Thermodynamic signatures of quantum criticality in cuprate superconductors.

The three central phenomena of cuprate (copper oxide) superconductors are linked by a common doping level p*-at which the enigmatic pseudogap phase ends and the resistivity exhibits an anomalous linear dependence on temperature, and around which the superconducting phase forms a dome-shaped area in the phase diagram1. However, the fundamental nature of p* remains unclear, in particular regarding whether it marks a true quantum phase transition. Here we measure the specific heat C of the cuprates Eu-LSCO and Nd-LSCO at low temperature in magnetic fields large enough to suppress superconductivity, over a wide doping range2 that includes p*. As a function of doping, we find that Cel/T is strongly peaked at p* (where Cel is the electronic contribution to C) and exhibits a log(1/T) dependence as temperature T tends to zero. These are the classic thermodynamic signatures of a quantum critical point3-5, as observed in heavy-fermion6 and iron-based7 superconductors at the point where their antiferromagnetic phase comes to an end. We conclude that the pseudogap phase of cuprates ends at a quantum critical point, the associated fluctuations of which are probably involved in d-wave pairing and the anomalous scattering of charge carriers.

Emergence of long-range order in sheets of magnetic dimers.

Quantum spins placed on the corners of a square lattice can dimerize and form singlets, which then can be transformed into a magnetic state as the interactions between dimers increase beyond threshold. This is a strictly 2D transition in theory, but real-world materials often need the third dimension to stabilize long-range order. We use high pressures to convert sheets of Cu(2+) spin 1/2 dimers from local singlets to global antiferromagnet in the model system SrCu2(BO3)2. Single-crystal neutron diffraction measurements at pressures above 5 GPa provide a direct signature of the antiferromagnetic ordered state, whereas high-resolution neutron powder and X-ray diffraction at commensurate pressures reveal a tilting of the Cu spins out of the plane with a critical exponent characteristic of 3D transitions. The addition of anisotropic, interplane, spin-orbit terms in the venerable Shastry-Sutherland Hamiltonian accounts for the influence of the third dimension.

Continuous and discontinuous quantum phase transitions in a model two-dimensional magnet.

The Shasty-Sutherland model, which consists of a set of spin 1/2 dimers on a 2D square lattice, is simple and soluble but captures a central theme of condensed matter physics by sitting precariously on the quantum edge between isolated, gapped excitations and collective, ordered ground states. We compress the model Shastry-Sutherland material, SrCu(2)(BO(3))(2), in a diamond anvil cell at cryogenic temperatures to continuously tune the coupling energies and induce changes in state. High-resolution X-ray measurements exploit what emerges as a remarkably strong spin-lattice coupling to both monitor the magnetic behavior and the absence or presence of structural discontinuities. In the low-pressure spin-singlet regime, the onset of magnetism results in an expansion of the lattice with decreasing temperature, which permits a determination of the pressure-dependent energy gap and the almost isotropic spin-lattice coupling energies. The singlet-triplet gap energy is suppressed continuously with increasing pressure, vanishing completely by 2 GPa. This continuous quantum phase transition is followed by a structural distortion at higher pressure.

Order by disorder spin wave gap in the XY pyrochlore magnet Er2Ti2O7.

The recent determination of a robust spin Hamiltonian for the antiferromagnetic XY pyrochlore Er2Ti2O7 reveals a most convincing case of the "Order-by-Quantum-Disorder" mechanism for ground state selection. This mechanism relies on quantum fluctuations to remove an accidental symmetry of the magnetic ground state, and selects a particular ordered spin structure below TN=1.2 K. The removal of the continuous degeneracy results in an energy gap in the spectrum of spin wave excitations, long wavelength pseudo-Goldstone modes. We have measured the Order-by-Quantum-Disorder spin wave gap at a zone center in Er2Ti2O7, using low incident energy neutrons and the time-of-flight inelastic scattering method. We report a gap of Δ=0.053±0.006 meV, which is consistent with upper bounds placed on it from heat capacity measurements and roughly consistent with the theoretical estimate of ∼0.02 meV, further validating the spin Hamiltonian that led to that prediction. The gap is observed to vary with the square of the order parameter, and goes to zero for T∼TN.

Thermal conductivity of Ho2Ti2O7 along the [111] direction.

