Spinons, solitons and random singlets in the spin-chain compound copper benzoate

Abstract

The $S=1/2$ antiferromagnetic Heisenberg chain is a paradigmatic quantum system hosting exotic excitations such as spinons and solitons, and forming random singlet state in the presence of quenched disorder. Realizing and distinguishing these excitations in a single material remains a significant challenge. Using nuclear magnetic resonance (NMR) on a high-quality single crystal of copper benzoate, we identify and characterize all three excitation types by tuning the magnetic field at ultra-low temperatures. At a low field of 0.2 T, a temperature-independent spin-lattice relaxation rate ($1/T_1$) over more than a decade confirms the presence of spinons. Below 0.4 K, an additional relaxation channel emerges, characterized by $1/T_1 \propto T$ and a spectral weight growing as $-\ln(T/T_0)$, signaling a random-singlet ground state induced by weak quenched disorder. At fields above 0.5 T, a field-induced spin gap $Δ\propto H^{2/3}$ observed in both $1/T_1$ and the Knight shift signifies soliton excitations. Our results establish copper benzoate as a unique experimental platform for studying one-dimensional quantum integrability and the interplay of disorder and correlations.

Figures (4)

• Figure 1: Schematic $H$-$T$ phase diagram of copper benzoate. The diagram shows three regimes with different dominant low-energy excitations: spinons, solitons, and RS state. The characteristic scaling behavior in each regime is indicated.
• Figure 2: Field- and temperature-dependent $^1$H NMR response.a NMR spectra acquired at 2 K with increasing magnetic fields. Peaks are labeled P$_1$–P$_5$. Red and blue arrows indicate peaks assigned to the H(1) and H(2) nuclear sites, respectively; green arrows denote peaks from the H(3) site or benzoate groups. b NMR spectra measured at 2 T with various temperatures down to 0.04 K. c Knight shift $K_{\rm n}$ measured at the H(1) site for magnetic fields of 2 T. The peak position in $K_{\rm n}$ is indicated by the downward arrow.
• Figure 3: Low-field spin-lattice relaxation rates.a Nuclear magnetization $M(t)$ as a function of time, measured at the P$_5$ peak under a magnetic field of 0.2 T at various temperatures. Solid lines represent fits to the data using a single-exponential function (at 0.4 K and above) and a double-exponential function (at 0.1 K and below) to extract the spin-lattice relaxation time $T_1$. b The fast component of $1/T_1$ as a function of temperature, measured at different magnetic fields. The dashed line is a guide to the eye, indicating a constant $1/T_1$ at low temperatures. c Spectral distribution $P(1/T_1)$ as a function of temperature obtained from the ILTA of $M(t)$ at 0.2 T. The color scale represents the relative spectral weight. The dotted and solid lines indicate constant $1/T_1$ and $1/T_1 \sim T$ behaviors for the fast and slow components, respectively. d Temperature dependence of the relative weight $w_2$ of the slow $T_1$ component in $M(t)$, derived from $P(1/T_1)$ at 0.2 T. The solid line is a fit to the function $w_2 = -a \ln(T/T_0)$ with $T_0 \approx 0.4$ K. e Correlation length $\xi$ (in units of the lattice constant $a$) as a function of temperature, obtained from quantum Monte Carlo (QMC) simulations for the RS state. The solid line is a fit to $\xi/a \propto -\ln(T/T_0)$.
• Figure 4: Field-induced gap and soliton excitations.a NMR spectra measured at 5 T as a function of temperature. b Knight shift $K_{\rm{n}}$ as a function of temperature at selected magnetic fields. Solid lines are fits using a thermal activation form (see text). c Spin-lattice relaxation rate $1/T_1$ at the P$_5$ site as a function of temperature. Solid lines are fits based on a gapped excitation spectrum (see text). d Magnetic field dependence of the excitation gap $\Delta$ extracted from different measurements. Data from specific heat ($C_{\rm P}$) and INS 1997_PRL_Dender are included for comparison. The solid line is a fit to $\Delta = 1.85 J (h/J)^{2/3}$. e Dipolar hyperfine field $H_{\rm hf}$ at the H(1) site calculated using the tanTRG method. Solid triangles and squares denote results with and without the staggered magnetization component included in addition to the uniform magnetization.