The Experts below are selected from a list of 303 Experts worldwide ranked by ideXlab platform
Pierre Pujol - One of the best experts on this subject based on the ideXlab platform.
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from quantum mechanics to classical statistical physics generalized rokhsar kivelson hamiltonians and the Stochastic Matrix form decomposition
Annals of Physics, 2005Co-Authors: Claudio Castelnovo, Christopher Mudry, Claudio Chamon, Pierre PujolAbstract:Abstract Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar–Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with Stochastic classical systems described by a Master equation of the Matrix type, hence their name. It then follows that the equilibrium partition function of the Stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the Stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct Stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.
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From quantum mechanics to classical statistical physics: Generalized Rokhsar–Kivelson Hamiltonians and the “Stochastic Matrix Form” decomposition
Annals of Physics, 2005Co-Authors: Claudio Castelnovo, Christopher Mudry, Claudio Chamon, Pierre PujolAbstract:Abstract Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar–Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with Stochastic classical systems described by a Master equation of the Matrix type, hence their name. It then follows that the equilibrium partition function of the Stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the Stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct Stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.
Claudio Castelnovo - One of the best experts on this subject based on the ideXlab platform.
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from quantum mechanics to classical statistical physics generalized rokhsar kivelson hamiltonians and the Stochastic Matrix form decomposition
Annals of Physics, 2005Co-Authors: Claudio Castelnovo, Christopher Mudry, Claudio Chamon, Pierre PujolAbstract:Abstract Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar–Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with Stochastic classical systems described by a Master equation of the Matrix type, hence their name. It then follows that the equilibrium partition function of the Stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the Stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct Stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.
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From quantum mechanics to classical statistical physics: Generalized Rokhsar–Kivelson Hamiltonians and the “Stochastic Matrix Form” decomposition
Annals of Physics, 2005Co-Authors: Claudio Castelnovo, Christopher Mudry, Claudio Chamon, Pierre PujolAbstract:Abstract Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar–Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with Stochastic classical systems described by a Master equation of the Matrix type, hence their name. It then follows that the equilibrium partition function of the Stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the Stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct Stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.
Claudio Chamon - One of the best experts on this subject based on the ideXlab platform.
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from quantum mechanics to classical statistical physics generalized rokhsar kivelson hamiltonians and the Stochastic Matrix form decomposition
Annals of Physics, 2005Co-Authors: Claudio Castelnovo, Christopher Mudry, Claudio Chamon, Pierre PujolAbstract:Abstract Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar–Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with Stochastic classical systems described by a Master equation of the Matrix type, hence their name. It then follows that the equilibrium partition function of the Stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the Stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct Stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.
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From quantum mechanics to classical statistical physics: Generalized Rokhsar–Kivelson Hamiltonians and the “Stochastic Matrix Form” decomposition
Annals of Physics, 2005Co-Authors: Claudio Castelnovo, Christopher Mudry, Claudio Chamon, Pierre PujolAbstract:Abstract Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar–Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with Stochastic classical systems described by a Master equation of the Matrix type, hence their name. It then follows that the equilibrium partition function of the Stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the Stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct Stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.
Christopher Mudry - One of the best experts on this subject based on the ideXlab platform.
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from quantum mechanics to classical statistical physics generalized rokhsar kivelson hamiltonians and the Stochastic Matrix form decomposition
Annals of Physics, 2005Co-Authors: Claudio Castelnovo, Christopher Mudry, Claudio Chamon, Pierre PujolAbstract:Abstract Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar–Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with Stochastic classical systems described by a Master equation of the Matrix type, hence their name. It then follows that the equilibrium partition function of the Stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the Stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct Stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.
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From quantum mechanics to classical statistical physics: Generalized Rokhsar–Kivelson Hamiltonians and the “Stochastic Matrix Form” decomposition
Annals of Physics, 2005Co-Authors: Claudio Castelnovo, Christopher Mudry, Claudio Chamon, Pierre PujolAbstract:Abstract Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar–Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with Stochastic classical systems described by a Master equation of the Matrix type, hence their name. It then follows that the equilibrium partition function of the Stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the Stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct Stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.
Heng Huang - One of the best experts on this subject based on the ideXlab platform.
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structured doubly Stochastic Matrix for graph based clustering structured doubly Stochastic Matrix
Knowledge Discovery and Data Mining, 2016Co-Authors: Xiaoqian Wang, Feiping Nie, Heng HuangAbstract:As one of the most significant machine learning topics, clustering has been extensively employed in various kinds of area. Its prevalent application in scientific research as well as industrial practice has drawn high attention in this day and age. A multitude of clustering methods have been developed, among which the graph based clustering method using the affinity Matrix has been laid great emphasis on. Recent research work used the doubly Stochastic Matrix to normalize the input affinity Matrix and enhance the graph based clustering models. Although the doubly Stochastic Matrix can improve the clustering performance, the clustering structure in the doubly Stochastic Matrix is not clear as expected. Thus, post processing step is required to extract the final clustering results, which may not be optimal. To address this problem, in this paper, we propose a novel convex model to learn the structured doubly Stochastic Matrix by imposing low-rank constraint on the graph Laplacian Matrix. Our new structured doubly Stochastic Matrix can explicitly uncover the clustering structure and encode the probabilities of pair-wise data points to be connected, such that the clustering results are enhanced. An efficient optimization algorithm is derived to solve our new objective. Also, we provide theoretical discussions that when the input differs, our method possesses interesting connections with K-means and spectral graph cut models respectively. We conduct experiments on both synthetic and benchmark datasets to validate the performance of our proposed method. The empirical results demonstrate that our model provides an approach to better solving the K-mean clustering problem. By using the cluster indicator provided by our model as initialization, K-means converges to a smaller objective function value with better clustering performance. Moreover, we compare the clustering performance of our model with spectral clustering and related double Stochastic model. On all datasets, our method performs equally or better than the related methods.
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KDD - Structured Doubly Stochastic Matrix for Graph Based Clustering: Structured Doubly Stochastic Matrix
Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2016Co-Authors: Xiaoqian Wang, Feiping Nie, Heng HuangAbstract:As one of the most significant machine learning topics, clustering has been extensively employed in various kinds of area. Its prevalent application in scientific research as well as industrial practice has drawn high attention in this day and age. A multitude of clustering methods have been developed, among which the graph based clustering method using the affinity Matrix has been laid great emphasis on. Recent research work used the doubly Stochastic Matrix to normalize the input affinity Matrix and enhance the graph based clustering models. Although the doubly Stochastic Matrix can improve the clustering performance, the clustering structure in the doubly Stochastic Matrix is not clear as expected. Thus, post processing step is required to extract the final clustering results, which may not be optimal. To address this problem, in this paper, we propose a novel convex model to learn the structured doubly Stochastic Matrix by imposing low-rank constraint on the graph Laplacian Matrix. Our new structured doubly Stochastic Matrix can explicitly uncover the clustering structure and encode the probabilities of pair-wise data points to be connected, such that the clustering results are enhanced. An efficient optimization algorithm is derived to solve our new objective. Also, we provide theoretical discussions that when the input differs, our method possesses interesting connections with K-means and spectral graph cut models respectively. We conduct experiments on both synthetic and benchmark datasets to validate the performance of our proposed method. The empirical results demonstrate that our model provides an approach to better solving the K-mean clustering problem. By using the cluster indicator provided by our model as initialization, K-means converges to a smaller objective function value with better clustering performance. Moreover, we compare the clustering performance of our model with spectral clustering and related double Stochastic model. On all datasets, our method performs equally or better than the related methods.
