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Lukin IV, Sotnikov AG. Corner transfer matrix renormalization group approach in the zoo of Archimedean lattices. Phys Rev E 2024; 109:045305. [PMID: 38755853 DOI: 10.1103/physreve.109.045305] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2024] [Accepted: 03/22/2024] [Indexed: 05/18/2024]
Abstract
We develop a new methodology to contract tensor networks within the corner transfer matrix renormalization group approach for a wide range of two-dimensional lattice geometries. We discuss contraction algorithms on the example of triangular, kagome, honeycomb, square-octagon, star, ruby, square-hexagon-dodecahedron, and dice lattices. As benchmark tests, we apply the developed method to the classical Ising model on different lattices and observe a remarkable agreement of the results with the available from the literature. The approach also shows the necessary potential to be applied to various quantum lattice models in a combination with the wave-function variational optimization schemes.
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Affiliation(s)
- I V Lukin
- Karazin Kharkiv National University, Svobody Square 4, 61022 Kharkiv, Ukraine and Akhiezer Institute for Theoretical Physics, NSC KIPT, Akademichna 1, 61108 Kharkiv, Ukraine
| | - A G Sotnikov
- Karazin Kharkiv National University, Svobody Square 4, 61022 Kharkiv, Ukraine and Akhiezer Institute for Theoretical Physics, NSC KIPT, Akademichna 1, 61108 Kharkiv, Ukraine
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Makarov D, Volkov OM, Kákay A, Pylypovskyi OV, Budinská B, Dobrovolskiy OV. New Dimension in Magnetism and Superconductivity: 3D and Curvilinear Nanoarchitectures. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2022; 34:e2101758. [PMID: 34705309 DOI: 10.1002/adma.202101758] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2021] [Revised: 05/16/2021] [Indexed: 06/13/2023]
Abstract
Traditionally, the primary field, where curvature has been at the heart of research, is the theory of general relativity. In recent studies, however, the impact of curvilinear geometry enters various disciplines, ranging from solid-state physics over soft-matter physics, chemistry, and biology to mathematics, giving rise to a plethora of emerging domains such as curvilinear nematics, curvilinear studies of cell biology, curvilinear semiconductors, superfluidity, optics, 2D van der Waals materials, plasmonics, magnetism, and superconductivity. Here, the state of the art is summarized and prospects for future research in curvilinear solid-state systems exhibiting such fundamental cooperative phenomena as ferromagnetism, antiferromagnetism, and superconductivity are outlined. Highlighting the recent developments and current challenges in theory, fabrication, and characterization of curvilinear micro- and nanostructures, special attention is paid to perspective research directions entailing new physics and to their strong application potential. Overall, the perspective is aimed at crossing the boundaries between the magnetism and superconductivity communities and drawing attention to the conceptual aspects of how extension of structures into the third dimension and curvilinear geometry can modify existing and aid launching novel functionalities. In addition, the perspective should stimulate the development and dissemination of research and development oriented techniques to facilitate rapid transitions from laboratory demonstrations to industry-ready prototypes and eventual products.
