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| symmetry-protected topological order | |
|---|---|
| Name | Symmetry-protected topological order |
| Field | Condensed matter physics |
| Introduced | 2010s |
| Notable examples | Topological insulators; Haldane phase; Topological superconductors |
symmetry-protected topological order Symmetry-protected topological order is a phase concept in condensed matter physics describing gapped quantum phases that are nontrivial only in the presence of certain symmetries. It contrasts with spontaneous symmetry breaking and intrinsic topological order and connects to robust boundary phenomena and entanglement structure in many-body systems.
Symmetry-protected topological order is defined for gapped Hamiltonians on lattices or manifolds and characterized by short-range entanglement, a unique bulk ground state on closed manifolds, and nontrivial response when symmetries are enforced; canonical literature situates the concept alongside studies at Princeton University, Harvard University, Stanford University, University of California, Berkeley, Massachusetts Institute of Technology and research groups at IBM and Microsoft Research. The defining features include gap stability under symmetry-preserving perturbations, absence of fractionalization typical of Fractional quantum Hall effect systems studied at Bell Labs and CERN, and classification by group cohomology and other algebraic invariants developed in work associated with Institute for Advanced Study and collaborations involving Paul Dirac-influenced formalisms. Symmetry constraints commonly invoked include time-reversal symmetry examined in experiments at Argonne National Laboratory, particle-number conservation relevant to experiments at Oak Ridge National Laboratory, and crystalline symmetries considered in projects at National Institute of Standards and Technology.
Famous realizations include topological insulators and superconductors discovered and characterized in experiments at Hitachi, IBM Research – Almaden, Cornell University, University of Tokyo, École Normale Supérieure, and theoretical proposals from groups at California Institute of Technology and Yale University. One-dimensional examples include the Haldane phase in spin chains originally proposed in work associated with University of Cambridge and experimentally probed at Max Planck Institute for Solid State Research and ISIS Neutron and Muon Source. Two-dimensional examples include quantum spin Hall systems first observed at University of Würzburg and in heterostructures studied at Bell Labs. Three-dimensional topological insulators were found in materials investigated at University of Geneva and Tokyo Institute of Technology. Superconducting SPT phases connect to efforts at Los Alamos National Laboratory and National High Magnetic Field Laboratory to realize Majorana modes akin to proposals from Microsoft Station Q. Cold-atom realizations have been pursued at MIT and University of Oxford optical lattice setups inspired by research at JILA and Max Planck Institute of Quantum Optics.
Classification frameworks draw from group cohomology methods related to mathematical work at University of Cambridge and Princeton University and from K-theory techniques originating in studies at University of California, Berkeley and Harvard University. The tenfold way classification of free fermion systems ties to symmetry classes developed at Los Alamos National Laboratory and formalized in collaborations involving researchers at University of Chicago and ETH Zurich. Interacting approaches include group supercohomology linked to projects at Weizmann Institute of Science and cobordism theories pursued by teams at Institut des Hautes Études Scientifiques and Perimeter Institute for Theoretical Physics. Higher-order SPTs and crystalline SPT classifications have been developed by consortia including European Organization for Nuclear Research researchers and theorists at Max Planck Institute for the Physics of Complex Systems. Connections to tensor-network classification methods were advanced by groups at Google Quantum AI and Microsoft Research.
Boundary phenomenology is central: nontrivial SPT phases host protected edge or surface modes investigated in experiments at University of California, Santa Barbara and Tata Institute of Fundamental Research. The bulk–boundary correspondence is formalized in models influenced by methods from ETH Zurich and Princeton University and tested in spectroscopic studies at Argonne National Laboratory and National Renewable Energy Laboratory. For fermionic systems, boundary Majorana modes relate to theoretical proposals from Caltech and Microsoft Research, while spin-chain end states relate to analyses originating at University of Cambridge and MPI for the Physics of Complex Systems. Anomalies at boundaries are connected to field-theory descriptions studied at Institute for Advanced Study and Perimeter Institute.
Experimental probes include angle-resolved photoemission spectroscopy used in measurements at Stanford Synchrotron Radiation Lightsource, scanning tunneling microscopy employed at IBM Research – Almaden and Lawrence Berkeley National Laboratory, and transport experiments carried out at University of Maryland and University of Illinois at Urbana-Champaign. Signatures include robust surface conduction reported in studies at University of California, Los Angeles and quantized responses akin to those measured in quantum Hall experiments at Columbia University and Bell Labs. Cold-atom simulators at MIT and University of Cambridge enable detection via time-of-flight and Bragg spectroscopy, while neutron scattering work at Oak Ridge National Laboratory and ISIS Neutron and Muon Source probes spin SPT signatures. Interferometry and Josephson junction experiments at Stanford University and Université Pierre et Marie Curie target superconducting SPT signatures.
Models include the Affleck–Kennedy–Lieb–Tasaki chain developed in contexts linked to University of Oxford and analytical tools such as tensor-network states advanced at University of Tokyo and University of Geneva. Field-theory approaches employ nonlinear sigma models referenced in seminars at Institute for Advanced Study and renormalization group analyses from groups at University of Chicago and Columbia University. Computational studies utilize density matrix renormalization group methods from Max Planck Institute for the Physics of Complex Systems and quantum Monte Carlo techniques developed at Los Alamos National Laboratory. Exactly solvable lattice constructions and decorated domain wall models have been proposed by researchers at Harvard University and Weizmann Institute of Science.
SPT phases connect to fault-tolerant quantum information proposals explored at IBM, Google, and Microsoft Research, and to topological quantum computation ideas rooted in work at Perimeter Institute and Caltech. They relate to intrinsic topological order exemplified by Fractional quantum Hall effect systems studied at Bell Labs and to symmetry-breaking phases investigated at University of Chicago and Princeton University. Cross-disciplinary links extend to metamaterials research at Cornell University and photonic implementations pioneered at Harvard University and University of Pennsylvania. Category:Condensed matter physics