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arXiv:2305.13410 · Physical review. B./Physical review. B

Conformal field theories generated by Chern insulators under decoherence or measurement

Kaixiang Su, Nayan E. Myerson-Jain, Cenke Xu

Published Jan 19, 2024• 15 citations
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Abstract

We demonstrate that the fidelity between a pure state trivial insulator and the mixed state density matrix of a Chern insulator generated from decoherence or measurement can be mapped to a variety of two-dimensional conformal field theories (CFTs); more specifically, the quantity $\mathcal{Z}=\mathrm{tr}{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{c}^{D}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\Omega}}}}$ is mapped to the partition function of the desired CFT, where ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\Omega}}}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{c}^{D}$ are respectively the density matrix of a pure state trivial insulator and the mixed state density matrix generated from the Chern insulator. For a pure state Chern insulator with Chern number $2N$, the fidelity $\mathcal{Z}$ is mapped to the partition function of the $\mathrm{U}{(2N)}_{1}$ CFT; under decoherence or measurement, the Chern insulator density matrix can experience a certain instability, and the ``partition function'' $\mathcal{Z}$ can flow to other interacting CFTs with smaller central charges. The R\'enyi relative entropy $\mathcal{F}=\ensuremath{-}ln\mathrm{tr}{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{c}^{D}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\Omega}}}}$ is mapped to the free energy of the CFT, and we demonstrate that the central charge of the CFT can be extracted from the finite-size scaling of $\mathcal{F}$, analogous to the well-known finite-size scaling of $2d$ CFT.

