Bethe Quantization and Finite-Volume Spectra

In a large spatial circle, factorized scattering quantizes rapidities by requiring that taking one particle once around the circle reproduce the same many-body state. For diagonal elastic scattering this gives the Bethe–Yang equations

$$e^{i m_{a_i}L\sinh\theta_i} \prod_{j\ne i}S_{a_i a_j}(\theta_i-\theta_j)=1.$$

The equation is asymptotic: boundary twists, statistics and phase branches must be declared, while virtual wrapping processes and bound-state finite-size effects are exponentially small but not contained in the displayed product.

Required background. Exact S-matrix bootstrap, CDD freedom, and completeness supplies the elastic two-body amplitudes and particle spectrum. Helpful background. Relativistic scattering kinematics supplies rapidity and invariant-energy conventions.

One particle transported around the circle

Place a massive integrable theory on a circle of circumference $L$ and assume:

• stable massive particles with $m_{\rm gap}L\gg1$;
• elastic diagonal scattering, so no internal-space diagonalization is needed;
• well-separated wave packets and asymptotic factorization;
• periodic boundary conditions for the total many-body wavefunction; and
• no level at a singular rapidity or threshold where exponentially small effects are enhanced.

A particle of species $a_i$ and rapidity $\theta_i$ has momentum

$$p_i=m_{a_i}\sinh\theta_i.$$

Moving it freely once around the circle gives $e^{ip_iL}$. During this motion it passes every other particle once, producing the ordered scalar factor

$$\prod_{j\ne i}S_{a_i a_j}(\theta_i-\theta_j).$$

Single-valuedness gives

$$\boxed{ e^{i m_{a_i}L\sinh\theta_i} \prod_{j\ne i}S_{a_i a_j}(\theta_i-\theta_j)=1 }.$$

The large-volume quantization and its density form are developed in Zamolodchikov 1990, § 2, equations (2.16)–(2.24), pp. 701–703.

For a twist $\varphi_{a_i}$ in the boundary condition, the right-hand side is $e^{i\varphi_{a_i}}$. A different convention can move a statistics sign between this twist and $S(0)$; physical levels are unchanged only when the branch and state-counting rules are translated together.

Logarithmic phases and quantum numbers

On the real rapidity axis write

$$S_{ab}(\theta)=e^{i\delta_{ab}(\theta)}.$$

Choosing a continuous phase branch converts the equations into

$$m_{a_i}L\sinh\theta_i +\sum_{j\ne i}\delta_{a_i a_j}(\theta_i-\theta_j) =2\pi I_i+\varphi_{a_i},$$

with integers $I_i$ in the stated periodic convention. Shifting a phase by $2\pi$ shifts the corresponding $I_i$ and changes no level. More subtly, a constant exchange factor such as $S=-1$ contributes $\pi$ for each crossing, so the effective integers can become half-integral after that phase is moved to the right-hand side.

For identical particles, equal quantum numbers usually produce coincident rapidities and must be treated according to $S(0)$, statistics, and any internal degeneracy. One should not impose fermionic exclusion on a bosonic coordinate ansatz or count ordered rapidities as distinct physical states without specifying the convention.

The energy and total momentum at Bethe–Yang order are

$$E_{\rm BY}(L)=\sum_i m_{a_i}\cosh\theta_i, \qquad P_{\rm BY}(L)=\sum_i m_{a_i}\sinh\theta_i.$$

Summing the logarithmic equations cancels pair phases when $\delta_{ab}(\theta)=-\delta_{ba}(-\theta)$ on the chosen branch, leaving the expected total-momentum quantization.

Two particles with zero total momentum

For two identical particles of mass $m$, take rapidities $\theta$ and $-\theta$. The first Bethe–Yang equation becomes

$$mL\sinh\theta+\delta(2\theta)=2\pi I,$$

and the level is

$$E_{\rm BY}(L)=2m\cosh\theta.$$

This is an implicit but one-dimensional quantization problem. For weak phase shift and large $L$,

$$\theta = \operatorname{arsinh}\!\left(\frac{2\pi I}{mL}\right) -\frac{\delta(2\theta_0)} {mL\cosh\theta_0+2\delta'(2\theta_0)} +\cdots,$$

where

$$\theta_0=\operatorname{arsinh}\!\left(\frac{2\pi I}{mL}\right).$$

The denominator is the Jacobian of the quantization condition. Its smallness signals that naïve iteration is ill conditioned.

