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 nonanalyticity 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 $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$. 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 nonlocal measure on the quantum side, further indicating the nonlocal 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.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 nonanalyticity 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 $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$. 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 nonlocal measure on the quantum side, further indicating the nonlocal 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:1907.06216Physical review. B./Physical review. BMay 13, 20208 citations

Replica topological order in quantum mixed states and quantum error correction

Zhuan Li, Roger S. K. Mong

Topological phases of matter offer a promising platform for quantum computation and quantum error correction. Nevertheless, unlike its counterpart in pure states, descriptions of topological order in mixed states remain relatively underexplored. Our work gives two definitions for replica topological order in mixed states, which involve $n$ copies of density matrices of the mixed state. Our framework categorizes topological orders in mixed states as either quantum, classical, or trivial, depending on the type of information that can be encoded. For the case of the toric code model in the presence of decoherence, we associate for each phase a quantum channel and describes the structure of the code space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information, while in the classical-topological phase, the quantum information has decohere and cannot be fully recovered. We accomplish this by describing the mixed state as a projected entangled pairs state (PEPS) and identifying the symmetry-protected topological order of its boundary state to the bulk topology. Using this formalism, we enumerate all the possible mixed state phases which result from applying a local decoherence channel to the toric code. In addition to the classical-topological phases that have been previously reported on, we also find mixed states exhibiting chiral topological order. We discuss the extent that our findings can be extrapolated to $n\ensuremath{\rightarrow}1$ limit.Topological phases of matter offer a promising platform for quantum computation and quantum error correction. Nevertheless, unlike its counterpart in pure states, descriptions of topological order in mixed states remain relatively underexplored. Our work gives two definitions for replica topological order in mixed states, which involve $n$ copies of density matrices of the mixed state. Our framework categorizes topological orders in mixed states as either quantum, classical, or trivial, depending on the type of information that can be encoded. For the case of the toric code model in the presence of decoherence, we associate for each phase a quantum channel and describes the structure of the code space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information, while in the classical-topological phase, the quantum information has decohere and cannot be fully recovered. We accomplish this by describing the mixed state as a projected entangled pairs state (PEPS) and identifying the symmetry-protected topological order of its boundary state to the bulk topology. Using this formalism, we enumerate all the possible mixed state phases which result from applying a local decoherence channel to the toric code. In addition to the classical-topological phases that have been previously reported on, we also find mixed states exhibiting chiral topological order. We discuss the extent that our findings can be extrapolated to $n\ensuremath{\rightarrow}1$ limit.

arXiv:2402.09516Physical review. B./Physical review. BMar 5, 20258 citations

Classical Topology and Quantum Phases: Quantum Mechanics

A. P. Balachandran

Elsevier eBooksJan 1, 19897 citations

Quantum Geometry and Topology

Ravi Shankar

Texts and readings in physical sciencesJan 1, 20173 citations

Robust topological degeneracy of classical theories

Mohammad-Sadegh Vaezi, Gerardo Ortíz, Zohar Nussinov

We challenge the hypothesis that the ground states of a physical system whose degeneracy depends on topology must necessarily realize topological quantum order and display nonlocal entanglement. To this end, we introduce and study a classical rendition of the Toric Code model embedded on Riemann surfaces of different genus numbers. We find that the minimal ground state degeneracy (and those of all levels) depends on the topology of the embedding surface alone. As the ground states of this classical system may be distinguished by local measurements, a characteristic of Landau orders, this example illustrates that topological degeneracy is not a sufficient condition for topological quantum order. This conclusion is generic and, as shown, it applies to many other models. We also demonstrate that certain lattice realizations of these models, and other theories, display a ground state entropy (and those of all levels) that is ``holographic'', i.e., extensive in the system boundary. We find that clock and $U(1)$ gauge theories display topological (in addition to gauge) degeneracies.We challenge the hypothesis that the ground states of a physical system whose degeneracy depends on topology must necessarily realize topological quantum order and display nonlocal entanglement. To this end, we introduce and study a classical rendition of the Toric Code model embedded on Riemann surfaces of different genus numbers. We find that the minimal ground state degeneracy (and those of all levels) depends on the topology of the embedding surface alone. As the ground states of this classical system may be distinguished by local measurements, a characteristic of Landau orders, this example illustrates that topological degeneracy is not a sufficient condition for topological quantum order. This conclusion is generic and, as shown, it applies to many other models. We also demonstrate that certain lattice realizations of these models, and other theories, display a ground state entropy (and those of all levels) that is ``holographic'', i.e., extensive in the system boundary. We find that clock and $U(1)$ gauge theories display topological (in addition to gauge) degeneracies.

arXiv:1511.07867Physical review. B./Physical review. BMay 9, 20165 citations

H

is mapped to the heat capacity of the classical spin model defined on dual hypergraph $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$. 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 nonlocal measure on the quantum side, further indicating the nonlocal 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.

n

copies of density matrices of the mixed state. Our framework categorizes topological orders in mixed states as either quantum, classical, or trivial, depending on the type of information that can be encoded. For the case of the toric code model in the presence of decoherence, we associate for each phase a quantum channel and describes the structure of the code space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information, while in the classical-topological phase, the quantum information has decohere and cannot be fully recovered. We accomplish this by describing the mixed state as a projected entangled pairs state (PEPS) and identifying the symmetry-protected topological order of its boundary state to the bulk topology. Using this formalism, we enumerate all the possible mixed state phases which result from applying a local decoherence channel to the toric code. In addition to the classical-topological phases that have been previously reported on, we also find mixed states exhibiting chiral topological order. We discuss the extent that our findings can be extrapolated to $n\ensuremath{\rightarrow}1$ limit.

U(1)

gauge theories display topological (in addition to gauge) degeneracies.