Abstract
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.
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References
S.R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48 (1993) 10345 [INSPIRE].
D. Perez-Garcia, F. Verstraete, M.M. Wolf and J.I. Cirac, Matrix Product State Representations, Quant. Inf. Comput. 7 (2007) 401 [quant-ph/0608197].
Y.-Y. Shi, L.-M. Duan and G. Vidal, Classical simulation of quantum many-body systems with a tree tensor network, Phys. Rev. A 74 (2006) 022320 [quant-ph/0511070].
G. Vidal, Class of Quantum Many-Body States That Can Be Efficiently Simulated, Phys. Rev. Lett. 101 (2008) 110501 [quant-ph/0610099].
F. Verstraete, V. Murg and J.I. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57 (2008) 143 [arXiv:0907.2796].
V.P. Yurov and A.B. Zamolodchikov, Truncated conformal space approach to scaling Lee-Yang model, Int. J. Mod. Phys. A 5 (1990) 3221 [INSPIRE].
V.P. Yurov and A.B. Zamolodchikov, Truncated fermionic space approach to the critical 2-D Ising model with magnetic field, Int. J. Mod. Phys. A 6 (1991) 4557 [INSPIRE].
A.J.A. James, R.M. Konik, P. Lecheminant, N.J. Robinson and A.M. Tsvelik, Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-abelian bosonization to truncated spectrum methods, arXiv:1703.08421 [INSPIRE].
S.J. Brodsky, H.-C. Pauli and S.S. Pinsky, Quantum chromodynamics and other field theories on the light cone, Phys. Rept. 301 (1998) 299 [hep-ph/9705477] [INSPIRE].
E. Katz, G. Marques Tavares and Y. Xu, Solving 2D QCD with an adjoint fermion analytically, JHEP 05 (2014) 143 [arXiv:1308.4980] [INSPIRE].
E. Katz, G. Marques Tavares and Y. Xu, A solution of 2D QCD at Finite N using a conformal basis, arXiv:1405.6727 [INSPIRE].
S.S. Chabysheva, Light-front ϕ 41 + 1 theory using a many-boson symmetric-polynomial basis, Few Body Syst. 57 (2016) 675 [arXiv:1512.08770] [INSPIRE].
M. Burkardt, S.S. Chabysheva and J.R. Hiller, Two-dimensional light-front \( \phi \) 4 theory in a symmetric polynomial basis, Phys. Rev. D 94 (2016) 065006 [arXiv:1607.00026] [INSPIRE].
E. Katz, Z.U. Khandker and M.T. Walters, A Conformal Truncation Framework for Infinite-Volume Dynamics, JHEP 07 (2016) 140 [arXiv:1604.01766] [INSPIRE].
N. Anand, V.X. Genest, E. Katz, Z.U. Khandker and M.T. Walters, RG flow from \( \phi \) 4 theory to the 2D Ising model, JHEP 08 (2017) 056 [arXiv:1704.04500] [INSPIRE].
T.R. Klassen and E. Melzer, Spectral flow between conformal field theories in (1+1)-dimensions, Nucl. Phys. B 370 (1992) 511 [INSPIRE].
M. Hogervorst, S. Rychkov and B.C. van Rees, Truncated conformal space approach in d dimensions: A cheap alternative to lattice field theory?, Phys. Rev. D 91 (2015) 025005 [arXiv:1409.1581] [INSPIRE].
S. Rychkov and L.G. Vitale, Hamiltonian truncation study of the \( \phi \) 4 theory in two dimensions, Phys. Rev. D 91 (2015) 085011 [arXiv:1412.3460] [INSPIRE].
S. Rychkov and L.G. Vitale, Hamiltonian truncation study of the \( \phi \) 4 theory in two dimensions. II. The ℤ 2 -broken phase and the Chang duality, Phys. Rev. D 93 (2016) 065014 [arXiv:1512.00493] [INSPIRE].
