Phase transitions are abundant in nature and associated with them are certain critical phenomena which, even though the microscopic orders may be completely different, give rise to many fundamental characteristics [219]. Phase transitions appear due to the necessity to balance ordering energy against the entropy of thermal fluctu- ations. For example, in a ferromagnet the exchange interaction favours the alignment of spins reducing the internal energy but at higher temperatures thermal fluctuations can maximise the entropy and the system prefers the disordered paramagnetic state. At such a second-order phase transition the order parameter, in this case the magnetisationM, is on average zero in the disordered phase and
grows continuously from zero once the ordered phase is entered at the transition temperature. Even though the spatial average of the order parameter is zero in the disordered state, upon approaching
4.2Background physics for Sr3Ru2O7 101
the phase transition droplets of order start to grow and fluctuate in and out of existence, i.e. some short range order develops. In the case of a ferromagnet one can say the spins are correlated over a short range called the correlation length. As the critical point is approached the correlation length diverges,ξ∼ |(T−Tc)/Tc|−ν 11
11νis a critical exponent. It char-
acterises the nature of the phase transi- tion and is an experimental observable.
, and at the critical point the system becomes scale invariant. The critical nature of a thermal phase transition is only observed very close to the transition. The microscopic details become unimpor- tant once there is no length scale other than the correlation length and the system is averaged over large distances. Here the behaviour falls into a universality class which only depends on the dimension- ality and the symmetry of the order parameter, and the relevant statistical physics can be treated classically.
quantum critical
ordered
phase disorderedquantum p T QCP thermally disordered classical critical
Fig. 4.9: Quantum critical point.
Schematic phase diagram of a second- order phase transition giving rise to a quantum critical point. pis a non- thermal tuning parameter that sup- press a second-order phase transition, thick black line, to absolute zero at the quantum critical point. At finite tem- perature and sufficiently close to the transition a classical critical region ex- ists where the critical fluctuations are much lower in energy than the temper- ature and the thermal phase transition is universal. The effect of the quantum critical fluctuations, however, can be observed over a much larger region of phase space, see text for further details. All this changes though if there exists a non-thermal tuning
parameter, such as pressure, doping or magnetic field, that can suppress the transition temperature to absolute zero (see figure 4.9). A quantum phase transition is accessed at absolute zero and here it is no longer the thermal fluctuations that melt the order but rather the abrupt change of ground state is due to quantum fluctuations arising from Heisenberg’s uncertainty principle. Since the critical nature of a thermal phase transition is only observed very close to the transition, one might ask why a quantum phase transition should be any more than just an academic curiosity since absolute zero temperature is never a practically achievable temperature. The answer is because unlike a thermal phase transition the effects of quantum criticality can be felt over a surprisingly much larger range.
The time scale of quantum critical fluctuations also depend on the distance from the critical point, but here the distance is along the tuning axis, τ ∼ |p−pc|−νz 12
12zis the dynamical critical expo-
nent.
. The energy of these fluctuations goes to zero at the critical tuning, but even away from this tuning sufficiently high temperatures allow thermal population of finite-time modes associated with the quantum mechanically driven phase change, so the system can still look critical. In this scenario the dominant fluctuations are thermally driven but the fluctuations are those of a scale invariant quantum-critical ground state. This region defines the cone of quantum criticality shown in figure 4.9. low density high density QCEP first order critical end point p B T
Fig. 4.10: Quantum critical end point.
Schematic of a first-order transition giving rise to a quantum critical end point. A first-order transition with tun- ing parameterBhas a critical end point when there is no symmetry breaking. If a second non-thermal tuning param- eter, p, can suppress the critical end point to absolute zero a quantum criti- cal end point is produced. In the case of Sr3Ru2O7 p can be related to the angle of the applied field.
Pronounced effects due to quantum critical fluctuations have been observed experimentally and extensively studied, especially in heavy fermion materials [220,221]. Here, relatively low tempera- ture magnetic states are often found and these can be successfully suppressed to absolute zero by the application of magnetic field, doping, or pressure, leading to a quantum critical point (QCP). Some common behavioural traits are observed. The residual spe- cific heat coefficient γ diverges upon approaching the quantum
102 Quantum Criticality and Metamagnetism of Strained Sr3Ru2O7
critical point, and this along with the observation of the resistivity exhibiting a linear temperature dependence seems to imply that the mass of the quasiparticles is diverging, and their characteristic energy scale vanishing, leaving only temperature as the remaining energy scale [219].
The fermionic criticality that creates this strange metallic state is still not fully understood, but QCPs provide more than just an exciting opportunity for modern theory because they are also a breeding ground for new stable phases of matter. Rather than face the mass divergence close to the QCP, more often than not it is observed that the electrons reorganise themselves into novel forms of order. 2 6 10 14 5 10 15 20 25 30 µ0H(T) T emp er ature (K) 1.0 1.5 2.0 d ln( ρ − ρ0 ) /d ln T
Fig. 4.11: Resistivity power law. Tem- perature dependence of resistivity mea- surements of Sr3Ru2O7 near the meta-
magnetic transition for fields applied along thecaxis. The exponent of the temperature dependent part of the re- sistivity is plotted under the assump- tion that the resistivity varies asρ= ρ0+ATα. Reproduced from [222].
Sr3Ru2O7 also shows quantum critical behaviour, but through a slightly different route. The quantum critical point as introduced above was achieved by suppressing a classically critical second- order phase transition to absolute zero using a non-thermal tuning parameter. The situation in Sr3Ru2O7is slightly different. As we saw in section 4.2.1, Sr3Ru2O7 shows a first-order metamagnetic transition. Normally there are no critical fluctuations at the dis- continuous jump of a first-order transition but since there is no symmetry breaking there is generally a critical endpoint terminat- ing the line of first-order transitions and here critical fluctuations responsible for critical opalescence are observed. By suppressing the critical endpoint to absolute zero a quantum critical endpoint (QCEP) is obtained, exhibiting all the hallmarks of a quantum critical point. Hkc 0 T 7.7 T 9 T Temperature (K) Cel /T (J/R u-mol K 2) 0 5 10 15 20 25 0.06 0.08 0.10 0.12 0.14 0.16
Fig. 4.12: Electronic specific heat.
The low temperature electronic specific heat of Sr3Ru2O7 with magnetic field aligned along the sample’scaxis. Close to the metamagnetic transition field 7.9 T the specific heat diverges loga- rithmically. Reproduced from [203].
With the field directed along the sample’sc-axis, the QCEP is
reached with a field of ∼8 T, and the magnitude of the field acts
as a tuning parameter for the quantum critical fluctuations [222]. Measurements of the temperature-dependent resistivity at a series of applied fields spanning the quantum critical region show the classic behaviour of a QCP, see figure 4.11. At both low and high fields Fermi liquid T2 temperature dependence is observed but
over a smaller and smaller temperature window as the QCEP is approached and at the critical field T-linear resistivity is observed.
Thermodynamic measurements are also consistent with the quan- tum critical scenario. Measurements of the temperature-dependent electronic specific heat show that in the vicinity of the critical field the low temperature specific heat diverges logarithmically [203], see figure 4.12.
As pointed out earlier, bare QCPs are not often observed in clean systems. Instead, a phase transition usually preempts the QCP. When the sample quality is high enough in Sr3Ru2O7 this is also observed. There is often a tendency for superconductivity to form around a QCP [223] but the large magnetic fields used to reach the QCEP in Sr3Ru2O7 prohibit superconductivity and an
4.2Background physics for Sr3Ru2O7 103
alternative form of order develops. The next section will discuss the unusual properties of this novel phase.