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arXiv:cond-mat/9902028 · Physical Review Letters

Edge Logarithmic Corrections Probed by Impurity NMR

Vivien Brunel, Marc Bocquet, Th. Jolicœur

Published Oct 4, 1999• 22 citations
arXivHTMLPDF

Abstract

Semi-infinite quantum spin chains display spin autocorrelations near the boundary with power-law exponents that are given by boundary conformal field theories. We show that NMR measurements on spinless impurities that break a quantum spin chain lead to a spin-lattice relaxation rate $1/{T}_{1}^{\mathrm{edge}}$ that has a temperature dependence which is a direct probe of the anomalous boundary exponents. For the antiferromagnetic $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ spin chain, we show that $1/{T}_{1}^{\mathrm{edge}}\ensuremath{\propto}T{log}^{2}T$ instead of ${log}^{1/2}T$ for a bulk measurement. We show that, for a one-dimensional conductor described by a Luttinger liquid, a similar measurement leads to a rate $1/{T}_{1}^{\mathrm{edge}}\ensuremath{\propto}T$, independent of the exponent ${K}_{\ensuremath{\rho}}$.

DOI
10.1103/physrevlett.83.2821

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Semi-infinite quantum spin chains display spin autocorrelations near the boundary with power-law exponents that are given by boundary conformal field theories. We show that NMR measurements on spinless impurities that break a quantum spin chain lead to a spin-lattice relaxation rate 1/T1edge1/{T}_{1}^{\mathrm{edge}}1/T1edge​ that has a temperature dependence which is a direct probe of the anomalous boundary exponents. For the antiferromagnetic S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 spin chain, we show that 1/T1edge\ensuremath∝Tlog2T1/{T}_{1}^{\mathrm{edge}}\ensuremath{\propto}T{log}^{2}T1/T1edge​\ensuremath∝Tlog2T instead of log1/2T{log}^{1/2}Tlog1/2T for a bulk measurement. We show that, for a one-dimensional conductor described by a Luttinger liquid, a similar measurement leads to a rate 1/T1edge\ensuremath∝T1/{T}_{1}^{\mathrm{edge}}\ensuremath{\propto}T1/T1edge​\ensuremath∝T, independent of the exponent K\ensuremathρ{K}_{\ensuremath{\rho}}K\ensuremathρ​.
Quasi-one-dimensional (1D) antiferromagnets are good candidates to realize a Tomonaga-Luttinger liquid (TLL), describing 1D interacting quantum systems, provided 3D coupling remains small as compared to temperature. In systems such as NiCl2{}_{2}2​-4SC(NH2{}_{2}2​)2{}_{2}2​ (DTN), made of weakly coupled spin S=1S=1S=1 chains, interchain coupling plays an important role at low temperature as it leads to antiferromagnetic long-range order below 111 K, corresponding to a Bose-Einstein condensate of magnons. Using time-dependent numerical simulations at finite temperature, the authors compute here the nuclear magnetic resonance (NMR) relaxation rate 1/T11/{T}_{1}1/T1​, a key experimental quantity, and address the question of the low-temperature crossover between TLL predictions and the higher-temperature regime in purely 111D quantum spins chains. They also comment on the experimental observation of the TLL regime, possibly masked by 333D long-range order in materials such as DTN.
Dynamic spin correlation functions 〈Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​(t)Sjx{\mathit{S}}_{\mathit{j}}^{\mathit{x}}Sjx​〉 for the one-dimensional S=1/2 XX model H=-Jtsumi{\mathit{Jtsum}}_{\mathit{i}}Jtsumi​{Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​Si+1x{\mathit{S}}_{\mathit{i}+1}^{\mathit{x}}Si+1x​+Siy{\mathit{S}}_{\mathit{i}}^{\mathit{y}}Siy​Si{\mathit{S}}_{\mathit{i}}Si​ $_{+1}^{\mathit{y}}\mathrm{}}$ are calculated exactly for finite open chains with up to N=10 000 spins. Over a certain time range the results are free of finite-size effects and thus represent correlation functions of an infinite chain (bulk regime) or a semi-infinite chain (boundary regime). In the bulk regime, the long-time asymptotic decay as inferred by extrapolation is Gaussian at T=\ensuremath{\infty}, exponential at 0T\ensuremath{\infty}, and power law (\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}1/2}$) at T=0, in agreement with exact results. In the boundary regime, a power-law decay is obtained at all temperatures; the characteristic exponent is universal at T=0 (\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$) and at 0T\ensuremath{\infty}(\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}3/2}$), but is site dependent at T=\ensuremath{\infty}. In the high-temperature regime (T/J\ensuremath{\gg}1) and in the low-temperature regime (T/J\ensuremath{\ll}1), crossovers between different decay laws can be observed in 〈Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​(t)Sjx{\mathit{S}}_{\mathit{j}}^{\mathit{x}}Sjx​〉. Additional crossovers are found between bulk-tpe and boundary-type decay for i=j near the boundary, and between spacelike and timelike behavior for i\ensuremath{\ne}j.
We study the dynamics of the one-dimensional Ising model in the presence of a static symmetry-breaking boundary field via the two-time autocorrelation function of the boundary spin. We find oscillatory correlations that decay as power laws. We uncover a phase diagram of dynamical responses where, upon tuning the strength of the boundary field, we observe distinct power laws that directly correspond to changes in the number of edge modes as the boundary and bulk magnetic field are varied. We suggest how the universal physics can be demonstrated in current experimental setups, such as Rydberg chains.
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We investigate magnetic properties induced by a spinless impurity in strongly correlated electron systems, i.e. the Hubbard model in the spatial dimension D=1,2,D=1,2,D=1,2, and 3. For the 1D system exploiting the Bethe ansatz exact solution we find that the spin susceptibility and the local density of states in the vicinity of a spinless impurity show divergent behaviors. The results imply that the induced local moment is not completely quenched at any finite temperatures. On the other hand, the spin lattice relaxation rate obtained by bosonization and boundary conformal field theory satisfies a relation analogous to the Korringa law, 1/T1T∼χ21/T_1T \sim \chi^21/T1​T∼χ2. In the 2D and 3D systems, the analysis based upon the antiferromagnetically correlated Fermi liquid theory reveals that the antiferromagnetic spin fluctuation developed in the bulk is much suppressed in the vicinity of a spinless impurity, and thus magnetic properties are governed by the induced local moment, which leads to the Korringa law of 1/T11/T_11/T1​.
We investigate magnetic properties induced by a spinless impurity in strongly correlated electron systems, i.e., the Hubbard model in spatial dimension D=1,2,D=1,2,D=1,2, and 3.3.3. For the one-dimensional (1D) system exploiting the Bethe ansatz exact solution we find that the spin susceptibility and the local density of states in the vicinity of a spinless impurity show divergent behaviors. The results imply that the induced local moment is not completely quenched at any finite temperatures. On the other hand, the spin lattice relaxation rate obtained by bosonization and boundary conformal field theory satisfies a relation analogous to the Korringa law, ${1/T}_{1}T\ensuremath{\sim}{\ensuremath{\chi}}^{2}.$ In the 2D and 3D systems, the analysis based upon the antiferromagnetically correlated Fermi liquid theory reveals that the antiferromagnetic spin fluctuation developed in the bulk is much suppressed in the vicinity of a spinless impurity, and thus magnetic properties are governed by the induced local moment, which leads to the Korringa law of 1/T1.{1/T}_{1}.1/T1​.