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

Abstract

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.

No unitary bootstrap for the fractal Ising model

“…Recently the authors of [68] have noticed that free field theories, and in all likelihood even the Wilson-Fisher fixed point, is non-unitary for any fractional dimension. Could this be an (admitedly somewhat trivial) explanation for our results?…”

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

“…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…”

“…As shown in [17], the raw HT numerical spectrum is expected to converge with polynomial rate 1/E ρ T , with ρ = d − 2∆ V > 0 by our assumption (1). This polynomial convergence must compete with the exponential growth of states in the Hilbert space.…”

“…As a result the convergence is improved. Renormalized HT has been applied in several strongly coupled QFT studies in d = 2 [18][19][20][21]24] and in one study in d = 2.5 [17]. We hope that in the future Hamiltonian Truncation will develop into an accurate numerical method, applicable also in d 3.…”

“…Motivated by the need to mitigate the above limitations, the recent works [17][18][19][20] (following notably [21]; see also [22][23][24]) started developing the theory of renormalized HT, 1 In relativistic QFTs one can also quantize on surfaces of constant light-cone coordinate. This light front quantization [9] is also used in numerical solutions of strongly coupled QFTs via a version of HT; some recent work is [10][11][12][13][14][15].…”