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Jiang B, Bouhon A, Wu SQ, Kong ZL, Lin ZK, Slager RJ, Jiang JH. Observation of an acoustic topological Euler insulator with meronic waves. Sci Bull (Beijing) 2024:S2095-9273(24)00227-5. [PMID: 38641514 DOI: 10.1016/j.scib.2024.04.009] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2023] [Revised: 01/21/2024] [Accepted: 03/25/2024] [Indexed: 04/21/2024]
Abstract
Topological band theory has conventionally been concerned with the topology of bands around a single gap. Only recently non-Abelian topologies that thrive on involving multiple gaps were studied, unveiling a new horizon in topological physics beyond the conventional paradigm. Here, we report on the first experimental realization of a topological Euler insulator phase with unique meronic characterization in an acoustic metamaterial. We demonstrate that this topological phase has several nontrivial features: First, the system cannot be described by conventional topological band theory, but has a nontrivial Euler class that captures the unconventional geometry of the Bloch bands in the Brillouin zone. Second, we uncover in theory and probe in experiments a meronic configuration of the bulk Bloch states for the first time. Third, using a detailed symmetry analysis, we show that the topological Euler insulator evolves from a non-Abelian topological semimetal phase via. the annihilation of Dirac points in pairs in one of the band gaps. With these nontrivial properties, we establish concretely an unconventional bulk-edge correspondence which is confirmed by directly measuring the edge states via. pump-probe techniques. Our work thus unveils a nontrivial topological Euler insulator phase with a unique meronic pattern and paves the way as a platform for non-Abelian topological phenomena.
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Affiliation(s)
- Bin Jiang
- Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou 215123, China; School of Physical Science and Technology & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
| | - Adrien Bouhon
- TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK; NORDITA, Stockholm University and KTH Royal Institute of Technology, Stockholm SE-106 91, Sweden.
| | - Shi-Qiao Wu
- School of Physical Science and Technology & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
| | - Ze-Lin Kong
- School of Physical Science and Technology & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
| | - Zhi-Kang Lin
- School of Physical Science and Technology & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
| | - Robert-Jan Slager
- TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK.
| | - Jian-Hua Jiang
- Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou 215123, China; School of Physical Science and Technology & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China.
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Logan JA, Mushnoori S, Dutt M, Tkachenko AV. Symmetry-specific orientational order parameters for complex structures. J Chem Phys 2022; 156:054108. [DOI: 10.1063/5.0076915] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Jack A. Logan
- Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA
- Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA
| | - Srinivas Mushnoori
- Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854, USA
| | - Meenakshi Dutt
- Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854, USA
| | - Alexei V. Tkachenko
- Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA
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Mushnoori S, Logan JA, Tkachenko AV, Dutt M. Controlling morphology in hybrid isotropic/patchy particle assemblies. J Chem Phys 2022; 156:024501. [PMID: 35032996 DOI: 10.1063/5.0076914] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Brownian dynamics is used to study self-assembly in a hybrid system of isotropic particles (IPs), combined with anisotropic building blocks that represent special "designer particles." Those are modeled as spherical patchy particles (PPs) with binding only allowed between their patches and IPs. In this study, two types of PPs are considered: Octahedral PPs (Oh-PPs) and Square PPs (Sq-PPs), with octahedral and square arrangements of patches, respectively. The self-assembly is additionally facilitated by the simulated annealing procedure. The resultant structures are characterized by a combination of local correlations in cubatic ordering and a symmetry-specific variation of bond orientation order parameters (SymBOPs). By varying the PP/IP size ratio, we detected a sharp crossover between two distinct morphologies in both types of systems. High symmetry phases, NaCl crystal for Oh-PP and square lattice for Sq-PP, are observed for larger size ratios. For the smaller ones, the dominant morphologies are significantly different, e.g., Oh-PPs form a compact amorphous structure with predominantly face-to-face orientation of neighboring PPs. Unusually, for a morphology without a long-range order, it is still possible to identify well organized coherent clusters of this structure, thanks to the adoption of our SymBOP-based characterization.
