Field-driven phase transitions in a quasi-two-dimensional quantum antiferromagnet

M B Stone^1,2,3,7, C Broholm^2,4, D H Reich², P Schiffer³, O Tchernyshyov², P Vorderwisch⁵ and N Harrison¹

¹ Neutron Scattering Science Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
² Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA
³ Department of Physics, Pennsylvania State University, University Park, PA 16802, USA
⁴ National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
⁵ Hahn-Meitner Institut, D-14109 Berlin, Germany
⁶ National High Magnetic Field Laboratory, LANL, Los Alamos, NM 87545, USA

⁷ Author to whom any correspondence should be addressed.

Email: stonemb@ornl.gov and broholm@jhu.edu

Received 16 November 2006
Published 16 February 2007

Contents

• 1. Introduction
• 2. Experimental techniques
• 3. Experimental results
  • 3.1. Susceptibility and magnetization
  • 3.2. Specific heat
  • 3.3. Elastic neutron scattering
  • 3.4. INS
• 4. Discussion
  • 4.1. Susceptibility and magnetization
  • 4.2. Specific heat
  • 4.3. ENS
    • 4.3.1. Ordered magnetic structure
    • 4.3.2. Phase diagram
    • 4.3.3. Critical exponents
  • 4.4. INS
    • 4.4.1. Gapped excitations
    • 4.4.2. Magnetic excitations in the high-field paramagnetic state
  • 4.5. Bose condensation of magnons
  • 4.6. Kosterlitz-Thouless (KT) scenario
• 5. Conclusions
• Acknowledgments
• References

1. Introduction

Molecular magnets form intricate quantum many body systems that are amenable to experimental inquiry. Their low energy degrees of freedom are quantum spins associated with transition metal ions such as copper (spin-1/2) or nickel (spin-1). The interactions between these spins can be described by a Heisenberg Hamiltonian of the form

Experiments on these materials can test existing theoretical predictions and also reveal entirely new cooperative phenomena [1]–[5].

If the generic ground state of transition metal oxides is the Néel antiferromagnet (AFM), it is singlet ground state magnetism for molecular magnets. When molecular units have an even number of spin-1/2 degrees of freedom they typically form a molecular singlet ground state. If inter-molecular interactions are sufficiently weak the result can be a quantum paramagnet. This is the state that Landau famously envisioned as a likely outcome of antiferromagnetic exchange interactions [6]. There are also indications that geometrical frustration on lattices containing multiple triangular plaquettes can suppress Néel order and/or produce singlet ground state magnetism as is seen for example in the 3D system Gd₃Ga₅O₁₂ (GGG) [7, 8] and possibly in CuHpCl [9].

While magnets with an isolated singlet ground state generally have no zero field phase transitions, there are at least two field-induced phase transitions that terminate in zero temperature quantum critical points (QCPs). The lowest lying excitations in such systems typically have spin S  =  1, and thus the lower field QCP, H_c1, occurs as the S_z  =  1 triplet branch is driven to degeneracy with the singlet ground state at a certain critical wavevector by the magnetic field. The system becomes magnetized even in the zero temperature limit for fields above H_c1. As the field is increased further, the system is ultimately ferromagnetically polarized at a second QCP characterized by an upper critical field H_c2 where the field energy overwhelms the spin–spin interactions [10]. Above H_c2, a second spin gap should open and increase linearly with applied field.

What happens as a function of field and temperature between these critical points depends critically on the inter-molecular connectivity. Three-dimensional (3D) systems such as Cs₃Cr₂Br₉ [11] and TlCuCl₃ [12]–[15] exhibit a phase transition to long-range antiferromagnetic order both as a function of applied field and applied pressure. For quasi-1D connectivity, the high-field phase should be an extended quantum critical phase. However, systems that have been examined thus far, such as Cu(NO₃)₂ · 2.5 H₂O [16],^Note1  and Cu(L-aspartato) (H₂O)₂ [19] exhibit long-range Néel order indicating that they may not be as 1D as originally envisioned. Another complication can be multiple molecules per unit cell which lead to staggered fields that break translation symmetry elements and preclude a critical phase. Several new spin ladder systems have recently been identified that may enable access to a critical phase above H_c1 [20, 21].

2D systems are of particular interest since they are at the lower critical dimension for Néel order, just as the parent compounds to high temperature superconductors. BaCuSi₂O₆ for example features spin-1/2 pairs coupled on a 2D square lattice [22]. An anomaly in the critical exponents and in the phase boundary at low temperatures has been interpreted as the result of a dimensional crossover from 3D to 2D close to quantum criticality [23]. In quasi-2D SrCu₂(BO₃)₂ competing interactions play a central role. The unique structure frustrates inter-dimer interactions, yielding effectively decoupled dimers [24]. While the singlet ground state of the material is known exactly, the three finite-magnetization plateaus and the critical phases that surround them are more challenging to understand. The purely organic quantum AFM F₂PNNO forms a quasi-2D network of spin dimers with a singlet ground state and a 1 meV gap to magnetic excitations [25]. The phase diagram includes a magnetization plateau that extends from μ₀H  =  15 T to μ₀H  =  25 T as well as regions with continuously varying magnetization that may indicate unique 2D quantum critical phases. Neutron scattering is required to disentangle these complex and unfamiliar high-field phases. Unfortunately few of these phases can be accessed with current high-field neutron scattering facilities, which are limited to fields below 17.5 Tesla.

The material examined here, piperazinium hexachlorodicuprate (PHCC), is a highly 2D quantum paramagnet with a lower critical field of 7.5 T that is well within reach of current high-field neutron scattering and an upper critical field of 37 T that can be reached in pulsed field magnetization measurements. The large difference between the critical fields indicates strong inter-dimer interactions and a potential for novel, strongly correlated 2D physics. PHCC, (Cu₄H₁₂N₂)Cu₂Cl₆, consists of Cu^2 +  spin-1/2 sites interacting predominantly within the crystallographic a–c-planes [26, 27], as illustrated in figure 1. Prior zero-field inelastic neutron scattering (INS) measurements probed the spin dynamics of the low-temperature quantum paramagnetic (QP) phase, and established the strongly 2D nature of magnetism in PHCC. The spin gap Δ  =  1.0 meV, and the magnetic excitations show dispersion with a 1.8 meV bandwidth in the (h0l) plane. There is less than 0.2 meV dispersion perpendicular to that plane, indicating a highly 2D spin system. The spin–spin interactions in the a–c-plane have the connectivity of frustrated tetragonal bilayers, as shown in figure 1(c). Bonds 6 and 3 form two extended 2D lattices linked by bond 1, which thus is J_⊥ in the bilayer description. Bonds 2 and 8 provide additional frustrated interlayer interactions.

