Tensor Network States and Effective Particles for Low-Dimensional Quantum Spin Systems

Vanderstraeten, Laurens

doi:10.1007/978-3-319-64191-1

Cited by 11 publications

(13 citation statements)

References 453 publications

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“…The blocks decrease exponentially as we go further from the diagonal, so we can, in order to solve the problem, consider them to be zero if |n − n | > M for a sufficiently large M . In this approximation, the coefficients c(n) obey From the special structure of the blocks A m [49] and their relation to the effective one-particle hamiltonian H eff (p), we already know a number of solutions to Eq. (219).…”

Section: Asymptotic Regimementioning

confidence: 99%

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Tangent-space methods for uniform matrix product states

Vanderstraeten

Haegeman

Verstraete

2019

SciPost Phys. Lect. Notes

Self Cite

186

182

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In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.

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Section: Asymptotic Regimementioning

confidence: 99%

“…the relative group velocity in the center of mass frame. Much like the proof of conservation of particle current in one-particle quantum mechanics, it can be shown [49] that, if (230) is to be the asymptotic form of an eigenstate, the coefficients r γ i (P, ω) should obey γ r γ i (P, ω) 2 dω γ dp (p γ ) = 0.…”

Section: S Matrix and Normalizationmentioning

confidence: 99%

Tangent-space methods for uniform matrix product states

Vanderstraeten

Haegeman

Verstraete

2019

SciPost Phys. Lect. Notes

Self Cite

186

182

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“…Recently AD techniques have been applied to the optimization problem of iPEPS [10]. Considerable effort has been put into development of powerful yet efficient optimization algorithms for iPEPS in recent years, from very inexpensive [35] though generally less accurate or more balanced [1,3,31] imaginary time evolution methods to powerful though more expensive variational optimization approaches [36,37]. While these methods form a broad and widely applicable set of tools, AD offers a compelling alternative since it essentially does not require the implementation of any optimization algorithm at all, but only a routine for contracting the network and computing expectation values.…”

Section: Automatic Differentiationmentioning

confidence: 99%

Automatic differentiation applied to excitations with projected entangled pair states

2022

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The excitation ansatz for tensor networks is a powerful tool for simulating the low-lying quasiparticle excitations above ground states of strongly correlated quantum many-body systems. Recently, the two-dimensional tensor network class of infinite projected entangled-pair states gained new ground state optimization methods based on automatic differentiation, which are at the same time highly accurate and simple to implement. Naturally, the question arises whether these new ideas can also be used to optimize the excitation ansatz, which has recently been implemented in two dimensions as well. In this paper, we describe a straightforward way to reimplement the framework for excitations using automatic differentiation, and demonstrate its performance for the Hubbard model at half filling.

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“…In all cases, the correlations between such DoF underpin the low-energy properties of the model. We note that similar ideas have recently been used in the framework of matrix product state methods in order to construct quasiparticle excitations in various 1D models [39][40][41] .…”

Section: Fig 1 (A)mentioning

confidence: 99%

Interaction distance in the extended XXZ model

et al. 2019

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We employ the interaction distance to characterise the physics of a one-dimensional extended XXZ spin model, whose phase diagram consists of both integrable and non-integrable regimes, with various types of ordering, e.g., a gapless Luttinger liquid and gapped crystalline phases. We numerically demonstrate that the interaction distance successfully reveals the known behaviour of the model in its integrable regime. As an additional diagnostic tool, we introduce the notion of "integrability distance" and particularise it to the XXZ model in order to quantity how far the ground state of the extended XXZ model is from being integrable. This distance provides insight into the properties of the gapless Luttinger liquid phase in the presence of next-nearest neighbour spin interactions which break integrability. arXiv:1905.07239v1 [cond-mat.str-el]

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Tensor Network States and Effective Particles for Low-Dimensional Quantum Spin Systems

Cited by 11 publications

References 453 publications

Tangent-space methods for uniform matrix product states

Tangent-space methods for uniform matrix product states

Automatic differentiation applied to excitations with projected entangled pair states

Interaction distance in the extended XXZ model

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