For $S=+1$, $\delta=0$ and $p=2\pi I/L$. For the constant amplitude $S=-1$, choose $\delta=\pi$; then

$$mL\sinh\theta=2\pi\left(I-\frac12\right).$$

The same spectrum can be described with half-integer effective quantum numbers. This example makes the phase/statistics bookkeeping visible without introducing interaction-dependent corrections.

What lies beyond Bethe–Yang

The exact finite-volume energy has the schematic form

$$E(L)=E_{\rm BY}(L)+O(e^{-\mu L}),$$

where $\mu$ depends on the lightest virtual process and can be smaller than a particle mass when a bound-state pole controls the correction.

• F-terms describe virtual particles wrapping around the circle and scattering from the physical state.
• μ-terms arise when a particle can be viewed as a bound state of virtual constituents; the pole geometry fixes the exponent.
• Vacuum polarization shifts the ground-state and excited-state energies even when no physical particle winds in the Bethe–Yang picture.
• Non-diagonal scattering replaces scalar factors by eigenvalues of a transfer matrix and usually requires nested Bethe equations.

Lüscher derives universal leading exponential corrections for stable and two-particle states in a massive QFT Lüscher 1986a, §§ 2–4, pp. 181–199 and Lüscher 1986b, §§ 2–4, pp. 157–179. Near thresholds, avoided crossings, or weakly bound states, these terms can be numerically important even when $mL$ looks moderately large.

The thermodynamic limit of these equations leads to the thermodynamic Bethe ansatz. Detailed numerical comparison of spectra, continuum extrapolation, and finite-volume uncertainties belongs to the finite-volume methods of Volume 8.

A finite-volume reliability check

For a proposed level, report at least:

1. the particle species, boundary twist, statistics convention, and phase branch;
2. the ordered Bethe quantum numbers $I_i$;
3. the joint residual of every quantization equation;
4. the Gaudin Jacobian $\partial Q_i/\partial\theta_j$ and its conditioning;
5. stability of the solution under a larger $L$;
6. the lightest expected wrapping exponent $\mu L$; and
7. whether a bound-state pole permits an enhanced μ-term.

A solution with a tiny algebraic residual can still be physically inaccurate if $e^{-\mu L}$ is not small. Conversely, a phase-branch change that shifts $I_i$ but preserves the rapidities is not a physical discrepancy.

The integrability exact-data chain places Bethe–Yang levels after spectrum and pole classification. The exact and rigorous status comparison keeps an asymptotic finite-volume calculation distinct from a controlled all-$L$ result.

Common pitfalls

Dropping the statistics phase. State whether exchange signs sit in $S(0)$, a boundary twist, or the allowed quantum numbers. Mixing conventions changes the apparent spectrum.

Calling the asymptotic equation exact at finite $L$. Bethe–Yang omits virtual wrapping. Its error is exponentially small only after the relevant scale $\mu L$ is identified.

Solving particles independently. The rapidities satisfy a coupled system. Individual small residuals evaluated with stale values of the other roots do not certify the joint solution.

Exercises

1. Derive the zero-momentum two-particle equation from the two Bethe–Yang conditions and show that the second condition is not independent when $S(\theta)S(-\theta)=1$.

Solution

For $\theta_1=\theta$ and $\theta_2=-\theta$, the first equation is $e^{imL\sinh\theta}S(2\theta)=1$. The second is $e^{-imL\sinh\theta}S(-2\theta)=1$. Elastic unitarity makes the second the inverse of the first. Taking a continuous logarithm gives $mL\sinh\theta+\delta(2\theta)=2\pi I$.

2. For $S=+1$, compute the lowest nonzero zero-momentum two-particle level with periodic boundary conditions.

Solution

Choose $I=1$. Then $p=2\pi/L$,

$$\theta=\operatorname{arsinh}\!\left(\frac{2\pi}{mL}\right), \qquad E(L)=2\sqrt{m^2+\left(\frac{2\pi}{L}\right)^2}.$$

The $I=0$ state has coincident zero momenta and its admissibility depends on statistics and the normalization of identical-particle states.

References

• Lüscher, Martin. “Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. I. Stable Particle States.” Communications in Mathematical Physics 104 (1986): 177–206. DOI.
• Lüscher, Martin. “Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. II. Scattering States.” Communications in Mathematical Physics 105 (1986): 153–188. DOI.
• Yang, C. N. “Some Exact Results for the Many-Body Problem in One Dimension with Repulsive Delta-Function Interaction.” Physical Review Letters 19 (1967): 1312–1315. DOI.
• Zamolodchikov, Al. B. “Thermodynamic Bethe Ansatz in Relativistic Models: Scaling 3-State Potts and Lee–Yang Models.” Nuclear Physics B 342 (1990): 695–720. DOI.