Thermal transport measurements have been made on the spin-ice material Ho(2)Ti(2)O(7) in an applied magnetic field with both the heat current and the field parallel to the [111] direction for temperatures from 50 mK to 1.2 K. A large magnetic field >6 T is applied to suppress the magnetic contribution to the thermal conductivity in order to extract the lattice conductivity. The low field thermal conductivity thus reveals a magnetic field dependent contribution to the conductivity which both transfers heat and scatters phonons. We interpret these magnetic excitations as monopolelike excitations and describe their behavior via existing Debye-Hückel theory.

Vibrating-coil magnetometry of the spin liquid properties of Tb2Ti2O7.

We have explored the spin liquid state in Tb(2)Ti(2)O(7) with vibrating-coil magnetometry down to ~0.04 K under magnetic fields up to 5 T. We observe magnetic history dependence below T*~0.2 K reminiscent of the classical spin ice systems Ho(2)Ti(2)O(7) and Dy(2)TiTi(2)O(7). The magnetic phase diagram inferred from the magnetization is essentially isotropic, without evidence of magnetization plateaus as anticipated for so-called quantum spin ice, predicted theoretically for [111] when quantum fluctuations renormalize the interactions. Instead, the magnetization for T≪T* agrees semiquantitatively with the predictions of "all-in-all-out" (AIAO) antiferromagnetism. Taken together, this suggests that the spin liquid state in Tb(2)Ti(2)O(7) is akin to an incipient AIAO antiferromagnet.

Fractalization drives crystalline states in a frustrated spin system.

The fractalized Hofstadter butterfly energy spectrum predicted for magnetically confined fermions diffracted by a crystal lattice has remained beyond the reach of laboratory-accessible magnetic fields. We find the geometrically frustrated spin system SrCu(2)(BO(3))(2) to provide a sterling demonstration of a system in which bosons confined by a magnetic and lattice potential mimic the behavior of fermions in the extreme quantum limit, giving rise to a sequence of plateaus at all magnetization m(z)/m(sat) = 1/q ratios 9 > or = q > or = 2 and p/q = 2/9 (m(sat) is the saturation magnetization) in magnetic fields up to 85 T and temperatures down to 29 mK, within the sequence of previously identified plateaus at 1/8, 1/4, and 1/3 of the saturated magnetization. We identify this hierarchy of plateaus as a consequence of confined bosons in SrCu(2)(BO(3))(2) mimicking the high magnetic field fractalization predicted by the Hofstadter butterfly for fermionic systems. Such an experimental realization of the Hofstadter problem for interacting fermions has not been previously achieved in real materials, given the unachievably high magnetic flux densities or large lattice periods required. By a theoretical treatment that includes short-range repulsion in the Hofstadter treatment, stripe-like spin density-modulated phases are revealed in SrCu(2)(BO(3))(2) as emergent from a fluidic fractal spectrum.

Interplay between structural, electronic, and magnetic degrees of freedom in Sr(3)Cr(2)O(8).

We present a theoretical investigation of the orbital ordering occurring in Sr(3)Cr(2)O(8) based on density functional theory calculations. We demonstrate that the strong electron correlation arising within the Cr-3d shell can clearly explain both the phase transition leading to the stabilization of its monoclinic C2/c space-group symmetry and its spin-singlet magnetic ground state. The relevance of the electronic structure determined theoretically is further established by comparison to high-resolution electron energy loss spectroscopy measurements.

Magnetoelastics of a spin liquid: X-ray diffraction studies of Tb2Ti2O7 in pulsed magnetic fields.

We report high resolution single crystal x-ray diffraction measurements of the frustrated pyrochlore magnet Tb2Ti2O7, collected using a novel low temperature pulsed magnet system. This instrument allows characterization of structural degrees of freedom to temperatures as low as 4.4 K, and in applied magnetic fields as large as 30 T. We show that Tb2Ti2O7 manifests intriguing structural effects under the application of magnetic fields, including strongly anisotropic giant magnetostriction, a restoration of perfect pyrochlore symmetry in low magnetic fields, and ultimately a structural phase transition in high magnetic fields. It is suggested that the magnetoelastic coupling thus revealed plays a significant role in the spin liquid physics of Tb2Ti2O7 at low temperatures.