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Affiliation(s)
- Srinivas Mushnoori
- Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854, USA
| | - Jack A Logan
- Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA
| | - Alexei V Tkachenko
- Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA
| | - Meenakshi Dutt
- Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854, USA
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Rao N, Liu K, Pollet L. Inferring hidden symmetries of exotic magnets from detecting explicit order parameters. Phys Rev E 2021; 104:015311. [PMID: 34412223 DOI: 10.1103/physreve.104.015311] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2020] [Accepted: 06/13/2021] [Indexed: 11/07/2022]
Abstract
An unconventional magnet may be mapped onto a simple ferromagnet by the existence of a high-symmetry point. Knowledge of conventional ferromagnetic systems may then be carried over to provide insight into more complex orders. Here we demonstrate how an unsupervised and interpretable machine-learning approach can be used to search for potential high-symmetry points in unconventional magnets without any prior knowledge of the system. The method is applied to the classical Heisenberg-Kitaev model on a honeycomb lattice, where our machine learns the transformations that manifest its hidden O(3) symmetry, without using data of these high-symmetry points. Moreover, we clarify that, in contrast to the stripy and zigzag orders, a set of D_{2} and D_{2h} ordering matrices provides a more complete description of the magnetization in the Heisenberg-Kitaev model. In addition, our machine also learns the local constraints at the phase boundaries, which manifest a subdimensional symmetry. This paper highlights the importance of explicit order parameters to many-body spin systems and the property of interpretability for the physical application of machine-learning techniques.
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Affiliation(s)
- Nihal Rao
- Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstrasse 37, 80333 München, Germany.,Munich Center for Quantum Science and Technology, Schellingstrasse 4, 80799 München, Germany
| | - Ke Liu
- Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstrasse 37, 80333 München, Germany.,Munich Center for Quantum Science and Technology, Schellingstrasse 4, 80799 München, Germany
| | - Lode Pollet
- Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstrasse 37, 80333 München, Germany.,Munich Center for Quantum Science and Technology, Schellingstrasse 4, 80799 München, Germany.,Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
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Greitemann J, Liu K, Pollet L. The view of TK-SVM on the phase hierarchy in the classical kagome Heisenberg antiferromagnet. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:054002. [PMID: 33017805 DOI: 10.1088/1361-648x/abbe7b] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2020] [Accepted: 10/05/2020] [Indexed: 06/11/2023]
Abstract
We illustrate how the tensorial kernel support vector machine (TK-SVM) can probe the hidden multipolar orders and emergent local constraint in the classical kagome Heisenberg antiferromagnet. We show that TK-SVM learns the finite-temperature phase diagram in an unsupervised way. Moreover, in virtue of its strong interpretability, it identifies the tensorial quadrupolar and octupolar orders, which define a biaxialD3hspin nematic, and the local constraint that underlies the selection of coplanar states. We then discuss the disorder hierarchy of the phases, which can be inferred from both the analytical order parameters and an SVM bias parameter. For completeness we mention that the machine also picks up the leading3×3correlations in the dipolar channel at very low temperature, which are however weak compared to the quadrupolar and octupolar orders. Our work shows how TK-SVM can facilitate and speed up the analysis of classical frustrated magnets.
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Affiliation(s)
- Jonas Greitemann
- Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstr. 37, 80333 München, Germany
- Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany
| | - Ke Liu
- Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstr. 37, 80333 München, Germany
- Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany
| | - Lode Pollet
- Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstr. 37, 80333 München, Germany
- Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany
- Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China
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Xu J, Chen JZY. General liquid-crystal theory for anisotropically shaped molecules: Symmetry, orientational order parameters, and system free energy. Phys Rev E 2021; 102:062701. [PMID: 33466056 DOI: 10.1103/physreve.102.062701] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2020] [Accepted: 11/12/2020] [Indexed: 11/07/2022]
Abstract
A general theory of liquid crystals is presented, starting from the group-theory symmetry analysis of the constituting molecules. A particular attention is paid to the type of elastic free-energies and their relationships with the molecular symmetries. The orientational order-parameter tensors are identified for each molecular symmetry, in a consideration of consistently keeping the leading, characteristic elastic free energies in a model. The order parameters are expressed in terms of symmetric traceless tensors, some of high orders, for all major molecular symmetries, including seven groups of axial symmetries and seven groups of polyhedral symmetries. For spatially inhomogeneous liquid crystals, the couplings of these tensors in the elastic energies are derived by expanding the interaction energies between these molecules. The aim is to provide a general view of the molecular symmetries of individual molecules, orientational order parameters characterizing the orientational distribution functions, and the elastic free energies, all under one single group-theory approach.
