Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-Abelian bosonization to truncated spectrum methods

James, Andrew; Konik, Robert; Lecheminant, P.; Robinson, Neil J.; Tsvelik, A. M.

doi:10.1088/1361-6633/aa91ea

Cited by 109 publications

(166 citation statements)

References 591 publications

(1,749 reference statements)

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“…This method and these results closely resemble those of Truncated Spectrum Approximation (TSA) methods [40][41][42], especially when considering the actual procedure for computing the dynamical structure factors. TSA methods aim to provide a description of a non-integrable Hamiltonian by identifying a nearby integrable Hamiltonian and constructing the matrix elements of the original Hamiltonian in a restricted eigenbasis of the integrable one.…”

Section: The Integrability-broken Xy Modelsupporting

confidence: 53%

Variational Schrieffer-Wolff transformations for quantum many-body dynamics

2020

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Building on recent results for adiabatic gauge potentials, we propose a variational approach for computing the generator of Schrieffer-Wolff transformations. These transformations consist of block diagonalizing a Hamiltonian through a unitary rotation, which leads to effective dynamics in a computationally tractable reduced Hilbert space. The generator of these rotations are computed variationally and thus go beyond standard perturbative methods; the error is controlled by the locality of the variational ansatz. The method is demonstrated on two models. First, in the attractive Fermi-Hubbard model with on-site disorder, we find indications of a lack of observable many-body localization in the thermodynamic limit due to the inevitable mixture of different spinon sectors. Second, in the low-energy sector of the XY spin model with a broken U (1)-symmetry, we analyze ground state response functions by combining the variational SW transformation with the truncated spectrum approach. arXiv:1910.11889v1 [cond-mat.str-el]

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Section: The Integrability-broken Xy Modelsupporting

confidence: 53%

Variational Schrieffer-Wolff transformations for quantum many-body dynamics

2020

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“…For a detailed review of truncated space methods the interested reader is referred to Ref. [95]. The Hamiltonian matrix corresponding to the action Eq.…”

Section: Methods and Notationsmentioning

confidence: 99%

Perturbative post-quench overlaps in quantum field theory

Hódsági

Kormos

Takács

2019

J. High Energ. Phys.

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In analytic descriptions of quantum quenches, the overlaps between the initial pre-quench state and the eigenstates of the time evolving Hamiltonian are crucial ingredients. We construct perturbative expansions of these overlaps in quantum field theories where either the pre-quench or the post-quench Hamiltonian is integrable. Using the E 8 Ising field theory for concrete computations, we give explicit expressions for the overlaps up to second order in the quench size, and verify our results against numerical results obtained using the Truncated Conformal Space Approach. We demonstrate that the expansion using the post-quench basis is very effective, but find some serious limitations for the alternative approach using the pre-quench basis.

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“…Zamolodchikov [6,7] and has since been applied in many contexts. See [8] for a recent extensive review and the bibliography.…”

Section: Abstract: Field Theories In Lower Dimensions Nonperturbativmentioning

confidence: 99%

“…Zamolodchikov [6,7] and has since been applied in many contexts. See [8] for a recent extensive review and the bibliography.The idea of HT is simple. The QFT Hamiltonian operator H is split as H 0 + V where H 0 is an exactly solvable Hamiltonian whose eigenstates form the basis of the Hilbert space.…”

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confidence: 99%

High-precision calculations in strongly coupled quantum field theory with next-to-leading-order renormalized Hamiltonian Truncation

Elias-Miró

Vitale

Rychkov

2017

J. High Energ. Phys.

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Hamiltonian Truncation (a.k.a. Truncated Spectrum Approach) is an efficient numerical technique to solve strongly coupled QFTs in d = 2 spacetime dimensions. Further theoretical developments are needed to increase its accuracy and the range of applicability. With this goal in mind, here we present a new variant of Hamiltonian Truncation which exhibits smaller dependence on the UV cutoff than other existing implementations, and yields more accurate spectra. The key idea for achieving this consists in integrating out exactly a certain class of high energy states, which corresponds to performing renormalization at the cubic order in the interaction strength. We test the new method on the strongly coupled two-dimensional quartic scalar theory. Our work will also be useful for the future goal of extending Hamiltonian Truncation to higher dimensions d 3. Keywords: Field Theories in Lower Dimensions, Nonperturbative EffectsArXiv ePrint: 1706.06121Open Access, c The Authors. Article funded by SCOAP 3 .https://doi.org/10.1007/JHEP10 (2017)213 JHEP10 (2017)213 Introduction. Many interesting strongly interacting Quantum Field Theories (QFTs) are not amenable to analytical treatment. Such theories are often studied via Lattice Monte Carlo (LMC) numerical simulations, starting from the discretized Euclidean action. However, LMC has some drawbacks, for example it cannot easily compute real-time observables, it is rather computationally expensive, and it cannot directly describe renormalization group (RG) flows starting from interacting fixed points. Therefore, it is worth exploring other numerical approaches to strongly interacting QFTs. One promising alternative is provided by the Hamiltonian methods, which look for the eigenstates of the quantum Hamiltonian. These methods use various finite-dimensional approximations to the full infinite-dimensional QFT Hilbert space. Notable examples are the methods using Matrix Product States [1, 2] and more general Tensor Networks [3] such as MERA [4] or PEPS [5]. In this paper we will be concerned with another representative of this group of methods -Hamiltonian Truncation (HT), also known as the Truncated Spectrum (or Space) Approach, which is a direct generalization of the variational Rayleigh-Ritz (RR) method from quantum mechanics. This method goes back to the seminal work of Yurov and Al. Zamolodchikov [6,7] and has since been applied in many contexts. See [8] for a recent extensive review and the bibliography.The idea of HT is simple. The QFT Hamiltonian operator H is split as H 0 + V where H 0 is an exactly solvable Hamiltonian whose eigenstates form the basis of the Hilbert space. One quantizes at surfaces of constant time and works in finite volume so that the spectrum is discrete. 1 The Hilbert space is then truncated to the low-lying eigenvectors of H 0 . The matrix of H in this truncated Hilbert space is diagonalized exactly on a computer, to find the low-energy spectrum of interacting eigenstates.As was understood early on [16], the numerical convergence of the HT...

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Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-Abelian bosonization to truncated spectrum methods

Cited by 109 publications

References 591 publications

Variational Schrieffer-Wolff transformations for quantum many-body dynamics

Variational Schrieffer-Wolff transformations for quantum many-body dynamics

Perturbative post-quench overlaps in quantum field theory

High-precision calculations in strongly coupled quantum field theory with next-to-leading-order renormalized Hamiltonian Truncation

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