Continuous Phase Transitions, Critical Phenomena, and the Renormalization Group

The Ising model that we've discussed extensively is the most prominant example of a special type of phase transition known as a continuous phase transition. Continuous phase transitions are sometimes described as "second-order" phase transitions though strictly speaking this is a narrower class.

First- and Second-Order Phase Transitions

This terminology relates back to an earlier classification scheme due to Ehrenfest that classified phase transitions according to the continuity class of the free energy. Phase transitions that had a continuous free energy, but a discontinuous first derivative (with respect to some thermodynamic variable) were classified as first-order. Examples of first-order phase transitions include the various solid/liquid/gas transitions because the order parameter—the change in density—is obtained from the first derivative of the free energy with respect to the chemical potential.

Second-order phase transitions and continuous phase transitions are synonymous in the modern era. They include situations where the free energy is continuous in its first two derivatives, but then potentially discontinuous at some higher derivative. In modern usage many other types of phase transitions fall under the heading of continuous phase transitions as well.

The key feature distinguishing a continuous phase transition is that as the tuning parameter is changed, the state of the system continuously enters a configuration with fewer symmetries than the underlying Hamiltonian. Here the tuning parameter (or control parameter) is a macroscopic control variable such as an external magnetic field, temperature, pressure, etc. The symmetry is measured with respect to an order parameter such as average net magnetization which captures the relevant features of the new symmetry as a macroscopic observable. As the tuning parameter is changed, the system undergoes spontaneous symmetry breaking and enters the configuration with reduced symmetry, as measured by the order parameter.

Let's see a few examples of this, starting with our favorite model, the Ising model.

The Ising ferromagnet, where the Hamiltonian is given by $-J\sum_{(i,j)\in\Lambda} s_i s_j$ with $J>0$ and is invariant under the two-fold symmetry of exchanging $s_i \rightarrow −s_i$. The tuning parameter is the temperature $T$, and the order parameter is the average net magnetization, $\avg{M}$. In the high-temperature (disordered) phase, the system maintains this two-fold symmetry, whereas in the low-temperature phase (and when $\Lambda$ is a two-dimensional lattice) the system obtains a spontaneous magnetization, and breaks the symmetry. Note that $\avg{M}$ goes to zero continuously as the temperature rises to $T_c$.
The Ising antiferromagnet, which we restrict to lie on a simple cubic lattice, has the same Hamiltonian but with $J\lt 0$. The symmetry is the same, but the order is the so-called Néel order given by the "staggered" average magnetization $\avg{M_{\pm}} = \frac{1}{N}\sum_j (−1)^j \avg{s_j}$, where we label sites in the lattice alternately with even or odd numbers.
The Heisenberg ferromagnet is similar to Ising models, but the spins are allowed to point in a continuum of directions. The parent symmetry is $O(3)$, the three-dimensional rotation group, and in three dimensions the system spontaneously magnetizes along a random direction in space when cooled below a critical temperature. Thus, the symmetry breaks from $O(3)$ to $O(2)$.
Many other examples besides magnetic transitions exist, such as some superconducting and superfluid transitions, percolation transitions, Bose-Einstein condensation, wetting transitions, the classic solid-liquid-gas transitions, as well as more exotic fare like topological phase transitions.

A cautionary word on terminology: the phase with a larger symmetry group is typically called the disordered phase, while the phase with the broken symmetry (smaller symmetry group) is called ordered. This seems a bit paradoxical at first, but a good way to remember it is that the order comes from fixing one of the symmetries, thus producing "order" from a larger sea of possible symmetry states.

In general, there can be many different broken-symmetry phases all with the same free energy, and each of these phases are macroscopically distinct in the thermodynamic limit. Therefore it is not possible for thermal fluctuations (at fixed temperature) to connect one phase to another.

This characterization of continuous phase transitions as arising from spontaneously broken symmetries is a bit misleading because it isn't the whole story. There are transitions known as infinite-order phase transitions that are continuous but break no symmetries. The Kosterlitz–Thouless transition in a two-dimensional model of rotating spins is perhaps the most famous example of this, but many quantum phase transitions have this property as well.

