Realizing Allso(N)1Quantum Criticalities in Symmetry Protected Cluster Models

Abstract

We show that all so(N )1 universality class quantum criticalities emerge when one-dimensional generalized cluster models are perturbed with Ising or Zeeman terms. Each critical point is described by a low-energy theory of N linearly dispersing fermions, whose spectrum we show to precisely match the prediction by so(N )1 conformal field theory. Furthermore, by an explicit construction we show that all the cluster models are dual to non-locally coupled transverse field Ising chains, with the universality of the so(N )1 criticality manifesting itself as N of these chains becoming critical. This duality also reveals that the symmetry protection of cluster models arises from the underlying Ising symmetries and it enables the identification of local representations for the primary fields of the so(N )1 conformal field theories. For the simplest and experimentally most realistic case that corresponds to the original one-dimensional cluster model with local three-spin interactions, our results show that the su(2)2 so(3)1 Wess-Zumino-Witten model can emerge in a local, translationally invariant and Jordan-Wigner solvable spin-1/2 model.A striking property of quantum phase transitions is that of universality. Near a critical regime separating distinct quantum states of matter, the system specific microscopic details are lost and the system acquires universal behavior that is characterized by scaling exponents of distinct observables near the transition [1]. A crucial step in putting order to the zoo of distinct universalities was made by noting that the diverging correlation length at the critical point implies conformal invariance, and thus a description by a conformal field theory (CFT) [2,3]. This approach is particularly powerful in one spatial dimension (1D), where the relevant CFT fully describes (the scaling of) the correlation functions.1D quantum models with critical points are thus the natural playgrounds for quantum criticality. The simplest and the most celebrated ones are the transverse field Ising (TFI) chain [4], that at criticality is described by the Ising CFT, and the XY chain [5] where the anisotropic transition is described by the so called so(2) 1 CFT. So ubiquitous are these universality classes that criticality beyond them is often filed under "exotic". Still, considering this ubiquitous nature these simple models have played in various fields of physics, it was only rather recently when the details of the criticality of the TFI chain were experimentally probed [6]. The challenge of probing quantum criticality beyond these simple models owes to the lack of tractable models that admit accessible experimental realization. To go beyond them, often higher spins [7][8][9][10][11][12], infinite range couplings [13,14], broken translational invariance [15], strongly interacting systems [16][17][18], or combinations there of [19,20], are required. Here we show that a hierarchy of local, exactly solvable and translationally invariant spin-1/2 models provides the simplest setting to generalize the Ising unive...

Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

“…Some portion of the phase diagram for −2 ≤ ω ≤ 2 is mapped out in reference [45] where order parameters are identified and calculated numerically. Several papers, for example [46,47], study spin models with competing 'large' cluster term and Ising term (i.e. non-zero t α , t −1 and t 0 ).…”

Recoverable information and emergent conservation laws in fracton stabilizer codes

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“…and N indicates the system size. We remark that the family of models in this parameterization includes many interesting ones, such as XY model with n-site interaction, the GHZ-cluster model, and the SPT-AFM models and other interesting ones that have been explored from different perspectives [47,48,50]. We discuss and analyze some of these in the following.…”