What spinon, soliton, and random-singlet-like excitations appear in copper benzoate?

Three excitation regimes in one spin chain

The paper asks whether a single, high-quality crystal of the S = 1/2 Heisenberg antiferromagnetic chain compound copper benzoate can host several distinct low-energy excitation types, and whether nuclear magnetic resonance (NMR) can tell them apart. The short answer is yes: the authors report spinon signatures at 0.2 T, an extra gapless relaxation channel below 0.4 K that they attribute to random-singlet-like behaviour, and a field-induced spin gap above 1 T that they identify with soliton excitations.

The separation matters because these regimes overlap in temperature but not in field. The spinon evidence comes from a nearly temperature-independent spin-lattice relaxation rate 1/T1 between 1.4 and 0.03 K at 0.2 T, together with a spin-spin relaxation rate 1/T2G that rises as T^(-1/2) below about 2 K. The random-singlet-like channel appears as a second, much slower relaxation component below 0.4 K. The soliton regime is marked by a Knight-shift upturn below about 2 K at fields of 2 to 10 T, fitted with a thermal activation gap.

The material choice is part of the argument. Copper benzoate orders, if at all, at or below 0.8 mK, so experiments above 30 mK should see a nearly pure one-dimensional chain. That one-dimensionality is what allows the intrinsic chain excitations and the disorder-driven low-temperature behaviour to be discussed as separate contributions rather than as one blurred response.

Why a flat relaxation rate signals spinons

1/T1 measures how quickly nuclear spins exchange energy with the surrounding electron spins, so it is sensitive to the density and character of low-energy magnetic excitations. In a spinon picture, the chain's spin-1/2 excitations disperse linearly, and the scaling relation 1/T1 proportional to T^(eta-1) with eta = 1 gives a relaxation rate that does not change with temperature. The observed plateau at 0.2 T is therefore the central spinon evidence, not merely a slow relaxation rate.

The second spinon indicator is 1/T2G, which reflects the spread of local fields from neighbouring spins. Its fitted T^(-1/2) upturn below 2 K matches the behaviour expected for spinons and previously reported in Sr2CuO3. The authors also note that above 4 K, 1/T1 follows a linear-in-T scaling consistent with spin fluctuations at zero wave vector, which is a different regime from the low-temperature plateau.

One expected feature is absent. A proposed logarithmic correction to 1/T1 at low temperature is not seen; instead, 1/T1f shows a weak downturn below 0.3 K that strengthens at 0.5 T and 1 T. The authors attribute that downturn to a field-induced effect rather than to the pure chain, which is why the spinon claim is tied specifically to the lowest-field data.

Two relaxation channels separate disorder from the pure chain

Below 0.2 K the nuclear magnetisation recovery can no longer be fitted with a single exponential. The authors use a double-exponential form with a fast and a slow T1 component, and they check that this two-component behaviour is intrinsic rather than an artefact of overlapping spectral lines. An inverse Laplace transform analysis then produces a distribution of relaxation rates P(1/T1) at each temperature, making the two channels visible as separate populations.

The fast component stays broadly temperature-independent before its field-induced suppression, matching the spinon channel. The slow component is about two orders of magnitude smaller, emerges below 0.4 K, and its rate decreases linearly with temperature on cooling. A linear 1/T1 is gapless behaviour that the pure Heisenberg chain model does not produce, so the authors assign it to regions affected by quenched disorder.

The relative spectral weight of the slow component, w2, grows logarithmically as -ln(T/T0) over 0.03 to 0.4 K, with fitted T0 = 0.40(2) K and c = 25(1). At 0.03 K, w2 reaches 65% and is still increasing, which the authors read as a crossover toward a distinct ground state rather than a phase transition at a sharp temperature.

A diverging Knight shift supports the random-singlet reading

The Knight shift Kn reports the local magnetic field seen by the proton, so it tracks the uniform spin susceptibility of the chain. At 0.2 T, Kn is fitted empirically to Kn proportional to T^(-1) ln^2(Omega/T), with Omega about 10(1) K, close to the exchange scale J of about 18 K. The authors present this as evidence for a crossover to random-singlet-like behaviour, and they stress that the functional form differs from a conventional impurity scenario, which would be gapped.

They also extract w2 from the low-frequency H(1) spectral component and compare it with w2 from the relaxation analysis. Both increase on cooling and both follow a logarithmic form, but the fitted T0 values differ. The authors use that difference to argue that T0 is a broad crossover temperature rather than a phase-transition temperature.

Two cautions belong with this result. The Kn fit is applied only within a narrow temperature window, and the authors state that the exact functional form needs further verification. They also treat the downturn of Kn below 0.2 K as likely an artefact of line broadening and reduced signal-to-noise, not as a physical feature.

