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Quantum criticality in the spin-1/2 Heisenberg chain system copper pyrazine dinitrate

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Science Advances  22 Dec 2017:
Vol. 3, no. 12, eaao3773
DOI: 10.1126/sciadv.aao3773


Low-dimensional quantum magnets promote strong correlations between magnetic moments that lead to fascinating quantum phenomena. A particularly interesting system is the antiferromagnetic spin-1/2 Heisenberg chain because it is exactly solvable by the Bethe-Ansatz method. It is approximately realized in the magnetic insulator copper pyrazine dinitrate, providing a unique opportunity for a quantitative comparison between theory and experiment. We investigate its thermodynamic properties with a particular focus on the field-induced quantum phase transition. Thermal expansion, magnetostriction, specific heat, magnetization, and magnetocaloric measurements are found to be in excellent agreement with exact Bethe-Ansatz predictions. Close to the critical field, thermodynamics obeys the expected quantum critical scaling behavior, and in particular, the magnetocaloric effect and the Grüneisen parameters diverge in a characteristic manner. Beyond its importance for quantum magnetism, our study establishes a paradigm of a quantum phase transition, which illustrates fundamental principles of quantum critical thermodynamics.


A quantum phase transition arises when the ground state of a quantum system changes as a function of an external parameter such as pressure or magnetic field. The quantum critical fluctuations associated with this instability often give rise to exotic behavior that is in stark contrast to the conventional properties of materials (1). They are suggested to be at the origin of the anomalous characteristics of a series of correlated electron systems like high-Tc cuprates, Fe-based superconductors, or heavy-fermion compounds (26). However, these systems are so complex that the underlying quantum phase transitions are often hard to identify. In this context, exactly solvable models provide an important guidance for the analysis of enigmatic quantum phase transitions in more complex systems.

Such solvable model systems can be realized by low-dimensional quantum magnets when the exchange interactions J between localized spins are effectively restricted to one or two dimensions. A particularly important model is the spin-1/2 XXZ chainEmbedded Image(1)Here, Embedded Image is the spin-1/2 operator on site i, g is the electronic g factor, μB is the Bohr magneton, Δ describes an anisotropy of the exchange coupling J, and a quantum phase transition can be induced by the magnetic field H. For Δ ≫ 1 or ≪ 1, the model decribed by Eq. 1 covers Ising or XY spin chains, respectively, and in both cases, a transverse magnetic field, that is, Embedded Image, induces the Ising quantum phase transition, which is the most prominent textbook example of quantum criticality (7). Experimental realizations are, for example, the materials LiHoF4 (8), CoNb2O6 (9) with Ising-type anisotropy, and Cs2CoCl4 (10, 11) with XY-type anisotropy.

A quantum phase transition belonging to a different universality class arises in spin-1/2 chains with isotropic Heisenberg exchange (Δ = 1). This model represents one of the most fundamental strongly correlated quantum systems, and the exact solution of its ground state was pioneered by Bethe in 1931 (12). Much later (13, 14), this was extended to finite-temperature calculations of the free energy Embedded Image(T, H) and then further improved (15), which now allows for a precise quantitative prediction of all thermodynamic properties. Up to a critical field Embedded Image, the ground state constitutes a gapless Tomonaga-Luttinger spin liquid, whereas at HHc, an excitation gap opens and the magnetization is fully saturated. Close to this quantum critical field Hc, the free energy per spin obeys the asymptotic expansionEmbedded Image(2)with the scaling functions Embedded Image and Embedded Image (see the Supplementary Materials). The first term in Eq. 2 is the ground-state energy of the field-polarized state that linearly decreases with increasing magnetic field. The second term defines the asymptotic scaling behavior close to quantum criticality. It arises from the thermal population of noninteracting spinon excitations, which are single spin-flip excitations of the polarized ground state that obey Fermi statistics. Their dispersion can be approximated by Embedded Image with mass m = ħ2/(a2J), chemical potential Embedded Image, and lattice constant a. The free energy of these free spinons can be expressed in terms of the function f0 after substituting Embedded Image in the momentum integral with the thermal wavelength Embedded Image. The resulting linear scaling T ~ |HHc|νz implies that νz = 1, with the correlation-length exponent ν = 1/2 and the dynamical exponent z = 2. The third term in Eq. 2 is attributed to the interaction between spinons and describes the leading correction to scaling.

