General Quantum Criticality

As explored in the previous chapter, a lot of interesting physics can be observed in the vicinity of second order phase transitions in materials at finite temperatures, where the thermody- namic properties obey universal power-law behaviour dependent only on the symmetries of the system. Quantum critical systems are systems which are proximate to a second order phase transitions at zero temperature. They are also characterised by unconventional power- law properties in the vicinity of the phase transition, but the power laws are different from those associated with the corresponding finite-temperature transitions [1, 2]. One way of gen- erating a quantum critical system is as follows. Suppose we have a material with a second order phase transition in it at a critical temperature Tc, and we choose to do something to

the system. We may, for example, consider physically squeezing the system and investigating its properties under pressure. In general we should expect the microscopic properties of the system, such as equilibrium inter-atomic spacing and magnetic interaction strengths between neighbouring atoms, to be different to the original properties of the system. This change of microscopic energy scales within the system in turn changes the critical temperature of the phase transition, which now becomes a function of pressure Tc æ Tc(P ). In many systems

it is experimentally found that the transition temperature decreases under the application of pressure, and in some of these systems a pressure Pú can be experimentally reached such that

the critical temperature of the transition is suppressed to absolute zero, Tc(Pú) = 0. This is

shown in Figure 2.1.

While in the vicinity of finite temperature phase transitions the thermodynamic properties are dominated by thermal fluctuations of an order parameter, at zero temperature no thermal fluctuations are present. If we consider the system at zero temperature and tune through the critical pressure Pú, we go from an ordered phase to a disordered phase via a phase transition

at absolute zero temperature. In this second order transition, it is quantum mechanical fluctu- ations of the order parameter which take the role of thermal fluctuations in finite temperature transitions and cause novel power laws to be observed in such systems.

One might argue that absolute zero temperature is never experimentally reachable, but just as the physics in the vicinity of a finite temperature phase transition is dominated by the order parameter fluctuations, the quantum critical point influences a wide region of the phase diagram. Effects of the quantum critical point can frequently be observed over a wide range of experimentally accessible temperatures. Above zero temperature, there is a whole quantum critical region where the physical properties are dominated by the interplay between quantum mechanical and thermal effects.

The argument presented above is not only valid for pressure, but any other possible tuning parameter. Another common parameter which is used is doping of the material with a partic- ular element. Magnetic fields can also be used in some situations, but are known to lead to a

Figure 2.2: The effective resistivity exponent in Sr2Ru3O7 calculated by fitting fl (T ) = fl0+

AT–), taken from Ref. [23]. Sr

2Ru3O7 has a metamagnetic quantum critical end-point tuned

to by magnetic field, at H = 7.8T . An approximately linear resistivity is seen in the quantum critical region, and a T2 _{resistivity in the Fermi liquid region outside of the ‘V’ shape.}

variety of complications [22]. These complications will become important in Chapter 5 when we deal with metamagnetism, and we discuss these complications there. For the remainder of Chapter 2 and Chapter 3, unless explicitly stated, we do not consider magnetic fields as a tuning parameter for quantum criticality.

A typical experimental signature of a quantum critical point on the phase diagram is a ‘V’ shaped quantum critical region where the unusual critical exponents are observed. An example of a material where this is clearly seen is Sr2Ru3O7, as shown in Figure 2.2. This

can be understood by considering a characteristic energy scale in the system , associated with the critical behaviour, which tends to zero at the quantum critical point. For example in second order phase transitions this may be the ‘mass’ of the order parameter fluctuations.

The origin of the ‘V’ shape is the crossover between the regimes kBT 7 . The quantum

critical regime is the region within the ‘V’ shape, where < kBT [2]. The fluctuations have

been made energetically cheap by the tuning parameter, and then become thermally excited in the quantum critical regime. These fluctuations lead to the observed unconventional power laws. The outside of the ‘V’ shape regime is commonly referred to as a ‘quantum disordered’ regime, and for the metallic systems considered in this thesis it is the Fermi liquid region.

A major difference between classical and quantum phase transitions is the role of the dynamical properties of the order parameter. When mathematically describing a quantum mechanical problem, a natural basis of states to work in is the set of eigenstates of the Hamil- tonian. In the types of systems that exhibit quantum phase transitions however, the eigenstates of the Hamiltonian are not usually known. Moreover, often the underlying Hamiltonian of the system is not known. Instead we must use the only description we can, and discuss the physics near the transition in terms of an order parameter. In this thesis we shall be considering sys- tems where the order parameter is a spatially dependent field „, and the key quantity which allows us to calculate physical properties of systems is the partition function Z, defined by

Z = Tr exp1≠— ˆH2. (2.1.1)

To represent the partition function in terms of the field „, we must use Feynman’s path integral formalism of quantum mechanics to write the partition function as

Z =

⁄

D„ (x, ·) exp (≠—S [„]) , (2.1.2)

where the action S is given by an integral over the Lagrangian

S =

⁄ —

0 d·

⁄

dx_{L („) .} (2.1.3)

same as the time-evolution operator in quantum mechanics, exp1≠it ~Hˆ

2

. The Lagrangian is dependent on · in a non-trivial manner precisely because we are using a description in terms of states which are not eigenstates of the Hamiltonian.

We must now consider the order parameter as a function of both space and imaginary time - it is as if we have gained an extra dimension. The dependence of the action on imaginary time may not be the same as the spatial dependence of the action. This anisotropy can be characterised by the dynamical exponent z, which plays a crucial role in determining many physical properties of quantum critical systems [2, 1].

The importance of dynamical effects near a quantum phase transition also leads to another interpretation of the quantum critical regime observed at finite temperatures above the quan- tum critical point. The length of the imaginary time dimension L· is inversely proportional

to temperature L· ≥ 1/kBT, and so for any finite temperature it is not infinite. When tuning

to a quantum critical point at r = 0 the correlation length diverges as › ≥ r≠‹, and corre-

spondingly the correlation ‘time’ (the correlation length along the imaginary time dimension) diverges as ›· ≥ ›z. At finite temperatures above such a quantum critical point it becomes

meaningful to compare the correlation length along the imaginary time dimension ›· to the

length of this dimension. The quantum critical region is the region where ›· > L·, and the

fluctuations are completely correlated along the imaginary time dimension. The crossover to the so-called ‘quantum disordered’ regime is when ›· ≥ L·, or equivalently T ≥ r‹z [1].

In most examples of quantum phase transitions, the behaviour is characterised by a single dynamical exponent. The aim of this thesis is to analyse quantum multicritical points, which are examples of situations where multiple dynamical exponents are important at the phase transition. We examine other examples of situations featuring multiple dynamical exponents in Section 2.6.