Universality and Critical Exponents

Continuous phase transitions are characterized by additional distinguishing features, namely divergent susceptibility, power-law decay of correlations, and other power-law behavior of macroscopic thermodynamic quantities near the critical point. What's more, these features seem to be dependent only on a few simple properties of the system and not on the microscopic details of the underlying Hamiltonian. This is the phenomenon of universality. All empirical evidence (and numerous mathematical proofs in special cases) seem to show that the behavior only depends on three simple quantities:

the dimension of the system,
the range of the interaction,
the spin dimension.

Moreover, when the interactions are finite-ranged, we can always trade-off range for spin dimension, so there are effectively only two quantites.

The power-law scaling of thermodynamic quantities can be characterized by the critical exponents. We will consider temperature as the tuning parameter, and use a dimensionless temperature \[\tau=\frac{T-T_c}{T_c}\,.\] To calculate any thermodynamic quantity, we can first calculate the free energy density (free energy per particle) via the equation \[f = \frac{F}{N} = -\frac{1}{N\beta}\log Z,\] where $Z$ is the partition function and $N$ is the total number of spins. From here it is easy to derive expressions for the physical quantities of interest that give us scaling laws. It is useful to allow our model to depend on an external field $h$, so that in the case of the Ising model the Hamiltonian is \[ H = -J \sum_{(i,j)\in\Lambda} s_i s_j - h \sum_{i\in \Lambda} s_i \] Then some examples of critical exponents and the physical quantities that they are associated with are as follows.

$\alpha$, Heat capacity: $C = \frac{\partial^2 f}{\partial\beta^2} \sim \lvert\tau\rvert^{-\alpha}$;
$\hat{\beta}$, Magnetization: $M = -\frac{\partial f}{\partial h}\sim \lvert\tau\rvert^{\hat{\beta}}$;
$\gamma$, Magnetic susceptibility: $\chi = \frac{\partial M}{\partial h} = -\frac{\partial^2 f}{\partial h^2} \sim \lvert\tau\rvert^{-\gamma}$;
$\delta$, Magnetic response: $M \sim \lvert h\rvert^{1/\delta}$ as $h \to 0$, $h$ on-site field strength.
$\nu$, Correlation length: $\xi \sim \lvert\tau\rvert^{-\nu}$.
$\eta$, Correlation function at $T_c$: $\langle s_0s_x\rangle \sim \frac{1}{|x|^{d-2+\eta}}$ as $|x|\to \infty$.

Not all of these are independent, in fact, and there are simple relationships between them called scaling relations. We will not be able to derive the scaling relations in this course, but to give you a sense of their power and elegance, here are some known scaling relations: \[\alpha + 2 \hat{\beta} + \gamma = 2\] \[\hat{\beta} \delta = \hat{\beta} + \gamma\] \[d \nu = 2-\alpha\] \[2-\eta = d \frac{\delta - 1}{\delta + 1}\] The latter two of these are known as hyperscaling laws because they depend also on the dimension $d$ of the space. These are believed to only be valid in sufficiently few dimensions (and fortunately, the 2 or 3 dimensions where we live is often small enough for them to be valid).

We would like to know how to compute these exponents from first principles. The method of mean-field theory which you discussed in lecture is one such method, but it has a major drawback: it only works in 4 or more spatial dimensions! Nonetheless, it can give us insights into critical behavior more generally, and by adding in calculations of fluctuations about the mean-field and using the renormalization group, we can often derive useful predictions.

The Renormalization Group

Continuing our tour of unfortunate terminology, we come to "the" "renormalization" "group". This is a general framework for understanding critical phenomena, and not a unique mathematical object as the definite article would suggest. In fact, insofar as it is rigorously defined, it is certainly not a group. And the term "renormalization" was originally coined in the 1960's by particle physicists, and it is misleading because renormalized fields do not play any essential role in the framework.

The basic idea of the renormalization group is to take the enormous number of degrees of freedom in a typical many-body system and reduce it to a smaller number of variables that leave the essential physics that we're interested in intact. This is easier said than done, but an example will certainly illustrate things more clearly.