Figure 1. Crystal structure of PHCC, (C4H12N2)(Cu2Cl6). (a) View along the a-axis, showing well-separated Cu–Cl planes. (b) View of a single Cu–Cl plane. Dotted lines between Cl sites show potential halide-halide contacts. (c) Second view of a single Cu–Cl plane, showing only the interacting Cu2 +  spin sites (solid circles). Significant antiferromagnetic spin–spin interactions are depicted as light blue ( frustrated ) and black ( satisfied ) lines with the same bond numbering used in [26]. Line thickness is proportional to the product of exchange constant and spin–spin correlation function, . Vectors represent the ordered spin structure determined by ENS at T  =  1.65 K and μ0H  =  13.7 T. Dashed lines are the a–c plane unit cell with room temperature lattice constants a  =  7.984(4) Å, b  =  7.054(4) Å, c  =  6.104(3) Å, and α  =  111.23(8)°, β  =  99.95(9)°, and γ  =  81.26(7)°, space group , from [28].

In the current study, we employed INS and elastic neutron scattering (ENS) as well as specific heat and magnetic susceptibility measurements to characterize the low-temperature phases of PHCC. Unlike 3D coupled-dimer systems, such as TlCuCl₃ [12]–[14], [29], we find a significant separation between the phase boundary to long-range order (LRO) and the crossover line delineating the high-field and high temperature boundary of the QP phase. This allows access to a disordered renormalized classical two-dimensional (RC–2D) regime near the lower QCP. Specific heat measurements in the disordered low-field QP phase agree well with series expansions from a quantum bilayer model. Combining ENS and field-dependent specific heat results allows a determination of the critical exponents associated with the transition to LRO. We have previously reported a re-entrant phase transition at low temperature near H_c1 wherein the magnetic LRO can melt into the QP phase as the temperature is either increased or decreased [30]. We proposed that coupling to auxiliary degrees of freedom is important near the QCP and can effect the phase diagram there. We also recently examined the QP to LRO phase transition and found that it can be described in terms of magnon Bose–Einstein condensation. In this communication, we provide further experimental and theoretical information regarding these phenomena, including field-dependent neutron scattering, magnetization, and specific heat data, and a more detailed theoretical discussion of magnon Bose–Einstein condensation in PHCC.

2. Experimental techniques

Single crystal samples of deuterated and hydrogenous PHCC were prepared using previously described methods [26]. Magnetic susceptibility measurements were performed at the National High Magnetic Field Laboratory pulsed field facility at the Los Alamos National Laboratory using a compensated coil susceptometer. The sample environment consisted of a pumped ³He cryostat located in the bore of a 50 Tesla pulsed field resistive magnet. Measurements were performed with the magnetic field along the b-, a*- and c-axes on hydrogenous samples with masses between m  =  0.7 mg and m  =  1.4 mg. An empty coil measurement was performed for each individual pulsed field measurement of the sample to determine the sample-independent background. Specific heat measurements were performed as a function of magnetic field, μ₀H ≤ 14 T, and temperature, T ≥ 1.8 K, on a deuterated single crystal sample, m  =  6.76 mg, with H || a* using a commercial calorimeter (Quantum Design PPMS).

Neutron scattering measurements were performed using the FLEX triple axis spectrometer at the cold neutron facility of the Hahn-Meitner Institut in Berlin, Germany. The sample was mounted on an aluminium holder and attached to the cold finger of a dilution refrigerator within a split coil μ₀H  =  14.2 T superconducting magnet. A magnetic field in excess of the superconducting critical field for aluminium was always present to ensure good thermal contact to the sample. The sample consisted of two deuterated single crystals with a total mass of m  =  1.75 g co-aligned within 0.5. The sample was oriented in the (h0l) scattering plane with H || b. Measurements were performed using a vertically focusing pyrolytic graphite (PG(002)) monochromator followed by a 60 ' horizontal collimator. Inelastic measurements used a liquid nitrogen cooled Be filter before the sample and a fixed final energy of E_f  =  3 meV. A horizontally focusing PG(002) analyser was used for inelastic measurements. Elastic measurements employed a flat PG(002) analyser with 60 ' collimation before and after the analyser. Measurements characterizing the temperature and magnetic field dependence of Bragg peaks used a liquid nitrogen cooled Be filter before the sample and neutron energy E_i  =  E_f  =  2.5 meV. Measurements of magnetic and nuclear Bragg peaks to determine the ordered magnetic structure were performed using the same collimation with a room temperature PG filter before the sample and E_i  =  E_f  =  14.7 meV selected by a vertically focusing PG(004) monochromator and a flat PG(002) analyser.

3. Experimental results

3.1. Susceptibility and magnetization

Pulsed field measurements of the magnetic susceptibility χ(H) up to μ₀H  =  50 T with H || b are shown in figure 2. For T ≤ 4 K, the susceptibility has two sharp features. The lower field feature marks the closure of the spin gap while the upper feature indicates the field where M(H) saturates, and the system enters the fully polarized (FP) regime. M(H) at T  =  0.46 K, obtained by integrating the corresponding χ(H) data, is shown as a solid line in figure 2. At low temperatures, these features in χ(H) indicate the location of the critical fields μ₀H_c1 ≈ 7.5 T and μ₀H_c2 ≈ 37 T. We note that although there is a local minimum in χ(H) at μ₀H  =  23.9 T near , and hence a point of inflection in the magnetization, there is no indication of fractional magnetization plateaus for T ≥ 0.46 K.

As T increases, the two features in χ(H) broaden, and above T  =  9 K, they merge into a single rounded maximum that moves to lower fields as the temperature is further increased. For T  >  20 K, χ(H) decreases monotonically with H. Despite the thermal broadening above T  =  5 K, several regimes may be identified from χ(H, T). These are most clearly seen in figure 3, which shows χ(H, T) at fixed field. The rounded peak and exponential decrease in χ(T) for low-magnetic fields indicate a quantum paramagnet and these features are similar to prior results obtained at H  =  0 for PHCC [26, 28, 31]. As the field increases, the peak in χ(T) shifts to lower T, and the low-temperature activated behaviour vanishes. This indicates suppression of the gap in the magnetic excitation spectrum by an applied field. However, as exemplified by the μ₀H  =  10 T data in figure 3(a), for  T the rounded maximum that signals the onset of 2D antiferromagnetic spin correlations is still visible (near T  =  12 K for μ₀H  =  10 T), before a sharp rise in χ(T) as the system enters the strongly magnetizable regime at lower T. As shown in figures 3(a) and (b) the boundary of this regime continues to move to higher T for μ₀H  <  25 T, before moving back to lower T as H approaches H_c2. In the FP region above H_c2 activated behaviour returns, as shown for μ₀H  =  40 T and μ₀H  =  47 T in figure 3(c). This indicates a second spin gap regime. Note that as the field increases above H_c2, the onset of the activated susceptibility moves to higher T, indicating that this second spin gap increases with field. An overview of χ(H, T) is given in the colour contour plot in figure 11(a) and will be discussed in more detail below.