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Affiliation(s)
- Denys Makarov
- Helmholtz-Zentrum Dresden - Rossendorf e.V., Institute of Ion Beam Physics and Materials Research, 01328, Dresden, Germany
| | - Oleksii M Volkov
- Helmholtz-Zentrum Dresden - Rossendorf e.V., Institute of Ion Beam Physics and Materials Research, 01328, Dresden, Germany
| | - Attila Kákay
- Helmholtz-Zentrum Dresden - Rossendorf e.V., Institute of Ion Beam Physics and Materials Research, 01328, Dresden, Germany
| | - Oleksandr V Pylypovskyi
- Helmholtz-Zentrum Dresden - Rossendorf e.V., Institute of Ion Beam Physics and Materials Research, 01328, Dresden, Germany
- Kyiv Academic University, Kyiv, 03142, Ukraine
| | - Barbora Budinská
- Superconductivity and Spintronics Laboratory, Nanomagnetism and Magnonics, Faculty of Physics, University of Vienna, Vienna, 1090, Austria
| | - Oleksandr V Dobrovolskiy
- Superconductivity and Spintronics Laboratory, Nanomagnetism and Magnonics, Faculty of Physics, University of Vienna, Vienna, 1090, Austria
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Lopez JH, Schwarz JM. Constraint percolation on hyperbolic lattices. Phys Rev E 2018; 96:052108. [PMID: 29347694 DOI: 10.1103/physreve.96.052108] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2015] [Indexed: 11/07/2022]
Abstract
Hyperbolic lattices interpolate between finite-dimensional lattices and Bethe lattices, and they are interesting in their own right, with ordinary percolation exhibiting not one but two phase transitions. We study four constraint percolation models-k-core percolation (for k=1,2,3) and force-balance percolation-on several tessellations of the hyperbolic plane. By comparing these four different models, our numerical data suggest that all of the k-core models, even for k=3, exhibit behavior similar to ordinary percolation, while the force-balance percolation transition is discontinuous. We also provide proof, for some hyperbolic lattices, of the existence of a critical probability that is less than unity for the force-balance model, so that we can place our interpretation of the numerical data for this model on a more rigorous footing. Finally, we discuss improved numerical methods for determining the two critical probabilities on the hyperbolic lattice for the k-core percolation models.
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Affiliation(s)
- Jorge H Lopez
- Department of Civil Engineering, Universidad Mariana, Pasto 520002, Colombia
| | - J M Schwarz
- Department of Physics, Syracuse University, Syracuse, New York 13244, USA.,Syracuse Biomaterials Institute, Syracuse, New York 13244, USA
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Brandão R, Miranda JA. Viscous fluid fingering on a negatively curved surface. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:013018. [PMID: 26274280 DOI: 10.1103/physreve.92.013018] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2015] [Indexed: 06/04/2023]
Abstract
Viscous fingering formation in flat Hele-Shaw cells is a classical and widely studied fluid mechanical problem. We examine the development of viscous fluid fingering on a two-dimensional surface of constant negative Gaussian curvature, the hyperbolic plane H(2). A perturbative mode-coupling formalism is applied to study the influence of the negative surface curvature on the two most important pattern formation mechanisms of the system: fingertip splitting and finger competition. We also report on a time-dependent control strategy placed on the injection rate, which is able to minimize viscous fingering growth on H(2).
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Affiliation(s)
- Rodolfo Brandão
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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Gendiar A, Daniška M, Krčmár R, Nishino T. Mean-field universality class induced by weak hyperbolic curvatures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:012122. [PMID: 25122266 DOI: 10.1103/physreve.90.012122] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2014] [Indexed: 06/03/2023]
Abstract
Order-disorder phase transition of the ferromagnetic Ising model is investigated on a series of two-dimensional lattices that have negative Gaussian curvatures. Exceptional lattice sites of coordination number seven are distributed on the triangular lattice, where the typical distance between the nearest exceptional sites is proportional to an integer parameter n. Thus, the corresponding curvature is asymptotically proportional to -n(-2). Spontaneous magnetization and specific heat are calculated by means of the corner transfer matrix renormalization group method. For all the finite n cases, we observe the mean-field-like phase transition. It is confirmed that the entanglement entropy at the transition temperature is linear in (c/6)ln n, where c = 1/2 is the central charge of the Ising model. The fact agrees with the presence of the typical length scale n being proportional to the curvature radius.
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Affiliation(s)
- Andrej Gendiar
- Institute of Physics, Slovak Academy of Sciences, SK-845 11, Bratislava, Slovakia
| | - Michal Daniška
- Institute of Physics, Slovak Academy of Sciences, SK-845 11, Bratislava, Slovakia
| | - Roman Krčmár
- Institute of Physics, Slovak Academy of Sciences, SK-845 11, Bratislava, Slovakia and Department of Physics, Graduate School of Science, Kobe University, Kobe 657-8501, Japan
| | - Tomotoshi Nishino
- Department of Physics, Graduate School of Science, Kobe University, Kobe 657-8501, Japan
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Gendiar A, Krcmar R, Andergassen S, Daniška M, Nishino T. Weak correlation effects in the Ising model on triangular-tiled hyperbolic lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:021105. [PMID: 23005721 DOI: 10.1103/physreve.86.021105] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/16/2012] [Indexed: 06/01/2023]
Abstract
The Ising model is studied on a series of hyperbolic two-dimensional lattices which are formed by tessellation of triangles on negatively curved surfaces. In order to treat the hyperbolic lattices, we propose a generalization of the corner transfer matrix renormalization group method using a recursive construction of asymmetric transfer matrices. Studying the phase transition, the mean-field universality is captured by means of a precise analysis of thermodynamic functions. The correlation functions and the density-matrix spectra always decay exponentially even at the transition point, whereas power-law behavior characterizes criticality on the Euclidean flat geometry. We confirm the absence of a finite correlation length in the limit of infinite negative Gaussian curvature.
