# Quantum phase transitions

In class we gave a number of examples of quantum phase transitions, including many quantum critical points whose theory is not yet well understood. Sometimes quantum Monte Carlo and other methods provide insight, and we had a guest lecture by Prof. Ribhu Kaul explaining some aspects of these approaches. Below are some notes on a different way to describe the quantum-classical mapping, along with some general thoughts on quantum phase transitions.

The main topic of this course has been new quantum phases of matter that emerge at low energy, usually because of the Coulomb interaction. That naturally raises the question of how transitions between quantum-mechanical phases differ from ordinary phase transitions in classical statistical mechanics. We will briefly cover the topic of "quantum phase transitions", following the textbook of Sachdev.

As we saw last time, a single material can support many phases. An example is the heavy-fermion material (like those we discussed last time) CeCu$_{6-x}$Au$_x$, which has a transition in doping $x$ between an incommensurate spin-density-wave and a strongly renormalized Fermi liquid; very similar materials also show superconductivity and commensurate magnetic order.

A surprising fact is that the universal properties of finite-temperature phase transitions, such as critical exponents, can be described by classical physics, even if one or both phases exist only because of quantum mechanics. For example, the best critical scaling seen in any transition is for the superfluid transition in He$^4$, and the critical exponents are well described by the classical "XY model" universality class. The concept of "universality class" we use here is defined precisely in Physics 212: for now, think of a universality class as the set of all phase transitions which differ only in details that do not affect critical exponents and other scaling behaviors. A heuristic argument for why classical physics applies is that close enough to a phase transition at nonzero temperature, the long-wavelength fluctuations are described by classical physics since the length scale is much larger than the scale over which phase coherence is significant.

Quantum phase transitions are driven by a nonthermal coupling such as magnetic field and occur at zero temperature. Our main technical goal here will be to show that some quantum phase transitions can be mapped onto {\it classical} phase transitions in one higher dimensions: this means in particular that their universal properties can be obtained from this quantum-classical (QC) mapping. It may be worth quickly reviewing some of the phases we have seen in the course to explain what sort of quantum phases may occur.

The temperature-driven transition into a superconductor or superfluid is described by classical physics; in practice the superfluid is much better than the superconductor for critical phenomena measurements because of the large Cooper pair size and long-range interactions in the latter. There are actual quantum phase transitions in some superconducting systems at $T=0$, as we discuss below.

The Kondo effect is quite amazing in that it generates a nonperturbative energy scale $T_K$ with no phase transition. We also talked about localization/delocalization transitions at zero temperature in systems with quenched disorder. These are bona fide quantum phase transitions, and a major success of the 1980s was in understanding how transitions like the Anderson localization transition can be connected to more conventional (e.g., magnetic) phase transitions. This is still a very active area, and the recent textbook of Efetov contains a good introduction to the major techniques.

Now we focus specifically on transitions where there is a local degree of freedom, like a spin or superconducting phase Importance of symmetry and dimensionality

Quantum Ising model. First consider the ordinary Ising model in one dimension,

\begin{equation}
H = -K \sum_i \sigma^z_i \sigma^z_{i+1} - h \sigma^z_i.
\end{equation}

We can either think of the spins as classical Ising spins or as Heisenberg spins, since even if the spins in the above are Heisenberg spins, the above calculation gives the same result since we can go to a . Also, even if the spins point along

The quantum or transverse-field Ising model is just

\begin{equation}
H = -K \sum_i \sigma^z_i \sigma^z_{i+1} - g \sigma^x_i.
\end{equation}

It turns out that having the magnetic field (now denoted by $g$) point in a different direction than the easy-axis of the spins dramatically changes the physics. It turns out that this quantum Ising model has a zero-temperature phase transition that is in the universality class of the classical 2D Ising model, which was solved by Onsager.

The classical Ising partition function can be written as, with $M$ the number of spins,

\begin{equation}
Z = \sum_{\sigma_i^z} \prod_{i=1}^M T_1(\sigma^z, \sigma^z_{i+1}) T_2(\sigma^z_i)
\end{equation}

where $T_1(\sigma^z_1,\sigma^z_2) = \exp(K \sigma^z_1 \sigma^z_2)$, $T_2(\sigma^z_i) = \exp(h \sigma^z_i$. But this product is equivalent to a matrix product (as shown in 212, and easy to convince yourself)

\begin{equation}
Z = Tr (T_1 T_2 T_1 T_2 \ldots) = Tr (T_1 T_2)^M.
\end{equation}

Using this it is easy to work through the full transfer-matrix solution of classical limit, including, for example, the correlation between two spins.

\begin{equation}
\langle \sigma^z_i \sigma^z_j \rangle = (\tanh K)^{|j-i|}
\end{equation}

which gives the result for the correlation length

\begin{equation}
\xi^{-1} = a^{-1} \log(\coth K).
\end{equation}

In the low-temperature limit $K \gg 1$, $\xi \approx a e^{2 K} / 2$. We will be interested in scaling properties when $\xi \gg a$, as only then is the physics "universal" and independent of short-length-scale details.

