Thermodynamics of the interplay between magnetism and high-temperature superconductivity

Copper-oxide-based high-temperature superconductors have complex phase diagrams with multiple ordered phases. It even appears that the highest superconducting transition temperatures for certain cuprates are found in samples that display simultaneous onset of magnetism and superconductivity. We show here how the thermodynamics of fluid mixtures-a touchstone for chemistry as well as hard and soft condensed matter physics-accounts for this startling observation, as well as many other properties of the cuprates in the vicinity of the instability toward "striped" magnetism.     

From the Cover: Using thermal boundary conditions to engineer the quantum state of a bulk magnet

The degree of contact between a system and the external environment can alter dramatically its proclivity to quantum mechanical modes of relaxation. We show that controlling the thermal coupling of cubic-centimeter–sized crystals of the Ising magnet LiHo_xY_1-xF₄ to a heat bath can be used to tune the system between a glassy state dominated by thermal excitations over energy barriers and a state with the hallmarks of a quantum spin liquid. Application of a magnetic field transverse to the Ising axis introduces both random magnetic fields and quantum fluctuations, which can retard and speed the annealing process, respectively, thereby providing a mechanism for continuous tuning between the destination states. The nonlinear response of the system explicitly demonstrates quantum interference between internal and external relaxation pathways.The coupling of a sample to its environment is both a fundamental theoretical concept and a powerful experimental tool in classical thermodynamics. For quantum systems, contact between the internal degrees of freedom and the external world, often referred to as the “bath,” can change the measured outcome completely. Typically, such experiments involve a small number of particles sensitive to subtle changes in the external incoherent environment, such as ultracold atoms confined in precisely controlled optical potentials (1–3). With the search for viable solid-state qubits for quantum computing, the control of bath-induced decoherence in solids also has become an important topic for engineers and condensed-matter physicists. Approaches have centered on the nuclear spin bath (4–6), modifying it either with isotopic substitution (7) or radio frequency pulses (8), and on electrical control of the exchange interaction between electron spins in coupled quantum dots (9). The question of the importance of coupling to an external bath, as provided by a cryostat, has not been researched as intensively. Here, we show that by engineering the thermal boundary conditions for a macroscopic magnetic crystal, it is possible to select distinct low temperature states. Conditions of constant energy, as opposed to constant temperature, yield relatively fewer low energy contributions to the fluctuation spectrum and decouple the spin excitations responsible for that spectrum into separate oscillators. The experiments show the importance of thermal heat sinking for quantum annealing, also referred to as adiabatic quantum computation (10–13), as well as new protocols for generating quantum cluster states (14).The LiHo_xY_1-xF₄ family of insulating magnetic salts provides a physical manifestation of the simplest quantum mechanical spin model, the Ising model in transverse field (15). Pure LiHoF₄ (16, 17) is a ferromagnet with Curie temperature, T_C = 1.53 K. External magnetic fields can produce the longitudinal and transverse fields in the model, chemical substitution of Ho³⁺ ions by the nonmagnetic species Y³⁺ provides quenched disorder, and the anisotropy of the dipolar coupling produces random internal transverse fields (18–21) as well as competing ferromagnetic and antiferromagnetic interactions. The combination of site dilution and external fields yields a wide variety of collective magnetic states, ranging from random field ferromagnet at x = 0.44 (22) to quantum spin glass at x = 0.167 (23). We focus here on the dilute limit of x = 0.045, for which there have been seemingly contradictory findings concerning the ground state.The primary diagnostic of the ground state has been the AC magnetic susceptibility, whose imaginary part χ″(f) is the quotient of the long-wavelength magnetic fluctuation spectrum, S(f), and the Bose factor, (n(hf) + 1) = 1/(1-exp-hf/kT), where h and k are Planck’s and Boltzmann’s constants, respectively. For our experiments, hf<<kT and hence χ″(f) = hf/kT S(f). The frequency at which the imaginary part peaks indicates the characteristic relaxation rate of the system, which for spin dynamics dominated by thermal activation over energy barriers will vary in accord with the Arrhenius law, (15, 24, 25), precisely what we see for temperatures 0.15 K < T < 1 K. Below T ∼ 0.15 K, deviations from Arrhenius behavior emerge (15, 24, 25); however, the nature of the deviations and their interpretation has been contested (26). One class of experiments found a low-frequency narrowing of the spectrum with decreasing T (24, 27, 28), accompanied by the magnetic equivalent of optical hole burning in the nonlinear response, where effectively isolated, mesoscopic clusters of spins can be addressed and manipulated using a pump/probe technique (24). A magnetic field applied transverse to the Ising axis introduces quantum fluctuations, and can influence the relaxation pathways of the coherent clusters (28). Moreover, muon spin-relaxation (µSR) studies have shown that the persistent spin-fluctuation rate remains constant down to T = 0.02 K, consistent with a spin-liquid ground state (29). By contrast, a second class of magnetic susceptibility studies found that LiHo_0.045Y_0.955F₄ behaved as a paramagnet approaching a spin–glass transition, which extrapolation suggests to occur at T_g ∼0.04 K, with a magnetic fluctuation spectrum that broadened symmetrically as the temperature was lowered (25). In this picture, the characteristic dissipative response moves more quickly to low frequency as the system as a whole freezes.The most significant distinction between the two classes of susceptibility experiments is the heat sinking of the sample to the cryostat. For the measurements yielding a spin liquid, a single crystal measuring (5 × 5 × 10) mm³ was heat sunk by sapphire rods pressed against the sample on either end of the long axis (24); in the spin–glass case a (0.57 × 0.77 × 7.7) mm³ sample was glued to a sapphire rod running along its length (25). The sapphire rods are then thermally anchored to the mixing chamber of the dilution refrigerator, coupling them to the environmental heat bath. If the thermal boundary conditions of the sample change appreciably, then the internal state of the system also may be expected to change. Just as the application of a transverse magnetic field affects the spin cluster dynamics and their coupling to the external world in this system (28), thermal boundary conditions can enhance or destroy isolated spin degrees of freedom, tune the system between classical and quantum mechanical limits (30), and alter the relative energies of competing ground states.