Krylov spaces for truncated spectrum methodologies

We propose herein an extension of truncated spectrum methodologies, a nonperturbative numerical approach able to elucidate the low energy properties of quantum field theories. TSMs, in their various flavors, involve a division of a computational Hilbert space, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="script">H</a:mi></a:math>, into two parts, one part, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msub><d:mi mathvariant="script">H</d:mi><d:mn>1</d:mn></d:msub></d:math> that is “kept” for the numerical computations, and one part, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi mathvariant="script">H</g:mi><g:mn>2</g:mn></g:msub></g:math>, that is discarded or “truncated.” Even though <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msub><j:mi mathvariant="script">H</j:mi><j:mn>2</j:mn></j:msub></j:math> is discarded, truncated spectrum methodologies will often try to incorporate the effects of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi mathvariant="script">H</m:mi><m:mn>2</m:mn></m:msub></m:math> in some effective way. In these terms, we propose to keep the dimension of <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msub><p:mi mathvariant="script">H</p:mi><p:mn>1</p:mn></p:msub></p:math> small. We pair this choice of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi mathvariant="script">H</s:mi><s:mn>1</s:mn></s:msub></s:math> with a Krylov subspace iterative approach able to take into account the effects of <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msub><v:mi mathvariant="script">H</v:mi><v:mn>2</v:mn></v:msub></v:math>. This iterative approach can be taken to arbitrarily high order and so offers the ability to compute quantities to arbitrary precision. In many cases it also offers the advantage of not needing an explicit UV cutoff. To compute the matrix elements that arise in the Krylov iterations, we employ a Feynman diagrammatic representation that is then evaluated with Monte Carlo techniques. Each order of the Krylov iteration is variational and is guaranteed to improve upon the previous iteration. The first Krylov iteration is akin to the next-to-leading order approach of Elias-Miró [NLO renormalization in the Hamiltonian truncation, ]. To demonstrate this approach, we focus on the (<y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mn>1</y:mn><y:mo>+</y:mo><y:mn>1</y:mn><y:mi>d</y:mi></y:math>)-dimensional <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msup><ab:mi>ϕ</ab:mi><ab:mn>4</ab:mn></ab:msup></ab:math> model and compute the bulk energy and mass gaps in both the <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi mathvariant="double-struck">Z</cb:mi><cb:mn>2</cb:mn></cb:msub></cb:math>-broken and unbroken sectors. We estimate the critical <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:msup><fb:mi>ϕ</fb:mi><fb:mn>4</fb:mn></fb:msup></fb:math> coupling in the broken phase to be <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:msub><hb:mi>g</hb:mi><hb:mi>c</hb:mi></hb:msub><hb:mo>=</hb:mo><hb:mn>0.2645</hb:mn><hb:mo>±</hb:mo><hb:mn>0.002</hb:mn></hb:math>.