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International Journal of Modern Physics B

NON-HERMITIAN TRICRITICALITY IN THE BLUME-CAPEL MODEL WITH IMAGINARY FIELD

G von Gehlen

Published Nov 1, 1994• 24 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979294001494

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Non-hermitian tricriticality in the Blume-Capel model with imaginary field

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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 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 $c=-22/5$). 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 ${\cal M}_{2,7}$-conformal theory with $c=-68/7$.

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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.
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.
The critical curve with central charge c=-22/5 of the Ising quantum chain in an imaginary longitudinal field is calculated by finite-size extrapolation of the Yang-Lee square root branch point position. Various power laws are found. The field dependence of the off-criticality mass scale and the four lowest off-criticality scaling functions are calculated. The latter turn out to be universal along the critical line and agree very well with the results of the truncated Hilbert space method for the c=-22/5 continuum theory. The author obtains the universal bulk scaling coefficient in qualitative agreement with the thermodynamic Bethe ansatz result. The scattering phaseshift calculated by Luscher's method (1988) reproduces the Cardy-Mussardo minimal solution (1989) for the S-matrix. If a boundary defect is introduced, the only non-trivial primary field decouples, so that the spectrum consists only of the tower of the unit operator. In the antiferromagnetic regime with a non-staggered imaginary field the author finds boundary-dependent lines where the gap vanishes.
The tricritical Ising universality class emerges at the end point of a line of continuous Ising phase transitions, above which the transition is first order. In 1+1 dimensions, it is described by the second minimal model of conformal field theory (CFT) with central charge c=7/10c=7/10c=7/10. Recently, there has been a revival of interest in the tricritical Ising model due to its emergent space-time supersymmetry. Combining numerical (infinite density-matrix renormalization group) and field theoretical (bosonization and CFT) techniques, there authors demonstrate here that Ising tricriticality occurs in a paradigmatic extended one dimensional Hubbard model with explicit bond dimerization. The low-energy sector can be described by a triple sine-Gordon model, which, in the transition region, reduces to the tricritical Ising CFT. The theoretically predicted decay of various two-point correlation functions is found to be in excellent agreement with the density-matrix renormalization group data.
We study many-body phase transitions in a one-dimensional ferromagnetic transversed field Ising model with an imaginary field, and show that the system exhibits three phase transitions: one second-order phase transition and two PT\mathcal{PT}PT phase transitions. The second-order phase transition occurring in the ground state is investigated via biorthogonal and self-normal entanglement entropy, for which we develop an approach to perform finite-size scaling theory to extract the central charge for small systems. Compared with the second-order phase transition, the first PT\mathcal{PT}PT transition is characterized by the appearance of an exceptional point in the full energy spectrum, while the second PT\mathcal{PT}PT transition only occurs in specific excited states. Furthermore, we interestingly show that both exceptional points are second-order in terms of scalings of imaginary parts of the energy. This work provides an exact solution for many-body phase transitions in non-Hermitian systems.
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.
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.