Now we will show that the above can really be viewed as the quantum mechanics of a single Ising spin. We can write

\begin{equation}
T_1 = e^K (1 + e^{-2K} {\hat \sigma}^x) \approx e^K (1 + (a/2 \xi) {\hat \sigma}^x),
\quad T_2 = \exp(a \sigma^z (h / a)).
\end{equation}

The so-called scaling limit will be taken as $a \rightarrow 0$ and the quantity ${\tilde h} = h/a$ finite, which is why we have formed this combination in the exponent.

Now the transfer matrix is just the quantum evolution operator of the Hamiltonian

\begin{equation}
H_Q = E_0 - {\Delta \over 2} {\hat \sigma}^x - {\tilde h} {\hat \sigma}^z
\end{equation}

with $E_0 = -K/a$, $\Delta = {1 \over \xi}$, over a short time $\tau = a$:

\begin{equation}
T_1 T_2 \approx \exp(-a H_Q).
\end{equation}

So

\begin{equation}
Z = Tr \exp(-H_Q / T)
\end{equation}

with $1/T = L_\tau = M a$.

Note that the energy $\Delta$ is the gap between the two states of the single spin in zero applied field:

\begin{equation}
E = E_0 \pm \sqrt{(\Delta / 2)^2 + {\tilde h}^2}.
\end{equation}

The gap is just the inverse of the correlation length in the classical model. So two important relations in this quantum-classical mapping are that the {\bf inverse temperature} $\beta$ in the quantum model becomes the {\bf system size} in the classical model, and the {\bf gap} $\Delta$ in the quantum model becomes the {\bf inverse correlation length} in the classical model.

In the same way, coupling a chain of spins becomes, via the above QC mapping, a 2D classical Ising model. A question you might think about Is whether there a signal of a truly quantum phase transition, i.e., one that cannot be related to a classical critical point via the above QC mapping.

We build on this formal relationship between the long-wavelength limit of the classical Ising chain in one dimension and the quantum dynamics of a single quantum spin, with Hamiltonian

\begin{equation}
H_Q = E_0 - {\Delta \over 2} {\hat \sigma}^x - {\tilde h} {\hat \sigma}^z.
\end{equation}

Our goals in the first part of the lecture will be to make the nature of this "quantum-classical mapping" clearer and to understand how it can be used to understand the phase diagram of these magnetic systems.

The steps in what we did last time were to write the partition function of the classical chain in the form

\begin{equation}
Z = Tr (T_1 T_2)^M
\end{equation}

where $T_1 T_2$ is the transfer matrix. For large correlation length (low temperature) the transfer matrix could be rewritten as $T_1 T_2 \approx e^{-a H_Q}$, so that the partition function is effectively that of the single quantum spin:

\begin{equation}
Z= Tr \exp(-H_Q / T)
\end{equation}

with $1/T = L_\tau = M a$. Note that as $M \rightarrow \infty$, the effective temperature in the quantum model goes to 0.

Recall that in deriving the above results we had to assume that the correlation length of the classical model was much larger than the correlation length. When this is true, the physics is largely independent of short-distance details: for instance, adding a next-nearest-neighbor interaction would change the problem quantitatively by changing the effective splitting $\Delta$ in the quantum model, but not change the qualitative picture. This means that the quantum Hamiltonian $H_Q$ does not correspond to just one classical model in one higher dimension; it corresponds to the {\bf scaling limit} of many different classical Hamiltonians, since $H_Q$ is only sensitive to the long-length-scale physics of the classical problem. One usually just tries to choose the simplest classical model from the set of classical models that relate to a given $H_Q$.

An intuitive picture of the above quantum-classical (QC) mapping is that a quantum model in $d$ dimensions at temperature $T$ is equivalent to the scaling limit of a classical model in a strange "slab" geometry: the classical model has $d$ infinite dimensions and a $d+1$th dimension of length proportional to $\beta = (kT)^{-1}$. Note also that the scaling limit depended on taking certain combinations in the classical model to be finite and correspond to the parameters $E_0$, ${\tilde h}$, and $\Delta$ of the quantum model.

