Thermodynamic Bethe Ansatz and Finite-Size Ground-State Energy

The thermodynamic Bethe ansatz converts factorized scattering into a nonlinear equation for rapidity occupation and, after exchanging Euclidean space and time, into the finite-size ground-state energy. The continuum equation is exact for the declared integrable scattering theory, statistics convention, and spectrum; a finite numerical solution still depends on rapidity cutoff, discretization, iteration, and branch choices. Infrared particle physics and the ultraviolet effective central charge provide independent checks.

Required background. Bethe quantization and finite-volume spectra supplies the finite-volume phase equations and their statistics conventions.

Root densities from the Bethe equations

Consider one stable particle of mass $m$ with diagonal elastic amplitude $S(\theta)$. Choose a continuous real-axis phase

$$S(\theta)=e^{i\delta(\theta)}$$

and define the kernel

$$\boxed{ \varphi(\theta) =-i\frac{\mathrm d}{\mathrm d\theta}\log S(\theta) =\delta'(\theta) }.$$

The equality fixes the logarithm branch; adding a constant $2\pi$ to $\delta$ changes no kernel, while a discontinuous branch can introduce a spurious delta function.

In a large spatial length $L$, let $\rho_{\rm p}(\theta)$ and $\rho_{\rm h}(\theta)$ be occupied-particle and hole densities per unit length and rapidity. Differentiating the logarithmic Bethe equation gives

$$\rho_{\rm p}(\theta)+\rho_{\rm h}(\theta) = \frac{m}{2\pi}\cosh\theta +\int_{-\infty}^{\infty} \frac{\mathrm d\theta'}{2\pi}\, \varphi(\theta-\theta')\rho_{\rm p}(\theta').$$

The sign of the convolution follows from the convention $mL\sinh\theta_i+\sum_j\delta(\theta_i-\theta_j)=2\pi I_i$. Changing the definition of $\varphi$ requires changing every later sign.

For the fermionic Bethe-occupation convention, the entropy density is

$$\begin{aligned} s={}& \int\mathrm d\theta\, \bigl[ (\rho_{\rm p}+\rho_{\rm h}) \log(\rho_{\rm p}+\rho_{\rm h}) \\ &\hspace{5em} -\rho_{\rm p}\log\rho_{\rm p} -\rho_{\rm h}\log\rho_{\rm h} \bigr]. \end{aligned}$$

“Fermionic” here describes exclusion of Bethe quantum numbers. It need not equal the microscopic spin-statistics label. A bosonic occupation convention changes the entropy, logarithm, and sometimes a zero-rapidity phase; these changes must be made as one package.

Pseudoenergy equation

Let $R$ be the inverse temperature in the thermal channel, or the spatial circumference after the Euclidean channel exchange, and set $r=mR$. Minimize energy minus $R^{-1}$ times entropy subject to the density constraint. With

$$\epsilon(\theta) =\log\frac{\rho_{\rm h}(\theta)}{\rho_{\rm p}(\theta)},$$

the stationary condition is

$$\boxed{ \epsilon(\theta) =r\cosh\theta -\int_{-\infty}^{\infty} \frac{\mathrm d\theta'}{2\pi}\, \varphi(\theta-\theta') \log\!\left(1+e^{-\epsilon(\theta')}\right) }.$$

The bulk-subtracted ground-state energy on the circle is

$$\boxed{ E_0^{\rm sub}(R) = -\frac{m}{2\pi} \int_{-\infty}^{\infty}\mathrm d\theta\, \cosh\theta\, \log\!\left(1+e^{-\epsilon(\theta)}\right) }.$$

The un-subtracted energy is

$$E_0(R)=\mathcal E_{\rm bulk}R+E_0^{\rm sub}(R),$$

where $\mathcal E_{\rm bulk}$ is scheme dependent. These equations and their relativistic finite-size interpretation are derived in Zamolodchikov 1990, § 2, equations (2.22)–(2.39), pp. 703–706. The density-and-entropy variational method descends from Yang and Yang 1969, §§ 2–3, pp. 1117–1122.

For several species, $\epsilon_a$, $m_a$, and $\varphi_{ab}=-i\partial_\theta\log S_{ab}$ become a coupled system. With non-diagonal scattering, one first diagonalizes the transfer problem and may need auxiliary or magnonic pseudoenergies. The one-equation formula must not be applied by replacing a matrix S matrix with an arbitrary eigenvalue.

Sinh-Gordon kernel check

For the pole-free scalar fixture

$$S_B(\theta) = \frac{\sinh\theta-i\sin(\pi B)} {\sinh\theta+i\sin(\pi B)}, \qquad 0<B<1,$$

write $s_B=\sin(\pi B)$. Direct differentiation gives

$$\varphi_B(\theta) = \frac{2s_B\cosh\theta} {\sinh^2\theta+s_B^2}.$$

The kernel is real, even, positive, and exponentially decaying at large $\lvert\theta\rvert$. Those properties are independent checks on the sign and logarithm branch. They do not alone prove that a numerical iteration has found the correct pseudoenergy.