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Affiliation(s)
- Jie Xu
- LSEC & NCMIS, Institute of Computational Mathematics and Scientific/Engineering Computing (ICMSEC), Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences, Beijing 100190, China
| | - Jeff Z Y Chen
- Department of Physics and Astronomy, University of Waterloo, Ontario, Canada N2L 3G1
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Ünal FN, Bouhon A, Slager RJ. Topological Euler Class as a Dynamical Observable in Optical Lattices. PHYSICAL REVIEW LETTERS 2020; 125:053601. [PMID: 32794847 DOI: 10.1103/physrevlett.125.053601] [Citation(s) in RCA: 22] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2020] [Revised: 06/21/2020] [Accepted: 07/06/2020] [Indexed: 05/22/2023]
Abstract
The last years have witnessed rapid progress in the topological characterization of out-of-equilibrium systems. We report on robust signatures of a new type of topology-the Euler class-in such a dynamical setting. The enigmatic invariant (ξ) falls outside conventional symmetry-eigenvalue indicated phases and, in simplest incarnation, is described by triples of bands that comprise a gapless pair featuring 2ξ stable band nodes, and a gapped band. These nodes host non-Abelian charges and can be further undone by converting their charge upon intricate braiding mechanisms, revealing that Euler class is a fragile topology. We theoretically demonstrate that quenching with nontrivial Euler Hamiltonian results in stable monopole-antimonopole pairs, which in turn induce a linking of momentum-time trajectories under the first Hopf map, making the invariant experimentally observable. Detailing explicit tomography protocols in a variety of cold-atom setups, our results provide a basis for exploring new topologies and their interplay with crystalline symmetries in optical lattices beyond paradigmatic Chern insulators.
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Affiliation(s)
- F Nur Ünal
- TCM Group, Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - Adrien Bouhon
- Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
- Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 21 Uppsala, Sweden
| | - Robert-Jan Slager
- TCM Group, Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
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Rey M. Group-theoretical formulation of an Eckart-frame kinetic energy operator in curvilinear coordinates for polyatomic molecules. J Chem Phys 2019; 151:024101. [DOI: 10.1063/1.5109482] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Michael Rey
- Groupe de Spectrométrie Moléculaire et Atmosphérique, UMR CNRS 7331, BP 1039, F-51687 Reims Cedex 2, France
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Liu K, Greitemann J, Pollet L. Generic first-order phase transitions between isotropic and orientational phases with polyhedral symmetries. Phys Rev E 2018; 97:012706. [PMID: 29448474 DOI: 10.1103/physreve.97.012706] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/10/2017] [Indexed: 06/08/2023]
Abstract
Polyhedral nematics are examples of exotic orientational phases that possess a complex internal symmetry, representing highly nontrivial ways of rotational symmetry breaking, and are subject to current experimental pursuits in colloidal and molecular systems. The classification of these phases has been known for a long time; however, their transitions to the disordered isotropic liquid phase remain largely unexplored, except for a few symmetries. In this work, we utilize a recently introduced non-Abelian gauge theory to explore the nature of the underlying nematic-isotropic transition for all three-dimensional polyhedral nematics. The gauge theory can readily be applied to nematic phases with an arbitrary point-group symmetry, including those where traditional Landau methods and the associated lattice models may become too involved to implement owing to a prohibitive order-parameter tensor of high rank or (the absence of) mirror symmetries. By means of exhaustive Monte Carlo simulations, we find that the nematic-isotropic transition is generically first-order for all polyhedral symmetries. Moreover, we show that this universal result is fully consistent with our expectation from a renormalization group approach, as well as with other lattice models for symmetries already studied in the literature. We argue that extreme fine tuning is required to promote those transitions to second-order ones. We also comment on the nature of phase transitions breaking the O(3) symmetry in general cases.
