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arXiv:2503.11788 · Physical review. B./Physical review. B

Effective conformal field theory generated from pure and dephased Chern insulator

Abhijat Sarma, Yimu Bao, Nayan E. Myerson-Jain, Thomas G. Kiely, Cenke Xu

Published Aug 4, 2025• 4 citations
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Abstract

We demonstrate that the fidelity between two states with different Chern numbers $\mathcal{Z}=\mathrm{tr}{\ensuremath{\rho}{\ensuremath{\rho}}^{\ensuremath{'}}}$ serves as a generating theory for an effective conformal field theory (CFT) at the $(2+0)$-dimensional temporal interface. $\ensuremath{\rho}$ can be chosen to be a pure trivial insulator, and ${\ensuremath{\rho}}^{\ensuremath{'}}$ can be taken as a pure or dephased Chern insulator density matrix. More specifically, we obtain the following results: (1) Through evaluation of the effective central charge, stiffness, and correlation function (the ``strange correlators''), we demonstrate that the fidelity between a trivial insulator and an insulator with Chern number $C=1$ maps to a CFT with effective central charge ${c}_{\mathrm{eff}}=1$, while the fidelity between two Chern insulators with Chern numbers $C=\ifmmode\pm\else\textpm\fi{}1$ maps to a CFT with ${c}_{\mathrm{eff}}=2$. (2) The density matrix of the Chern insulator becomes a quantum spin Hall insulator in the doubled Hilbert space, and the dephasing acts as an interaction between the two spin species. (3) In the limit of infinite dephasing the Chern insulator becomes a superconductor in the doubled Hilbert space, featuring the ``strong-weak'' U(1) spontaneous symmetry breaking. An analysis based on the Laughlin wave function and previous studies of the projected wave function of the quantum spin Hall insulator suggest this is a power-law superconductor. (4) With increasing strength of dephasing, the amplitude of the single-particle strange correlator is suppressed, while the Cooper pair strange correlator is enhanced, consistent with the trend of emerging superconductivity.