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Affiliation(s)
- Andrej Gendiar
- Institute of Physics, Slovak Academy of Sciences, SK-845 11 Bratislava, Slovakia
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Wu ZX, Holme P. Majority-vote model on hyperbolic lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:011133. [PMID: 20365349 DOI: 10.1103/physreve.81.011133] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/01/2009] [Indexed: 05/29/2023]
Abstract
We study the critical properties of a nonequilibrium statistical model, the majority-vote model, on heptagonal and dual heptagonal lattices. Such lattices have the special feature that they only can be embedded in negatively curved surfaces. We find, by using Monte Carlo simulations and finite-size analysis, that the critical exponents 1/nu , beta/nu , and gamma/nu are different from those of the majority-vote model on regular lattices with periodic boundary condition, which belongs to the same universality class as the equilibrium Ising model. The exponents are also from those of the Ising model on a hyperbolic lattice. We argue that the disagreement is caused by the effective dimensionality of the hyperbolic lattices. By comparative studies, we find that the critical exponents of the majority-vote model on hyperbolic lattices satisfy the hyperscaling relation 2beta/nu+gamma/nu=D(eff), where D(eff) is an effective dimension of the lattice. We also investigate the effect of boundary nodes on the ordering process of the model.
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Affiliation(s)
- Zhi-Xi Wu
- Department of Physics, Umeå University, Umeå, Sweden.
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Sakaniwa Y, Shima H. Survival of short-range order in the Ising model on negatively curved surfaces. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:021103. [PMID: 19792073 DOI: 10.1103/physreve.80.021103] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2009] [Indexed: 05/28/2023]
Abstract
We examine the ordering behavior of the ferromagnetic Ising lattice model defined on a surface with a constant negative curvature. Small-sized ferromagnetic domains are observed to exist at temperatures far greater than the critical temperature, at which the inner-core region of the lattice undergoes a mean-field phase transition. The survival of short-range order at such high temperatures can be attributed to strong boundary-spin contributions to the ordering mechanism as a result of which boundary effects remain active even within the thermodynamic limit. Our results are consistent with the previous finding of disorder-free Griffiths phase that is stable at temperatures lower than the mean-field critical temperature.
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Affiliation(s)
- Yasunori Sakaniwa
- Department of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan
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Baek SK, Minnhagen P, Shima H, Kim BJ. Phase transition of q-state clock models on heptagonal lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:011133. [PMID: 19658679 DOI: 10.1103/physreve.80.011133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2009] [Indexed: 05/28/2023]
Abstract
We study the q-state clock models on heptagonal lattices assigned on a negatively curved surface. We show that the system exhibits three classes of equilibrium phases; in between ordered and disordered phases, an intermediate phase characterized by a diverging susceptibility with no magnetic order is observed at every q>or=2. The persistence of the third phase for all q is in contrast with the disappearance of the counterpart phase in a planar system for small q, which indicates the significance of nonvanishing surface-volume ratio that is peculiar in the heptagonal lattice. Analytic arguments based on Ginzburg-Landau theory and generalized Cayley trees make clear that the two-stage transition in the present system is attributed to an energy gap of spin-wave excitations and strong boundary-spin contributions. We further demonstrate that boundary effects break the mean-field character in the bulk region, which establishes the consistency with results of clock models on boundary-free hyperbolic lattices.
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Affiliation(s)
- Seung Ki Baek
- Department of Physics, Umeå University, 901 87 Umeå, Sweden
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