Chandra Varma: Derivation of Quantum Critical Fluctuation spectra for Orbital Currents

Chandra started by drawing a phase diagram of high T_c cuprates with the quantum critical point inside the superconducting dorm and finite temperature transition line terminating at that critical point. He said he will describe the formulation of the microscopic theory that gives this phase diagram and explains the physics almost everywhere in the phase diagram.

He briefly mentioned the marginal Fermi liquid phenomenology that says many things can be explained if one assumes a q-independent dynamical response function,

\chi (\omega, q) = \chi_0 tanh (\omega/2T) for \omega < \omega_c, where \omega_c is some cutoff. In the putative quantum critical region, he said that

"Lots of things have been predicted, and no alternative has been found".

The question is "how does this spectrum arise?." Chandra began with the description of the ordered phase associated with the quantum critical point. He said 1) the cuprates are charge-transfer insulators and large U is present both at the Cu and O site. 2) The interaction between Cu and O is responsible for new phases. He wrote down the interaction term that involves the repulsive interaction between the density at the Cu site and that at the O site, namely V \sum_i n^{d}_i (n^p_{i+x} + other three neighboring O sites). He said this interaction can be rewritten in terms of quadratic forms of the currents, -V/4 \sum_i (J^2_{ix} + J^2_{iy}) + ... Here J_{ix} and J_{iy} describe the currents through the O and Cu sites in the horizontal and vertical directions. He said the mean field theory gives a local minimum that does not break translational symmetry. This solution is also characterized by finite expectation values of J_{ix} and J_{iy}.

He calls these currents "coherent" parts or the currents that order. He draw an example of such a (translationally symmetric) ordered current pattern. He said that there is now evidence for such a current-carrying state in the pseudogap region of the phase diagram.

He said that he is now going to present the derivation of the marginal Fermi liquid spectra starting from this picture of the current-carrying state and the corresponding quantum critical point. He pointed out that there are four possible current-carrying states with the broken time reversal symmetry and they are characterized by four possible directions of "staggered" magnetization within the unit cell. He then claimed that the effective model describing these four states are the so-called Ashkin-Teller model or two-coupled Ising models. This model has two kinds of terms; J_2 describing the spin-spin interaction for each Ising spin degrees of freedom \sigma and \tau, and J_4 that involves energy-density-energy-density interaction of two kinds of Ising spins, namely

H_{AT} = J_2 (\sigma_i \sigma_j + \tau_i \tau_j) + J_4 (\sigma_i \tau_i \sigma_j \tau_j).

He said that in some range of J_2/J_4, basically a Gaussian theory is valid. Here the model is supposed to be equivalent to an XY model with a four-fold anisotropy;

H = \sum_{ij} \kappa (J_2,J_4) cos(\theta_i-\theta_j) + h \sum_i cos(4 \theta_i).

Here the XY degree of freedom correspons to the direction of the "staggered" magnetization within the unit cell. Thus the ordered phase of this model is supposed to correspond to the current-carrying state mentioned above. He claimed that the four-fold anisotropy is irrelevant in the fluctuating regime (disordered state) while it is relevant in the ordered phase. He then said that, according to the analysis of the Ashkin-Teller model, the specific heat is completely smooth across the finite temperature ordering transition (to the current-carrying state or the Ising-symmetry-broken phase); thus it is expected that there will be no anomaly in the specific heat across the transition.

He went on to describe the effective model in the quantum fluctuation regime. He added a simple dynamic term for \theta and the Ohmic dissipation term proportional to |\omega| (which is supposed to arise after integrating out underlying fermions). He calls the strength of this Ohimic dissipation term, \alpha. He said this model has a quantum phase transition at \alpha_c = 4\pi and described the finite temperature phase diagram where the finite temperature transition line terminates at the critical point at zero temperature. Then he said this model allows the "exact" computation of the correlation functions. He said this is achieved by some clever trick that leads to the separation of two degrees of freedom that depend only
on space and time, respectively. As a result, the total partition function can be written as the product of two parts; each one involves only either the space or time fluctuations. The consequence, he said, is that the "staggered" magnetization, M (now it became an XY degree of freedom in the disordered regime) within the unit acquires a peculiar form of the correlator that is completely local in space;

\delta (r-r') 1/(\tau - \tau').

The Fourier transform of this correlator gives the marginal Fermi liquid spectrum in the frequency-momentum space.

He then turned to the question of superconductivity. He said the Ising degrees of freedom would couple to the underlying fermions; this coupling has the form of the current-current
interaction where the "coherent" part of the current (or the collective part) couples to the fermion current. He said the effective four-fermion interaction arising from this current-current interaction is strongly momentum dependent and gives rise to an attractive interaction in the d-wave channel.

Several questions were asked after his 15-20 mins presentation.

Andrey Chubukov asked how the strongly-momentum dependent effective interaction can give rise to the momentum-independent self-energy expected in the marginal Fermi liquid. Chandra said if one works with the circular Fermi surface and the q-independent correlator, the self-energy turns out to be momentum independent.

Catherine Pepin asked whether there is anything one should worry about the transport coefficient because after all this is a q=0 fluctuations. Chandra said there is no vertex correction.

Piers Coleman asked how and why the partition function can be written as the product of two contributions that only depend on space or time. Chandra started with a Villan form of his effective model and said that a clever choice of two orthogonal degrees of freedom (integer fields in the Villan action) leads to this construction.



Philip Anderson said there must be some peculiar response to the magnetic field; the "staggered" magnetization would become asymmetric within the unit cell and it will lead to some kind of distortion. Chandra said such a piezo-magnetic effect does not arise in his state because of some symmetry reasons.

Mike Norman asked whether there is any consequence from some kind of chirality (in the current) fluctuations. He mentined the case of MnSi where some kind of chirality effect has been discussed. Chandra said this may be a different issue.