Field-induced Bose-Einstein condensation of triplons up to 8 K in Sr3Cr2O8.

Single crystals of the spin dimer system Sr(3)Cr(2)O(8) have been grown for the first time. Magnetization, heat capacity, and magnetocaloric effect data up to 65 T reveal magnetic order between applied fields of H(c1) approximately 30.4 T and H(c2) approximately 62 T. This field-induced order persists up to T(c)(max) approximately 8 K at H approximately 44 T, the highest observed in any quantum magnet where H(c2) is experimentally accessible. We fit the temperature-field phase diagram boundary close to H(c1) using the expression T(c) = A(H-H(c1))(nu). The exponent nu = 0.65(2), obtained at temperatures much smaller than T(c)(max), is that of the 3D Bose-Einstein condensate (BEC) universality class. This finding strongly suggests that Sr(3)Cr(2)O(8) is a new realization of a triplon BEC where the universal regimes corresponding to both H(c1) and H(c2) are accessible at (4)He temperatures.

Two-dimensional kagome correlations and field induced order in the ferromagnetic XY pyrochlore Yb2Ti2O7.

Neutron scattering measurements show the ferromagnetic XY pyrochlore Yb2Ti2O7 to display strong quasi-two-dimensional (2D) spin correlations at low temperature, which give way to long range order (LRO) under the application of modest magnetic fields. Rods of scattering along 111 directions due to these 2D spin correlations imply a magnetic decomposition of the cubic pyrochlore system into decoupled kagome planes. A magnetic field of approximately 0.5 T applied along the [110] direction induces a transition to a 3D LRO state characterized by long-lived, dispersive spin waves. Our measurements map out a complex low temperature-field phase diagram for this exotic pyrochlore magnet.

Ground state selection under pressure in the quantum pyrochlore magnet Yb2Ti2O7.

A quantum spin liquid is a state of matter characterized by quantum entanglement and the absence of any broken symmetry. In condensed matter, the frustrated rare-earth pyrochlore magnets Ho2Ti2O7 and Dy2Ti2O7, so-called spin ices, exhibit a classical spin liquid state with fractionalized thermal excitations (magnetic monopoles). Evidence for a quantum spin ice, in which the magnetic monopoles become long range entangled and an emergent quantum electrodynamics arises, seems within reach. The magnetic properties of the quantum spin ice candidate Yb2Ti2O7 have eluded a global understanding and even the presence or absence of static magnetic order at low temperatures is controversial. Here we show that sensitivity to pressure is the missing key to the low temperature behaviour of Yb2Ti2O7. By combining neutron diffraction and muon spin relaxation on a stoichiometric sample under pressure, we evidence a magnetic transition from a disordered, non-magnetic, ground state to a splayed ferromagnetic ground state.

Crystallization of spin superlattices with pressure and field in the layered magnet SrCu2(BO3)2.

An exact mapping between quantum spins and boson gases provides fresh approaches to the creation of quantum condensates and crystals. Here we report on magnetization measurements on the dimerized quantum magnet SrCu2(BO3)2 at cryogenic temperatures and through a quantum-phase transition that demonstrate the emergence of fractionally filled bosonic crystals in mesoscopic patterns, specified by a sequence of magnetization plateaus. We apply tens of Teslas of magnetic field to tune the density of bosons and gigapascals of hydrostatic pressure to regulate the underlying interactions. Simulations help parse the balance between energy and geometry in the emergent spin superlattices. The magnetic crystallites are the end result of a progression from a direct product of singlet states in each short dimer at zero field to preferred filling fractions of spin-triplet bosons in each dimer at large magnetic field, enriching the known possibilities for collective states in both quantum spin and atomic systems.

Field-stepped broadband NMR in pulsed magnets and application to SrCu2(BO3)2 at 54T.

