Numerical renormalization group integrated Hamiltonian truncation: Toward generic deformation of integrable lattice models

Abstract

We present a hybrid lattice Hamiltonian truncation method that integrates the numerical renormalization group (NRG) with a truncated lattice integrable spectrum. The technique is tailored for generic deformations of integrable lattice models, where the NRG enables a controlled incorporation of high-energy states. The method extends the basis set more effectively and efficiently than brute-force truncation, meanwhile significantly reducing errors. We show its capability on two paradigmatic models: an Ising chain in a magnetic field and a quantum Ising ladder. The resulting dynamical structure factors accurately capture the essential low-energy physics, including the $E_8$ and $\mathcal{D}_8^{(1)}$ excitations of the former and later models, respectively, demonstrating the approach's computational efficiency and high performance.

Figures (3)

• Figure 1: An illustration of the NRG procedure to diagonalize $H$ from $m$-th step to $m+1$-th step. With $H_{\Phi \Phi}^{(m)},~H_{\Phi \phi}^{(m)},~H_{\phi \Phi}^{(m)}~\text{and}~H_{\phi \phi}^{(m)}$ being the matrices block of $\mathcal{H}$ under certain bases.
• Figure 2: (a) Ground state energy of $H_{{}_\text{ICMF}}^{{}^\text{OBC}}$ at $h_z = 0$ for $L$ from 30 to 60, under different transverse fields with $g = 0.8$ and $g = 1.0$. The inset: $\delta E$ vs $L$ with different $g$. (b) DSF $S^{xx}(\omega \leq 4 m_1,q = 0)$ with $L = 100,~g = 1,~h_z = 0.05$. Red dashed lines mark masses of seven single $E_8$-particles. Green and blue dashed lines label the two-particle and three-particle threshold energies, respectively.
• Figure 3: (a) Ground state energy of $H_{\text{II}}$ with OBC calculated by DMRG and NRG-TLISA for size $L = 20$ to $L=60$ with $g = 1,~~g_i = 0.1$. Inset: $\delta E$ vs $L$. (b) DSF $S^{xx(1)}(\omega<3m_{B_1},q = 0)$ with $L = 100,~g = 1,~g_i = 0.1$. Red dashed lines denote the breather $m_{B_2}$ and soliton $m_A$ masses and green dashed lines denote the two-particle ($2 m_{B_1}$) threshold energy.