The Experts below are selected from a list of 51405 Experts worldwide ranked by ideXlab platform
Frank Verstraete - One of the best experts on this subject based on the ideXlab platform.
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matrix Product Operator symmetries and intertwiners in string nets with domain walls
arXiv: Quantum Physics, 2020Co-Authors: Laurens Lootens, Jutho Haegeman, Jurgen Fuchs, Christoph Schweigert, Frank VerstraeteAbstract:We provide a description of virtual non-local matrix Product Operator (MPO) symmetries in projected entangled pair state (PEPS) representations of string-net models. Given such a PEPS representation, we show that the consistency conditions of its MPO symmetries amount to a set of six coupled equations that can be identified with the pentagon equations of a bimodule category. This allows us to classify all equivalent PEPS representations and build MPO intertwiners between them, synthesising and generalising the wide variety of tensor network representations of topological phases. Furthermore, we use this generalisation to build explicit PEPS realisations of domain walls between different topological phases as constructed by Kitaev and Kong [Commun. Math. Phys. 313 (2012) 351-373]. While the prevailing abstract categorical approach is sufficient to describe the structure of topological phases, explicit tensor network representations are required to simulate these systems on a computer, such as needed for calculating thresholds of quantum error-correcting codes based on string-nets with boundaries. Finally, we show that all these string-net PEPS representations can be understood as specific instances of Turaev-Viro state-sum models of topological field theory on three-manifolds with a physical boundary, thereby putting these tensor network constructions on a mathematically rigorous footing.
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galois conjugated tensor fusion categories and nonunitary conformal field theory
Physical Review Letters, 2020Co-Authors: Laurens Lootens, Jutho Haegeman, Robijn Vanhove, Frank VerstraeteAbstract:We provide a generalization of the matrix Product Operator formalism for string-net projected entangled pair states (PEPS) to include nonunitary solutions of the pentagon equation. These states provide the explicit lattice realization of the Galois conjugated counterparts of (2+1)-dimensional topological quantum field theories, based on tensor fusion categories. Although the parent Hamiltonians of these renormalization group fixed point states are gapless, these states can still be the topological ground states of a gapped non-Hermitian Hamiltonian. We show by example that the topological sectors of the Yang-Lee theory (the nonunitary counterpart of the Fibonacci fusion category) can be constructed, even in the absence of closure under Hermitian conjugation of the basis elements of the Ocneanu tube algebra. The topological sector construction is demonstrated by applying the concept of strange correlators to the Yang-Lee model, giving rise to a nonunitary version of the classical hard hexagon model in the Yang-Lee universality class and obtaining all generalized twisted boundary conditions on a finite cylinder of the Yang-Lee edge singularity. Finally, we construct the PEPS transfer matrix and show that taking the Hermitian conjugate changes the topological phase for these nonunitary string-net models.
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diagonalizing transfer matrices and matrix Product Operators a medley of exact and computational methods
Annual Review of Condensed Matter Physics, 2017Co-Authors: Jutho Haegeman, Frank VerstraeteAbstract:Transfer matrices and matrix Product Operators play a ubiquitous role in the field of many-body physics. This review gives an idiosyncratic overview of applications, exact results, and computational aspects of diagonalizing transfer matrices and matrix Product Operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and new results. Topics discussed are exact solutions of transfer matrices in equilibrium and nonequilibrium statistical physics, tensor network states, matrix Product Operator algebras, and numerical matrix Product state methods for finding extremal eigenvectors of matrix Product Operators.