Low-temperature pulsed field susceptibility measurements were also performed at T  =  0.46 K for H || c and H || a*. These are shown together with the H || b data near H_c1 and H_c2 in figure 4. Although the overall shape of χ(H) is similar for all three orientations, the differences in the onset fields shown in figure 4 indicate that the critical fields are orientation-dependent. Because the ratio of the critical fields is nearly the same for each of the three single crystal orientations, [ 0.203(3), , and , differences in the magnitude of the critical fields are likely due to g-factor anisotropy associated with the coordination of the Cu^2 +  ions in PHCC.

3.2. Specific heat

Measurements of the specific heat as a function of temperature and magnetic field are shown in figure 5. Prior measurements were performed on protonated samples in the same orientation, H || a*, but were restricted to T ≤ 2 K and μ₀H ≤ 9 T [26]. The total specific heat C_p is shown for representative fields in figure 5(a). In zero field, C_p shows a broad rounded maximum at T ≈ 7 K, with an exponentially activated region at lower T associated with the formation of the spin gap. Above H_c1 the form of the high-temperature specific heat changes, and as shown for μ₀H  =  14 T, a lambda-type anomaly can be seen that is associated with the phase transition to magnetic LRO at T  =  T_N. In figure 5(b), we plot the magnetic portion of the specific heat, C_m  =  C_p – C_lattice, where C_lattice T ³ is the lattice/phonon contribution to the specific heat determined by comparison of the data to the bilayer model discussed later in the text. This lattice contribution is shown as a dashed line in figure 5(a). Figure 5(b) shows the changes in the low-field activated behaviour of C_m as H increases and the spin gap closes, as well as the increase in magnitude of the lambda peaks as H increases above H_N. The area under the peak is greater for larger applied fields indicating that the T  =  0 ordered moment increases with magnetic field. The field dependence of C_p is plotted in figure 5(c) for selected temperatures. We observe a sharp peak at the transition to LRO at H  =  H_N, the location of which has the same field and temperature dependence as that seen in C_p(T) at fixed H.

3.3. Elastic neutron scattering

Neutron diffraction was used to explore the nature of the field-induced LRO transition indicated by the heat capacity data. Field and temperature-dependent antiferromagnetic Bragg peaks were observed at wavevector transfer , 0, for integer n and m throughout the (h0l) plane. This indicates a magnetic structure that doubles the unit cell along the a- and c-axes.

Figure 6 depicts a wavevector scan through the Bragg peak as a function of (a) magnetic field and (b) temperature. The Bragg peak intensity at T  =  1.7 K increases as a function of applied field. At μ₀H  =  14.2 T (figure 6(b)) there is only a flat background at T  =  10.1 K; however, for T  =  3.7 K the broad maximum indicates short-range AFM correlations. At T  =  1.7 K the resolution limited peak indicates long-range antiferromagnetic order. Scans through this peak along the and directions were also resolution limited. Magnetic Bragg peaks were found at locations that were indistinguishable from the nuclear half-wavelength, λ/2, Bragg peaks, which were observed upon removal of the Be filter to admit λ/2 neutrons from the PG(004) monochromator Bragg reflection. No indication of incommensurate order was found along the and directions through the entire Brillouin zone at T  =  1.9 K and μ₀H  =  11.8 T.

Figure 6. Wavevector dependent elastic scattering from PHCC as measured through the Bragg peak along the direction. The energy resolution was 0.047 meV full width at half maximum (FWHM). (a) Magnetic field, H || b, dependence at T  =  1.7 K. (b) Temperature dependence at μ0H  =  14.2 T.

To determine the phase boundary and the critical properties of the phase transition, the Bragg peak was measured as a function of magnetic field at 16 temperatures for 0.065 ≤ T ≤ 2.85 K. Data were obtained by continuously sweeping the magnetic field (μ₀ dH/dt  =  0.01 or 0.03 T min^–1 while simultaneously counting diffracted neutrons over a fixed time interval of 0.5 min per point. At T  =  0.06 K the measured field trace was indistinguishable for the slower and faster rates. The results are shown for select temperatures in figure 7. The temperature dependence of the Bragg peak at μ₀H  =  14.2 T was measured by integrating over a series of rocking curves at 37 temperatures as shown in figure 8. The Bragg intensity increases continuously upon cooling below a distinct critical temperature as expected for a second-order phase transition.

Figure 7. Magnetic field, H || b, dependence of elastic scattering intensity of the Bragg peak at select temperatures. The energy resolution was 0.047 meV FWHM. Temperatures presented increase according to the figure legend from left to right. Solid lines are fits described in the text over the region μ0H  <  μ0HN  +  0.75 T. Inset: magnetic order parameter critical exponent, β, determined by fitting a power law I(H) (H – HN)2β to magnetic field and temperature-dependent scattering from the antiferromagnetic Bragg peak. The highest temperature data point at T  =  3.69 K corresponds to the temperature scan shown in figure 8.

Figure 8. Temperature dependence of antiferromagnetic Bragg peak elastic scattering intensity for PHCC at μ0H  =  14.2 T with H || b. The data points were obtained from the integrated intensities of rocking curves measured at each temperature. Inset: log–log plot of integrated intensity as a function of the difference between the determined Néel temperature, TN  =  3.705(3) K, and the sample temperature. The solid lines are fits over the range T  >  TN – 0.33 K as described in the text.

Figures 9(a) and (c) show low-temperature magnetic field sweeps of the magnetic Bragg intensity plotted as a contour plot and as intensity versus field in the vicinity of the critical field for the transition to Néel order (H_N). The data indicate that the ordered phase is thermally re-entrant for μ₀H ≈ 7.5 Tesla. Upon heating from 0.065 K, H_N initially decreases with increasing temperature. However, there is an apparent minimum in the critical field curve at T  =  0.20 K, above which temperature H_N increases on heating. The same data are plotted as a function of temperature at fixed fields in figure 9(b). These data show a peak in the Bragg scattering intensity at T  =  0.20 K for fields between μ₀H ≈ 7.45 T and μ₀H ≈ 7.9 T. The peak location remains at T ≈ 0.2 K until vanishing for fields above μ₀H ≈ 8.1 T.

Figure 9. Experimental indications of re-entrant LRO in PHCC as a function of magnetic field, H || b, and temperature. (a) Elastic scattering intensity of the antiferromagnetic Bragg peak as a function of applied magnetic field and temperature. False colour image was obtained by combining all individual magnetic field order parameter scans. (b) Elastic scattering intensity of the antiferromagnetic Bragg peak as a function of temperature. The data were obtained by combining all of the individual magnetic field order parameter scans and averaging over μ0ΔH  =  0.1 T for each magnetic field shown. The symbols on the vertical axis in (a) correspond to constant temperature data shown in (b). (c) Magnetic field order parameter scans of the antiferromagnetic Bragg peak performed in the region of the re-entrant phase transition. Solid lines are fits described in the text.