The most famous example of a renormalization group transformation is called block-spin renormalization. In its simplest instantiation, we simply start with an Ising lattice model in two dimensions and course-grain the spins into blocks. For concreteness, we can do the following procedure. Start with a $3N\times3N$ lattice, and course-grain into $3\times3$ blocks. For each block, replace the spins in that block by a single spin whose state is given by the majority vote of the spins in the block.

Modify the code in the ising.zip files to implement the block-spin transformation, and return a plot of the new $N\times N$ (blocked) lattice.
Run the code (Metropolis or Wolff) until the system has reached a typical (metastable) state, then apply the block transformation. This should look like the original system in equillibrium, but at a different temperature. What effective temperature do you see, and how does it depend on the original temperature?

The idea that this exercise is getting to is the notion of fixed points of a renormalization group flow. The transformation determines a transformation of the thermodynamic variables like magnetization, and we can iterate this transformation and ask for its fixed points. The critical point of the Ising model is a fixed point of the block-spin renormalization group flow. The remarkable implication of this is that the physics of the Ising model is scale invariant at the critical point!

Quantum Phase Transitions and the Transverse-Field Ising Model

The phase transitions discussed above all work with temperature as the tuning parameter. But there is another class of phase transitions that only occur at zero temperature. Here the tuning parameters are the coupling constants of various terms in a Hamiltonian.

In class you discussed how there could be no phase transition in a one-dimensional Ising model at temperature $T\gt0$. This is still true, but only for thermal phase transitions. When we are at zero temperature for a given Hamiltonian, there is still a unique thermal equillibrium state, it is just given by the ground state (lowest energy eigenstate) of the Hamiltonian. The key feature that determines the boundary of two phases is the spectral gap between the ground state(s) and the first excited state(s). When this energy gap is large (a positive number that doesn't decrease as the number of spins increases) then you are in a stable quantum phase. Conversely, when this gap closes, this is a signature of a quantum phase transition. By varying the coupling constants of a Hamiltonian at zero temperature we can move between two different gapped phases through such a quantum phase transition.

Quantum phase transitions can be much more difficult to study than thermal phase transition because the obvious way to study them involves computing the ground state of the Hamiltonian. There are by now very sophisticated methods for that avoid this brute-force method of exact diagonalization, but we can still use it to check our work for small examples.

The paradigmatic example of a quantum phase transition is the transverse-field Ising model. Here there are two competing interactions: a transverse field pointing in the $x$ direction and a ferromagnetic Ising-like coupling along the $z$ axis. The Hamiltonian for the system is given by \[H = -J \sum_{j=1}^{n} Z_j Z_{j+1} -B\sum_{j=1}^{n} X_j\,,\] where for simplicity we assume periodic boundary conditions. We can use the following code to compute the Hamiltonian and the lowest energy eigenstates.

X=[0 1;1 0]; Z=[1 0;0 -1]; % Pauli X and Z matrices
n=12; % length of the chain
J=1; B=1; % coupling strengths
H=-J*nnchainp(Z,Z,n)-B*nnchainp(X,eye(2),n); % Hamiltonian

[v,lambda] = eigs(H,6,'sa'); % lowest energy eigenvalues and eigenstates

The matlab file nnchainp.m contains a method for building the transverse-field Ising model Hamiltonian as a large sparse matrix. It outputs the matrix \[\sum_{j=1}^n a_jb_{j+1}\] with periodic boundary conditions, where $a_j$ is a spin operator acting on site $j$ and $b_{j+1}$ acts on site $j+1$ (with all other sites acting like the identity operator).

A convenient parameterization of this model is to divide the Hamiltonian by $\lvert J\rvert$ and study the behavior with respect to $b = B/\lvert J\rvert$ and the sign of $J$.

Compute the average magnetization along the $x$ and $z$ directions as a function of $b$. What happens?
Now compute the average spin-spin correlation $\avg{ZZ}$ as a function of $b$.
Plot the spectral gap as a function of b.
Where is the quantum phase transition in this model?