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anyons and matrix Product Operator algebras
Annals of Physics, 2017Co-Authors: Nick Bultinck, Jutho Haegeman, Frank Verstraete, Dominic J. Williamson, Mehmet B. Şahinoğlu, Michaël MariënAbstract:Abstract Quantum tensor network states and more particularly projected entangled-pair states provide a natural framework for representing ground states of gapped, topologically ordered systems. The defining feature of these representations is that topological order is a consequence of the symmetry of the underlying tensors in terms of matrix Product Operators. In this paper, we present a systematic study of those matrix Product Operators, and show how this relates entanglement properties of projected entangled-pair states to the formalism of fusion tensor categories. From the matrix Product Operators we construct a C ∗ -algebra and find that topological sectors can be identified with the central idempotents of this algebra. This allows us to construct projected entangled-pair states containing an arbitrary number of anyons. Properties such as topological spin, the S matrix, fusion and braiding relations can readily be extracted from the idempotents. As the matrix Product Operator symmetries are acting purely on the virtual level of the tensor network, the ensuing Wilson loops are not fattened when perturbing the system, and this opens up the possibility of simulating topological theories away from renormalization group fixed points. We illustrate the general formalism for the special cases of discrete gauge theories and string-net models.
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matrix Product Operator representations
New Journal of Physics, 2010Co-Authors: B Pirvu, Valentin Murg, J I Cirac, Frank VerstraeteAbstract:We show how to construct relevant families of matrix Product Operators (MPOs) in one and higher dimensions. These form the building blocks for the numerical simulation methods based on matrix Product states and projected entangled pair states. In particular, we construct translationally invariant MPOs suitable for time evolution, and show how such descriptions are possible for Hamiltonians with long-range interactions. We show how these tools can be exploited for constructing new algorithms for simulating quantum spin systems.
Garnet Kin-lic Chan - One of the best experts on this subject based on the ideXlab platform.
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conversion of projected entangled pair states into a canonical form
Physical Review B, 2019Co-Authors: R Haghshenas, Matthew J Orourke, Garnet Kin-lic ChanAbstract:We propose an algorithm to convert a projected entangled pair state (PEPS) into a canonical form, analogous to the well-known canonical form of a matrix Product state. Our approach is based on a variational gauging ansatz for the QR tensor decomposition of PEPS columns into a matrix Product Operator and a finite depth circuit of unitaries and isometries. We describe a practical initialization scheme that leads to rapid convergence in the QR optimization. We explore the performance and stability of the variational gauging algorithm in norm calculations for the transverse-field Ising and Heisenberg models on a square lattice. We also demonstrate energy optimization within the PEPS canonical form for the transverse-field Ising and Heisenberg models. We expect this canonical form to open up improved analytical and numerical approaches for PEPS.
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Matrix Product Operators, matrix Product states, and ab initio density matrix renormalization group algorithms
Journal of Chemical Physics, 2016Co-Authors: Garnet Kin-lic Chan, Anna Keselman, Naoki Nakatani, Zhendong Li, Steven R. WhiteAbstract:Current descriptions of the ab initio density matrix renormalization group (DMRG) algorithm use two superficially different languages: an older language of the renormalization group and renormalized Operators, and a more recent language of matrix Product states and matrix Product Operators. The same algorithm can appear dramatically different when written in the two different vocabularies. In this work, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the ab initio DMRG sweep using a matrix Product Operator based code, and the equivalence to the original renormalized Operator implementation. Next we describe how to implement the general matrix Product Operator/matrix Product state algebra within a pure renormalized Operator-based DMRG code. Finally, we discuss two improvements of the ab initio DMRG sweep algorithm motivated by matrix Product Operator language: Hamiltonian compression, and a sum over Operators representation that allows for perfect computational parallelism. The connections and correspondences described here serve to link the future developments with the past and are important in the efficient implementation of continuing advances in ab initio DMRG and related algorithms.