A field-induced gap points to solitons

At fields from 2 to 10 T, Kn turns up below about 2 K and saturates at the lowest temperatures. The authors fit this with a thermally activated form Kn = a exp(-Delta/kBT) + b, where Delta is an excitation gap and b comes from a zero-temperature spin-canting effect produced by the staggered field. The gap extracted from these fits increases monotonically with field from 1 to 10 T.

The field dependence is the discriminating evidence. A power-law fit to the NMR data gives Delta proportional to J(h/J)^0.8(1), where h is the effective staggered field, while the theoretical curve for soliton excitations is Delta proportional to J(h/J)^(2/3). The authors compare their gap values with those from specific heat and inelastic neutron scattering and describe the overall behaviour as consistent with soliton excitations arising from the field-induced staggered magnetisation.

The same gapped picture appears in 1/T1 at the P5 site, where the data are fitted with a gapped excitation spectrum. The authors connect the low-energy solitons to an integrable sine-Gordon description and note that the model's coupling constant varies with field, which they present as an opening for studying other predicted excitations such as breathers.

How strong is the disorder behind the random-singlet-like state

The paper does not intentionally dope the crystal, so the authors argue the disorder is likely domain-type: small dirtier regions isolated by much cleaner ones, possibly from stress or from freezing of benzoate-group rotations during growth or cooling. They estimate the disorder level from three independent energy scales: the onset of nonzero random-singlet-like volume fraction below T0 about 0.4 K, a broadening upturn below 0.8 K, and robustness of the gapless behaviour against fields below 0.5 T, corresponding to about 0.4 K.

Combined with J about 18 K, these scales point to a disorder strength of roughly Delta J/J of 0.02 to 0.05. The authors contrast this weak level with the stronger Ge-doped disorder reported in BaCu2(Si0.5Ge0.5)2O7. They also run quantum Monte Carlo simulations of a disordered chain with a bimodal coupling distribution, in which 10% of bonds carry Delta J = 0.03, to model the correlation length of such a weakly disordered chain.

The authors use the random-singlet picture to explain an earlier observation in this material: the suppression of thermal conductivity below 0.3 K. In their account, localised spin excitations inside randomly distributed singlets impede itinerant spinons, and in a strictly one-dimensional chain the percolation threshold sits at zero disorder, so even weak disorder can interrupt transport.

Where the random-singlet picture stops applying

The random-singlet scenario is defined in the zero-temperature and zero-field limit, yet the key evidence here is collected at 0.2 T. The authors address this tension directly: a field polarises long-distance singlets but also reduces the triplet gap for pairs whose exchange coupling is comparable to the Zeeman energy, and pairs with Delta near g muB H become effectively gapless. That combination, they argue, preserves a finite low-energy density of states at low field.

Above 0.5 T, the authors say the Zeeman effect partially destroys the random-singlet-like state, producing coexistence of gapped and gapless excitations through phase separation. This is why the random-singlet-like claim is anchored to the 0.2 T data and does not extend to the higher fields used for the soliton analysis.

The authors also flag what remains unmeasured. A direct measurement of the correlation length and its exact functional form is called for in future work, and local modifications of interchain coupling near rare defects cannot be fully excluded. The random-singlet-like identification therefore rests on a combination of relaxation, spectral-weight, and Knight-shift signatures rather than on a direct measurement of singlet pairing.

The field axis, not temperature alone, separates the phases

The most useful way to read this paper is as a field-dependent map rather than a single claim about copper benzoate. At 0.2 T, the constant 1/T1 plateau and the T^(-1/2) rise of 1/T2G identify spinons. Below 0.4 K at the same low field, the slow relaxation channel with 1/T1 proportional to T and the logarithmically growing w2 identify a random-singlet-like contribution that reaches 65% spectral weight at 0.03 K. Above 1 T, the activated Knight-shift upturn and the gap growing roughly as H^(2/3) identify solitons.

The comparison that matters is between the measured gap exponent and the theoretical one: the NMR fit gives 0.8(1) against the predicted 2/3, and the authors present the soliton identification as consistent with, rather than exactly confirming, that prediction. Similarly, the random-singlet-like state is supported by three signatures but is described as a finite-temperature crossover, with T0 treated as a broad crossover scale rather than a transition temperature.

For readers deciding how far to carry these results, the paper's own boundary is the field axis: the random-singlet-like behaviour is argued to survive below 0.5 T, the soliton regime is established from 1 to 10 T, and the spinon plateau is cleanest at 0.2 T. The distinctive contribution is that one material, probed locally by NMR at millikelvin temperatures, shows all three regimes and lets disorder be separated from the intrinsic chain response.

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