3.4. INS

INS was used to probe spin-dynamics as a function of magnetic field and temperature. In particular, constant wavevector scans at were performed as a function of magnetic field at T  =  0.06 and 4.5 K. A portion of these data is shown in figure 10. The low-temperature scan at H  =  0 T shows a resolution limited excitation at ω ≈ 1 meV, consistent with the previous zero field measurements [26]. In an applied field, this excitation splits into three components indicating a zero field triplet 1 meV above a singlet ground state. The intensity distribution is consistent with the upper and lower peaks resulting from transitions to S_z  =  ±1 states while the central highest intensity peak corresponds to a transition to an S_z  =  0 state. The lower energy, S_z  =   + 1, peak is observable until μ₀H ≈ 6 T, where it is obscured by elastic nuclear incoherent scattering. Both remaining gapped excitations increase in energy continuously as the field is increased above H_c1. The higher energy excitation increases in energy and decreases in intensity at a faster rate than the excitation which evolves out of the S_z  =  0 component of the triplet. There are also subtle differences in the temperature- and field-dependence of the width of the excitations, which we shall return to later.

Figure 10. (a) Constant wavevector INS from PHCC at T  =  0.06 K (open symbols) and T  =  4.5 K (solid symbols) for different values of applied magnetic field, H || b. Solid (dashed) lines are fits to Gaussians for the low (high) temperature data as described in the text. Data are offset vertically for clarity. (b) Peak locations derived from data such as those shown in (a) as a function of applied magnetic field. Solid lines are a fit to the T  =  0.06 K data as described in the text. Dashed lines extend these lines to the origin.

4. Discussion

4.1. Susceptibility and magnetization

A second view of the magnetic susceptibility data shown in figures 2 and 3 is provided by the colour contour map in figure 11(a). Since the sharp features in χ(H) at the lowest-T herald the onset of finite magnetization at H_c1 and the entry into the FP regime at H_c2, we identify the continuation of these features for T ≤ 5 K as experimental signatures of the closing of the low-field spin gap and the opening of the high-field spin gap with increasing field in the vicinity of the critical fields H_c1 and H_c2, respectively. These features provide a starting point in constructing an H–T phase diagram for PHCC, and are shown as open circles in figure 11(b). By extending these points through inclusion of the sharp features visible in χ(T) for T  >  5 K, they are seen to demarcate the entire large-susceptibility region in figure 11(a). At nonzero T, these points delineate a crossover, rather than a phase transition. Near the lower QCP H_c1, for example, this boundary marks a crossover from the non-magnetized quantum paramagnet (spin gap) regime to a quasi-2D renormalized classical (RC–2D) region with finite magnetization.

Figure 11. (a) Colour contour map of χ(H,T) for PHCC. (b) Magnetic field versus temperature phase diagram of PHCC. Hc1 and Hc2 are zero-temperature QCPs described in the text. Data points were obtained from pulsed field susceptibility (χ(T, H)), INS, elastic neutron scattering (ENS), and specific heat measurements (CP (T, H)). Specific heat measurements were performed with H || a*; all other measurements were made with H || b. Values from CP(T, H) have been renormalized as discussed in the text. LRO indicates a LRO phase; QP is a quasi-2D QP phase; FP corresponds to a ferromagnetic polarized phase; RC–2D indicates a quasi-2D renormalized classical phase; and P corresponds to the semi-classical paramagnetic region. Dashed lines are guides to the eye. Open diamonds represent the rounded peak locations in χ(T) delineating a higher temperature classical paramagnetic regime from the correlated low-temperature regimes, QP and FP. Open circles represent locations of sharp features in χ(T) and χ(H). Closed diamonds correspond to the closing of the spin gap as determined from the INS data in figure 15. Closed squares represent transitions to LRO as determined from the field and temperature dependence of the antiferromagnetic Bragg peak shown in figures 7 and 8. (c) Log–log plot of the low-temperature re-entrant behaviour of the LRO phase.

There are two additional crossovers in the H–T plane of PHCC marked by rounded maxima in χ(T) for H  <  H_c1 and H  >  H_c2. These are indicated by open diamonds in figure 11(b). At low field, the resulting boundary is associated with the development of 2D antiferromagnetic correlations as the system is cooled toward the spin gap regime. This process has been well-studied in other spin gap systems, such as Haldane spin-1 chains [32]. The intersection of this crossover with the zero-field axis in figure 11(b) marks the onset of a high-temperature semi-classical paramagnetic regime (P) where the system has not yet attained significant antiferromagnetic correlations and the magnetic susceptibility is that of a disordered ensemble of exchange-coupled spins with one energy scale, i.e. Curie–Weiss behaviour.

The high-field crossover (open diamonds above H  =  H_c2 in figure 11(b)) is not as well-characterized experimentally, but should mark the onset of appreciable ferromagnetic correlations as the system is cooled at high-field towards the FP state. The excitations in the FP state are expected to be ferromagnetic spin waves. As for a conventional ferromagnet in a field, these excitations will show a field-dependent gap, although here the energy scale for the gap should be set by . Fits of the form χ(T) e^–Δ/k_BT, such as those shown in figure 3(c), yield estimates for the gap in the FP regime. Values of Δ_FP as determined from a simultaneous fit of χ(T) in the temperature range 0.75 ≤ T ≤ 6.5 K for 40 ≤ μ₀H ≤ 47 T are shown in the inset of figure 3(c) and are consistent with a linear increase of Δ_FP with field above H_c2.

The evolution of the magnetic susceptibility as a function of magnetic field and temperature has been examined numerically for the two-leg spin-ladder [33]. The calculations find a low-temperature field-dependent magnetization, which is symmetric about with square root singularities near the upper and lower critical fields. These results are consistent with general expectations for 1D spin gap systems [34]. These results are not dissimilar to our findings for PHCC. The spin-ladder calculations also report similar susceptibility anomalies: a double peaked function at low temperatures, a single rounded peak that moves to lower fields as the temperature is increased, and a monotonically decreasing field dependence of χ as the temperature is increased beyond a certain temperature, T_cl. A crossover temperature marking the boundary between the classical and quantum regimes, was identified as for the two-leg spin ladder. For PHCC, this corresponds to  K which agrees well with the zero-field lower temperature bound of the crossover regime defined in figure 11 where we find T_cl ≈ 15.4 K based upon susceptibility measurements.

Close to the critical fields the low-temperature field-dependent magnetization should follow power laws of the form , where H_c is a critical field [35]. Fitting of the magnetization in the vicinity of the critical fields to this form was done simultaneously for all T  =  0.46 K single crystal orientations with a common value of δ₁ (δ₂) over ranges of magnetic field between 0.5 and 3 Tesla above (below) H_c1 (H_c2) in 0.25 Tesla increments. Overall amplitudes were allowed to vary between the different orientations and critical fields were fixed to values found for fits with a 5 Tesla range. The resulting mean value and the corresponding power-law for a range of one Tesla above the critical field is shown in figures 2 and 4. The errors quoted for the δ₁ parameter correspond to one standard deviation greater and less than the respective maximum and minimum fitted values for the range of magnetic fields used in the fits. The analogous analysis in the vicinity of H_c2 yields . While the fits plotted in terms of magnetization versus field appear appropriate (figure 4(c) and (d)), the limitations of the analysis become apparent when comparing field derivatives of the fit and data in a differential susceptibility plot (figure 4(a) and (b)). It appears that non-critical effects may be important over the range of fitting and this could certainly affect the extracted values of δ. Antiferromagnets are expected to have δ  =  2 in d  =  1 dimension [35] and δ  =  1 with logarithmic corrections in d  =  2 dimensions [36]–[38]. Above the upper critical dimension, d  >  2, one expects mean-field behaviour with δ  =  1 and no logarithmic corrections. An interpretation of H_c1 as magnon Bose–Einstein condensation yields δ₁  =  δ₂  =  1 (see section 4.5) with logarithmic corrections which may in fact be consistent with the data. We note that anomalous critical behaviour may also result from a dimensional cross over as recently found in the frustrated antiferromagnet BaCuSi₂O₆ [39].

