Universal Relations in Long-range Quantum Spin Chains

Abstract

Understanding the emergence of novel collective behaviors in strongly interacting systems lies at the heart of quantum many-body physics. Valuable insight comes from examining how few-body correlations manifest in many-body systems, embodying the ``from few to many'' philosophy. An intriguing example is the set of universal relations in ultracold atomic gases, which connect a wide range of observables to a single quantity known as the contact. In this Letter, we demonstrate that universal relations manifest in a distinct class of quantum many-body systems, long-range quantum spin chains, which belong to a completely new universality class. Using effective field theory and the operator product expansion, we establish connections between the asymptotic behavior of equal-time spin correlation functions, the dynamical structure factor, and the contact density. The theoretical predictions for equal-time correlators are explicitly verified through numerical simulations based on matrix product states. Our results could be readily tested in state-of-the-art trapped-ion systems.

Figures (5)

• Figure 1: We present a schematic of our main results. We consider a one-dimensional long-range quantum spin chain with spin-rotation symmetry along the $z$-direction, focusing on $\alpha\in(3/2,2)$, where the system is strongly interacting near the two-magnon resonance. In the dilute magnon regime, we derive universal relations that connect equal-time spin correlation functions and the dynamical structure factor to the contact density. Although the system lacks full spin-rotation symmetry, the spin correlators of different type share the same contact constant $C$.
• Figure 2: Feynman diagrams for (a) the self-energy of the dimer field $d$, (b) the matrix element of $\bar{d}d(x)$, (c) the matrix element of $\bar{\psi}\psi(x_1)\bar{\psi}\psi(x_2)$, and (d) the matrix element of $\bar{\psi}(x_1)\psi(x_2)$. In all diagrams, solid and dashed lines denote the bare propagators of $\psi$ and $d$, respectively. Double dashed lines denote the dressed propagator of the dimer field $d$.
• Figure 3: Numerical results for equal-time correlators with $\alpha=1.9$ and $J_z/J=1.9$ in a system of size $L=200$ with open boundary conditions, evaluated in the ground state of the three-magnon sector ($N=3$). The correlators are shown as functions of $|i-j|$ for different values of $X=(i+j)/2$. Data points represent numerical results, while solid lines correspond to fits over $|i-j| \in [4,30]$, as elaborated in the main text.
• Figure 4: Results for the contact density $c(X)$, extracted from fits to correlators in the ground state of the system with $N=2,3,4$ magnons. We set $\alpha=1.9$, $J_z/J=1.9$, and consider a system of size $L=200$ with open boundary conditions. Error bars are estimated by varying the fitting range $|i-j| \in [L_s,30]$ with $L_s \in \{2,4,6,8\}$.
• Figure 5: We plot the universal behavior of the dynamical structure factor as a function of $\omega / (u k^z)$ for various $\alpha$, which is non-vanishing only for $\omega/uk^z>1/2^{z-1}$. The integral expression are provided in the supplementary material SM.