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Matrix Product Operators, Matrix Product States, and ab initio Density Matrix Renormalization Group algorithms
arXiv: Chemical Physics, 2016Co-Authors: Garnet Kin-lic Chan, Anna Keselman, Naoki Nakatani, Zhendong Li, Steven R. WhiteAbstract:Current descriptions of the ab initio DMRG algorithm use two superficially different languages: an older language of the renormalization group and renormalized Operators, and a more recent language of matrix Product states and matrix Product Operators. The same algorithm can appear dramatically different when written in the two different vocabularies. In this work, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the ab-initio DMRG sweep using a matrix Product Operator based code, and the equivalence to the original renormalized Operator implementation. Next we describe how to implement the general matrix Product Operator/matrix Product state algebra within a pure renormalized Operator-based DMRG code. Finally, we discuss two improvements of the ab initio DMRG sweep algorithm motivated by matrix Product Operator language: Hamiltonian compression, and a sum over Operators representation that allows for perfect computational parallelism. The connections and correspondences described here serve to link the future developments with the past, and are important in the efficient implementation of continuing advances in ab initio DMRG and related algorithms.
Semiha Bahceli - One of the best experts on this subject based on the ideXlab platform.
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Product Operator descriptions of the 2d dept j resolved nmr experiment for weakly coupled isn n 1 2 3 spin systems
Physica Scripta, 2010Co-Authors: Ahmet Tokatli, Semiha BahceliAbstract:There are a variety of multi-pulse nuclear magnetic resonance (NMR) experiments for spectral assignment of complex molecules in a solution. The two-dimensional (2D) distortionless enhancement by polarization transfer (DEPT) J-resolved NMR experiment is a 13C-detected, spectral editing polarization transfer technique. The Product Operator theory is widely used for an analytical description of the multi-pulse NMR experiment for weakly coupled spin systems. In this study, analytical descriptions of the 2D DEPT J-resolved NMR experiment for weakly coupled ISn (, ; n=1, 2, 3) spin systems using the Product Operator theory have been introduced for the first time. The calculated intensities and positions of the observable signals are simulated for molecules containing [13C , 81Br )] nuclei by using a MAPLE program on a computer. Finally, we present a theoretical discussion and experimental suggestions.
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Product Operator descriptions of inept and rinept nmr spectroscopies for isn i 1 2 s 3 2 spin systems
Journal of Magnetic Resonance, 2004Co-Authors: Ahmet Tokatli, Azmi Gencten, Ozden Tezel, Mukerrem şahin, Semiha BahceliAbstract:Abstract The Product Operator descriptions of INEPT and reverse INEPT (RINEPT) NMR experiments are introduced for weakly coupled ISn (I=1/2, S=3/2 with n=1,2,3) spin systems. Explicit expressions for polarization transfer from spin-3/2 quadrupolar nuclei to spin-1/2 nuclei (and reversed polarization transfer) are given in detail by using the evolutions of Product Operators under the spin–spin coupling Hamiltonian. The results calculated for the intensities and positions of the observable signals are simulated in the molecules containning the 119Sn (I=1/2) and 35Cl (S=3/2) nuclei at the coupling constant of JSn–Cl=375 Hz by using the Maple programme on computer.
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Product Operator theory for spin 5 2 nuclei application to 2d j resolved nmr spectroscopy
Spectroscopy Letters, 2003Co-Authors: Ahmet Tokatli, Azmi Gencten, Semiha BahceliAbstract:Product Operator theory is a simple quantum mechanical method that has often been used to analytically describe multi‐pulse NMR experiments for weakly coupled spin systems. Considering the existence of 2D‐J resolved NMR spectra of aqueous solutions containing S = 5/2 nuclear spins, the Product Operator formalism has been extended to the weakly coupled IS (I = 1/2, S = 5/2) spin system. The evolution of Ix, Iy, IxSz and IySz Product Operators under spin–spin coupling Hamiltonian are given here. The analytical results obtained are applied to the well‐known gated decoupler pulse sequence for heteronuclear 2D‐J resolved NMR spectroscopy.
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Product Operator theory for spin 3 2 nuclei and application to 2d j resolved nmr spectroscopy
Chemical Physics Letters, 2002Co-Authors: Azmi Gencten, Ozden Tezel, Semiha BahceliAbstract:Abstract The detailed description of the Product Operator formalism for a weakly coupled IS (I=1/2, S=3/2) spin system has been presented and the shorthand notations for the evolutions of Ix, Iy, IxSz and IySz Product Operators under the spin–spin coupling Hamiltonian have been obtained in a different form. Furthermore, as an application and a verification, the Product Operator formalism has been used for the analytical description of a 2D J-resolved nuclear magnetic resonance (NMR) experiment for IS (I=1/2, S=3/2) spin systems.
