice Implementation and Numerical Methods

The study uses the state-operator correspondence and finite-size scaling analysis to numerically demonstrate that each model has a YL critical point under a particular deformation For the Schwinger model, bosonization is used For the other models, the Polyakov-Hubbard transformation is used Numerical computations are performed using Density Matrix Renormalization Group (DMRG) and Bulirsch–Stoer (BST) algorithm

Criticality Criteria

The pseudo-critical point is defined by the condition (N + 2)Ek=0 for the Hamiltonian (3.4), which produces a sequence of pseudo-critical points hstz,c that converges to the thermodynamic critical point hstz,c in N → ∞ limit The scaling dimensions are calculated using the formula ∆k n = Ek n(N) − Ek=0(N) / (Ek=10(N) − Ek=0(N)) → N→∞ ∆k n - ∆I - ∆∂ϕ - ∆ϕ = ∆k n

Scaling Dimension Agreement

The extrapolated values for the scaling dimensions in the k = 0 sector are found to be approximately 10−4 and 10−2 away from the exact theoretical scaling dimensions ∆ϕ = −2/5, ∆ϕ = 8/5, and ∆2ϕ = 18/5 The effective central charge is within 10−4 of the exact value ceff = 2/5 in 2D YL CFT

Final Findings on Critical Points

The thermodynamic critical point for the antiferromagnetic quantum Ising model (6.2) is found to be hstz,c = 0.69092 for hx = 2.5 and hz = 1 The YL pseudo-critical point for the three-state clock model is found at hc = 0.71406 The numerical results confirm that the critical point belongs to the 2D YL universality class The first spin-zero scaling dimension corresponds to ∆ϕ in the YL CFT, with a difference of 10−4 from the exact value ∆ϕ = −2/5

How it works

The LGW action for the three-state quantum clock model is obtained by applying the Polyakov-Hubbard transformation The potential U has the form U[ϕ1, ϕ2] = − ln eϕ1−βhλ + 2e− 1/2(βhλ−ϕ1)cosh √3/2(ϕ2 − iβh) For h > 0, the action has the PT symmetry: ϕ2 → −ϕ2 and i → -i The YL field ϕ2 fully decouples from the massive field ϕ1, and its action coincides with the action (2.13) for the classical Ising model The numerical spectrum of the Hamiltonian HCMYL shows massless YL states together with massive states when hλ > 0

How it works

The deformation of the three-state quantum clock model (9.6) breaks the Z3 and C symmetries but preserves the combination CT, which is identified with the Yang-Lee PT symmetry The YL field ϕ2 fully decouples from the massive field ϕ1 when βhλ > 0 and sufficiently large h The numerical results for these correlation functions approach the expected shapes in (10.10) and (10.12) as N → ∞ <ref:

1. Improved ability to predict Yang-Lee critical points in various 1+1D quantum models by leveraging state-operator correspondence and finite-size scaling analysis to identify critical values, as demonstrated by Using the state-operator correspondence and the finite-size scaling analysis, we numerically demonstrate that each of these models has a YL critical point under a particular deformation that preserves PT symmetry.

2. Enhanced capacity for characterizing Conformal Field Theory (CFT) structures at Yang-Lee critical points by accurately computing scaling dimensions and structure constants, as shown by the paper's we identify the YL critical point, compute the scaling dimensions of the lowest operators in each model, and find perfect agreement with the exact results for the YL criticality in two dimensions.

3. Refined simulation capabilities for non-unitary systems by enabling high-accuracy numerical computation of correlation functions using methods like Density Matrix Renormalization Group (DMRG) and Bulirsch–Stoer (BST) algorithms, allowing precise extrapolation to the thermodynamic limit, as evidenced by The results are shown in Figure 3 and The results are shown in Table 5.

4. Advanced understanding of universality classes through spectral analysis of lattice Hamiltonians, enabling the identification of critical points belonging to specific CFTs (e.g., M(2, 5) or Ising), as implied by the comparison between the calculated scaling dimensions and theoretical predictions like the exact results for the YL CFT.

5. Specific prediction of phase diagrams for complex deformations in quantum systems, allowing researchers to map out regions where Yang-Lee criticality exists based on symmetry preservation, such as showing that the Landau-GinzburgWilson (LGW) description for the three-state quantum clock model explains why the YL phase transition appears only for the positive sign of the PT-symmetric deformation.

In this article, we present four different 1+1 D quantum models that realize the Yang-Lee (YL) phase transition under a deformation that preserves PT symmetry. These are the antiferromagnetic Ising spin chain in transverse and longitudinal magnetic fields, the massive Schwinger model, the Blume-Capel model, and the three-state quantum clock model. Using the state-operator correspondence, we identify the YL critical point, compute the scaling dimensions of the lowest operators in each model, and find perfect agreement with the exact results for the YL criticality in two dimensions. Using bosonization for the Schwinger model and the Polyakov-Hubbard transformation for the other models, we show that in all of these quantum models the YL critical point is described, as expected, by a massless bosonic field with an i ϕ cubed interaction. In the quantum clock model, this critical field interacts with a massive bosonic field, and we identify the massless and massive states in the Hamiltonian spectrum. In addition, we numerically compute the two-point function of ϕ at the Yang-Lee critical point and show that it grows with distance, in agreement with theoretical expectations.