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arXiv:2404.04324 · Journal of Physics A Mathematical and Theoretical

Critical behavior of dirty free parafermionic chains

Akshat Pandey, Aditya Cowsik

Published Jul 24, 2024• 0 citations
arXivHTMLPDF

Abstract

Abstract A family of Z n -symmetric non-Hermitian models of Baxter was shown by Fendley to be exactly solvable via a parafermionic generalization of the Clifford algebra. We study these models with spatially random couplings, and obtain several exact results on thermodynamic singularities as the distributions of couplings are varied. We find that these singularities, independent of n, are identical to those in the random transverse-field Ising chain; correspondingly the models host infinite-randomness critical points. Similarities in structure to exact methods for random Ising models, a strong-disorder renormalization group, and generalizations to other models with free spectra, are discussed.

DOI
10.1088/1751-8121/ad6723

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Abstract A family of Z n -symmetric non-Hermitian models of Baxter was shown by Fendley to be exactly solvable via a parafermionic generalization of the Clifford algebra. We study these models with spatially random couplings, and obtain several exact results on thermodynamic singularities as the distributions of couplings are varied. We find that these singularities, independent of n, are identical to those in the random transverse-field Ising chain; correspondingly the models host infinite-randomness critical points. Similarities in structure to exact methods for random Ising models, a strong-disorder renormalization group, and generalizations to other models with free spectra, are discussed.
Certain spin chains, such as the quantum Ising chain, have free fermion spectra which can be expressed as the sum of decoupled two-level fermionic systems. Free parafermions are a generalisation of this idea to \mathbb{Z}_N ℤ N -symmetric clock models. In 1989 Baxter discovered a non-Hermitian but \mathcal{PT} 𝒫 𝒯 -symmetric model directly generalising the Ising chain, which was later described by Fendley as a free parafermion spectrum. By extending the model’s magnetic field parameter to the complex plane, it is shown that a series of exceptional points emerges, where the quasienergies defining the free spectrum become degenerate. An analytic expression for the locations of these points is derived, and various numerical investigations are performed. These exceptional points also exist in the Ising chain with a complex transverse field. Although the model is not in general \mathcal{PT} 𝒫 𝒯 -symmetric at these exceptional points, their proximity can have a profound impact on the model on the \mathcal{PT} 𝒫 𝒯 -symmetric real line. Furthermore, in certain cases an exceptional point may appear on the real line (with negative field).
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
The spectrum of the quantum Ising chain can be found by expressing the spins in terms of free fermions. An analogous transformation exists for clock chains with symmetry, but is of less use because the resulting parafermionic operators remain interacting. Nonetheless, Baxter showed that a certain non-Hermitian (but PT -symmetric) clock Hamiltonian is 'free', in the sense that the entire spectrum is found in terms of independent energy levels, with the striking feature that there are n possibilities for occupying each level. Here I show this directly explicitly finding shift operators obeying a generalization of the Clifford algebra. I also find higher Hamiltonians that commute with Baxter's and prove their spectrum comes from the same set of energy levels. This thus provides an explicit notion of a 'free parafermion'. A byproduct is an elegant method for the solution of the Ising/Kitaev chain with spatially varying couplings.
Abstract In this article we outline the historical development and key results obtained to date for free parafermionic spin chains. The concept of free parafermions provides a natural N-state generalization of free fermions, which have long underpinned the exact solution and application of widely studied quantum spin chains and their classical counterparts. In particular, we discuss the Baxter-Fendley free parafermionic Z(N) spin chain, which is a relatively simple non-Hermitian generalization of the Ising model.
We study quantum phase transitions in transverse-field Ising spin chains in which the couplings are random but hyperuniform, in the sense that their large-scale fluctuations are suppressed. We construct a one-parameter family of disorder models in which long-wavelength fluctuations are increasingly suppressed as a parameter $\ensuremath{\alpha}$ is tuned. For $\ensuremath{\alpha}=0$, one recovers the familiar infinite-randomness critical point. For $0<\ensuremath{\alpha}<1$, we find a line of infinite-randomness critical points with continuously varying critical exponents; however, the Griffiths phases that flank the critical point at $\ensuremath{\alpha}=0$ are absent at any $\ensuremath{\alpha}>0$. When $\ensuremath{\alpha}>1$, randomness is a dangerously irrelevant perturbation at the clean Ising critical point, leading to a state we call the critical Ising insulator. In this state, thermodynamics and equilibrium correlation functions behave as in the clean system. However, all finite-energy excitations are localized, thermal transport vanishes, and autocorrelation functions remain finite in the long-time limit. We characterize this line of hyperuniform critical points using a combination of perturbation theory, renormalization-group methods, and exact diagonalization.
The critical behavior of the random transverse-field Ising model in finite-dimensional lattices is governed by infinite disorder fixed points, several properties of which have already been calculated by the use of the strong disorder renormalization-group (SDRG) method. Here we extend these studies and calculate the connected transverse-spin correlation function by a numerical implementation of the SDRG method in d=1,2d=1,2d=1,2, and 3 dimensions. At the critical point an algebraic decay of the form $\ensuremath{\sim}{r}^{\ensuremath{-}{\ensuremath{\eta}}_{t}}$ is found, with a decay exponent being approximately ${\ensuremath{\eta}}_{t}\ensuremath{\approx}2+2d$. In d=1d=1d=1 the results are related to dimer-dimer correlations in the random antiferromagnetic XXXXXX chain and have been tested by numerical calculations using free-fermionic techniques.