An almost ideal material to study this quantum phase transition is copper pyrazine dinitrate (CuPzN), Cu(C4H4N2)(NO3)2. It comprises spin-1/2 chains of Cu2+ ions along the a axis, which interact via the pyrazine rings (C4H4N2) (see inset of Fig. 1), with an antiferromagnetic exchange J/kB = 10.6 K (16, 17). The electronic g factor is weakly anisotropic (18), with gb = 2.27 for Embedded Image, resulting in a critical field μ0Hc ≃ 13.9 T that is accessible by laboratory magnets. Typical signatures of quantum criticality have been reported for CuPzN (19) based on measurements of magnetization (2022), nuclear magnetic relaxation (23), thermal expansion (24), and magnetic heat transport (25). Deviations from the Heisenberg spin chain model, Eq. 1 with Δ = 1, may result from interchain couplings or anisotropies due to staggered g tensor components, and both perturbations are present in CuPzN. However, the spin chain’s symmetry remains preserved for Embedded Image (26), and the interchain couplings are so weak that long-range antiferromagnetic order only develops below TN = 107 mK (27) in zero magnetic field. Close to the critical field, no indication of this could be detected even down to 80 mK (20). Here, we compare a comprehensive set of thermodynamic data of CuPzN to the analytic Bethe-Ansatz solutions of the Heisenberg chain model.

Fig. 1 Phase diagram of CuPzN with a field-induced quantum critical point.

The color code reflects the measured specific heat C(H,T), and symbols indicate the positions of various thermodynamic signatures that obey the quantum critical scaling T ~ |HHc|νz, with νz = 1 and μ0Hc = 13.9 T. Also shown is the basic structure of the Cu2+ spin chains with S = 1/2 that are exchange-coupled via pyrazine rings C4H4N2.


Figure 2 gives an overview of the experimental data. The a axis thermal expansion α = (∂La ∕ ∂T) ∕ La and magnetostriction λ = (∂La ∕ ∂(μ0H)) ∕ La, in part presented already in a preliminary report (24), are displayed as open symbols in Fig. 2 (A and B, respectively). Below 10 K, α is almost entirely of magnetic origin; phonons hardly contribute anymore, as is illustrated in the inset. In zero field, the magnetic contribution results in a broad maximum around 5 K, which shifts to lower temperature with increasing field. A characteristic sign change of α is observed close to Hc, reflecting entropy accumulation close to the quantum phase transition (28). Close to the critical field, the magnetostriction λ exhibits a strong anomaly that sharpens with decreasing temperature. The magnetic contributions of the Heisenberg chain (1) to α and λ result from a pressure-dependent J(p) (for uniaxial p||a) and are given by Embedded Image and Embedded Image, respectively. Here, VS = 202 Å3 is the volume per spin in CuPzN and Embedded Image. The resulting fits based on Embedded Image(T, H) of the Bethe-Ansatz solution are shown by the solid lines in Fig. 2 (A and B). For α(T, H) = α1D(T, H) + αphon(T), a field-independent phononic background based on the Debye model (see inset) has been included. Both α(T, H) and λ(T, H) are well reproduced, apart from some minor deviations of the low-field α(T) around 5 K, which may, in part, arise from an improper description of αphon(T). The quantum critical signatures around Hc are perfectly reproduced, although there is essentially only one adjustable parameter Embedded Image because J/kB = 10.6 K and gb = 2.27 are known from previous studies (16, 18). Note that this pressure dependence is more than one order of magnitude smaller than the corresponding values of the spin-Peierls system CuGeO3 (29). This suggests that the magnetoelastic coupling in CuPzN is small enough that the magnetic order at 107 mK preempts a spin-Peierls transition, which is an inherent instability of half-integer spin chains toward a combined lattice and spin dimerization (30, 31).

Fig. 2 Thermodynamic quantities of the Heisenberg spin chain compound CuPzN.

