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Scaling Lee–Yang: a negative central charge and a positive particle gap

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All tutorials · Model and numerical guide · Sine-Gordon comparison

The scaling Lee–Yang model is a particularly clean nonunitary example: it has one massive particle, yet its ultraviolet conformal theory has negative central charge. Here we calculate the periodic vacuum energy across scales, then follow a selected one-particle gap toward its infinite-volume mass.

These are continuum field-theory calculations. The tools do not construct a finite non-Hermitian spin chain, compute left/right eigenvectors, or measure matrix elements. Real energies in this example do not imply a unitary theory.

1. Which energy does the vacuum tool return?

Use particle mass m=1, circumference L and velocity=1:

bethe-lee-yang-vacuum --mass 1 --length 0.1 \
  --csv-table vacuum=ly-vacuum.csv

The useful dimensionless variable is \(r=mL\). The output separates the bulk-subtracted energy and its scaling function,

\[ E_{0,C}=E_0-L e_{\mathrm{bulk}},\qquad Y(r)=LE_{0,C},\qquad c_{\mathrm{eff}}(r)=-\frac{6Y(r)}{\pi}. \]

The tool does not supply an absolute bulk energy density. At r=0.1, it returns \(Y\simeq-0.20835016\) and \(c_{\mathrm{eff}}\simeq0.39791949\). Since L=0.1 here, the energy itself is approximately -2.08350158, not -0.20835. The columns casimir_energy and scaling_function deliberately keep these quantities separate.

2. Why does the plot approach +0.4 instead of -4.4?

The periodic Lee–Yang vacuum effective charge approaches two fifths at small mL and zero at large mL. The periodic Lee–Yang vacuum effective charge approaches two fifths at small mL and zero at large mL.

The ultraviolet CFT is \(M(2,5)\), with central charge \(c=-22/5\) and lowest left/right conformal weight \(h_{\min}=\bar h_{\min}=-1/5\). On a periodic circle, the lowest energy samples that negative-weight sector, not the identity sector:

\[ E_{0,C}\sim\frac{2\pi}{L}\left(2h_{\min}-\frac{c}{12}\right) =-\frac{\pi c_{\mathrm{eff}}^{\mathrm{UV}}}{6L},\qquad c_{\mathrm{eff}}^{\mathrm{UV}}=c-24h_{\min}=\frac25. \]

Thus the positive coefficient extracted from the lowest energy is the effective central charge. To recover c, one also needs the conformal-sector information: \(c=2/5+24(-1/5)=-22/5\). Neither the sign of the vacuum energy nor a fit to its coefficient alone determines c. The finite-r curve is a computed scaling diagnostic, not a central charge at every circumference.

The model and finite-volume conventions are developed by Bajnok–el Deeb–Pearce, especially Eqs. (151)–(152), (212) and (218). In the small-r limit, \(Y\to-\pi/15\); at large r, the bulk-subtracted vacuum energy tends to zero. The markers are eleven separately converged TBA solves, not values from an interpolation formula. At r=0.001 the effective charge is within \(3\times10^{-7}\) of 0.4; that remaining finite-r difference is not solver noise.

3. Follow one massive excitation

The excited-state tool is a separate TBA calculation with quantized sources. For now it supports one zero-momentum regular branch at \(5\le mL\le30\):

bethe-lee-yang-excited --mass 1 --length 5 \
  --csv-table levels=ly-levels.csv --csv-table source=ly-source.csv \
  --csv-table gap=ly-gap.csv

The levels table contains the vacuum and one-particle Casimir energies. The quantity to compare with an excitation calculation is instead

\[ \Delta E=E_{1,C}-E_{0,C},\qquad \Delta E/m\longrightarrow1 \quad\text{as }mL\longrightarrow\infty. \]

At m=1, L=5, the vacuum Casimir energy is about -0.0012816882 and the one-particle Casimir energy is 1.0293562331. Their difference is 1.0306379213. Replacing the gap by the excited row would lose a visible vacuum correction.

The one-particle level and vacuum-relative gap approach the mass, while the finite-size gap correction stays much larger than the numerical error estimate. The one-particle level and vacuum-relative gap approach the mass, while the finite-size gap correction stays much larger than the numerical error estimate.

The right panel separates the physical finite-size correction from the estimated numerical error. At mL=20 the gap is still above m by roughly \(8.8\times10^{-8}m\), while the exported error estimate is about \(8.7\times10^{-14}m\). The estimate is not a rigorous interval bound, but these scales are well separated. Connecting lines merely guide the eye; no exponential fit or extrapolated mass is needed for this comparison.

Dorey–Tateo, Eqs. (2.3)–(2.7), derive the excited-state source construction by analytic continuation. Our implementation uses its regular infrared branch. The source table reports the imaginary source coordinate \(\beta\), its displacement from \(\pi/6\), and its quantization diagnostics. This beta is not sine-Gordon's coupling.

Do not continue the plotted curve toward the UV by ignoring the tool's domain check. The source configuration changes at smaller volume; the current solver does not implement that continuation. The vacuum UV calculation above is supported independently. Nor does changing a parameter in the source-free vacuum equation generate an excited level.

4. Match a non-Hermitian MPS calculation

The first figure is a finite-size energy benchmark, and the second a gap benchmark. For a lattice realization approaching this scaling theory:

  • Identify the periodic sector and the lowest conformal weight before extracting c from finite-size energies. The coefficient alone gives an effective value.
  • Match the physical mass, velocity and circumference. The exported theory uses velocity=1; a lattice energy scale and correlation length must be converted consistently before identifying r.
  • Subtract the same ground-state energy when forming a gap. A fitted bulk energy density cancels between levels, but the finite-size vacuum term does not.
  • Treat lattice corrections, finite-bond-dimension errors and TBA numerical estimates as separate sources of discrepancy. These energy tools supply no biorthogonal spectral weights or ordinary positive-norm interpretation.

A simple units test is m=2, L=1/2 versus m=1, L=1: r is unchanged, Y is unchanged, and the Casimir energy doubles. Likewise, the one-particle gap at m=2, L=2.5 is twice that at m=1, L=5. Saved exports check both transformations. The frontends support fp64, native long-double and optional MPLAPACK fp128; the tutorial uses fp64 throughout.

5. Reproduce and validate

Download the vacuum scans: 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 1, 2, 5, 10, 20, and the m=2, L=1/2 units check.

mL (m=1) Levels Source Gap
5 CSV CSV CSV
6 CSV CSV CSV
8 CSV CSV CSV
10 CSV CSV CSV
15 CSV CSV CSV
20 CSV CSV CSV

The rescaled one-particle example also has separate levels, source and gap tables.

From the source checkout, with the plotting dependencies:

python3 scripts/plot_field_theory_tutorial.py --check
python3 scripts/plot_field_theory_tutorial.py
# Optional: regenerate both field-theory tutorials using native executables.
python3 scripts/plot_field_theory_tutorial.py --solver-dir /path/to/bethe/bin
python3 -m unittest discover -s scripts -p 'test_tutorial_field_theory.py'

The shared script preserves the individual runs' metadata, joins levels/source/gap from the same invocation, checks convergence and refinement diagnostics, and reconstructs the gap to catch energy-reference mistakes. Missing observables, non-finite values and failed statuses are rejected. The tests check UV/IR behaviour, mass scaling, a value from the independent vacuum oracle, and corrupted exports. The reference guide describes the independent quadrature and the meaning of each error estimate.