Classical criticality establishes quantum topological order

Abstract

We establish an important duality correspondence between topological order in quantum many body systems and criticality in ferromagnetic classical spin systems. We show how such a correspondence leads to a classical and simple procedure for characterization of topological order in an important set of quantum entangled states, namely the Calderbank-Shor-Steane (CSS) states. To this end, we introduce a particular quantum Hamiltonian which allows us to consider the existence of a topological phase transition from quantum CSS states to a magnetized state. We study the ground state fidelity in order to find non-analyticity in the wave function as a signature of a topological phase transition. Since hypergraphs can be used to map any arbitrary CSS state to a classical spin model, we show that fidelity of the quantum model defined on a hypergraph $H$ is mapped to the heat capacity of the classical spin model defined on dual hypergraph $\tilde{H}$. Consequently, we show that a ferromagnetic-paramagnetic phase transition in a classical model is mapped to a topological phase transition in the corresponding quantum model. We also show that magnetization does not behave as a local order parameter at the transition point while the classical order parameter is mapped to a non-local measure on the quantum side, further indicating the non local nature of the transition. Our procedure not only opens the door for identification of topological phases via the existence of a local and classical quantity, i.e. critical point, but also offers the potential to classify various topological phases through the concept of universality in phase transitions.

arXiv comment6 + 2 pages, 2 figure, Accepted for publication in Physical Review B

References in corpus (42)

• Fault-tolerant quantum computation by anyons
• Good Quantum Error-Correcting Codes Exist
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Topological quantum memory
• Topological orders and Edge excitations in FQH states
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Stabilizer Codes and Quantum Error Correction
• Topological Quantum Distillation
• Identifying Topological Order by Entanglement Entropy
• Fidelity approach to quantum phase transitions
• Criticality, the area law, and the computational power of PEPS
• Experimental Quantum Computations on a Topologically Encoded Qubit
• Topological quantum order: stability under local perturbations
• Quantum memories at finite temperature
• Robustness of a perturbed topological phase
• Quantum Hypergraph States
• Fidelity susceptibility and long-range correlation in the Kitaev honeycomb model
• From quantum mechanics to classical statistical physics: generalized Rokhsar-Kivelson Hamiltonians and the "Stochastic Matrix Form" decomposition
• A quantum topological phase transition at the microscopic level
• Entanglement, fidelity and topological entropy in a quantum phase transition to topological order
• Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity
• Quantum correlations in topological quantum phase transitions
• X-Cube Fracton Model on Generic Lattices: Phases and Geometric Order
• A Quantum Approach to Classical Statistical Mechanics
• Feasibility of self-correcting quantum memory and thermal stability of topological order
• Correlation Length, Universality Classes, and Scaling Laws Associated with Topological Phase Transitions
• Fisher information approach to non-equilibrium phase transitions in quantum XXZ spin chain with boundary noise
• Completeness of the classical 2D Ising model and universal quantum computation
• Encoding Hypergraphs into Quantum States
• Classical spin models and the quantum stabilizer formalism
• Multi-partite entanglement and quantum phase transition in the one-, two-, and three-dimensional transverse field Ising model
• On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers
• Global entanglement and quantum phase transitions in the transverse XY Heisenberg chain
• On the robustness of topological quantum codes: Ising perturbation
• Classical Ising model test for quantum circuits
• Dual correspondence between classical spin models and quantum CSS states
• An algorithmic proof for the completeness of two-dimensional Ising model
• On thermal stability of topological qubit in Kitaev's 4D model
• A strong-weak coupling duality between two perturbed quantum many-body systems: CSS codes and Ising-like systems
• Phase transition in a noisy Kitaev toric code model
• Noisy Toric code and random bond Ising model: The error threshold in a dual picture

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