Markus Garst (Köln) -- Multiscale quantum criticality: Nematic instability in metals

References for what Markus will talk about:
M. Zacharias, P. Wölfle and M. Garst, PRB 80, 165116 (2009)
M. Garst and A. Chubukov, PRB 81, 235105 (2010)


Markus began by apologizing to the experimentalists for giving a (possibly) technical theoretical talk, but there was no mass exit from the lecture theatre so all should be OK.

Introduction: Quantum Phase Transitions (QPT) and multiple scales

A 2nd order QPT is an instability in the ground state at T=0, as a function of some control parameter we will call r (e.g. magnetic field, pressure, doping, etc...) While this transition happens strictly only at T=0, it has a strong effect on finite-T properties due to an abundance of low-energy fluctuations. So on phase diagram, should have T too - region in phase space where properties controlled by Quantum Critical Point (QCP)

Some technical things:
--Correlation length exponent \nu : \xi \sim |r|^{-\nu} (\xi = correlation length, r=control parameter)
--dynamical exponent z : spectrum of critical fluctuations goes as \omega \sim k^z which gives a vanishing characteristic energy scale \epsilon\sim\xi^-z
--enhanced dimensionality: correlation volume in space: \xi^d and time \xi^z gives you an effective dimensionality d_eff = d + z

This set of exponents will give you a scaling ansatz of the critical free energy
F(r,T) = b^{-(d+z)} F ( r b^{1/\nu}, T b^z)
for some arbitrary scaling variable b.
Such scaling behavior are widely used to interpret a wide range of experiments where you don't know exactly the microscopic details of what happens, but the scaling may still work.

Quantum-to-classical crossover: we have the correlation volume in time \xi^z, but this is limited by the temperature \xi^z < t="0),">0). However, the flow is such that for T>0, a crossover temperature (\xi_T) may be defined where the flow leaves the T=0 path to divert to the classical critical point.
(see e.g. Nelson 1975, Millis, 1993)

As a brief summary: the relation between the thermal length \xi_T and correlation length, \xi gives us a crossover in the phase diagram.

In many systems, there is a coexistence of low-energy fluctuations.
For example, near a magnetic instability, have critical magnetic fluctuations as well as ballistic electrons - so can find sometimes two dynamical exponents z_1 and z_2.

With two dynamical exponents, this raises a lot of points:
--coexistence and interacting fluctuations -> entanglement!
--identification of proper critical degrees of freedom?
--two dynamical exponents-> breakdown of scaling and power laws?
--crossovers in phase diagram
--fluctuation driven first order transitions
Markus tells us that these are the `big questions' in this topic, many of which are unanswered. The remainder of this talk will be about a simple specific case.

Nematic Instability of Fermi liquid

Pomeranchuk instability: instability of FS towards development of a quadropole moment
Order parameter is a tensor object, similar to nematics. The traceless part of the strain tensor gives us the shear modes of the Fermi surface. In d=2, there are two shear modes.

We will look at a simple model Hamiltonian introduced by Oganesyan, Kivelson and Fradkin (2001). They developed an effective bosonic model which acts as the Ginzburg-Landau theory in the usual way by Hubbard-Stratonovich and integrating out fermions. More complicated than regular GL due to tensor order parameter, but otherwise standard. We note that in d=2, no cubic term is allowed, which means a 2nd order transition is possible.

OKF analyzed this model, finding a Pomeranchuk instability, with a criterion basically identical to the Stoner criterion as a function of some control parameter r depending on the density of states and the coupling constant (a control parameter Piers referred to as pretentious, but this story can wait for a rainy day...)

Markus then says the most interesting point about the instability is the dynamics - which he will now discuss.

One can have polarization of the excitations both longitudinal and transverse to the quadrupolar momentum q tensor - but because of different phase spaces for exciting particle hole pairs, one finds
--longitudinal polarization, Landau damping z=3 dynamics
--transverse polarization, ballistic z=2 dynamics.
This is the multi-scale property that was introduced early.

Two energy scales, two modes. Which mode is more important?

Naive answer: longitudinal z=3 mode has larger phase space: \Omega_n \sim q^z, so dominates the specific heat. This led OKF to claim that the z=2 mode plays no role in the critical theory.

However in reality, things are much more interesting. Transverse d=2 mode has smaller effective dimension d+z=4 - so generates logarithmic singularities in loop corrections, and it is the interplay of both modes that determines the critical properties.


