General Relationship Between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States (1103.5437v1)

Abstract: We consider (2+1)-dimensional topological quantum states which possess edge states described by a chiral (1+1)-dimensional Conformal Field Theory (CFT), such as e.g. a general quantum Hall state. We demonstrate that for such states the reduced density matrix of a finite spatial region of the gapped topological state is a thermal density matrix of the chiral edge state CFT which would appear at the spatial boundary of that region. We obtain this result by applying a physical instantaneous cut to the gapped system, and by viewing the cutting process as a sudden "quantum quench" into a CFT, using the tools of boundary conformal field theory. We thus provide a demonstration of the observation made by Li and Haldane about the relationship between the entanglement spectrum and the spectrum of a physical edge state.

• The paper establishes that the low-energy entanglement spectrum directly maps onto the chiral edge state spectrum in (2+1)-D topological phases.
• It employs boundary CFT and quantum quench methods to analytically validate numerical findings by Li and Haldane.
• Findings highlight the potential of using entanglement measurements as a diagnostic tool for exploring novel topological quantum states.

Relationship Between Entanglement Spectrum and Edge State Spectrum in Topological Quantum States

Topological phases of matter encompass a set of quantum states characterized by global topological features, rather than local order parameters or symmetry breaking. These states can host protected edge states and reveal unique ground state degeneracies depending on the topology of the space they inhabit. Such phases have been observed in systems like integer and fractional quantum Hall states and in topological insulators. The paper "General Relationship Between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States" by Xiao-Liang Qi, Hosho Katsura, and Andreas W. W. Ludwig explores an intriguing relationship between the entanglement spectrum of a topological phase and the spectrum of its physical edge states.

The authors focus on (2+1)-dimensional topological quantum states that exhibit edge states described by (1+1)-dimensional chiral Conformal Field Theories (CFTs), a typical instance being quantum Hall systems. A key contribution of this work is the demonstration that the reduced density matrix of a finite spatial region within a gapped topological state can be expressed as a thermal density matrix of the chiral CFT that operates at the region's boundary.

Through a theoretical framework, the authors relate the entanglement Hamiltonian, derived from a reduced density matrix due to tracing out environmental degrees of freedom, with the physical Hamiltonian governing edge states. Notably, the entanglement spectrum’s low-energy states correspond to those most entangled with the remaining system and critically correlate with the low-lying spectrum at the boundary.

The methodology leverages the notion of a quantum quench and employs boundary CFT techniques to analyze the system. An analytical argument is constructed to support Li and Haldane's numerical observations on this spectrum correspondence, surpassing previous numerical validations with analytical rigor.

Numerical and Analytical Insights

Significant insights arise from formulating the entanglement Hamiltonian $H_{\rm E}$ in relation to the physical Hamiltonian of the edge $H_L$ as: $H_{\rm E} \propto H_L$ This relationship holds within a fixed topological sector under consideration, implying that the thermal density matrix of the entangled region mimics that of a thermal state governed by the chiral edge Hamiltonian.

An additional detailed example involves free fermion systems, like the integer quantum Hall effect, where the authors verify this relation using many-body frameworks and entanglement studies. The mappings and comparisons presented here serve to illustrate and strengthen the generative arguments made for interacting systems.

Broader Implications and Future Directions

This work contributes significantly to our understanding of how quantum entanglement can serve as a bridge between bulk and boundary theories in topological states. The established correspondence between entanglement and edge spectra not only extends the utility of entanglement measurements in probing exotic phases but also may inform the design of quantum information processes using topological phases.

The framework may apply to higher-dimensional topological states and enriched topological insulators, yet this necessitates adequate treatment beyond (2+1)-dimensional constructs, likely involving complex topological structures and entanglements. As new phases are discovered and experimental techniques advance, the theoretical insights from this paper can immensely guide the exploration of topologically ordered systems and their potential applications in quantum computing and beyond.