Free-Majorana infrared and ultraviolet checks

For a free massive Majorana particle in the fermionic convention,

$$S(\theta)=-1, \qquad \varphi(\theta)=0.$$

The pseudoenergy is exactly

$$\epsilon(\theta)=r\cosh\theta,$$

and

$$E_0^{\rm sub}(R) = -\frac{m}{2\pi} \int_{-\infty}^{\infty}\mathrm d\theta\, \cosh\theta \log\!\left(1+e^{-r\cosh\theta}\right).$$

For $r\gg1$, expand the logarithm. Since

$$\int_{-\infty}^{\infty}\mathrm d\theta\, \cosh\theta\,e^{-r\cosh\theta} =2K_1(r),$$

the leading infrared term is

$$E_0^{\rm sub}(R) =-\frac{m}{\pi}K_1(r)+O(e^{-2r}).$$

This checks the sign, normalization, and one-particle wrapping behavior.

In the ultraviolet, define

$$c_{\rm eff}(R) = -\frac{6R}{\pi} \left[ E_0(R)-\mathcal E_{\rm bulk}R \right].$$

The free-Majorana solution approaches

$$\lim_{R\to0}c_{\rm eff}(R)=\frac12.$$

For a unitary theory with the identity vacuum this equals the CFT central charge. In a nonunitary theory it is $c_{\rm eff}=c-24h_{\min}$, so calling it $c$ without identifying the lowest conformal weight is incorrect. The infrared Bessel expansion and ultraviolet effective-central-charge analysis are given in Zamolodchikov 1990, § 3, pp. 708–711.

Numerical solution and error separation

A trustworthy calculation records:

1. the S-matrix branch and kernel sign;
2. the statistics or occupation convention;
3. rapidity cutoff $\Theta$, grid or quadrature rule, and interpolation;
4. iteration, damping, and stopping criterion;
5. the supremum or weighted norm of the nonlinear residual;
6. stability under increasing $\Theta$ and refining the grid;
7. energy stability beyond the pseudoenergy residual;
8. the $r\gg1$ particle or Bessel limit; and
9. ultraviolet convergence of $c_{\rm eff}$ after bulk subtraction.

Cutoff, discretization, and nonlinear-solver errors are distinct. Agreement with the ultraviolet endpoint does not by itself prove the proposed RG trajectory: different kernels can share an endpoint while differing at intermediate $r$.

The integrability exact-data chain places the infrared and ultraviolet checks after the scattering and spectrum inputs. The exact and rigorous status comparison separates the exact continuum equation from its discretized solution and from a constructive local-QFT result.

Common pitfalls

Changing the kernel sign in isolation. The density equation, pseudoenergy equation, and phase definition must transform together. Recomputing $-i\partial_\theta\log S$ is the fastest check.

Equating microscopic spin with TBA statistics. Bethe-state exclusion and the convention for $S(0)$ determine the entropy factor. State the package used.

Extracting a central charge before subtracting bulk energy. The extensive term $\mathcal E_{\rm bulk}R$ is scheme dependent and contaminates the $1/R$ coefficient if left in place.

Exercises

1. Differentiate the sinh-Gordon fixture and derive the displayed kernel.

Solution

With $s_B=\sin(\pi B)$,

$$\begin{aligned} -i\frac{\mathrm d}{\mathrm d\theta}\log S_B &= -i\cosh\theta \left[ \frac{1}{\sinh\theta-is_B} -\frac{1}{\sinh\theta+is_B} \right] \\ &= \frac{2s_B\cosh\theta} {\sinh^2\theta+s_B^2}. \end{aligned}$$

The final expression is even and real for real $\theta$.

2. Derive the leading free-Majorana infrared energy from the logarithm expansion and state its control parameter.

Solution

For $r\gg1$, $\log(1+e^{-r\cosh\theta})=e^{-r\cosh\theta}+O(e^{-2r\cosh\theta})$. Using the integral representation of $K_1$ gives

$$E_0^{\rm sub}(R) =-\frac{m}{2\pi}\,2K_1(r)+O(e^{-2r}) =-\frac{m}{\pi}K_1(r)+O(e^{-2r}).$$

The expansion is controlled by $e^{-r}=e^{-mR}$.

References

• Yang, C. N., and C. P. Yang. “Thermodynamics of a One-Dimensional System of Bosons with Repulsive Delta-Function Interaction.” Journal of Mathematical Physics 10 (1969): 1115–1122. 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.