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Phase transitions take place in our universe very frequently. The high Tcexample discussed

above may sound distant from our daily life. The most familiar example of phase transitions may be boiling water, which we face almost everyday. As we increase the temperature, thermal fluctuations among water molecules increase as well. Once the temperature reaches the boiling temperature T = 373K, thermal fluctuations are strong enough to overcome the Van der Waals bonds between water molecules and thereby release molecules into gas form. The transition from liquid to gas is a classical first-order phase transition, because the first derivative of the Gibbs free energy in this process is discontinuous. In Fig. 1.1(a), we use solid lines to label coexistence curves, which consist of first-order transition points in the phase diagram. The liquid–gas coexistence curve ends at fixed T = Tc when there is

no separation between liquid and gas. The temperature and pressure above which liquid water and gas become indistinguishable is called the critical temperature (Tc= 647K) and

the critical pressure (PC = 22 MPa) respectively. Exactly at the point (Tc, Pc) the phase

transition is continuous because the first derivative of Gibbs free energy is continuous. A similar phase transition also exists in the Ising model, which is used among other things to describe some ferromagnets. The phase diagram of the Ising model is depicted in Fig. 1.1(b). In this case, there is a special line H = 0, where the system has a higher symmetry. Under the Currie temperature, the phase transition between two symmetry-broken ferromagnetic states is a first-order phase transition. At the Currie temperature Tc, this transition becomes

continuous.

Phase transitions can be characterized by order parameters. An order parameter is a physical quantity that measures the degree of order in a state. In many phase transitions,

3 Liquid Gas Solid

T

P

H

T

(a)

(b)

(T

, P

)

T

Figure 1.1: Schematic phase diagrams of (a) water molecules H2O, and (b) the Ising model

in a magnetic field. Both diagrams have first-order phase transitions between (a) liquid and gas, and (b) between two ferromagnetic phases. Solid green lines are coexistence curves of first-order transition points. Continuous phase transitions take place at the critical point, shown here as the dark solid circles.

such as the Ising Model at H = 0, a symmetry is broken when entering the ordered state, therefore the order parameter will change from zero to a nonzero value in this transition. In the case of the liquid–gas transition, there is no symmetry breaking but the density can be used as the order parameter to distinguish two phases. Order parameters can be calculated by taking the first derivative of the free energy with respect to some external fields, such as the magnetic field in the Ising model. As shown in Fig. 1.2, the sudden change of the density ρ under Tc and Pc indicates a first-order phase transition, and the continuous but singular

change exactly at Tc and Pc implies a second-order phase transition. The magnetization

M in Ising model corresponds to the density ρ in water. However, please note that, these two phase transitions are not exactly equivalent: at the critical point, there is no symmetry broken in the liquid-gas transition, while there is a spontaneous spin rotational symmetry broken in ferromagnets. Nevertheless, with the order parameter of the gas-liquid defined as the difference in density between the liquid and the gas, and that of the Ising model being the magnetization M, the symmetries of these order parameters are the same and, according to universality, the critical points are therefore associated with the same kinds of singularities (the same critical exponents).

P

M

(a)

(b)

Gas

Liquid

T > T

T = T

T < T

H

Figure 1.2: (a) The density of water molecules ρ changes continuously when T ≥ Tc and

P ≥ Pc, and changes discontinuously below Tc and Pc. Similarly, (b) the magnetic order

parameter (magnetization M ) changes continuously above Tcand discontinuously below Tc.

The discontinuous change corresponds to first-order transitions, which end at the critical point. At Tcthe behavior is still singular, while for T > Tc, the phase transition is replaced

by smooth changes.

Similar to the classical phase transitions, there are first order and second order phase transitions in quantum systems as well. A Quantum Phase Transition (QPT) is a phase transition taking place between two distinct ground states (i.e., at temperature T = 0) with tuning parameters like pressure, magnetic field, chemical substitution and so on at zero temperature [6–9]. A schematic phase diagram is shown in Fig.1.3, where g is the tuning parameter. Classical phase transitions are driven by thermal fluctuations. Analogously, the driving source of QPTs are provided by quantum fluctuations, which arise from competing (non-commuting) interactions in the Hamiltonian of the system. Strictly speaking, QPTs take place only at zero temperature, however, the quantum fluctuations still affect system properties at a certain range of finite temperatures, where the energy scale of the quantum fluctuations is similar to the thermal fluctuations. Quantum effects are still dominant when mode frequencies of the system are below the quantum-to-classical crossover frequency

kBT

~ [10].

The established theoretical framework for explaining both classical phase transitions and QPTs is Landau-Ginzburg-Wilson (LGW) paradigm. The central concept of LGW theory

5

PHASE I

T

gc

g

Quantum Critical

Classical

PHASE II

Figure 1.3: A schematic phase diagram for Quantum Phase Transition (QPT). A QPT is driven by quantum fluctuations tuned by some parameter g at zero temperature. The parameter could be pressure, magnetic field, chemical substitution etc. The curved lines together with the critical point gc form the quantum critical “fan” region below certain

temperature, where quantum fluctuations still affect finite temperature properties of the system.

is to use order parameter fields and their gradients to construct an effective free energy and obtain solutions using various approximations (such as mean-field theory) [11]. For studying the second order phase transitions, Renormalization Group methods (RG) [12] can be used to explain and study the singularities. LGW and RG methods have been successfully used to explain many QPTs.

In this thesis, we are particularly interested in continuous phase transitions, where the order parameter vanishes at some critical point gc at T = 0. In addition to the vanishing

order parameter, some physical properties, such as the characteristic length ξ (e.g. the correlation length), the susceptibility and the specific heat (in some cases) diverge at the critical point as |g − gc|−p, where p is the corresponding critical exponent. The critical expo-

nents have universal values — values depending only on symmetries and the dimensionality of the system. We will consider quantum spin models in which the non-magnetic phase has an excitation gap. The characteristic energy, for example, the energy gap between the first excited state and the ground state (for the gapped system), vanishes as a power-law decay of the characteristic length with the dynamic exponent z, as ∆ ∝ ξ−z. The critical state abides at or near gc, where the competition between two phases is the strongest. In

some quantum models, this critical state might be the long-searched-for Spin Liquid [13–16], which might be related to the superconducting states in high Tc superconductors and can

also exist in certain classes of insulators with localized spin [5, 17, 18]. We will discuss spin liquid states more in Sec.1.4.