DOI
10.1103/physrevb.109.035146

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We demonstrate that the fidelity between a pure state trivial insulator and the mixed state density matrix of a Chern insulator generated from decoherence or measurement can be mapped to a variety of two-dimensional conformal field theories (CFTs); more specifically, the quantity Z=tr\ensuremathρ\ifmmode^\elseˆ\ficD\ensuremathρ\ifmmode^\elseˆ\fi\ensuremathΩ\mathcal{Z}=\mathrm{tr}{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{c}^{D}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\Omega}}}}Z=tr\ensuremathρ\ifmmode^\elseˆ\fi​cD​\ensuremathρ\ifmmode^\elseˆ\fi​\ensuremathΩ​ is mapped to the partition function of the desired CFT, where \ensuremathρ\ifmmode^\elseˆ\fi\ensuremathΩ{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\Omega}}}\ensuremathρ\ifmmode^\elseˆ\fi​\ensuremathΩ​ and \ensuremathρ\ifmmode^\elseˆ\ficD{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{c}^{D}\ensuremathρ\ifmmode^\elseˆ\fi​cD​ are respectively the density matrix of a pure state trivial insulator and the mixed state density matrix generated from the Chern insulator. For a pure state Chern insulator with Chern number 2N2N2N, the fidelity Z\mathcal{Z}Z is mapped to the partition function of the U(2N)1\mathrm{U}{(2N)}_{1}U(2N)1​ CFT; under decoherence or measurement, the Chern insulator density matrix can experience a certain instability, and the ``partition function'' Z\mathcal{Z}Z can flow to other interacting CFTs with smaller central charges. The R'enyi relative entropy F=\ensuremath−lntr\ensuremathρ\ifmmode^\elseˆ\ficD\ensuremathρ\ifmmode^\elseˆ\fi\ensuremathΩ\mathcal{F}=\ensuremath{-}ln\mathrm{tr}{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{c}^{D}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\Omega}}}}F=\ensuremath−lntr\ensuremathρ\ifmmode^\elseˆ\fi​cD​\ensuremathρ\ifmmode^\elseˆ\fi​\ensuremathΩ​ is mapped to the free energy of the CFT, and we demonstrate that the central charge of the CFT can be extracted from the finite-size scaling of F\mathcal{F}F, analogous to the well-known finite-size scaling of 2d2d2d CFT.
Modern quantum platforms naturally realize many-body states shaped by measurement, decoherence, and dissipation. The appropriate object is often not a wave function or ground state, but a density matrix. This thesis asks how phases and phase transitions can be defined for such mixed states, especially when universal structure appears only through nonlinear, replicated, or doubled diagnostics. The thesis develops this perspective using nonlinear diagnostics, doubled-Hilbert- space representations, statistical-mechanics mappings, and generalized symmetry. It shows that weak measurement can suppress higher-form symmetry breaking and drive mixed-state transitions; maps fidelity diagnostics of decohered Chern insulators to effective conformal-field-theory partition functions; and maps Rényi diagnostics of noisy stabilizer and CSS codes to dual statistical-mechanics models connected to decodability. It also shows that measurement and Lindbladian evolution can prepare density matrices whose doubled-space representations realize spin-liquid-like states and superconducting order. Finally, it identifies strong-to-weak symmetry breaking as a route to universal mixed-state behavior connected to hydrodynamic emergence. Together, these results show that openness is not only a perturbation that degrades pure-state physics. Measurement, decoherence, and dissipation can hide, preserve, trans- form, or generate structure in density matrices. The resulting phases and transitions are tied to information processing: what can be decoded, distinguished, recovered, or lost to local probes.
Conformal invariance powerfully constrains the critical behavior of two-dimensional classical systems with short-range interactions and the critical theories in two-dimensions are parametrized by a dimensional number, termed central charge or conformal anomaly ccc. However, experimental determination of the central charge of a conformal field theory has not been done before. Here we propose to extract the central charge of the conformal field theory corresponding to a critical point of a two-dimensional lattice models from the quantum decoherence measurement of a probe spin which is coupled to the two-dimensional lattice models. Conformal invariance predicts that the leading finite-size correction to the free energy for a two-dimensional system at a conformal invariant critical point is linearly related to the conformal anomaly for various boundary conditions. A recent discovery of thermodynamic holography allows us to obtain the free energy of many-body system from central spin decoherence measurement. Thus the central charge of the conformal field theory could be extracted from central spin decoherence measurement. We have applied the method to the two-dimensional Ising model and extracted the central charge of two-dimensional Ising model with good precision. This work provides a useful approach to extracting the central charge of conformal field theory in various two-dimensional lattice systems.
Decoherence inevitably happens when a quantum state is exposed to its environment, which can affect quantum critical points (QCPs) in a nontrivial way.As was pointed out in the recent literature on (1 + 1)d conformal field theory (CFT) [Garratt et al.Measurements conspire nonlocally to restructure critical quantum states, arXiv:2207.09476(2022)], the effect of weak measurement can be mathematically mapped to the problem of boundary CFT.In this work, we focus on the (2 + 1)d QCPs, whose boundary and defect effects have attracted enormous theoretical and numerical interests very recently.We focus on decoherence caused by weak measurements with and without postselecting the measurement outcomes.Our main results are as follows: (1) for an O(N ) Wilson-Fisher QCP under weak measurement with postselection, an observer would, in general, observe two different types of boundary and defect criticality with very different behaviors from the well-known Wilson-Fisher fixed points; in particular, it is possible to observe the recently proposed exotic "extraordinary-log" correlation.(2) An extra quantum phase transition can be driven by decoherence, if we consider quantities nonlinear with the decohered density matrix, such as the Renyi entropy.We demonstrate the connection between this transition to the information-theoretic transition driven by an error in the toric code model.(3) When there is no postselection, though correlation functions between local operators remain the same as the undecohered pure state, nonlocal operators such as the "disorder operator" would have qualitatively distinct behaviors; and we also show that the decoherence can lead to confinement.
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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.
Conformal field theory provides universal predictions for Loschmidt amplitudes following quenches from product states to critical Hamiltonians. Building on this observation, we develop a route to extracting conformal data from real-time dynamics without preparing critical low-energy states. After analytic continuation, the Loschmidt amplitude is described by a boundary-CFT partition function on a strip, whose transverse transfer matrix encodes both the boundary operator spectrum and the central charge. Local space-time perturbations of the amplitude are governed by equilibrium correlation functions, and therefore provide access to critical exponents. In parallel, generalized temporal entropies exhibit scaling with time analogous to the equilibrium scaling of spatial entanglement entropy. We show that the low-lying boundary spectrum can be reconstructed from the system-size dependence of finite-chain Loschmidt echoes, whose damped oscillations encode differences of boundary scaling dimensions. Finally, we propose a finite-size scaling protocol that can extract these quantities from simulations or experiments on state-of-the-art quantum platforms.