4.2. Specific heat

We compare our specific heat measurements to the frustrated bilayer model of Gu and Shen [40]. Their model contains a strong interlayer bond J₀, a weaker intralayer bond J₁, and a frustrating interlayer bond J₂. This exchange topology is similar to that in PHCC. We include a magnetic field-dependent term in their thermodynamic expansion via a linear Zeeman splitting of the triplet excited state as a function of applied magnetic field. We carried out a global fit of all our specific heat data with μ₀H ≤ 9 T and T ≤ 30 K using only the free energy terms up to powers of 1/E^–2 in Gu and Shen's expansion, one independent prefactor for each magnetic field, and global terms consisting of the three exchange constants, a g-factor and a lattice contribution T³. The results of this fit are shown as solid lines in figure 5 and account very well for the temperature and field dependence of the specific heat with J₀  =  2.08(1) meV, J₁  =  0.34(3) meV and J₂  =  –0.13(3) meV.

The exchange parameters for the frustrated bilayer model determined from the specific heat data for PHCC can be used to evaluate the corresponding dispersion relation, equation (4) of [40]. Figure 12 depicts the calculated dispersion for PHCC along several high-symmetry directions in reciprocal space, compared to the zero-field triplon dispersion measured by INS [26, 27]. The values derived from the fits to C_m for the spin gap Δ  =  1.11(1) meV and bandwidth of 1.92(2) meV agree quite well with the measured spin gap, Δ  =  0.99 meV and bandwidth, 1.73 meV. Furthermore, the frustrated bilayer model reproduces the shape of the dispersion relation remarkably well given that specific heat data provided the only experimental input.

The lambda-type peaks in the magnetic heat capacity yield an additional measure of the phase boundary between the quasi-2D RC–2D disordered phase and the LRO phase. However, C_p was measured with H || a*, while all the other data included in figure 11(b) were measured with H || b. As noted above, differences in the critical fields measured through χ(H) for different field orientations are accounted for by g-factor anisotropy. After rescaling by the ratio determined from χ(H), the phase boundary derived from C_p(H, T) is seen to agree very well with that determined from ENS as shown in figures 11(b) and (c) and discussed below.

The magnetic field-dependent specific heat shown in figure 5(c) also provides a sensitive measure of critical scaling in the vicinity of the LRO transition. C_m(H) is shown in figure 13 as a function of reduced magnetic field, near the transition field H_N. Power law behaviour is clearly seen for H  >  H_N, while below H_N, the magnetic specific heat asymptotically approaches the same power law behaviour as the H  >  H_N portion of the data. In addition, there is a region of reduced field for H  <  H_N, 0.02 ≤ h ≤ 0.2, where we observe a different power-law behaviour, i.e. a different slope on the log–log scale plot as indicated by the dashed line in figure 13. The two different power-laws observed for H  <  H_N, as shown by the fits for 0.02 ≤ h ≤ 0.2 and the change in power-law as one approaches H_N indicate a distinct non-QP regime for H  <  H_N. Comparison of the magnetic susceptibility and neutron scattering measurements reported in figures 11(b) and (c) are consistent with such a regime as we shall discuss later in the text.

Figure 13. Magnetic heat capacity as a function of reduced magnetic field, h, in the vicinity of the phase transition to LRO. H  <  HN data (solid symbols) are plotted against . H  >  HN data (open symbols) are plotted against . Black solid lines correspond to a global fit of H  >  HN data to a power law as discussed in the text. Coloured solid lines correspond to individual power law fits for H  >  HN. Individual fit results are very similar to the global fit result. Dashed lines correspond to individual power law fits for H  <  HN for 0.025 ≤ h ≤ 0.2 in the RC–2D regime.

We fit the H  >  H_N data in figure 13 to a power law of the form

where α is the specific heat exponent. A global fit for T  =  2 and 3 K and H  >  H_N results in α  =  0.097(2) while individual fits yield α(T  =  2K)  =  0.104(3) and α(T  =  3K)  =  0.072(2). Individual fits to equation (2) for H  <  H_N over the range 0.02 ≤ h ≤ 0.2 yield α(T  =  2K)  =  0.175(2) and α(T  =  3K)  =  0.2626(7). These values are larger than those obtained for H  >  H_N, which indicates that the fits include data beyond the critical regime. Table 1 summarizes theoretical and experimental results for various types of 3D.

4.3. ENS

4.3.1. Ordered magnetic structure To determine the field-induced magnetic structure we measured 14 magnetic Bragg peaks in the (h0l) plane. The measurements were carried out at T  =  1.65 K and μ₀H  =  13.7 T which is within the ordered phase as established by heat capacity measurements. Table 2 shows raw experimental results in the form of wavevector integrated intensities, I_m(τ_m), for transverse (rocking) scans. I_m(τ_m) is related to the magnetic structure of the material as follows

Here is a normalization factor that depends on aspects of the instrument such as the incident beam flux that are not considered in the elastic resolution function  [45] which is sharply peaked around q  =  0. N_m and v_m are the number of magnetic unit cells and the magnetic unit cell volume respectively. The magnetic Bragg intensity is given by

where r₀  =  5.38 fm [46],

is the magnetic structure factor, g_d is the Lande g-factor for the magnetic ion on site d within the magnetic unit cell, and F_d(Q) is the magnetic form factor of the Cu^2 +  ion [46, 47]. The magnetic structure enters through the expectation values for the spin vector, S_d in units of .

Table 2. Measured and calculated relative magnetic Bragg peak scattering intensities at the indicated magnetic reciprocal lattice points . The parentheses indicate statistical errors only. The minimum in the crystallographic reliability index is found for the magnetic structure depicted in figure 1 and described in the text.

Table 2 compares the measured scattering intensity I_m(τ_m) to the calculated intensity for the uniaxial magnetic structure that best fits the data and is shown in figure 1(c). The ordered moment lies within ~7(4) of the a–c-plane and the in-plane component is within ~6(4) of the a-axis. For T  =  1.65 K and μ₀H  =  13.7 T the sub-lattice magnetization is , determined by normalization to incoherent scattering from the sample. Assuming antiferromagnetic interactions, the ordered structure is consistent with the sign of the bond energies determined on the basis of zero-field INS [26].

