The Bethe Ansatz

Interacting particles in one dimension c.l.i

Let us consider \(C\) different "colors" of particles living on a continuum interval in one dimension, with kinetic enegy given by free dynamics, each type of particle having mass \(m_c\), \(c = 1, ... ,C\). Let there be \(N_c\) particles of color \(c\), so that the total number of particles is \(N = \sum_c N_c\). Let the particles also interact with each other according to a color-dependent potential \(V^{c_1 c_2} (x)\) corresponding to a density-density interaction. A generic first-quantized Hamiltonian can be written

\begin{equation*} H = \sum_{c=1}^{C} \sum_{j_c = 1}^{N_c} \frac{-\hbar^2}{2m_c} \frac{\partial^2}{\partial_{x_{c,j_c}}^2} + \frac{1}{2} \sum_{c_1 \leq c_2 = 1}^{C} \sum_{j_1 = 1}^{N_{c_1}} \sum_{j_2 = 1}^{N_{c_2}} V^{c_1 c_2} (x_{c_1 j_1} - x_{c_2 j_2}). \end{equation*}

Introducing a set of canonical field operators \(\psi_c (x,t)\) obeying the standard equal-time commutation relations

\begin{equation*} \left[ \psi_{c_1} (x_1), \psi^{\dagger}_{c_2} (x_2) \right]_\zeta = \delta_{c_1 c_2} \delta (x_1 - x_2), \hspace{1cm} \left[ \psi_{c_1} (x_1), \psi_{c_2} (x_2) \right]_\zeta = 0 = \left[ \psi^{\dagger}_{c_1} (x_2), \psi^{\dagger}_{c_2} (x_2) \right]_\zeta \end{equation*}

in which \(\zeta = +\) for bosons and \(\zeta = -\) for fermions and \([,]_{\pm}\) denote the commutator (bosons) or anticommutator (fermions), the Hamiltonian is written in operator form (neglecting boundary terms) as

\begin{align*} H =& \sum_{c=1}^{C}\int dx ~\psi^\dagger_c (x) \left(-\frac{\hbar^2}{2m_c} \partial_x^2 \right) \psi_c(x) + \nonumber \\ &+ \frac{1}{2} \sum_{c_1 \leq c_2 =1}^{C}\int dx_1 dx_2~ V^{c_1 c_2} (x_1 - x_2) \psi^\dagger_{c_1} (x_1) \psi^\dagger_{c_2}(x_2) \psi_{c_2} (x_2) \psi_{c_1} (x_1). \end{align*}

Interactions affect bosonic and fermionic systems in dramatically different ways. Spinless fermions develop a ground state Fermi segment (i.e. one-d sphere) of width \(2k_F\) (where \(k_F = \pi n\), \(n\) being their average density) with total energy density \(e_0 = \frac{k_F^3}{3\pi} = \frac{\pi^2}{3} n^3\), even if they do not interact with each other. A single species of fermions will moreover be completely insensitive to ultralocal (\(V(x_1 - x_2) \propto \delta (x_1 - x_2)\)) interactions due to the (real space) Pauli principle. On the other hand, the ground state of noninteracting bosons is a Bose-Einstein condensate-like state where all bosons simultaneously occupy the lowest-energy single-particle state allowed by the quantization conditions. Adding any form of interaction will dramatically alter this picture. Repulsive interactions will lead to the formation of an effective ground state Fermi segment; attractive interactions will cause a collapse to a new ground state where all particles are bound to each other.

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24