Pulsed magnets generate the highest magnetic fields as brief transients during which the observation of NMR is difficult, however, this is the only route to unique insight into material properties up to the regime of 100T. Here, it is shown how rather broad NMR spectra can be assembled in a pulsed magnet during a single field pulse by using the inherent time dependence of the field for the recording of field-stepped free induction decays that cover a broad frequency range. The technique is then applied to (11)B NMR of the spin-dimer system SrCu2(BO3)2, a magnetic insulator known to undergo a series of field-driven changes of the magnetic ground state. At peak fields of about 54T at the Dresden High Magnetic Field Laboratory, (11)B NMR spectra spanning a total of about 9MHz width are reconstructed. The results are in good accordance with a change from a high-temperature paramagnetic state to a low-temperature commensurate superstructure of field-induced spin-dimer triplets.

Spin waves and quantum criticality in the frustrated XY pyrochlore antiferromagnet Er2Ti2O7.

We report detailed measurements of the low temperature magnetic phase diagram of Er2Ti2O7. Heat capacity and time-of-flight neutron scattering studies of single crystals reveal unconventional low-energy states. Er3+ magnetic ions reside on a pyrochlore lattice in Er2Ti2O7, where local XY anisotropy and antiferromagnetic interactions give rise to a unique frustrated system. In zero field, the ground state exhibits coexisting short and long-range order, accompanied by soft collective spin excitations previously believed to be absent. The application of finite magnetic fields tunes the ground state continuously through a landscape of noncollinear phases, divided by a zero temperature phase transition at micro{0}H{c} approximately 1.5 T. The characteristic energy scale for spin fluctuations is seen to vanish at the critical point, as expected for a second order quantum phase transition driven by quantum fluctuations.

Structural Fluctuations in the spin-liquid state of Tb2Ti2O7.

High-resolution x-ray scattering measurements on single crystal Tb2Ti2O7 reveal finite structural correlations at low temperatures. This geometrically frustrated pyrochlore is known to exhibit a spin-liquid or cooperative paramagnetic state at temperatures below approximately 20 K. Parametric studies of structural Bragg peaks appropriate to the Fd3[over ]m space group of Tb2Ti2O7 reveal substantial broadening and peak intensity reduction in the temperature regime 20 K to 300 mK. We also observe a small, anomalous lattice expansion on cooling below a density maximum at approximately 18 K. These measurements are consistent with the development of fluctuations above a cooperative Jahn-Teller, cubic-tetragonal phase transition at very low temperatures.

In-gap spin excitations and finite triplet lifetimes in the dilute singlet ground state system SrCu(2-x)Mgx(BO3)2.

High resolution neutron scattering measurements on a single crystal of SrCu(2-x)Mgx(BO3)2 with x approximately 0.05 reveal the presence of new spin excitations within the gap of this quasi-two-dimensional, singlet ground state system. The application of a magnetic field induces Zeeman-split states associated with S=1/2 unpaired spins which are antiferromagnetically correlated with the bulk singlet. Substantial broadening of both the one- and two-triplet excitations in the doped single crystal is observed, as compared with pure SrCu2(BO3)2. Theoretical calculations using a variational algorithm and a single quenched magnetic vacancy on an infinite lattice are shown to qualitatively account for these effects.

Field-induced order and spin waves in the pyrochlore antiferromagnet Tb2Ti207.

High resolution time-of-flight neutron scattering measurements on Tb(2)Ti(2)0(7) reveal a rich low temperature phase diagram in the presence of a magnetic field applied along [110]. In zero field at T = 0.4 K, Tb(2)Ti(2)0(7) is a highly correlated cooperative paramagnet with disordered spins residing on a pyrochlore lattice of corner-sharing tetrahedra. Application of a small field condenses much of the magnetic diffuse scattering, characteristic of the disordered spins, into a new Bragg peak characteristic of a polarized paramagnet. At higher fields, a magnetically ordered phase is induced, which supports spin wave excitations indicative of continuous, rather than Ising-like, spin degrees of freedom.

High-resolution study of spin excitations in the singlet ground state of SrCu2(BO3)2.

High-resolution, inelastic neutron scattering measurements on SrCu2(BO3)2, a realization of the Shastry-Sutherland model for two-dimensional Heisenberg antiferromagnets, reveal the dispersion of the three single triplet excitations continuously across the (H,0) direction within its tetragonal basal plane. These measurements also show distinct Q dependencies for the single and multiple triplet excitations, and that these excitations are largely dispersionless perpendicular to this plane. The temperature dependence of the intensities of these excitations is well described as the complement of the dc susceptibility of SrCu2(BO3)2.