We argued in class, and can see explicitly using the methods developed in 212, that it is the long-length-scale physics of the scaling limit that controls the universal properties near a continuous (a. k. a. second-order) phase transition. This means that quantum phase transitions in the quantum model can be related using the QC mapping to classical transitions in the classical analogue. The case we studied so far, the classical Ising chain in one dimension and single Ising spin, has no interesting phase transition. In one more dimension, we start to see interesting results: the quantum (transverse-field) Ising chain in one dimension at zero temperature has a transition as a function of $g$ (the ratio of magnetic field to Ising coupling) that is equivalent to the 2D classical Ising model, the simplest nontrivial phase transition. The QC mapping can also be used to understand how this transition develops as $T \rightarrow 0$ in the quantum model, which corresponds to strips in the classical case of larger and larger size. Methods known as "finite-size scaling" can be used to understand the asymptotic behavior of such systems, studied numerically.

You may well wonder how, since we said earlier that quantum phase transitions exist only at zero temperature, their properties appear in real $T>0$ experiments. There are two cases we need to worry about: in one, there is only a true transition at zero temperature (there is no classical phase transition in the system). This is the case for the quantum Ising chain. At $T=0$ there is a transition between a quantum disordered phase and a "renormalized classical" phase, and this phase transition is in the universality class of the 2D Ising model. At nonzero temperature, there is no thermodynamic transition (no singularity in any derivative of the free energy), but there is a crossover between the two phases, and there is an additional regime above the critical point called the "quantum critical" regime: here the physics is not well described by either simple limit but by interacting dynamics of critical fluctuations.

The $O(2)$ and $O(3)$ rotor models are often used to model quantum phase transitions where the symmetry is different than in the quantum Ising case discussed above. The $O(2)$ rotor model, with quantum Hamiltonian for a single rotor

\begin{equation}
H_Q = - \Delta {\partial \over \partial \theta}^2 - {\bar h} \cos \theta,
\end{equation}

has a transition in the universality class of the classical XY in one higher dimension (cf. Sachdev). The coupling between two rotors is just proportional to $\cos(\theta_i - \theta_j)$. Here the quantum model describes the physics of a particle moving freely on the circle, plus a potential tending to put the particle at one particular point. The $O(3)$ rotor model has a transition in the universality class of the classical Heisenberg in one higher dimension. Note that, as shown before, the quantum Heisenberg model in $d$ dimensions {\it does not} map onto the classical Heisenberg model in $d+1$ dimensions: the details of the $SU(2)$-symmetric Heisenberg model wind up giving "Berry phases" in its classical analog.

Let me give a quick hand-waving argument that the model of coupled superconducting grains with a charging energy is essentially similar to (and in the universality class of) the $O(2)$ quantum rotor model. We start with the charging-energy part of the model

\begin{equation}
H_C = E_C (n_i - n_0)^2.
\end{equation}

This is written in terms of the number density $n_i$, which we would like to eliminate in favor of the superconducting phase $\theta_i$. Recall that $n_i$ is the conjugate variable to the local phase $\theta_i$ (to be precise, let us define $n_i$ as a local charge density; then there is a factor $2 e$ required to make $n_i$ dimensionless, so that $n_i / 2 e$ is the number of Cooper pairs). In the imaginary-time partition function we should have the Lagrangian, which is obtained from the Hamiltonian in the usual fashion:

\begin{equation}
Z = \int\,d\theta_i\,e^{- \int\,d\tau\,i {n_i \over 2 e} {d \theta_i \over d\tau} + E_C (n_i - n_0)^2}.
\end{equation}

Now this is quadratic, and integrating out the $n_i$ variable will give a term ${\dot \theta}^2$. This means essentially that we have found a classical version of the problem

that is described by the classical $XY$ model in one higher dimension. Since the $O(2)$ quantum rotor also maps onto the classical $XY$ model in one higher dimension, as shown explicitly in Sachdev's book, the $O(2)$ rotor and the superconducting grain model must have the same universal properties.

Another example of a quantum phase transition relates to our previous discussion of the Haldane gap in spin-one chains. One way to confirm the conjectured gap is via numerical techniques, but a nice piece of analytic insight was provided by Affleck, Kennedy, Lieb, and Tasaki. By adding a biquadratic interaction $\lambda ({\bf s}_1 \cdot {\bf s}_2)^2$ in the spin-one chain, AKLT found a solvable point at $\lambda=1/3$ where the Haldane gap vanishes. For larger values of the coupling, the system is gapless, while for smaller values, there is a nonzero gap and finite correlation length. So the AKLT point is essentially a critical point between two different one-dimensional phases.