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Physical review. B./Physical review. B

Conformal anomaly in finite-temperature magnetic response of one-dimensional spin systems

Christian Northe, Chao Zhang, R. Wawrzyńczak, Johannes Gooth, Stanisław Gałeski, E. M. Hankiewicz

Published Sep 4, 2024• 0 citations
Publisher

Abstract

The conformal anomaly indicates the breaking of conformal symmetry (angle-preserving transformations) in quantum theory by quantum fluctuations, and is a close cousin of the gravitational anomaly. We show that the conformal anomaly controls the variance of the local magnetization ${M}_{\mathrm{loc}}$ at finite temperatures in spin chains and spin ladders. This effect is predicted to appear at constant and variable temperature across the sample. The change of ${M}_{\mathrm{loc}}$ induced by the conformal anomaly is of the order of 3--5% of angular momentum $\ensuremath{\hbar}/2$ at 1 K for ${({\mathrm{C}}_{7}{\mathrm{H}}_{10}\mathrm{N})}_{2}{\mathrm{CuBr}}_{4}$ (DIMPY) or copper pyrazine nitrate (CuPzN) and increases linearly with temperature. Further, for a temperature gradient of 10% across the sample, the time relaxation of the nonequilibrium ${M}_{\mathrm{loc}}$ is of the order of nanoseconds. Thus, we believe that experimental techniques such as neutron scattering, nuclear magnetic resonance (NMR), spin noise, and ultrafast laser pumping should determine the presence of the conformal anomaly. Therefore, we outline a method to detect the conformal anomaly in spin observables of strongly interacting low-dimensional magnets.