4.3.2. Phase diagram Figures 7, 8 and 9(c) compare the temperature and field dependence of the magnetic Bragg intensity at to power laws and I(T) (T_N – T)^2β in order to extract the order parameter critical exponent, β. Rounding in the vicinity of the phase transition was accounted for as a distribution in the experimental critical field. Possible origins of the 3.6% distribution width include inhomogeneities in the applied field and chemical impurities. Least squares fitting included only data within 0.75 T and 0.33 K of criticality. The corresponding fits appear as solid lines in the figures.

Values of (T_N, H_N) extracted from the analysis appear as filled squares in figure 11. There is a clear distinction between the onset of a state of high differential susceptibility as determined from χ(H, T) and the boundaries of the LRO phase extracted from neutron diffraction. We denote the state of the system between these boundaries as 2D renormalized classical. Spin correlations in this phase were examined by quasi-ENS measurements along the and directions through the wavevector at T  =  3 K and μ₀H  =  9 T. No magnetic Bragg peaks were encountered. Instead the broad peak centred at indicates short-range correlations with the same local structure as the LRO that sets in at slightly higher fields.

4.3.3. Critical exponents The order parameter critical exponent, β, determined for each of the fits to the magnetic field and temperature-dependent scans is shown versus temperature in the inset to figure 7. For T ≥ 0.5 K, the average value is β  =  0.34(2) which is indistinguishable from the critical exponent for the 3D Heisenberg [42], Ising (β  =  0.326), and XY (β  =  0.345) models [41]. However, the positive specific heat critical exponent, α, uniquely identifies the phase transition as being in the 3D Ising universality class. This is consistent with expectations for a phase transition to long-range magnetic order in space group . We also calculate the remaining critical exponents based upon the Rushbrooke, α  +  2β  +  γ  =  2, Fisher, γ  =  ν(2 – η), Josephson, 2 – α  =  νd, and Widom, , scaling laws, assuming d  =  3 Ds [41, 48]. The values are again consistent only with those of 3D Ising transitions as summarized in table 1. Recalling the exponents δ₁ and δ₂ in based on the magnetic susceptibility data, note that these values are different from δ derived from the scaling relations. This is because δ₁ and δ₂ are associated with the uniform susceptibility while δ in table 1 should characterize the response to a staggered field conjugate to the Néel order parameter.

The apparent order parameter critical exponent increases as temperature is reduced, such that β  =  0.40(1) for T ≤ 0.43 K (see inset to figure 7). A possible interpretation of this finding is that the system enters a crossover regime associated with the approach to quantum criticality. Indeed, in the T  =  0 limit the system should be above its upper critical dimension, such that mean field behaviour with a critical exponent β_mf  =  0.5 is expected.

The field dependence of the magnetic Bragg peak intensity shown in figures 9(a)–(c), indicate a re-entrant phase transition. The H–T phase boundary obtained by fitting these data (figure 11(b)) show a local minimum in the phase boundary between the LRO and QP phases. A possible origin of this anomaly is coupling to non-magnetic degrees of freedom such as nuclear spin or lattice distortions that are unimportant at higher temperatures but may become important sufficiently close to the QCP [30].

4.4. INS

4.4.1. Gapped excitations Figure 10 shows the magnetic field and temperature dependence of the excitation spectrum at the critical wavevector for 0 ≤ H  <  H_c2. We fit each spectrum to a sum of Gaussian peaks to account for the magnetic excitations with a constant background and a field independent Gaussian+Lorentzian lineshape centred at zero-energy to account for incoherent elastic nuclear scattering. The field dependence of the peak locations are plotted for the T  =  0.06 and 4.5 K data in figure 10(b). The data demonstrate the triplet nature of the excited state through its Zeeman splitting. Above H_c1, the rate of change of the peak position as a function of field for the higher energy peak is twice that of the lower energy excitation, indicating a magnetized ground state. We fit the low-T field-dependent peak locations to the spectrum of an antiferromagnetically coupled spin-1/2 pair with singlet–triplet splitting Δ subject to a magnetic field described by . Results of this two-parameter fit to the low-temperature peak locations are plotted as solid lines in figure 10(b), and yield values of Δ  =  1.021(2) meV, g  =  2.33(1), and  T, which are in agreement with zero-field INS and bulk magnetization data.

The average resolution-corrected half width at half maximum, Γ, of the three Gaussians that were fitted to each spectrum for T  =  0.065 and 4.5 K are shown in figure 14. While Γ, considering systematic uncertainties in the energy resolution, is indistinguishable from zero at low T, it increases with applied field at elevated temperature. Also increasing with field and temperature is the magnon density. Hence the result is consistent with an increased rate of magnon collisions as their density increases. Self consistent mean field theories have been developed to account for this effect [29].

The inset to figure 14 shows the field-dependent ratio of the integrated intensities of the central Gaussian peak to that of the satellite peaks. Based upon the polarization factor and the Wigner–Eckart theorem as applied to a Zeeman split triplet, this ratio should be 0.5 for H  <  H_c1 [49]. Both the low- and high-temperature constant wavevector scans are consistent with this result. Above H_c1, Bose condensation modifies the ground state and the experiment shows a decrease of intensity for the highest energy mode.

We use INS to examine the crossover between the low-field gapped (QP) and gapless (RC–2D) portions of the phase diagram. The T  =  0.06 K lower critical field determined from the field-dependent low-temperature constant wavevector scans described earlier is plotted in figures 11(b) and (c) (solid diamond at T  =  0.06 K). With wavevector transfer at and energy transfer fixed at ω  =  0.3 meV the data in figure 15 show the field-dependent intensity for various fixed temperatures. At the lowest temperature a sharp peak indicates the field at which the S_z  =  1 mode passes through the 0.3 meV energy window of the detection system. With increasing temperature and hence increasing magnon density not only does the peak broaden as expected from an increased relaxation rate, it also moves to higher fields indicating a finite slope for the crossover line in the H–T plane. Just as for the finite magnon lifetime, the origin of this effect is magnon repulsion. The increasing background is likely due to population of low-energy gapless excitations which develop for magnetic fields above the lower critical field. By extrapolation to the ω  =  0 line in figure 15(b), these data were used to determine the temperature dependence of the field required to close the spin gap and the corresponding data points are included in figures 11(a) and (b) (solid diamonds). The coincidence between the crossover field determined in this manner and the fields associated with differential susceptibility anomalies establish a not unexpected association between effectively closing the gap at the critical wavevector and maximizing the differential uniform susceptibility.

Figure 15. (a) INS intensity at ω  =  0.3 meV and as a function of applied magnetic field, H || b. Solid lines are the result of a global fit to a Gaussian peak with a sloping `background' above the magnetic field H0 and a constant background below the magnetic field H0. Values of the constant background, H0, and the slope were global parameters in the fit. (b) Energy transfer versus magnetic field for peak locations as determined from (a) and the T  =  4.5 K and T  =  0.06 K data in figure 10(a). Symbols used in (b) correspond to those used in (a). Solid lines are fits to lines as described in the text. The extrapolated intercepts define an effective temperature-dependent crossover field.