(A) Thermal expansion α, (B) magnetostriction λ, (C) magnetic moment m per spin and susceptibility χ = ∂m/∂(μ0H) (insets), (D and E) molar specific heat C as a function of temperature and magnetic field, respectively, and (F) the field derivative of the molar entropy ∂S/∂H. In (D) and (E), the data for increasing field and temperature are offset with respect to each other by 1 and 0.1 J mol−1 K−1, respectively. Symbols are experimental data, and solid lines are fits obtained via the Bethe-Ansatz solution of the Heisenberg spin chain model, with the parameter set J/kB = 10.6 K, Embedded Image, and gb = 2.27. For α and C, field-independent phononic background contributions are included, which are calculated by the Debye formula and become relevant above 5 K [see dashed lines in (A) and (D)]. For C, an additional contribution Cnuc from the nuclear spins of copper becomes relevant at lowest temperatures and high fields. For T = 0.3 K, the calculated Cnuc is shown by the dotted line in (E), whereas the bare Heisenberg contribution C1D is displayed by the dashed line and the sum of both (solid line) reproduces the experimental data. The corresponding nuclear contribution ∂Snuc/∂H is negligibly small as shown by the dotted line in (F).

The magnetization of CuPzN is compared to the Bethe-Ansatz solutions in Fig. 2C. At 0.3 K, the magnetic moment per spin m has a relatively sharp kink close to μ0Hc ≃ 13.9 T and reaches saturation above about 15 T, which causes an asymmetric peak in the differential susceptibility χ = ∂m/∂(μ0H) shown in the upper inset. With increasing temperature, the critical signatures broaden systematically, and the data are well described by the Heisenberg model (solid lines), although the agreement at lowest temperature is not as good as that of λ. The lower inset shows that the model also reproduces χ(T, μ0H = 1 T) up to high temperature.

The molar specific heat C as a function of temperature and field is displayed in Fig. 2 (D and E, respectively). At zero field, C(T) strongly resembles the thermal expansion. This is rooted in the single energy scale J of the Heisenberg model (1) that implies a Grüneisen scaling Embedded Image, with the molar volume Vm = NAVS. The low-temperature C(H) is characterized by a slightly asymmetric double-peak structure centered at Hc that broadens with increasing temperature. Such a double peak is generic for metamagnetic quantum criticality (32). The positive curvature ∂2C/∂H2 at Hc is linked via a Maxwell relation to the curvature of the susceptibility ∂2χ/∂T2 that is positive because of the diverging χ(T → 0, H = Hc).

The behavior of C(T, H) is dominated by the magnetic contribution of the Heisenberg chains, but for its quantitative description, we also have to consider contributions from phonons and from nuclear spins. Although the phonons start to contribute above about 5 K, the nuclear contribution is relevant at lowest temperatures and high fields only, as is shown exemplarily for T = 0.3 K by the dotted line in Fig. 2E. The calculated total specific heat is then given by the solid lines, which perfectly reproduce the experimental data up to 1 K, whereas some systematic deviations on the order of 10% are found around 2.5 K, whose origin remains unclear (see the Supplementary Materials).

Finally, in Fig. 2F, we present the isothermal magnetocaloric effect, that is, the field derivative of the molar entropy ∂S/∂H. Similar to α, this quantity shows a characteristic sign change approaching a divergence on decreasing temperature, which directly reflects the entropy accumulation close to Hc. Again, the experimental data are fully reproduced by the Bethe-Ansatz solution of Eq. 1 (solid lines in Fig. 2F). Note that there is also a contribution from the nuclear spin entropy, shown by the dotted line, but it is so small that it can be safely neglected.


Now, we turn to a discussion of the field-induced quantum criticality and compare the data with the scaling predictions of Eq. 2. For this, we confine ourselves to data obtained below 2 K in the field range μ0Hc ± 4 T. To extract the bare magnetic properties of the Heisenberg spin chains, phononic and/or nuclear background contributions are subtracted. From the full fits of Fig. 2, however, it is inferred that Cnuc causes the only relevant correction in this low-temperature range. According to Eq. 2, susceptibility χ1D, specific heat coefficient C1D/T, thermal expansion α1D, and magnetostriction λ1D are all predicted to diverge as Embedded Image at the critical field. After multiplying by Embedded Image, these quantities are described by universal scaling functions asymptotically close to the quantum critical point when plotted versus the scaling variable gbμBμ0(HHc)/(kBT). These scaling functions are directly related to f0 of Eq. 2 and are shown as solid black lines in Fig. 3 (A to D). We find a very good scaling collapse for C1D and α1D, but substantial deviations are observed for χ1D and are even more pronounced for λ1D. These deviations arise from corrections to scaling that, depending on their relative magnitude, can spoil a full scaling collapse in an extended parameter range. This is confirmed by the blue and red dashed lines that display the Bethe-Ansatz solutions for temperatures 0.25 and 2 K, respectively. As discussed in the Supplementary Materials, these corrections to scaling are only negligible in the limit Embedded Image, whereas Embedded Image is still sizeable even at T = 0.25 K. The good scaling collapse observed for C1D and α1D is attributed to numerical factors that are small, although formally of order one.