Steven R. White - One of the best experts on this subject based on the ideXlab platform.
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the itensor software library for tensor network calculations
arXiv: Mathematical Software, 2020Co-Authors: Matthew P Fishman, Steven R. White, Miles E StoudenmireAbstract:ITensor is a system for programming tensor network calculations with an interface modeled on tensor diagram notation, which allows users to focus on the connectivity of a tensor network without manually bookkeeping tensor indices. The ITensor interface rules out common programming errors and enables rapid prototyping of tensor network algorithms. After discussing the philosophy behind the ITensor approach, we show examples of each part of the interface including Index objects, the ITensor Product Operator, tensor factorizations, tensor storage types, algorithms for matrix Product state (MPS) and matrix Product Operator (MPO) tensor networks, quantum number conserving block-sparse tensors, and the NDTensors library. We also review publications that have used ITensor for quantum many-body physics and for other areas where tensor networks are increasingly applied. To conclude we discuss promising features and optimizations to be added in the future.
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Matrix Product Operators, matrix Product states, and ab initio density matrix renormalization group algorithms
Journal of Chemical Physics, 2016Co-Authors: Garnet Kin-lic Chan, Anna Keselman, Naoki Nakatani, Zhendong Li, Steven R. WhiteAbstract:Current descriptions of the ab initio density matrix renormalization group (DMRG) algorithm use two superficially different languages: an older language of the renormalization group and renormalized Operators, and a more recent language of matrix Product states and matrix Product Operators. The same algorithm can appear dramatically different when written in the two different vocabularies. In this work, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the ab initio DMRG sweep using a matrix Product Operator based code, and the equivalence to the original renormalized Operator implementation. Next we describe how to implement the general matrix Product Operator/matrix Product state algebra within a pure renormalized Operator-based DMRG code. Finally, we discuss two improvements of the ab initio DMRG sweep algorithm motivated by matrix Product Operator language: Hamiltonian compression, and a sum over Operators representation that allows for perfect computational parallelism. The connections and correspondences described here serve to link the future developments with the past and are important in the efficient implementation of continuing advances in ab initio DMRG and related algorithms.
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Matrix Product Operators, Matrix Product States, and ab initio Density Matrix Renormalization Group algorithms
arXiv: Chemical Physics, 2016Co-Authors: Garnet Kin-lic Chan, Anna Keselman, Naoki Nakatani, Zhendong Li, Steven R. WhiteAbstract:Current descriptions of the ab initio DMRG algorithm use two superficially different languages: an older language of the renormalization group and renormalized Operators, and a more recent language of matrix Product states and matrix Product Operators. The same algorithm can appear dramatically different when written in the two different vocabularies. In this work, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the ab-initio DMRG sweep using a matrix Product Operator based code, and the equivalence to the original renormalized Operator implementation. Next we describe how to implement the general matrix Product Operator/matrix Product state algebra within a pure renormalized Operator-based DMRG code. Finally, we discuss two improvements of the ab initio DMRG sweep algorithm motivated by matrix Product Operator language: Hamiltonian compression, and a sum over Operators representation that allows for perfect computational parallelism. The connections and correspondences described here serve to link the future developments with the past, and are important in the efficient implementation of continuing advances in ab initio DMRG and related algorithms.
Azmi Gencten - One of the best experts on this subject based on the ideXlab platform.