4.4.2. Magnetic excitations in the high-field paramagnetic state The previous results indicate a well defined crossover in PHCC from a low field and low temperature state dominated by a gap in the excitation spectrum to a high-field state with an excitation spectrum that has finite spectral weight at zero energy. The crossover clearly occurs at lower fields than required to induce long-range magnetic order. There are possible analogies to certain S  =  1/2 1D Heisenberg antiferromagnetic systems including the linear chain, the next nearest-neighbour frustrated chain, and the alternating chain where a low-energy continuum of excitations is predicted to exist upon closure of the spin gap [49]–[51]. Calculations examining both gapped and gapless S  =  1/2 spin–orbital chains describe an incommensurate gapless continuum for large portions of exchange and orbital interaction parameter space [52]. A characteristic of these continua are soft incommensurate modes moving through the Brillouin zone from the antiferromagnetic zone centre to the ferromagnetic point as a function of applied magnetic field [34], [53]–[55]. Incommensurate magnetic field-dependent soft modes are also found in numerical calculations for the 2D antiferromagnetic square lattice with next nearest-neighbour interactions, a system with similar antiferromagnetic interactions as PHCC [38].

An important distinction between PHCC and the systems mentioned above is that for PHCC the putative gapless state so far was only observed at finite temperatures. Ascribing this to the inevitable effects of inter-plane coupling, the analogue of which occurs via inter-chain coupling in quasi-1D systems, there exists a potential for field-induced incommensurate correlations in PHCC. Here, we present preliminary experiments at the highest fields presently accessible to neutron scattering. The field was μ₀H  =  14.2 Tesla and the temperature chosen was T  =  4.5 K, which is 20% above the critical temperature at that field (T_C(μ₀H  =  14.2 T)  =  3.705(3) K). Shown in figure 16(a) are difference data indicating excess magnetic spectral weight at the antiferromagnetic zone centre as compared to . The solid line shows a fit to the following resolution convolved Lorentzian relaxation spectrum,

which obeys detailed balance and features a relaxation rate, Γ  =  0.025 ± 0.2 meV. The spectrum extends to and is approximately symmetric about zero energy, which is consistent with a 2D renormalized classical phase close to the phase transition to commensurate magnetic order.

Figure 16. (a) Magnetic inelastic scattering from PHCC at the critical wavevector for μ0H  =  14.2 Tesla and T  =  4.5 K which is immediately outside the LRO phase. A presumed non-magnetic background measured at has been subtracted. The solid line is a fit to a Lorentzian relaxation function as described in the text. (b) Constant energy scans along the direction at T  =  4.5 K and μ0H  =  14.2 T for ω  =  0.36 and 0.55 meV. Solid lines are fits to a Lorentzian with a sloping background.

Figure 16(b) shows two constant energy scans at ω  =  0.36 and 0.55 meV that indicate short-range commensurate magnetic correlations. The fitted Lorentzian peaks, which are much broader than the experimental resolution (solid bar) correspond to a dynamic correlation length scale of 3.2 ± 2.5 Å and 4.2 ± 2.1 Å for the ω  =  0.55 meV and ω  =  0.36 meV measurement respectively. The magnetization of PHCC is only 20% of the saturation magnetization in these measurements so any incommensurate shift, which should be a similar fraction of the distance to the Brillouin zone centre is not clearly visible in the present data.

4.5. Bose condensation of magnons

In this section, the phase transition to long-range magnetic order in PHCC is analysed from the perspective of magnon condensation [56]–[59]. Following Nikuni et al  [58], Misguich and Oshikawa [60] and Sherman et al  [61] we treat magnons as bosons with a density n that is proportional to the deviation of the longitudinal magnetization from its plateau value M_z0: . We note that in our discussion and derivation in this section and subsection 4.01, we use units in which  =  k_B  =  1. The low-density magnon gas is described by the Hamiltonian

where is the boson chemical potential and v₀ is a short-range repulsion. The global minimum of the dispersion relation ε_k is assumed to be at . For convenience, excitation energies are measured from the bottom of the band, so that .

At zero temperature the system acquires a condensate when μ  >  0. The energy density of the ground state in the presence of a condensate (neglecting quantum corrections) is , which is minimized when |α|²  =  μ/2v₀. In this approximation the boson density is

n is related to the longitudinal magnetization as follows

At a finite temperature, we obtain an effective Hamiltonian

where n is the density of thermally excited bosons. The bosons condense when the renormalized chemical potential vanishes. This yields the critical density

The critical field is then

The boson scattering v₀ can be extracted from low-temperature magnetization curves. The magnon density per unit cell n is equal to the ratio M/M_sat (there is exactly one magnon per unit cell when M  =  M_sat). Based on the magnetization curve at the lowest available temperature T  =  0.46 K we obtain v₀  =  1.91 meV × the unit-cell volume (ucv). In other words, if we model the magnons as bosons on a cubic lattice with anisotropic hopping and on-site interactions U, the interaction strength would be U  =  2 meV. The linear behaviour extends over a range of some 10 T, or 13 K in temperature units.

Using this value of the boson coupling, we can test the predicted relation between the critical field and critical magnon density (12) at finite temperatures from the data for n  =  M/M_sat in the temperature range 0.46 K ≤ T ≤ 6 K. From these we extracted pairs (M, H) with H(T) values sufficiently close to the line of phase transitions H_c(T). The resulting graph is shown in figure 17. The zero-temperature critical field H_c1  =  7.41 T was determined from the onset of Neel order at the lowest temperatures (T  <  0.3 K). The agreement with equation (12) is fair.

Figure 17. Critical field Hc versus critical magnon number per unit cell nc. The straight line is equation 12 with  T and v0  =  1.9 meV ucv.

The line H_c(T) is obtained from equations (11) and (12). The magnon dispersion is taken from Stone et al  [26] (the 6-term version). That dispersion relation is strictly 2D, which precludes Bose condensation at a finite temperature. We therefore added weak tunnelling between the planes:

The experimental data [17] give an upper bound for the total bandwidth in the [010] direction: 4τ  <  0.5 meV.

When T    τ, only the bottom of the magnon band has thermal excitations and one may replace the exact band structure with a parabolic dispersion to obtain

where is the 3D effective mass. In the limit of a very weak interplane tunnelling, we can also access a regime where τ    T    W, where W ≈ 2 meV is the magnon bandwidth. In this regime the in-plane dispersion can still be treated as parabolic. This yields the critical density for a quasi-2D Bose gas [62]

where and b is the interplane distance. In the intermediate regime the critical density first rises faster than T ^3/2 [60] before crossing over to the T log T behaviour.