We now become a bit more technical and look at the structure of the theory at T=0.
The transverse z=2 fluctuations allows us to write a logarithmic RG flow, giving mass renormalization. This flow is marginally irrelevant, but introduces a logarithmic scale dependence of the correlation length
\xi^{-2}(\epsilon) \sim r / [log(1/\epsilon)]^{4/9} for \epsilon>\xi^{-2} (z=2 energy scale)
This exponent 4/9 differs from Ising and XY universality, and is characteristic for Pomeranchuk.

What about theory at finite temperature?
multiple z -> multiple thermal lengths (\xi_T \sim T^{-1/z} )

For each thermal length, there is a quantum to classical crossover - so can have coexistence of quantum and classical fluctuations because of the multi-scale dynamics. (This is Markus' answer to Piers' question at the beginning).

This overlap regime in fact controls the thermodynamics in a wide range of the phase diagram.

Lets look at this again, in terms of RG: on the RG phase diagram, one mode wants to push the system away from the primary quantum critical point, while the other competes with it pushing in the other direction; and it is this wide range of scales that is important for thermodynamics in a wide range of the phase diagram.

Now look at the temperature boost of the RG flow -> some technical calculation, but the answer ultimately is that there is a universal correlation length at criticality (r=0):
\xi^{-2} = cT
which is universal in that it is generated, but independent of, the bare quartic coupling, u. This feature is entirely due to the competition of these different z=2 and z=3 modes.
In other words: multi-scale dynamics leads to a new kind of universality.

These multiple scales are also present in the phase diagram:
two thermal lengths -> two crossover lines in the phase diagram.
The sensitivity of something on the crossover depends strongly on specific thermodynamic quantity that you are looking at. Means that there is no scaling in terms of a single dynamical exponent; things are just a bit more complicated.

Markus then briefly shows us the results for a few thermodynamic quantities (specific heat, etc..) , which all have `funny' logs in them.


Electron spectral function

One-loop self-energy from z=3 mode (see Oganesyan Kivelson and Fradkin, 2001; Metzner, Rohe and Andergassen 2003, etc...) gives singular correction.
Also look at one-loop self-energy from the transverse z=2 mode; which gives interesting contribution to off-mass shell part, and singular correction to Z.

Sum up more logs, RG, ... find that the combination \Gamma Z = 1 is invariant. Hence the polarizations are unaffected by the electrons, and the critical dynamics are preserved.

Question (Piers): Is this a Ward identity? Answer: Not strictly speaking (but unfortunately I missed why - the slides are getting mode and more technical and difficult to blog...).

Hence the electron propagator has three important parts:
-- non-Fermi liquid frequency dependence at z=3 energy scale
--anomalous dimension at z=1 energy scale (z=1 is electrons)
--interesting correlation length dependence at z=2 energy scale.

This is still no the full story though, as there are further logarithms, including some that appear only in 3rd loop order (see Mross, McGreevy, Liu and Senthil, 2010)

Summary

--Nematic quantum criticality in metals: multiple energy scales
--extended quantum-to -classical crossover where quantum and classical fluctuations coexist and interact, and may lead to new forms of universality.
--In thermodynamics, all energy scales can appear, depending on quantity in question

Questions:

Q) Does this theory obey sum rules?
A) It should, no good reason why it shouldn't although must take into account all correct crossover scales to make them work. Chubukov extended this by commenting that this is a low energy calculation, high energy modes will adjust to make sum rules work

Q (Nevidomsky) In many cases, longitudinal modes don't couple to things, is that also true here?
A) (mostly given by Chubukov) - longitudinal and transverse for this quadropolar order are a different notation to what we are used to - should be careful trying to draw analogies

Q) In which (real) materials might you expect this to occur?
A) Crystal lattice may make big differences in this theory e.g. z=2 mode may become gapped (did I hear that correctly?), so at the moment, this work is without reference to real materials.

Q) (missed)

Q (Schofield) In the Stoner criterion, the Pomeranchuk instability condensed around q=0, rather like a ferro-magnet. If the condensation was about finite q (like SDW), would the structure of the theory change?
A) Yes, dramatically. No z=3 mode, lots of other stuff

Comment (Nersesyan) There are also cases where this multi-scale criticality can arise, even without different z's (e.g. in spin-charge separation in Luttinger-Liquid)
Answer: Absolutely true, although these sorts of single z cases are somewhat simpler, as you know how to rescale momenta, etc... Lots of new stuff when more than one z present.