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a complete Product Operator theory for is i s 1 spin system and application to dept hmqc nmr experiment
Molecular Physics, 2006Co-Authors: Azmi Gencten, I şakaAbstract:There exist a variety of multi-pulse NMR experiments for spectral assignment of complex molecules in solution. The DEPT−HMQC NMR experiment is a combination of DEPT and HMQC experiments. The Product Operator theory is widely used for analytical description of multi-pulse NMR experiment for weakly coupled spin systems. In this study, a complete Product Operator theory for the IS (I = ½, S = 1) spin system is presented by obtaining the evolutions of some Product Operators under the spin–spin coupling Hamiltonian and the evolutions of some angular momentum Operators under the chemical shift and radio frequency (r.f.) pulse Hamiltonians. As an application, Product Operator theory has been used for the analytical description of DEPT−HMQC NMR experiment for CDn groups. Theoretical discussions and experimental suggestions for the sub-spectral editing of CDn groups are presented for this experiment.
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Product Operator descriptions of inept and rinept nmr spectroscopies for isn i 1 2 s 3 2 spin systems
Journal of Magnetic Resonance, 2004Co-Authors: Ahmet Tokatli, Azmi Gencten, Ozden Tezel, Mukerrem şahin, Semiha BahceliAbstract:Abstract The Product Operator descriptions of INEPT and reverse INEPT (RINEPT) NMR experiments are introduced for weakly coupled ISn (I=1/2, S=3/2 with n=1,2,3) spin systems. Explicit expressions for polarization transfer from spin-3/2 quadrupolar nuclei to spin-1/2 nuclei (and reversed polarization transfer) are given in detail by using the evolutions of Product Operators under the spin–spin coupling Hamiltonian. The results calculated for the intensities and positions of the observable signals are simulated in the molecules containning the 119Sn (I=1/2) and 35Cl (S=3/2) nuclei at the coupling constant of JSn–Cl=375 Hz by using the Maple programme on computer.
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Product Operator theory for spin 5 2 nuclei application to 2d j resolved nmr spectroscopy
Spectroscopy Letters, 2003Co-Authors: Ahmet Tokatli, Azmi Gencten, Semiha BahceliAbstract:Product Operator theory is a simple quantum mechanical method that has often been used to analytically describe multi‐pulse NMR experiments for weakly coupled spin systems. Considering the existence of 2D‐J resolved NMR spectra of aqueous solutions containing S = 5/2 nuclear spins, the Product Operator formalism has been extended to the weakly coupled IS (I = 1/2, S = 5/2) spin system. The evolution of Ix, Iy, IxSz and IySz Product Operators under spin–spin coupling Hamiltonian are given here. The analytical results obtained are applied to the well‐known gated decoupler pulse sequence for heteronuclear 2D‐J resolved NMR spectroscopy.
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Product Operator theory for spin 3 2 nuclei and application to 2d j resolved nmr spectroscopy
Chemical Physics Letters, 2002Co-Authors: Azmi Gencten, Ozden Tezel, Semiha BahceliAbstract:Abstract The detailed description of the Product Operator formalism for a weakly coupled IS (I=1/2, S=3/2) spin system has been presented and the shorthand notations for the evolutions of Ix, Iy, IxSz and IySz Product Operators under the spin–spin coupling Hamiltonian have been obtained in a different form. Furthermore, as an application and a verification, the Product Operator formalism has been used for the analytical description of a 2D J-resolved nuclear magnetic resonance (NMR) experiment for IS (I=1/2, S=3/2) spin systems.
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a Product Operator theory of 13c spin echo j modulation nmr spectroscopy for cdn n 1 2 3 groups
Spectroscopy Letters, 2001Co-Authors: Azmi Gencten, Ozden TezelAbstract:Product Operator formalism is widely used for analytical description of multiple–pulse NMR experiments for a weakly coupled spin systems. 13C spin-echo J-modulation NMR spectroscopy for CHn ve CDn groups is used for identification of different carbon groups. In this study, by using the Product Operator technique, the analytical description of 13C spin-echo J-modulation NMR spectroscopy for CDn (n = 1,2,3) groups is presented and the experimental identifications of 13C NMR signals of CD3, CD2 and CD groups and also quaternary carbons are discussed.