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arXiv:2606.19732

Quantum models with the Yang-Lee phase transition

Erick Arguello Cruz, Grigory Tarnopolsky

hep-thcond-mat.stat-mechcond-mat.str-elquant-phSubmitted Jun 18, 2026• Announced Jun 19, 2026• 0 citations
v1 · Jun 18, 2026v2 · Oct 15, 2026
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Abstract

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 $i φ^3$ 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.

Comments
33 pages + appendices, 16 figures

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In this article, we present four different 1+11+11+1D quantum models that realize the Yang-Lee (YL) phase transition under a deformation that preserves PTPTPT 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 iφ3i φ^3iφ3 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 φφφ at the Yang-Lee critical point and show that it grows with distance, in agreement with theoretical expectations.
Lee-Yang theory offers a unifying framework for understanding classical phase transitions and dynamical quantum phase transitions through the analysis of partition functions and Loschmidt echoes. Recently, this framework was extended to characterize quantum phase transitions of quantum Ising models by introducing the concepts of non-Hermitian parity-symmetry breaking and fidelity zeros. Here, we generalize the theory by studying a broad class of quantum models, including the XY, the XXZ, the XYZ, and the Z_{p} clock models in one dimension, subject to a complex magnetic field. For the XY, XXZ, and XYZ models, we find that the complex field breaks parity symmetry and induces oscillations of the ground state between the two parity sectors, giving rise to fidelity zeros within the ordered phases. For the Z_{3} clock model, the complex field splits the real part of the ground-state energy between the neutral sector (q=0) and the charged sectors (q=1,2), while preserving the degeneracy within the charged sector. Fidelity zeros arise only after projecting out one of the charged sectors. For the Z_{4} and Z_{5} clock models, the ground states are instead projected to oscillate between the neutral sector (q=0) and the charged sectors q=2 and q=1, respectively, giving rise to fidelity zeros. Finite-size scaling of these zeros yields critical exponents in full agreement with analytical predictions, demonstrating that this approach is applicable not only to the Ising model with Z_{2} symmetry, but also to more general Heisenberg-type models and systems with higher discrete symmetries.
Using finite-size-scaling methods, we study the quantum chain version of the spin-111-Blume-Capel model coupled to an imaginary field. The aim is to realize higher order non-unitary conformal field theories in a simple Ising-type spin model. We find that the first ground-state level crossing in the high-temperature phase leads to a second-order phase transition of the Yang-Lee universality class (central charge c=−22/5c=-22/5c=−22/5). The Yang-Lee transition region ends at a line of a new type of tricriticality, where the {\em three} lowest energy levels become degenerate. The analysis of the spectrum at two points on this line gives good evidence that this line belongs to the universality class of the M2,7{\cal M}_{2,7}M2,7​-conformal theory with c=−68/7c=-68/7c=−68/7.
Lee-Yang theory offers a unifying framework for understanding classical phase transitions and dynamical quantum phase transitions through the analysis of partition functions and Loschmidt echoes. Recently, this framework is extended to characterize quantum phase transitions of quantum Ising models by introducing the concepts of non-Hermitian parity-symmetry breaking and fidelity zeros. Here, we generalize the theory by studying a broad class of quantum models, including the XY, the XXZ, the XYZ, and the Zp\mathbb{Z}_pZp​ clock models in one dimension, subject to a complex magnetic field. For the XY, XXZ and XYZ models, we find that the complex field breaks parity symmetry and induces oscillations of the ground state between the two parity sectors, giving rise to fidelity zeros within the ordered phases. For the Z3\mathbb{Z}_3Z3​ clock model, the complex field splits the real part of the ground-state energy between the neutral sector (q=0q=0q=0) and the charged sectors (q=1,2q=1,2q=1,2), while preserving the degeneracy within the charged sector. Fidelity zeros arise only after projecting out one of the charged sectors. For the Z4\mathbb{Z}_4Z4​ and Z5\mathbb{Z}_5Z5​ clock models, the ground states are instead projected to oscillate between the neutral sector (q=0q=0q=0) and the charged sectors q=2q=2q=2 and q=1q=1q=1, respectively, giving rise to fidelity zeros. Finite-size scaling of these zeros yields critical exponents in full agreement with analytical predictions, demonstrating that this approach is applicable not only to the Ising model with Z2\mathbb{Z}_2Z2​ symmetry, but also to more general Heisenberg-type models and systems with higher discrete symmetries.
Using finite-size-scaling methods, we study the quantum chain version of the spin-1-Blume-Capel model coupled to an imaginary field. The aim is to realize simple non-unitary minimal conformal field theories in an Ising-type spin model. We find that the first ground-state level crossing in the high-temperature phase leads to a second-order phase transition of the Yang-Lee universality class (central charge c=−22/5). The Yang-Lee transition region ends at a line of a new type of tricriticality, where the three lowest energy levels become degenerate. The analysis of the spectrum at two points on this line gives good evidence that this line realizes the universality class of the ℳ2,7-conformal theory with c=−68/7.
The quantum phase transition of a one-dimensional transverse field Ising model in an imaginary longitudinal field is studied. A new order parameter M is introduced to describe the critical behaviors in the Yang-Lee edge singularity (YLES). The M does not diverge at the YLES point, a behavior different from other usual parameters. We term this unusual critical behavior around YLES as the pseudo-YLES. To investigate the static and driven dynamics of M, the (1+1) dimensional ferromagnetic-paramagnetic phase transition ((1+1) D FPPT) critical region, (0+1) D YLES critical region and the (1+1) D YLES critical region of the model are selected. Our numerical study shows that the (1+1) D FPPT scaling theory, the (0+1) D YLES scaling theory and (1+1) D YLES scaling theory are applicable to describe the critical behaviors of M, demonstrating that M could be a good indicator to detect the phase transition around YLES. Since M has finite value around YLES, it is expected that M could be quantitatively measured in experiments.
Predicting the phase diagram of interacting quantum many-body systems is a central problem in condensed matter physics and related fields. A variety of quantum many-body systems, ranging from unconventional superconductors to spin liquids, exhibit complex competing phases whose theoretical description has been the focus of intense efforts. Here, we show that neural network quantum states can be combined with a Lee-Yang theory of quantum phase transitions to predict the critical points of strongly correlated spin lattices. Specifically, we implement our approach for quantum phase transitions in the transverse-field Ising model on different lattice geometries in one, two, and three dimensions. We show that the Lee-Yang theory combined with neural network quantum states yields predictions of the critical field, which are consistent with large-scale quantum many-body methods. As such, our results provide a starting point for determining the phase diagram of more complex quantum many-body systems, including frustrated Heisenberg and Hubbard models.