DOI
10.1103/nn2m-w4vk

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We demonstrate that the fidelity between two states with different Chern numbers Z=tr\ensuremathρ\ensuremathρ\ensuremath′\mathcal{Z}=\mathrm{tr}{\ensuremath{\rho}{\ensuremath{\rho}}^{\ensuremath{'}}}Z=tr\ensuremathρ\ensuremathρ\ensuremath′ serves as a generating theory for an effective conformal field theory (CFT) at the (2+0)(2+0)(2+0)-dimensional temporal interface. \ensuremathρ\ensuremath{\rho}\ensuremathρ can be chosen to be a pure trivial insulator, and \ensuremathρ\ensuremath′{\ensuremath{\rho}}^{\ensuremath{'}}\ensuremathρ\ensuremath′ can be taken as a pure or dephased Chern insulator density matrix. More specifically, we obtain the following results: (1) Through evaluation of the effective central charge, stiffness, and correlation function (the strange correlators''), we demonstrate that the fidelity between a trivial insulator and an insulator with Chern number $C=1$ maps to a CFT with effective central charge ${c}_{\mathrm{eff}}=1$, while the fidelity between two Chern insulators with Chern numbers $C=\ifmmode\pm\else\textpm\fi{}1$ maps to a CFT with ${c}_{\mathrm{eff}}=2$. (2) The density matrix of the Chern insulator becomes a quantum spin Hall insulator in the doubled Hilbert space, and the dephasing acts as an interaction between the two spin species. (3) In the limit of infinite dephasing the Chern insulator becomes a superconductor in the doubled Hilbert space, featuring the strong-weak'' U(1) spontaneous symmetry breaking. An analysis based on the Laughlin wave function and previous studies of the projected wave function of the quantum spin Hall insulator suggest this is a power-law superconductor. (4) With increasing strength of dephasing, the amplitude of the single-particle strange correlator is suppressed, while the Cooper pair strange correlator is enhanced, consistent with the trend of emerging superconductivity.
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 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 $\mathcal{F}=\ensuremath{-}ln\mathrm{tr}{{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{c}^{D}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\Omega}}}}$ 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 F\mathcal{F}F, analogous to the well-known finite-size scaling of 2d2d2d CFT.
We investigate signatures of chirality at Chern insulator transitions in the presence of a boundary. The transition between a trivial insulator and a Chern insulator with Chern number C=1C=1C=1 is described by a massless Dirac fermion whose parity anomaly gives a critical Hall conductivity σxy=12e2hσ_{xy}=\frac{1}{2}\frac{e^2}{h}σxy​=21​he2​. Using a Dirac mass domain wall construction, we show that the chiral edge mode delocalizes into the bulk at criticality, but the boundary fermion correlation function retains a chiral structure and acquires the scaling dimension of the bulk fermion. We compute current correlation functions and demonstrate that the anomaly of the bulk Hall response is matched by delocalized chiral modes near the boundary. Using only the residual conformal symmetry of a (2+1)d conformal field theory (CFT) in a half-space, we identify parity-odd terms in current and energy-momentum tensor correlation functions that encode these modes and determine their electromagnetic and gravitational anomaly coefficients. Our analysis therefore applies to general time-reversal breaking (2+1)d CFTs, beyond the free Dirac transition. In particular, the analysis of the gravitational anomaly is also applicable to the free Majorana CFT governing the transition between a trivial superconductor and a topological superconductor. We also extend our results to more general Chern number changing transitions and to the transition between a (3+1)d topological insulator and a trivial insulator.
We show that synthetic spin-orbit coupling for ultracold atoms in optical Raman potentials can be exploited to build versatile quantum simulators of correlated Chern insulators connected to strongly coupled four-Fermi field theories similar to the Gross-Neveu model in (2 + 1) dimensions.Exploiting this multidisciplinary perspective, we identify a large-N quantum anomalous Hall (QAH) effect in absence of any external magnetic field, and use it to delimit regions in parameter space where these correlated topological phases appear, the boundaries of which are controlled by strongly coupled fixed points of these four-Fermi relativistic field theories.We further show how, for strong interactions, the QAH effect gives way to magnetic phases described by a two-dimensional quantum compass model in a transverse field.We present a detailed description of the phase diagram using the large-N effective potential, and variational techniques such as projected entangled pairs.
We study topological insulators under dephasing noise. With examples of both a 2d2d2d Chern insulator and a 3d3d3d topological insulator protected by time-reversal symmetry, we demonstrate that there is a phase transition at finite dephasing strength between phases with nontrivial and trivial topological indices. Here, the topological index is defined through the correlation matrix. The transition can be diagnosed through the spectrum of the whole correlation matrix or of a local subsystem. Interestingly, even if the topological insulator is very close to the topological-trivial critical point in its Hamiltonian, it still takes finite strength of dephasing to change the topological index, suggesting the robustness of topological insulators under dephasing. We further consider Chern insulators in the presence of real-space disorder, which exhibit a ground-state transition between topological and Anderson insulating phases. We find that even strongly disordered Chern insulators, close to the critical disorder strength, exhibit robustness with respect to dephasing.
A flat band with Chern number C = 0 , well isolated from the rest of Hilbert space by a gap much larger than interaction strength, is an unexpected context for fractional quantum Hall physics. In this work, we demonstrate the emergence of the fractional Chern insulator (FCI) states in such a trivial flat band using large-scale exact diagonalization and infinite density matrix renormalization group simulations. Hosted by an anisotropic lattice, this band has a sharp peak for the quantum geometry. At 1/3 hole filling, we find the unexpected FCI with 3-fold ground-state degeneracy and σ H = − 1 / 3 ( e 2 / h ) . The momentum-space carrier distribution shows that the quantum geometry peak tends to push the interacting holes away, toward other regions of the Brillouin zone with the nearly "ideal" quantum geometry, underlying the formation of the FCI in the C = 0 flat band. In comparison, we find only trivial ground states when the quantum geometry peak becomes less avoidable. Our work not only shows that FCl states could exist without the single-particle topology, relying on the more intrinsic quantum geometry, but also suggests a possibly generic scenario to explore FCIs in otherwise trivial flat bands.
The interplay among symmetry, topology and condensed matter systems has deepened our understandings of matter and lead to tremendous recent progresses in finding new topological phases of matter such as topological insulators, superconductors and semi-metals. Most examples of the aforementioned topological materials are free fermion systems, in this thesis, however, we focus on their strongly correlated counterparts where electron-electron interactions play a major role. With interactions, exotic topological phases and quantum critical points with fractionalized quantum degrees of freedom emerge. In the first part of this thesis, we study the problem of resonant tunneling through a quantum dot in a spinful Luttinger liquid. It provides the simplest example of a (0+1)d system with symmetry-protected phase transitions. We show that the problem is equivalent to a two channel SU(3) Kondo problem and can be mapped to a quantum Brownian motion model on a Kagome lattice. Utilizing boundary conformal field theory, we find the universal peak conductance and compute the scaling behavior of the resonance line-shape.\nFor the second part, we present a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory is based on a SO(7)1=SO(7)1/(G2)1 x (G2)1 coset construction and implemented by a solvable two-dimensional network model. In addition, we predict a closely related ''anti-Fibonacci'' phase, whose topological order is characterized by the tricritical Ising model. Finally, we propose an interferometer that generalizes the Z2 Majorana interferometer and directly probes the Fibonacci non-Abelian statistics.\nFor the third part, we argue that a correlated fluid of electrons and holes can exhibit fractional quantum Hall effects at zero magnetic field. We first show that a Chern insulator can be realized as a free fermion model with p-wave(m=1) excitonic pairing. Its ground state wavefunction is then worked out and generalized to m>1. We give several pieces of evidence that this conjectured wavefunction correctly describes a topological phase, dubbed ''fractional excitonic insulator'', within the same universality class as the corresponding Laughlin state at filling 1/m. We present physical arguments that gapless states with higher angular momentum pairing between energy bands are conducive to forming the fractional excitonic insulator in the presence of repulsive interactions. Without interactions, these gapless states appear at topological phase transitions which separate the trivial insulator from a Chern insulator with higher Chern number. Since the nonvanishing density of states at these higher angular momentum band inversion transitions can give rise to interesting many-body effects, we introduce a series of minimal lattice models realizations in two dimensions. We also study the effect of rotational symmetry broken electron-hole exciton condensation in our lattice models using mean field theory.