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Affiliation(s)
- Ke Liu
- Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstrasse 37, 80333 Munich, Germany
| | - Jonas Greitemann
- Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstrasse 37, 80333 Munich, Germany
| | - Lode Pollet
- Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstrasse 37, 80333 Munich, Germany
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Liu K, Nissinen J, de Boer J, Slager RJ, Zaanen J. Hierarchy of orientational phases and axial anisotropies in the gauge theoretical description of generalized nematic liquid crystals. Phys Rev E 2017; 95:022704. [PMID: 28297940 DOI: 10.1103/physreve.95.022704] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2016] [Indexed: 06/06/2023]
Abstract
The paradigm of spontaneous symmetry breaking encompasses the breaking of the rotational symmetries O(3) of isotropic space to a discrete subgroup, i.e., a three-dimensional point group. The subgroups form a rich hierarchy and allow for many different phases of matter with orientational order. Such spontaneous symmetry breaking occurs in nematic liquid crystals, and a highlight of such anisotropic liquids is the uniaxial and biaxial nematics. Generalizing the familiar uniaxial and biaxial nematics to phases characterized by an arbitrary point-group symmetry, referred to as generalized nematics, leads to a large hierarchy of phases and possible orientational phase transitions. We discuss how a particular class of nematic phase transitions related to axial point groups can be efficiently captured within a recently proposed gauge theoretical formulation of generalized nematics [K. Liu, J. Nissinen, R.-J. Slager, K. Wu, and J. Zaanen, Phys. Rev. X 6, 041025 (2016)2160-330810.1103/PhysRevX.6.041025]. These transitions can be introduced in the model by considering anisotropic couplings that do not break any additional symmetries. By and large this generalizes the well-known uniaxial-biaxial nematic phase transition to any arbitrary axial point group in three dimensions. We find in particular that the generalized axial transitions are distinguished by two types of phase diagrams with intermediate vestigial orientational phases and that the window of the vestigial phase is intimately related to the amount of symmetry of the defining point group due to inherently growing fluctuations of the order parameter. This might explain the stability of the observed uniaxial-biaxial phases as compared to the yet to be observed other possible forms of generalized nematic order with higher point-group symmetries.
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Affiliation(s)
- Ke Liu
- Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, PO Box 9506, NL-2300 RA Leiden, The Netherlands
| | - Jaakko Nissinen
- Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, PO Box 9506, NL-2300 RA Leiden, The Netherlands
| | - Josko de Boer
- Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, PO Box 9506, NL-2300 RA Leiden, The Netherlands
| | - Robert-Jan Slager
- Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, PO Box 9506, NL-2300 RA Leiden, The Netherlands
| | - Jan Zaanen
- Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, PO Box 9506, NL-2300 RA Leiden, The Netherlands
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Romano S. Computer simulation study of a mesogenic lattice model based on long-range dispersion interactions. Phys Rev E 2016; 94:042702. [PMID: 27841513 DOI: 10.1103/physreve.94.042702] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2016] [Indexed: 06/06/2023]
Abstract
In contrast to thermotropic biaxial nematic phases, for which some long sought for experimental realizations have been obtained, no experimental realizations are yet known for their tetrahedratic and cubatic counterparts, involving orientational orders of ranks 3 and 4, respectively, also studied theoretically over the last few decades. In previous studies, cubatic order has been found for hard-core or continuous models consisting of particles possessing cubic or nearly cubic tetragonal or orthorhombic symmetries; in a few cases, hard-core models involving uniaxial (D_{∞h}-symmetric) particles have been claimed to produce cubatic order as well. Here we address by Monte Carlo simulation a lattice model consisting of uniaxial particles coupled by long-range dispersion interactions of the London-De Boer-Heller type; the model was found to produce no second-rank nematic but only fourth-rank cubatic order, in contrast to the nematic behavior long known for its counterpart with interactions truncated at nearest-neighbor separation.
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Affiliation(s)
- Silvano Romano
- Physics Department, University of Pavia, via A. Bassi 6, 27100 Pavia, Italy
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