Quantum spin model for Reggeon field theory

Abstract

We study the Hamiltonian form of Reggeon field theory on a lattice in the two-dimensional $D = 2$ transverse or impact-parameter space. The Hamiltonian formalism allows naturally for the privileged character of the longitudinal variable or rapidity, which is kept continuous. Based on recent results for the one-dimensional theory, we argue that we may truncate the single-site basis of the Hamiltonian by retaining the lowest two states only, and we arrive at a lattice spin model. In terms of Pauli spin matrices ${\sigma }_k^{\overrightarrow{n}}$ at each site $\overrightarrow{n} = (n_1,n_2)$, our Reggeon quantum spin model has the Hamiltonian $H=\Sigma {\overrightarrow{n}}^{}\left(\frac{\delta }{2}\right)(1-{\sigma }_x^{\overrightarrow{n}})+\Sigma {\overrightarrow{n}·\hat{i}}^{}A[1-2\mathrm{}{\sigma }_{+}^{\overrightarrow{n}+\hat{i}})(1-2\mathrm{}{\sigma }_{+}^{\overrightarrow{n}})-{\sigma }_z^{\overrightarrow{n}+\hat{i}}{\sigma }_z^{\overrightarrow{n}}]$, where $\hat{i}$ represents the lattice unit vectors (1,0) and (0,1). The parameters $\delta$ and $A$ are related to those of the original field theory. All the approximations are valid for small Regge slope ${\alpha }_0^{\prime }$ in the region of the phase transition at output Regge intercept $\alpha \mathrm{}\left(0\right)=1\mathrm{}$. The critical exponents of the original system can be determined by the properties of the low-energy states of $H$ in strict analogy with the established relation between the ${\phi }^4$ theory in $d = 2$ and the ground state of the quantum Ising model with a transverse magnetic field in $D=d-1\mathrm{}=1$ dimension. The general properties of the Reggeon quantum spin model are exhibited, and several new approximation schemes for Reggeon field-theoretic calculations are suggested. While the above Hamiltonian is adequate, by universality, to describe the critical Pomeron, systematic improvements can be made to study the theory away from criticality by retaining more states in the single-site basis of the Hamiltonian.