Fig. 3 Quantum critical scaling of thermodynamic quantities close to the critical field Hc.

Noncritical background contributions due to phonons and/or nuclear spins have been subtracted. Multiplication by Embedded Image and plotting versus the scaling parameter Embedded Image cause a collapse of (A) C1D/T, (B) α1D, (C) χ1D, and (D) λ1D toward critical scaling functions (solid black lines), which are derived from f0(x) of Eq. 2; symbol colors indicate different temperatures from 0.3 (blue) to 2.0 K (red). The corresponding Bethe-Ansatz results calculated for T = 2 K are shown as red dashed lines in (A) to (D) and, in addition, for T = 0.25 K as blue dashed lines in (C) and (D). The importance of corrections to scaling increases from (A) to (D), spoiling a complete scaling collapse. (E) The experimentally obtained magnetic field–dependent Grüneisen parameter ΓH,1D (symbols) is perfectly described by its critical behavior (solid lines) given by Eq. 3. The dashed lines show the universal divergences of Eq. 3 in the zero-temperature limit, and the inset compares the corresponding k/T divergence at H = Hc with the experimental data (symbols). (F) The pressure-dependent Grüneisen parameter Γp,1D is, according to Eq. 4, proportional to ΓH,1D, and consequently, both Grüneisen parameters collapse on the very same scaling function Φ(x) of Eq. 3, as shown in (G) and (H).

A quantity of particular interest close to field-induced quantum criticality is the adiabatic magnetocaloric effect defined by the magnetic field–dependent Grüneisen parameter Embedded Image that quantifies the ability of the system to adiabatically change the temperature upon a field change. General scaling considerations predict that ΓH diverges with characteristic exponents close to quantum criticality, which allows one to identify and classify the quantum critical point (28, 33). Figure 3E shows that with decreasing temperature, the obtained ΓH,1D(T, H) of CuPzN approaches a sign-change singularity at Hc, in agreement with the expected asymptotic quantum critical behaviorEmbedded Image(3)The scaling function is related to f0 of Eq. 2 via Embedded Image. The critical Embedded Image is plotted as solid lines in Fig. 3E and perfectly reproduces the experimental ΓH,1D(T, H). The asymptotics for x → ±∞ result in characteristic zero-temperature divergencies Embedded Image, with the universal prefactors Φ(x → ± ∞) = 1 and 1/2 (33), respectively, as shown by the dashed lines. The data at 0.4 K are already close to this universal behavior but only for H < Hc. Close to the critical field, Embedded Image with Embedded Image that results in the divergence Embedded Image, with Embedded Image K/T, which perfectly agrees with the data (see inset of Fig. 3E).

Closely related to ΓH is the pressure-dependent Grüneisen parameter Γp = Vmα/C (28, 33). The most singular contribution to α1D arises from the pressure-dependent Hc(p) so that, asymptotically, α1DVm = − ∂S1D/∂p = (∂S1D/∂H)(∂Hc/∂p). This yields the proportionalityEmbedded Image(4)close to quantum criticality. The experimentally obtained Γp,1D = Vmα1D/C1D is displayed in Fig. 3F and already indicates that, apart from the opposite signs, the field and temperature dependences of Γp,1D and ΓH,1D are identical. This is quantitatively confirmed in Fig. 3 (G and H), showing that both ΓH,1D(HHc) and −Γp,1D(HHc)/(∂Hc/∂p) perfectly collapse on the very same scaling function Φ(x) from Eq. 3.

