Today at the CQT, Paolo Zanardi from the University of Southern California is giving a talk on “Quantum Fidelity and the Geometry of Quantum Criticality”. Here are my rough notes…
The motto from the early days of quantum information theory was “Information is physical.” You need to care about the physical medium in which information is encoded. But we can also turn it around: “Physics is informational”.
In a “classical phase transition”, thermal fluctuations play a crucial role. At zero temperature these go away, but there can still be different phases depending on other parameters. A transition between phases at zero temperature is called a quantum phase transitions. One way to detect a quantum phase transition is simply to notice that ground state depends very sensitively on the parameters near such a point. We can do this mathematically using a precise way of measuring distances between states: the Fubini-Study metric, which I’ll define below.
Suppose that is a manifold parametrizing Hamiltonians for a quantum system, so each point gives a self-adjoint operator on some finite-dimensional Hilbert space, say . Of course in the thermodynamic limit (the limit of infinite volume) we expect our quantum system to be described by an infinite-dimensional Hilbert space, but let’s start out with a finite-dimensional one.
Furthermore, let’s suppose each Hamiltonian has a unique ground state, or at least a chosen ground state, say . Here does not indicate a point in space: it’s a point in , our space of Hamiltonians!
This ground state is really defined only up to phase, so we should think of it as giving an element of the projective space . There’s a god-given metric on projective space, called the Fubini-Study metric. Since we have a map from to projective space, sending each point to the state (modulo phase), we can pull back the Fubini-Study metric via this map to get a metric on .
But, the resulting metric may not be smooth, because may not depend smoothly on . The metric may have singularities at certain points, especially after we take the thermodynamic limit. We can think of these singular points as being ‘phase transitions’.
If what I said in the last two paragraphs makes no sense, perhaps a version in something more like plain English will be more useful. We’ve got a quantum system depending on some parameters, and there may be points where the ground state of this quantum system depends in a very drastic way on slight changes in the parameters.
But we can also make the math a bit more explicit. What’s the Fubini-Study metric? Given two unit vectors in a Hilbert space, say and , their Fubini-Study distance is just the angle between them:
This is an honest Riemannian metric on the projective version of the Hilbert space. And in case you’re wondering about the term ‘quantum fidelity’ in the title of Zanardi’s talk, the quantity
is called the fidelity. The fidelity ranges between 0 and 1, and it’s 1 when two unit vectors are the same up to a phase. To convert this into a distance we take the arc-cosine.
When we pull the Fubini-Study metric back to , we get a Riemannian metric away from the singular points, and in local coordinates this metric is given by the following cool formula:
where is the derivative of the ground state as we move in the th coordinate direction.
But Michael Berry came up with an even cooler formula for . Let’s call the eigenstates of the Hamiltonian , so that
And let’s rename the ground state , so
Then a calculation familiar to those you’d see in first-order perturbation theory shows that
This is nice because it shows is likely to become singular at points where the ground state becomes degenerate, i.e. where two different states both have minimal energy, so some difference becomes zero.
To illustrate these ideas, Zanardi did an example: the XY model in an external magnetic field. This is a ‘spin chain’: a bunch of spin-1/2 particles in a row, each interacting with their nearest neighbors. So, for a chain of length , the Hilbert space is a tensor product of copies of :
The Hamiltonian of the XY model depends on two real parameters and . The parameter describes a magnetic field pointing in the direction:
where the ‘s are the ever-popular Pauli matrices. The first term makes the components of the spins of neighboring particles want to point in opposite directions when is big. The second term makes components of neighboring spins want to point in the same direction when is big. And the third term makes all the spins want to point up (resp. down) in the direction when is big and negative (resp. positive).
What’s our poor spin chain to do, faced with such competing directives? At zero temperature it seeks the state of lowest energy. When is less than -1 all the spins get polarized in the spin-up state; when it’s bigger than 1 they all get polarized in the spin-down state. For in between, there is also some sort of phase transition at . What’s this like? Some sort of transition between ferromagnetic and antiferromagnetic?
We can use a transformation to express this as a fermionic system and solve it exactly. Physicists love exactly solvable systems, so there have been thousands of papers about the XY model. In the thermodynamic limit () the ground state can be computed explicitly, so we can explicitly work out the metric on the parameter space that has as coordinates!
I will not give the formulas — Zanardi did, but they’re too scary for me. I’ll skip straight to the punchline. Away from phase transitions, we see that for nearby values of parameters, say
the ground states have
for some constant . That’s not surprising: even though the two ground states are locally very similar, since we have a total of spins in our spin chain, the overall inner product goes like .
But at phase transitions, the inner product decays even faster with :
for some other constant .
This is called enhanced orthogonalization since it means the ground states at slightly different values of our parameters get close to orthogonal even faster as grows. Or in other words: their distance as measured by the metric grows even faster.
This sort of phase transition is an example of a “quantum phase transition”. Note: we’re detecting this phase transition not by looking at the ground state expectation value of a given observable, but by how the ground state itself changes drastically as we change the parameters governing the Hamiltonian.
The exponent of here — namely the 2 in — is ‘universal’: i.e., it’s robust with respect to changes in the parameters and even the detailed form of the Hamiltonian.
Zanardi concluded with an argument showing that not every quantum phase transition can be detected by enhanced orthogonalization. For more details, try:
• Silvano Garnerone, N. Tobias Jacobson, Stephan Haas and Paolo Zanardi, Fidelity approach to the disordered quantum XY model.
• Silvano Garnerone, N. Tobias Jacobson, Stephan Haas and Paolo Zanardi, Scaling of the fidelity susceptibility in a disordered quantum spin chain.
For more on the basic concepts, start here:
• Lorenzo Campos Venuti and Paolo Zanardi, Quantum critical scaling of the geometric tensors, 10.1103 Phys. Rev. Lett. 99.095701.
As a final little footnote, I should add that Paolo Zanardi said the metric defined as above was analogous to Fisher information metric. So, David Corfield should like this…