Two-dimensional universal behaviour

Plotting the magnetic heat capacity of Cu(pyz)(gly)ClO4 on a logarithmic scale

[Figs. 3.10(a) and (b)], a power law dependence ofCmag ∝Tn is observed when H

is approximately equal to Hc1 and Hc2 (yellow points in each figure). A universal

critical exponentnis expected for the heat capacity measurements of BEC systems performed at H =Hc1,2, where n=d/2 and dis the spatial-dimensionality of the

spin-exchange network [52,95]. For reliable exponents to be extracted fromCmag(T),

the data must be modelled in a temperature range for which the upper bound is less than 40% of the maximum temperature of the BEC phase [146]; and the lower bound exceeds the energy scale of uniaxial symmetry breaking perturbations in the spin Hamiltonian [52]. As no symmetry breaking terms are evident in the thermodynamic properties of Cu(pyz)(gly)ClO4for experimental investigations above 400 mK, linear

fits are made to the logCmagvs. logT data of Cu(pyz)(gly)ClO4for 0.4≤T ≤0.56 K

to revealn= 2.13(2) and 0.99(2) at Hc1 and Hc2, respectively.

The exponent ofn = 0.99(2) close toHc2 is consistent with d= 2, which is indicative of two-dimensional universal behaviour within this quantum spin network. This value for the exponent agrees with the analysis of the structure of this com- pound (above) that concluded that there is likely to be a large spatial-delocalization of spin-density within individual [Cu(pyz)(gly)]+_{layers of the material. However, it}

is important to stress that the bulk thermodynamic data presented in this chap- ter would not be consistent with interlayer interactions that are precisely zero. Spin-networks comprised of isolated layers of exchange-coupled antiferromagnetic

Figure 3.10: Logarithmic plot of the published [74] heat capacity as a function of temperature (points) for quasistatic applied magnetic fields: (a)µ0H ≤3 T; and(b) µ0H≥4 T. Solid lines are a linear fit to the data for applied fields close to each QCP.

The lattice contribution to the measurement has been subtracted. (c) The heat capacity vs. applied magnetic fieldµ0H exhibits maxima at H=Hc1,2. The peak

amplitude at the low-field QCP is reduced relative that observed at high-field QCP at the same temperature. TheT along each isotherm is stable to within _≈30 mK.

(d)The ratio of the high-field to low-field peaks in the measured heat capacity as a function of the sample temperature (normalized to the maximum temperature of the BEC phase Tmax) as derived from panel (c) for Cu(pyz)(gly)ClO4 (Tmax = 1.3 K)

(triangles) and from the published [52,94] heat capacity of the related BEC material NiCl2-4SC(NH2)2 (Tmax = 1.2 K [94]) (diamonds).

spin-dimers with uniaxial (or circular) symmetry are anticipated to adopt a low- temperature Berezinskii-Kosterlitz-Thouless (BKT) phase [17, 18, 145] and crossing a BKT phase boundary is not predicted to induce observable signatures in ther- modynamic quantities including heat capacity and differential susceptibility. For Cu(pyz)(gly)ClO4, measured peaks in bothCmag(T) and dM/dH mark the bound-

ary to a distinct magnetic phase that forms within a dome in theH₋T plane, which necessarily indicates that the interlayer spin-exchange interactions must be finite in this compound. Nevertheless, the n = 0.99(2) heat capacity exponent at Hc2 cor-

roborates the approximate two-dimensional character of the material because it is well below then= 1.5 expected for three-dimensional (3D) systems, where theoret- ical predictions derived from an analytical finite-scaling analysis of a 3D quantum critical systems suggest that this power law ought to be obeyed [148].

The separate heat capacity exponent of n _≈ 2 measured at Hc1 requires

further interpretation within the two-dimensional model, and the thermodynamic properties of related spin-gap materials offers one possible explanation. For in-

stance, the measured magnetic Gr¨uneisen parameter Γ of the spin-ladder compound (C4H12N)4CuCl4 exhibits a critical (linear) scaling with temperature at Hc2 for

measurements in the range 0.1< kbT /J < 0.4 (where J is the rung coupling con-

stant), while Γ vs. T only tends towards a linear responseHc1for sample tempera-

tures below≈0.1J/kb [147]. Given that an identical scaling of the thermodynamic

quantities of exchange-coupled dimer systems can occur atHc1 and Hc2, but only

when measured over separate T windows, one avenue for future investigations of Cu(pyz)(gly)ClO4 could be to followCmag(T) atHc1,2 down to dilution refrigerator

temperatures to determine whether the two exponents are equal at lowerT. An alternative strategy to explain the value of theHc1exponent is to consider

the effect that symmetry breaking perturbations in the spin Hamiltonian might have on the expected values of n. One such interaction, that cannot be ruled out from the structure alone [74], is a Dzyaloshinskii-Moriya (DM) interaction. While there is no evidence for a DM interaction from the magnetometry measurements, if their energy scale of a DM term is indeed greater than 400 mK then this would induce a cross over from a BEC phase (i.e. an XY spin model) to an Ising universality class at sufficiently low temperatures. For Ising systems, the critical exponent in heat capacity is expected to be n = d, in contrast to n = d/2 for an XY model. The measured exponent ofn = 2.13(2) atHc1, if it were indeed interpreted in the

Ising model, would be consistent with a two-dimensional spin network. However this conclusion would be difficult to reconcile with the result forn_≈1 at Hc2 and

so this model is not appealing as a means to unambiguously explain the discrepancy in the measured exponents. An independent argument, which might account for the difference innatHc1,2, is that zero-point fluctuations are a significant perturbation

to the quantum spin system close to Hc1, the QCP at the boundary between the

BEC and quantum disordered phase, but are less important atHc2for heat capacity

measurements at the same temperature.