DOI
10.1103/physrevb.110.104409

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The conformal anomaly indicates the breaking of conformal symmetry (angle-preserving transformations) in quantum theory by quantum fluctuations, and is a close cousin of the gravitational anomaly. We show that the conformal anomaly controls the variance of the local magnetization Mloc{M}_{\mathrm{loc}}Mloc​ at finite temperatures in spin chains and spin ladders. This effect is predicted to appear at constant and variable temperature across the sample. The change of Mloc{M}_{\mathrm{loc}}Mloc​ induced by the conformal anomaly is of the order of 3--5% of angular momentum \ensuremathℏ/2\ensuremath{\hbar}/2\ensuremathℏ/2 at 1 K for (C7H10N)2CuBr4{({\mathrm{C}}_{7}{\mathrm{H}}_{10}\mathrm{N})}_{2}{\mathrm{CuBr}}_{4}(C7​H10​N)2​CuBr4​ (DIMPY) or copper pyrazine nitrate (CuPzN) and increases linearly with temperature. Further, for a temperature gradient of 10% across the sample, the time relaxation of the nonequilibrium Mloc{M}_{\mathrm{loc}}Mloc​ is of the order of nanoseconds. Thus, we believe that experimental techniques such as neutron scattering, nuclear magnetic resonance (NMR), spin noise, and ultrafast laser pumping should determine the presence of the conformal anomaly. Therefore, we outline a method to detect the conformal anomaly in spin observables of strongly interacting low-dimensional magnets.
The conformal anomaly indicates the breaking of conformal symmetry (angle-preserving transformations) in the quantum theory by quantum fluctuations and is a close cousin of the gravitational anomaly. We show, for the first time, that the conformal anomaly controls the variance of the local magnetization MlocM_{loc}Mloc​ at finite temperatures in spin chains and spin ladders. This effect is perceived at constant and variable temperature across the sample. The change of MlocM_{loc}Mloc​ induced by the conformal anomaly is of the order of 3-5\% of the maximal spin at one Kelvin for DIMPY or CuPzN and increases linearly with temperature. Further, for a temperature gradient of 10\% across the sample, the time-relaxation of the non-equilibrium MlocM_{loc}Mloc​ is of the order of nanoseconds. Thus, we believe that experimental techniques such as neutron scattering, nuclear magnetic resonance~(NMR), spin noise and ultrafast laser pumping should pinpoint the presence of the conformal anomaly. Therefore, we pave the road to detect the conformal anomaly in spin observables of strongly interacting low-dimensional magnets.
The effect of thermal fluctuations on the dynamics of a gapped quantum magnet is studied using inelastic neutron scattering on copper nitrate, a model material for the spin-1/2, one-dimensional (1D) bond alternating Heisenberg chain. A large, highly deuterated, single-crystal sample of copper nitrate is produced using a solution growth method and measurements are made using the high-resolution backscattering spectrometer OSIRIS at the ISIS Facility. Theoretical calculations and numerical analysis are combined to interpret the physical origin of the thermal effects observed in the magnetic spectra. The primary observations are (1) a thermally induced central peak due to intraband scattering, which is similar to Villain scattering familiar from soliton systems in 1D, and (2) the one-magnon quasiparticle pole is seen to develop with temperature into an asymmetric continuum of scattering. We relate this asymmetric line broadening to a thermal strongly correlated state caused by hard-core constraints and quasiparticle interactions. These findings are a counter example to recent assertions of the universality of line broadening in 1D systems and are applicable to a broad range of quantum systems.
We show that the leading finite-size correction to lnZ for a two-dimensional system at a conformally invariant critical point on a strip of length L width \ensuremath{\beta}(\ensuremath{\beta}\ensuremath{\ll}L) is (\ensuremath{\pi}/6)c(L/\ensuremath{\beta}), where c is the conformal anomaly. Equivalently, the leading low-temperature correction to the free energy of a one-dimensional quantum system is -(\ensuremath{\pi}/6)cL(kT)2{)}^{2})2\ensuremath{\Elzxh}v, where v is the effective ``velocity of light.'' The latter formula is used to check recently derived critical theories of spin-s quantum chains against Bethe-Ansatz solutions.
This thesis is concerned with the numerical study of one-dimensional (1D) spin-1/2 quantum magnets and related method development. Its focus is on the calculation of dynamical spin correlation functions both at zero and finite temperature. This is motivated by the accessibility of dynamical quantities in experiments such as inelastic neutron scattering (INS) and electron spin resonance (ESR). The numerical methods used in this thesis are based on extensions of the density-matrix renormalization group (DMRG) and are formulated in the framework of matrix product states (MPS). While zero-temperature dynamical correlation functions are computed with existing MPS frequency-domain methods, an MPS frequency-domain approach for their calculation at finite temperature is developed in this thesis. The new method combines the Liouville-space formulation of the dynamics with a moment expansion of the dynamical correlation function. The majority of the results are obtained via MPS-based Chebyshev expansions. These numerical techniques are applied to two different model systems describing real materials. The first one is the material copper pyrimidine dinitrate (Cu-PM) which is modeled by a 1D spin-1/2 Heisenberg antiferromagnet with Dzyaloshinskii-Moriya interactions. The spin dynamics of this model is studied in an applied magnetic field and compared to ESR experiments. Zero-temperature calculations for momentum- and frequency-resolved dynamical quantities give direct access to the intensity of the elementary excitations and go beyond the low-energy description by the quantum sine-Gordon model. Thus, a deviation from the Lorentz invariant dispersion for the single-soliton resonance is found. The presence of the strongest boundary bound state previously predicted from a boundary sine-Gordon field theory is confirmed, while composite boundary-bulk excitations have too low intensities to be found in the numerical results. At finite temperature, there is a temperature-induced crossover of the soliton. Moreover, additional temperature effects such as interbreather transitions emerge, which is confirmed by accompanying ESR experiments on Cu-PM over a wide range of the applied field strength. The second system studied in this thesis is the compound BaCu2_22​V2_22​O8_88​. It is shown that the magnetic properties of this quasi-1D material can be described by a strongly alternating antiferromagnetic-ferromagnetic spin-1/2 Heisenberg chain. As found for other dimerized systems, the strong correlations in BaCu2_22​V2_22​O8_88​ persist even at elevated temperatures. Moreover, these correlations lead to an asymmetric lineshape broadening of the magnetic excitations at finite temperature. Upon raising the temperature, an increasingly asymmetric lineshape is observed in high-resolution INS experiments for BaCu2_22​V2_22​O8_88​. In this thesis, the lineshape is calculated by the MPS finite-temperature method developed as a part of this work. Comparing these results to the INS data and the lineshape obtained by a diagrammatic approach, excellent agreement is found over a broad temperature range. This demonstrates that coherent quantum behavior persists at elevated temperatures in BaCu2_22​V2_22​O8_88​ and that it can be predicted quantitatively.
We study dynamical properties at finite temperature (T)(T)(T) of Heisenberg spin chains with random antiferromagnetic exchange couplings, which realize the random singlet phase in the low-energy limit, using three complementary numerical methods: exact diagonalization, matrix-product-state algorithms, and stochastic analytic continuation of quantum Monte Carlo results in imaginary time. Specifically, we investigate the dynamic spin structure factor $S(q,\ensuremath{\omega})$ and its $\ensuremath{\omega}\ensuremath{\rightarrow}0$ limit, which are closely related to inelastic neutron scattering and nuclear magnetic resonance (NMR) experiments (through the spin-lattice relaxation rate 1/T1)1/{T}_{1})1/T1​). Our study reveals a continuous narrow band of low-energy excitations in $S(q,\ensuremath{\omega})$, extending throughout the qqq space, instead of being restricted to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as found in the uniform system. Close to $q=\ensuremath{\pi}$, the scaling properties of these excitations are well captured by the random-singlet theory, but disagreements also exist with some aspects of the predicted qqq dependence further away from $q=\ensuremath{\pi}$. Furthermore we also find spin diffusion effects close to q=0q=0q=0 that are not contained within the random-singlet theory but give non-negligible contributions to the mean 1/T11/{T}_{1}1/T1​. To compare with NMR experiments, we consider the distribution of the local relaxation rates 1/T11/{T}_{1}1/T1​. We show that the local 1/T11/{T}_{1}1/T1​ values are broadly distributed, approximately according to a stretched exponential. The mean 1/T11/{T}_{1}1/T1​ first decreases with TTT, but below a crossover temperature it starts to increase and likely diverges in the limit of a small nuclear resonance frequency ${\ensuremath{\omega}}_{0}$. Although a similar divergent behavior has been predicted and experimentally observed for the static uniform susceptibility, this divergent behavior of the mean 1/T11/{T}_{1}1/T1​ has never been experimentally observed. Indeed, we show that the divergence of the mean 1/T11/{T}_{1}1/T1​ is due to rare events in the disordered chains and is concealed in experiments, where the typical 1/T11/{T}_{1}1/T1​ value is accessed.
By means of inelastic neutron scattering we investigate finite temperature dynamics in the quantum spin ladder compound (C_{5}H_{12}N)_{2}CuBr_{4} (BPCB) near the magnetic field induced quantum critical point with dynamical exponent z=2. We observe universal finite-temperature scaling of the transverse local dynamic structure factor in spectacular quantitative agreement with long-standing theoretical predictions. At the same time, already at rather low temperatures, we observe strong nonuniversal longitudinal fluctuations. To separate the two, we make use of an intrinsic leg-exchange symmetry of the spin ladder. Complementary measurements of specific heat also reveal striking scaling behavior near the quantum critical point.