J. Elias-Miro, M. Montull and M. Riembau, The renormalized Hamiltonian truncation method in the large E T expansion, JHEP 04 (2016) 144 [arXiv:1512.05746] [INSPIRE].
P. Giokas and G. Watts, The renormalisation group for the truncated conformal space approach on the cylinder, arXiv:1106.2448 [INSPIRE].
G. Feverati, K. Graham, P.A. Pearce, G.Z. Toth and G. Watts, A renormalisation group for the truncated conformal space approach, J. Stat. Mech. 0803 (2008) P03011 [hep-th/0612203] [INSPIRE].
G.M.T. Watts, On the renormalisation group for the boundary Truncated Conformal Space Approach, Nucl. Phys. B 859 (2012) 177 [arXiv:1104.0225] [INSPIRE].
M. Lencsés and G. Takács, Excited state TBA and renormalized TCSA in the scaling Potts model, JHEP 09 (2014) 052 [arXiv:1405.3157] [INSPIRE].
J. Elias-Miro, S. Rychkov and L.G. Vitale, NLO Renormalization in the Hamiltonian Truncation, Phys. Rev. D 96 (2017) 065024 [arXiv:1706.09929] [INSPIRE].
R.M. Konik and Y. Adamov, A Numerical Renormalization Group for Continuum One-Dimensional Systems, Phys. Rev. Lett. 98 (2007) 147205 [cond-mat/0701605] [INSPIRE].
K.G. Wilson, The Renormalization Group: Critical Phenomena and the Kondo Problem, Rev. Mod. Phys. 47 (1975) 773 [INSPIRE].
A. Coser, M. Beria, G.P. Brandino, R.M. Konik and G. Mussardo, Truncated Conformal Space Approach for 2D Landau-Ginzburg Theories, J. Stat. Mech. 1412 (2014) P12010 [arXiv:1409.1494] [INSPIRE].
Z. Bajnok and M. Lajer, Truncated Hilbert space approach to the 2d \( \phi \) 4 theory, JHEP 10 (2016) 050 [arXiv:1512.06901] [INSPIRE].
M. Lüscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 1. Stable Particle States, Commun. Math. Phys. 104 (1986) 177 [INSPIRE].
T.R. Klassen and E. Melzer, On the relation between scattering amplitudes and finite size mass corrections in QFT, Nucl. Phys. B 362 (1991) 329 [INSPIRE].
T.R. Klassen and E. Melzer, The Thermodynamics of purely elastic scattering theories and conformal perturbation theory, Nucl. Phys. B 350 (1991) 635 [INSPIRE].
S.-J. Chang, The Existence of a Second Order Phase Transition in the Two-Dimensional \( \phi \) 4 Field Theory, Phys. Rev. D 13 (1976) 2778 [Erratum ibid. D 16 (1977) 1979] [INSPIRE].
D. Schaich and W. Loinaz, An Improved lattice measurement of the critical coupling in ϕ 42 theory, Phys. Rev. D 79 (2009) 056008 [arXiv:0902.0045] [INSPIRE].
A. Milsted, J. Haegeman and T.J. Osborne, Matrix product states and variational methods applied to critical quantum field theory, Phys. Rev. D 88 (2013) 085030 [arXiv:1302.5582] [INSPIRE].
P. Bosetti, B. De Palma and M. Guagnelli, Monte Carlo determination of the critical coupling in ϕ 42 theory, Phys. Rev. D 92 (2015) 034509 [arXiv:1506.08587] [INSPIRE].
A. Pelissetto and E. Vicari, Critical mass renormalization in renormalized \( \phi \) 4 theories in two and three dimensions, Phys. Lett. B 751 (2015) 532 [arXiv:1508.00989] [INSPIRE].
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Elias-Miró, J., Rychkov, S. & Vitale, L.G. High-precision calculations in strongly coupled quantum field theory with next-to-leading-order renormalized Hamiltonian Truncation. J. High Energ. Phys. 2017, 213 (2017). https://doi.org/10.1007/JHEP10(2017)213
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DOI: https://doi.org/10.1007/JHEP10(2017)213