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arXiv:2502.14556

Exploring the phase diagram of $SU(2)_4$ strange correlator

Shen, Ce

Published Feb 20, 2025• 3 citations
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Abstract

We investigate the phase diagram of a quantum many-body system constructed via the strange correlator approach, based on the non-Abelian $SU(2)_4$ fusion category, to probe topological phase transitions. Using tensor network methods, we numerically compute the half-infinite chain entanglement entropy derived from the dominant eigenvector of the transfer matrix and map the entropy across a spherical two-dimensional parameter space. Our results reveal a phase diagram significantly more complex than previously reported, including a gapless phase consistent with a conformal field theory (CFT) of central charge $c=1$. Critical lines separating distinct phases are identified, with one such line bounding the CFT phase exhibiting a higher central charge $c=2$, indicative of an unconventional critical regime.

DOI
10.48550/arxiv.2502.14556

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Exploring the phase diagram of SU(2)4SU(2)_4SU(2)4​ strange correlator
We investigate the phase diagram of a quantum many-body system constructed via the strange correlator approach, based on the non-Abelian SU(2)4SU(2)_4SU(2)4​ fusion category, to probe topological phase transitions. Using tensor network methods, we numerically compute the half-infinite chain entanglement entropy derived from the dominant eigenvector of the transfer matrix and map the entropy across a spherical two-dimensional parameter space. Our results reveal a phase diagram significantly more complex than previously reported, including a gapless phase consistent with a conformal field theory (CFT) of central charge c=1c=1c=1. Critical lines separating distinct phases are identified, with one such line bounding the CFT phase exhibiting a higher central charge c=2c=2c=2, indicative of an unconventional critical regime.
We use the formalism of strange correlators to construct a critical classical lattice model in two dimensions with the Haagerup fusion category H_{3} as input data. We present compelling numerical evidence in the form of finite entanglement scaling to support a Haagerup conformal field theory (CFT) with central charge c=2. Generalized twisted CFT spectra are numerically obtained through exact diagonalization of the transfer matrix, and the conformal towers are separated in the spectra through their identification with the topological sectors. It is further argued that our model can be obtained through an orbifold procedure from a larger lattice model with input Z(H_{3}), which is the simplest modular tensor category that does not admit an algebraic construction. This provides a counterexample for the conjecture that all rational CFT can be constructed from standard methods.
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Due to the absence of local order parameters, it is a challenging task to characterize the quantum topological phase transitions between topologically ordered phases in two dimensions. In this paper, we construct a topologically ordered tensor network wave function with one parameter $\ensuremath{\lambda}$, describing both the toric code state $(\ensuremath{\lambda}=1)$ and double semion state $(\ensuremath{\lambda}=\ensuremath{-}1)$. Via calculating the correlation length defined from the one-dimensional quantum transfer operator of the wave function norm, we can map out the complete phase diagram in terms of the parameter $\ensuremath{\lambda}$, and three different quantum critical points (QCPs) at $\ensuremath{\lambda}=0,\ifmmode\pm\else\textpm\fi{}1.73$ are identified. The first one separates the toric code phase and double semion phase, while latter two describe the topological phase transitions from the toric code phase or double semion phase to the symmetry-breaking phase, respectively. When mapping the quantum tensor network wave function to the exactly solved statistical model, the norm of the wave function is identified as the partition function of the classical eight-vertex model, and both QCPs at $\ensuremath{\lambda}=\ifmmode\pm\else\textpm\fi{}1.73$ correspond to the eight-vertex model at the critical point $\ensuremath{\lambda}=\sqrt{3}$, while the QCP at $\ensuremath{\lambda}=0$ corresponds to the critical six-vertex model. Actually such a quantum-classical mapping cannot yield the complete low-energy excitations at these three QCPs. We further demonstrate that the full eigenvalue spectra of the transfer operators without/with the flux insertions can give rise to the complete quantum criticalities, which are described by the (2+0)-dimensional free boson conformal field theories (CFTs) compactified on a circle with the radius R=6R=\sqrt{6}R=6​ at $\ensuremath{\lambda}=\ifmmode\pm\else\textpm\fi{}\sqrt{3}$ and R=8/3R=\sqrt{8/3}R=8/3​ at $\ensuremath{\lambda}=0$. From the complete transfer operator spectra, the finite-size spectra of the CFTs for the critical eight-vertex model are obtained, and the topological sectors of anyonic excitations are yielded as well. Furthermore, for the QCP at $\ensuremath{\lambda}=0$, no anyon condensation occurs, but the emerged symmetries of the matrix product operators significantly enrich the topological sectors of the CFT spectra. Finally, we provide our understanding of the (2+0)-dimensional conformal quantum criticalities and their possible connection with the generic (2+1)-dimensional CFTs for quantum topological phase transitions.
Conformal field theory (CFT) conventionally possesses positive central charges and describes continuous phase transitions in quantum many-body systems. Recently, CFT with complex-valued central charges has been proposed to relate to the discontinuous phase transitions of the Potts models. Here, the authors demonstrate that the scaling of complex-valued entanglement entropy characterizes complex CFT by applying the density matrix renormalization group method to the non-Hermitian five-state Potts model. This study suggests a way to investigate the entanglement properties of complex CFT.
We extend the concept of strange correlators, defined for symmetry-protected phases in You et al. [Phys. Rev. Lett. 112, 247202 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.247202], to topological phases of matter by taking the inner product between string-net ground states and product states. The resulting two-dimensional partition functions are shown to be either critical or symmetry broken, since the corresponding transfer matrices inherit all matrix product operator symmetries of the string-net states. For the case of critical systems, these nonlocal matrix product operator symmetries are the lattice remnants of topological conformal defects in the field theory description. Following Aasen et al. [J. Phys. A 49, 354001 (2016)JPAMB51751-811310.1088/1751-8113/49/35/354001], we argue that the different conformal boundary conditions can be obtained by applying the strange correlator concept to the different topological sectors of the string net obtained from Ocneanu's tube algebra. This is demonstrated on the lattice by calculating the conformal field theory spectra in the different topological sectors for the Fibonacci (hard-hexagon) and Ising string net. Additionally, we provide a complementary perspective on symmetry-preserving real-space renormalization by showing how known tensor network renormalization methods can be understood as the approximate truncation of an exactly coarse-grained strange correlator.