Numerical Renormalization Group for Continuum One-Dimensional Systems

Konik, Robert; Adamov, Yury

doi:10.1103/physrevlett.98.147205

Cited by 74 publications

(143 citation statements)

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“…Similar effects were observed for boundary form factors in [51] and based on the accumulated data we are inclined to think that there is something special about cutoff dependence of diagonal matrix elements. Convergence of the TCSA can be improved by renormalization group methods [48,49,50]; as we discussed in subsection 4.3 it turns out that the TCSA data are not yet in the regime where the leading RG behaviour is applicable, and the extrapolation fits are not reliable enough to determine the exponent of the cutoff dependence. This can be helped by applying the numerical RG technique proposed in [49]; however, developing a systematic program for that takes a substantial amount of effort and time, and work in this direction has just started.…”

Section: Discussionmentioning

confidence: 99%

“…In sine-Gordon theory, they have been observed to become smaller when decreasing ξ, so the two sources of deviations behave the opposite way when the sine-Gordon coupling is varied. Behaviour of truncation errors in the asymptotic regime of large values of the cutoff can be theoretically described by a Wilsonian renormalization group [48,49,50].…”

Section: Sources Of Deviationsmentioning

confidence: 99%

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Sine-Gordon multisoliton form factors in finite volume

2012

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Section: Discussionmentioning

confidence: 99%

Section: Sources Of Deviationsmentioning

confidence: 99%

Sine-Gordon multisoliton form factors in finite volume

2012

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“…It is natural to employ the ideas of renormalization in this context, and the current literature already contains several proposals. One approach [48] (described in more detail in [21]) is numerical RG, inspired by the namesake method used in Wilson's famous solution of the Kondo problem. The idea is to add states to the Hilbert space in manageable batches, and after each addition rediagonalize the Hamiltonian and throw out the least important states so that the total number of retained states never grows more than a few thousand, while the total number of explored states may be several orders of magnitude larger.…”

Section: Discussionmentioning

confidence: 99%

Truncated conformal space approach inddimensions: A cheap alternative to lattice field theory?

2015

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We show how to perform accurate, nonperturbative and controlled calculations in quantum field theory in d dimensions. We use the truncated conformal space approach, a Hamiltonian method which exploits the conformal structure of the UV fixed point. The theory is regulated in the IR by putting it on a sphere of a large finite radius. The quantum field theory Hamiltonian is expressed as a matrix in the Hilbert space of conformal field theory states. After restricting ourselves to energies below a certain UV cutoff, an approximation to the spectrum is obtained by numerical diagonalization of the resulting finite-dimensional matrix. The cutoff dependence of the results can be computed and efficiently reduced via a renormalization procedure. We work out the details of the method for the ϕ 4 theory in d dimensions with d being not necessarily integer. A numerical analysis is then performed for the specific case d ¼ 2.5, a value chosen in the range where UV divergences are absent. By going from weak to intermediate to strong coupling, we are able to observe the symmetry-preserving, symmetry-breaking, and conformal phases of the theory, and perform rough measurements of masses and critical exponents. As a byproduct of our investigations we find that both the free and the interacting theories in nonintegral d are not unitary, which however does not seem to cause much effect at low energies.

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“…The ellipsis in H in (26) refer to the Casimir energy and other exponentially suppressed corrections needed to correctly put the theory in finite volume. They are discussed in detail in [18] and defined in eqs.…”

Section: Jhep10(2017)213mentioning

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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Numerical Renormalization Group for Continuum One-Dimensional Systems

Cited by 74 publications

References 9 publications

Sine-Gordon multisoliton form factors in finite volume

Sine-Gordon multisoliton form factors in finite volume

Truncated conformal space approach inddimensions: A cheap alternative to lattice field theory?

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

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