A Cheap Alternative to the Lattice?

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 (TCSA), 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 QFT Hamiltonian is expressed as a matrix in the Hilbert space of CFT 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 phi^4 theory in d dimensions with d not necessarily integer. A numerical analysis is then performed for the specific case d = 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 non integral d are not unitary, which however does not seem to cause much effect at low energies.

56 pages, 22 figures; V2: defective figure file replaced; V3: minor corrections; refs added; V4: refs updated

References in corpus (13)

• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Conformal Field Theories in Fractional Dimensions
• Hamiltonian Truncation Study of the Phi^4 Theory in Two Dimensions
• Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions
• Hamiltonian Truncation Study of the Phi^4 Theory in Two Dimensions II. The Z_2-Broken Phase and the Chang Duality
• The renormalisation group for the truncated conformal space approach on the cylinder
• Solving 2D QCD with an adjoint fermion analytically
• Truncated Conformal Space Approach for 2D Landau-Ginzburg Theories
• Excited state TBA and renormalized TCSA in the scaling Potts model
• A Renormalisation group for TCSA
• A solution of 2D QCD at Finite $N$ using a conformal basis
• Ising Spectroscopy II: Particles and poles at T>Tc
• Interpolating between $a$ and $F$

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• Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-abelian bosonization to truncated spectrum methods
• Bootstrapping SCFTs with Four Supercharges
• Hamiltonian Truncation Study of the Phi^4 Theory in Two Dimensions II. The Z_2-Broken Phase and the Chang Duality
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