Curves for n_c(T) were obtained by integrating equation (11) numerically for several values of the interplane tunnelling τ. Since v₀ is already fixed, there are no further adjustable parameters. The data below 2.5 K (H_c  <  9 T, n_c  <  0.05 per unit cell) are best described by the curves for τ  =  0.03(1) meV. The results are shown in figure 18.

The theory based on the Hamiltonian (10) assumes that the boson interaction v₀ is weak. However, it is also applicable to strongly interacting bosons, as long as the interaction is short-range and the gas is sufficiently dilute. In that case v₀ is not the bare interaction strength (which can be large) but rather the scattering amplitude (the t-matrix) for a pair of low-energy bosons. For hard-core bosons on a cubic lattice with a nearest-neighbour hopping amplitude t, one obtains v₀ ≈ 8t [63]. In our case t ≈ W/8  =  0.25 meV giving v₀ ≈ 2 meV. Another estimate along these lines, using the `exact' magnon dispersion yields, via

a similar value of v₀  =  2.5 meV. These numbers demonstrate that the inferred interaction strength of 1.91 meV ucv is quite reasonable.

At higher temperatures the agreement worsens. At T  =  3.7 K—the highest temperature where the condensation is observed—the mean-field theory predicts H_c  =  11 T, well below the observed value of 14.2 T. The observed magnon density at the critical field is 0.19 per unit cell (the theoretical estimate is 0.10). It is possible that the mean-field theory of magnon condensation, appropriate for a dilute Bose gas of magnons, becomes unreliable due to overcrowding. (At a technical level, the assumption of pairwise interactions may break down: two particles no longer collide in a vacuum.) Another possible reason for this failure is an increased strength of (thermal) critical fluctuations.

4.6. Kosterlitz–Thouless (KT) scenario

If the planes in PHCC were fully decoupled from each other the Bose condensation would change into the KT vortex-unbinding transition. It does not seem likely that the data can be reconciled with this scenario: in d  =  2 spatial dimensions the crossover exponent  =  1, so that the line of KT transitions is expected to be linear.

The quantum criticality of the 2D Bose gas was recently examined by Sachdev and Dunkel [64] who used a combination of scaling arguments and numerics to determine H_c(T) for a weakly interacting Bose gas (10). (Analysing the RG flow alone does not solve the problem: the KT fixed point lies in a strongly coupled limit.)

The RG analysis focuses on two variables: the (renormalized) chemical potential and the two-particle scattering amplitude v. Unlike in d  =  3Ds, the scattering amplitude vanishes in the limit of zero energy. Therefore the role of the coupling constant is played by the t-matrix for particles with energies

where Λ is a high-energy cutoff of the same order as the magnon bandwidth.

While both and v flow under RG transformations, a certain combination of these variables, namely , remains unchanged. The value of this constant determines whether the system flows to the disordered phase or to the phase with long-range correlations. The KT transition corresponds to

At the KT transition, the renormalized chemical potential is

We see that is a bit below T (particularly if T    Λ), which means that there are many thermally excited particles.

The magnetic field is connected to the bare chemical potential

The density is computed for an ideal gas

This yields the final result for H_c(T):

which depends weakly on the ultraviolet cutoff Λ. For Λ  =  2.5 meV, we have μ ≈ 1.5T in the temperature range of interest. The curve is labelled `KT1' in figure 18.

The result (22) can also be obtained directly from equations (3) and (4) of Sachdev and Dunkel [64] if one takes the classical limit μ    T. The value μ ≈ 1.5T suggests that we are (just) outside the classical regime and one needs to solve the Sachdev–Dunkel equations numerically. The resulting curve (for Λ  =  2.5 meV) is shown in figure 18 as `KT2'. It is also approximately linear, with a slope dμ/dT ≈ 1.

Even with the refined approach, the KT theory does not seem to agree with the data. As anticipated, it produces a linear dependence H_c(T) (up to logarithmic corrections) instead of a higher power law. The rise of the data above the KT line at high temperatures may signal the inadequacy of the continuum approximation at high magnon concentrations.

5. Conclusions

Our measurements establish PHCC as a quasi-2D spin gap system. The gap is sufficiently small to allow experimental access to the lower critical field while the bandwidth is sufficiently large to produce several distinct phases.

The long-range Néel ordered phase is a commensurate AFM with moments perpendicular to the applied field direction. The critical exponents for the phase transition are similar to those of a 3D Ising model. The transition into this phase can be described as magnon Bose–Einstein condensation. Mean-field (Hartree–Fock) theory adequately describes the magnetization M(H) at the lowest temperatures (T  =  0.46 K) and the shape of the critical line H_c(T) for T  <  3 K. The theory predicts a small but nonzero magnon dispersion of 0.08 ≥ 4τ ≤ 0.16 meV perpendicular to the 2D planes. The increase of H_c(T) over T ^3/2 behaviour is caused by deviations of the magnon density of states from the E^1/2 behaviour of the continuum theory. The extra DOS (due to a van Hove singularity) is positive, hence an increase over the T ^3/2 law. The continuum KT theory gives an approximately linear dependence H_c(T) and thus does not seem to do a good job.

Close to the lower QCP there is an anomaly in the phase boundary that is not accounted for by the BEC theory (see figures 11(b) and 9(a)–(c)). Low T anomalies near the zero temperature termination of phase boundaries are also seen experimentally in the quantum liquids ³He [65] and ⁴He [66, 67], the frustrated spin system GGG [7, 8], and in numerical and theoretical results for the BCC Ising AFM [68], the Ising AFM on a cubic lattice [69], and S  =  1/2 XXZ 1D- [70] and 2D- [71] AFMs. In ⁴He the anomaly is associated with there being more low-energy degrees of freedom in the solid than the liquid and a low-T anomaly in the phase boundary of LiHoF₄ has been associated with nuclear spin degrees of freedom [72]. For PHCC, we speculate that features beyond a spin model such as the relatively soft lattice or nuclear spins become relevant sufficiently close to the QCP. Such a mechanism which could be rather general merits further exploration through experiments that probe potential auxiliary degrees of freedom directly.

The high-field paramagnetic phase is perhaps the most intriguing aspect of PHCC. The crossover from the quantum paramagnet to this phase remains remarkably sharp upon heating and the experiments show that a finite temperature pseudo spin gap closes there. Further analysis is required to understand why this crossover remains so distinct. A proper examination of the gapless high-field phase requires INS in magnetic fields of the order of 30 Tesla where the system is more heavily magnetized. This work must await development of a higher field facility for INS.

Acknowledgments

We acknowledge discussions with A Aharony, O Entin–Wohlman, A B Harris, S Sachdev and T Yildirim. Work at JHU was supported by the NSF through DMR-0074571, DMR-0306940, DMR-0348679 and by the BSF through grant no. 2000-073. Work at PSU was supported by the NSF through DMR-0401486. ORNL is managed for the US DOE by UT-Battelle Inc. under contract DE-AC05-00OR2272. We thank P Smeibidl and S Kausche for assistance with the sample environment at HMI. M B S thanks B G Ueland for assistance with heat capacity measurements at PSU.

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Notes

Note1  See also [17].