In summary, the low-temperature thermodynamics of CuPzN is excellently described by the Heisenberg spin-1/2 chain model after taking into account small phononic and/or nuclear background contributions. We have demonstrated the emergence of universal scaling behavior close to its field-induced quantum critical point. Comparison between experiment and the exact Bethe-Ansatz solution has elucidated the importance of corrections to scaling, which varies from quantity to quantity and might spoil a scaling collapse over an extended parameter regime. This unprecedented quantitative understanding establishes CuPzN as an instructive reference material for the emergence of quantum criticality.


Sample preparation and measurements

Single crystals of CuPzN were grown from an aqueous solution of pyrazine and Cu nitrate via slow evaporation. Typical crystals have a length along the a axis of about 10 mm. Perpendicular to the a axis, the crystals are usually smaller than 1 mm, with b being the shortest axis. Crystals of CuPzN are orthorhombic (Pmna), with the lattice constants a = 6.712 Å, b = 5.142 Å, and c = 11.73 Å (34). Magnetic fields were applied along the crystallographic b axis. Measurements of the thermal expansion and the magnetostriction were performed using a home-built capacitance dilatometer in a transverse configuration, that is, measuring the length change along the chain direction a, with the magnetic field applied along b. The uniaxial thermal expansion coefficient α and the magnetostriction coefficient λ of the a axis were obtained from the data by numerical differentiation, Embedded Image. The specific heat was measured using a home-built calorimeter based on the relaxation time method. The addenda were obtained in a separate run and subtracted from the obtained total specific heat. The magnetization was measured with a capacitive Faraday magnetometer that was previously calibrated in magnetic fields and matched to the data taken at temperatures larger than 2 K, with the vibrating sample magnetometer (VSM) option of a commercial physical property measurement system (PPMS) (Quantum Design). The magnetocaloric effect was measured in a continuous way, as described in the Supplementary Materials.

Theoretical modeling

For the calculation of the thermodynamical potential of the spin-1/2 Heisenberg chain, we used the method described by Klümper (15). This requires the numerical solution of a set of just two nonlinear integral equations (NLIEs) for two auxiliary functions. There are equivalent but numerically differently conditioned formulations of these NLIEs. Here, we used the formulation of Klümper and Scheeren (35).

The free energy per site for temperature T and magnetic field H is obtained as a contour integralEmbedded Image(5)Here, Embedded Image is a narrow closed contour around the entire real axis involving an auxiliary function a(x). This function satisfies the NLIEEmbedded Image(6)

In this formulation, the invariance of the free energy under a sign change of the magnetic field (H → −H) is not manifest but, of course, true. The NLIE can be solved iteratively with fast convergence for positive values of H. In numerical calculations, the integral over a function g(x) along the contour Embedded Image was replaced by integrals over two functions g(x + i/2) and g(xi/2) along the real axis. In this manner, the single-contour NLIE is equivalent to two coupled NLIEs. Convolutions were treated by fast Fourier algorithms.


Supplementary material for this article is available at

section S1. Magnetocaloric effect

section S2. Deviations of C around 2.5 K

section S3. Nuclear contributions

section S4. Quantum critical theory and corrections to scaling

fig. S1. Magnetocaloric effect measurement.

fig. S2. Theoretical prediction for the scaling of quantum critical thermodynamics in CuPzN.

fig. S3. Deviations from critical scaling.

References (3639)

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.


Acknowledgments: Funding: This work was supported by the Deutsche Forschungsgemeinschaft via FOR 960 (Quantum Phase Transitions), FOR 2316 (Correlations in Integrable Quantum Many-Body Systems), SFB 1143 (Correlated Magnetism: From Frustration to Topology), and CRC 1238 (Control and Dynamics of Quantum Materials; Project B01). O.B. acknowledges support from the Quantum Matter and Materials Program at the University of Cologne funded by the German Excellence Initiative. Author contributions: T.L. initiated the investigations on CuPzN and coordinated the project. M.M.T. provided the single crystals, and O.B. and J.R. performed the measurements. M.G. headed the analysis of the data. A.K. contributed the numerical solutions of the Bethe-Ansatz model, and O.B. further processed these solutions to adjust them to the experimental data. O.B., M.G., and T.L. wrote the manuscript with input from all authors. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

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