Hamiltonian truncation study of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:math>theory in two dimensions. II. The<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>-broken phase and the Chang duality

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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...

“…This is what was done in the previous works [17][18][19], where (11) was truncated to the leading order (LO) n = 2, and ∆H 2 was computed in an analytic local approximation…”

Section: Jhep10(2017)213mentioning

confidence: 99%

Section: Jhep10(2017)213mentioning

confidence: 99%

“…We will only study here the Z 2 -invariant phase. The Z 2 -broken phase at g > g c was studied previously in [19,28,29].…”

Section: Jhep10(2017)213mentioning

confidence: 99%

Section: Jhep10(2017)213mentioning

confidence: 99%

Section: Jhep10(2017)213mentioning

confidence: 99%

See 3 more Smart Citations

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

Elias-Miró

Vitale

Rychkov

2017

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Approaching the self-dual point of the sinh-Gordon model

Konik

Lajer

Mussardo

2021

One of the most striking but mysterious properties of the sinh-Gordon model (ShG) is the b → 1/b self-duality of its S-matrix, of which there is no trace in its Lagrangian formulation. Here b is the coupling appearing in the model’s eponymous hyperbolic cosine present in its Lagrangian, cosh(bϕ). In this paper we develop truncated spectrum methods (TSMs) for studying the sinh-Gordon model at a finite volume as we vary the coupling constant. We obtain the expected results for b ≪ 1 and intermediate values of b, but as the self-dual point b = 1 is approached, the basic application of the TSM to the ShG breaks down. We find that the TSM gives results with a strong cutoff Ec dependence, which disappears according only to a very slow power law in Ec. Standard renormalization group strategies — whether they be numerical or analytic — also fail to improve upon matters here. We thus explore three strategies to address the basic limitations of the TSM in the vicinity of b = 1. In the first, we focus on the small-volume spectrum. We attempt to understand how much of the physics of the ShG is encoded in the zero mode part of its Hamiltonian, in essence how ‘quantum mechanical’ vs ‘quantum field theoretic’ the problem is. In the second, we identify the divergencies present in perturbation theory and perform their resummation using a supra-Borel approximate. In the third approach, we use the exact form factors of the model to treat the ShG at one value of b as a perturbation of a ShG at a different coupling. In the light of this work, we argue that the strong coupling phase b > 1 of the Lagrangian formulation of model may be different from what is naïvely inferred from its S-matrix. In particular, we present an argument that the theory is massless for b > 1.

Solving the 2D SUSY Gross-Neveu-Yukawa model with conformal truncation

Fitzpatrick

Katz

Walters

et al. 2021

We use Lightcone Conformal Truncation to analyze the RG flow of the two-dimensional supersymmetric Gross-Neveu-Yukawa theory, i.e. the theory of a real scalar superfield with a ℤ2-symmetric cubic superpotential, aka the 2d Wess-Zumino model. The theory depends on a single dimensionless coupling $$ \overline{g} $$ g ¯ , and is expected to have a critical point at a tuned value $$ {\overline{g}}_{\ast } $$ g ¯ ∗ where it flows in the IR to the Tricritical Ising Model (TIM); the theory spontaneously breaks the ℤ2 symmetry on one side of this phase transition, and breaks SUSY on the other side. We calculate the spectrum of energies as a function of $$ \overline{g} $$ g ¯ and see the gap close as the critical point is approached, and numerically read off the critical exponent ν in TIM. Beyond the critical point, the gap remains nearly zero, in agreement with the expectation of a massless Goldstino. We also study spectral functions of local operators on both sides of the phase transition and compare to analytic predictions where possible. In particular, we use the Zamolodchikov C-function to map the entire phase diagram of the theory. Crucial to this analysis is the fact that our truncation is able to preserve supersymmetry sufficiently to avoid any additional fine tuning.