Automated label flows for excited states of correlation functions in lattice gauge theory

Extracting excited states from lattice gauge theory correlation functions can be achieved through ${\chi }^2$ minimization fits or algebraic approaches such as the variational method and Prony's method. Performing any kind of error analysis often leads to overlapping confidence regions of model parameters, even when the spectrum is not particularly dense. To correctly estimate errors, one must beware of mislabeling the states. In this work, we provide an algorithm that we call automated label flows which consistently and systematically identifies a deterministic labeling of states. This is a black-box approach in the sense that it gives a sensible set of labels without user guidance. As an example, we pair one black box method with another, analyzing a lattice correlation function from real data using automated label flows in the context of Prony's method.

Figure 1

Labeling $M$ tuples $(z_m,a_m)$ according to mass for 1000 bootstrap samples when data is relatively low in noise. Plots show real part of logs. Left: sets of $M=2$ tuples with $y_n(t=8)$. Right: sets of $M=3$ tuples with $y_n(t=1)$.

Figure 2

State labeling for noisy data. Plots show real part of logs. (a) Labeling according to mass for $M=3$, $t=5$. (b) Same as panel (a), but zoomed into a region where ground and first excited states lie.

Figure 3

Rescaled errors for $M=3$ at $t=5$ clusters with $\epsilon =0.8$ (left) and $\epsilon =0.1$ (right). Original clusters plotted in Fig. 2.

Figure 4

Example of labels flowing from $\epsilon =0.78$ to $\epsilon =0.79$ to $\epsilon =0.80$ for the $11\mathrm{th}$ bootstrap sample of the $M=3$, $t=16$ extraction. The blue arrows, red arrows, and maroon arrows represent the flow of the ground state, first excited state, and second excited states, respectively.

Figure 5

Label flows for first (left) and 33rd (right) bootstrap samples of $M=3$ at $t=5$ clusters, starting from $\epsilon =0.05$, denoted by the starred point, then flowing through $\epsilon =0.1,0.2,0.3,...,0.9,$ to $\epsilon =1.0$

Figure 6

Clusters with flowed labels for $M=5$ at $t=5$, zoomed out (left) and zoomed into where ground state and first excited state lie (right). Original clusters plotted in Fig. 2.

Figure 7

Label flow for $11\mathrm{th}$ bootstrap sample of $M=3,t=16$ extraction, exhibiting a label collision. Left and right plots depicting flow with arrows pointing in direction of increasing $\epsilon$. Right hand plot zoomed in to where the flow is ambiguous.

Figure 8

Generalized effective masses from Prony's method using automated label flows with $M=2$ (upper left), 3 (upper right), 4 (lower left), 5 (lower right). The $x$ axis indicates the time slice $t$ at the beginning of the Prony stencil of $2M$ time slices for, i.e., the time slices $t$ from a given stencil $y_n\left(t\right)$.

Figure 9

Comparison between effective masses of the ground state (left) and first excited state (right) obtained from $M=1,2,3,4,5$ state extractions from Prony's method. For easy comparison between different numbers of states $M$, the $x$ axis indicates the middle time slice used in the Prony stencil of $2M$ time slices.

Figure 10

Ground-state extraction from $M=2$ model compared to exponential fit to three (left) and four (right) states.

Figure 11

Ground-state extraction from $M=3$ model compared to exponential fit to three (left) and four (right) states.

Figure 12

Ground-state extraction from $M=4$ model compared to exponential fit to three (left) and four (right) states.

Figure 13

Ground-state extraction from $M=5$ model compared to exponential fit to three (left) and four (right) states.

Figure 14

First excited-state extraction from $M=2$ model compared to exponential fit to three (left) and four (right) states.

Figure 15

First excited-state extraction from $M=3$ model compared to exponential fit to three (left) and four (right) states.

Figure 16

First excited-state extraction from $M=4$ model compared to exponential fit to three (left) and four (right) states.

Figure 17

First excited-state extraction from $M=5$ model compared to exponential fit to three (left) and four (right) states.

Figure 18

Second excited-state extraction from $M=3$ model compared to exponential fit to three (left) and four (right) states.

Figure 19

Second excited-state extraction from $M=4$ model compared to exponential fit to three (left) and four (right) states.

Figure 20

Second excited-state extraction from $M=5$ model compared to exponential fit to three (left) and four (right) states.