292. Universal properties of the U(1) current at deconfined quantum critical points

This is a guest post by Flavio Nogueira from the U. of Berlin. Dmitry.

Before I start talking my recent preprint [http://arxiv.org/abs/0902.0364], let me thank Dmitry for inviting me to write this contribution in his blog.

I cannot talk about my preprint without first talking a little bit about the paper of Pawel Kovtun and Adam Ritz [http://arxiv.org/abs/0806.0110; Phys. Rev. D 78, 066009 (2008)] entitled “Universal conductivity and central charges”. For theories having a gravitational dual, Kovtun and Ritz have derived interesting relations between the central charges and the universal amplitudes of certain thermodynamic quantities. Before talking about it more concretely, we have to recall some basic aspects of a conformal field theory (CFT).

Let us consider two important conserved physical quantities in a CFT, namely, the energy-momentum tensor and the current . At zero temperature, a -dimensional CFT has correlation functions for these quantities given by

(1)

(2)

where and is the surface of a unit sphere in dimensions. The numbers and are the so called central charges of the operators and , respectively. Thus, we see that the above correlation functions of a CFT are simply determined by dimensional analysis and rotation invariance. This follows from the fact that a CFT is scale invariant. Interestingly, at finite temperature the pressure and the charge susceptibility of a CFT are also determined by dimensional analysis,

, .

This happens because and are related to and , respectively. For instance, , where is the conserved charge associated to the current and is the (infinite) volume.

For a two-dimensional CFT, we have the exact universal relations [I. Affleck, Phys. Rev. Lett. 56, 746 (1986); H. W. J. Bloete, J. L. Cardy, and M. P. Nightingale, Phys. Rev. Lett. 56, 742 (1986)],

, . (3)

The above result can be intuitively understood in the following way. To compute the correlation functions (1) and (2) at finite temperature, we need to consider the system in in a spacetime where the time direction is made periodic with period . In two dimensions this corresponds to map the plane into a cylinder and such a transformation is conformal. Thus, for a theory with conformal invariance we can expect that zero and finite temperature universal properties are related. In this case the central charges are determining the thermodynamics of the theory.

What Kovtun and Ritz have shown, is that for -dimensional theories with a gravitational dual the following relations should hold:

, (4)

. (5)

For the above formulas reduce correctly to the result in (3). Note, however, that if by chance we find a CFT at fulffilling the above equations, this would not necessarily mean that the corresponding theory has a gravitational dual. However, this may well be the case, but a proof would certainly not be easy.

Now, note that many interesting theories are actually not conformal. However, there is a class of theories which are conformal at a precise value of the coupling constant. This special value of the coupling constant defines what is called a quantum critical point (QCP). In field theory language it corresponds to a non-trivial fixed point of the renormalization group (RG) function. Thus, the result given in Eqs. (4) and (5) should also be valid for a theory whose QCP has a gravitational dual.

A QCP separates different phases of a theory and governs a second-order phase transition. In condensed matter physics there are many examples of theories having a QCP. Particularly interesting are some theories defined in dimensions. For example, there are phase transitions in magnetic Mott insulators which involve competing ordering states. A paradigmatic example is the quantum phase transition between a so called Neel state (i.e., an antiferromagnetically ordered state) and a crystal of singlet valence bonds, the so called valence-bond solid (VBS) state. Read and Sachdev [N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989); Phys. Rev. B 42, 4568 (1990)] have shown that the effective field theory governing this quantum phase transition is a gauge theory. The quantum critical point of such a system is governed by an Abelian Higgs model with two complex scalar fields, z and . These fields are the elementary constituents of the theory. They are called spinons. The spinons are the building blocks of the spin orientation field,

where , being a Pauli matrix, and the constraint must be satisfied. We can write a Lagrangian describing the dynamics of the field . From textbooks on solid state physics we learn that spin-waves in an antiferromagnet have a relativistic spectrum, i.e., . Therefore, the simplest Lagrangian for an antiferromagnet reads

.

In terms of the spinon fields this becomes

, (6)

with the additional constraint that , as required by . This is the model. It will be convenient to consider a version with complex fields, i.e., the model, in which case the constraint becomes , with the summation over running from to .

From the equations of motion we obtain

.

Thus, at the classical level the gauge field emerges from the spinon fields. If we specialize to and consider the dual field , we can easily show that the flux of through a closed surface is quantized:

,

where is an integer. Here it is useful to note the differences between the above flux quantization and the flux quantization in a superconductor. In a superconductor it is the flux through an open surface that is quantized, such that Stokes theorem can be applied to show that the circulation of the vector potential is quantized. In our case it is the flux through a closed surface that is quantized. Thus, in the superconducting case flux quantization leads us to consider lines as topological defects, i.e., vortex lines. For the model case, on the other and, the flux emerges from a point, a magnetic monopole in space time, the so called instanton.

The instantons play a crucial role in an antiferromagnetic Mott insulator, especially in the VBS phase. The reason for this is an important ingredient not yet mentioned here: the Berry phases. By quantizing a spin system with the path integral we always obtain a Berry phase in addition to the action describing the elementary excitations of the system. In an antiferromagnet this Berry phase is alternating and is actually responsible for the appearance of a non-trivial paramagnetic phase like the VBS one. As shown by Read and Sachdev, the Berry phases are not very important in the Neel phase, but are crucial in the VBS phase. Recently the important role of the interplay between the Berry phases and the intantons was discussed by Senthil and collaborators [T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science, 303, 1490 (2004), http://arxiv.org/abs/cond-mat/031132; T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Phys. Rev. B 70, 144407 (2004), http://arxiv.org/abs/cond-mat/0312617]. They realized that near the phase transition the instantons are actualy suppressed by the Berry phases, which leads to spinon deconfinement at the phase transition. Recent Monte Carlo simulations confirmed that this meachanism for instanton suppression works for the case of an easy-plane antiferromagnet [S. Kragset, E. Smorgrav, J. Hove, F. S. Nogueira, and A. Sudbo, Phys. Rev. Lett. 97, 247201 (2006), http://arxiv.org/abs/cond-mat/0609336], but in this case the phase transition was shown to be first-order [see also the simulations of A. B. Kuklov, N. V. Prokof'ev, B. V. Svistunov, and M. Troyer, Ann. Phys. (N.Y.) 321, 1602 (2006); http://arxiv.org/abs/cond-mat/0602466], i.e., no QCP in this case. In cases where a QCP exist, which is the interesting situation to us, the QCP of the instanton-suppressed theory governs a completely new universality class featuring a large anomalous dimension of the magnetic order parameter. Note that in a Landau-Ginzburg-Wilson (LGW) type of approach the magnetization and the VBS order parameters would compete and never lead to a QCP. Moreover, for a case where the paramagnetic case has no broken symmetry (i.e., no VBS) and only the magnetic order parameter is available, there is a second-order phase transition with a very small anomalous dimension. The reason why the anomalous dimension is large in the case of the instanton-suppressed antiferromagnet is simple. As we have mentioned earlier, the spinons are the building blocks of the magnetic order parameter. So, in order to obtain the anomalous dimension of it we have to calculate the anomalous dimension of a composite operator, and this leads to a large anomalous dimension for . This is impossible in the case of a LGW theory, since there is the elementary field.

The supression of the instantons by the Berry phases at the QCP can be consistently achieved by including a Maxwell term in the Lagrangian (6). We will also soften the constraint. This softening does not affect the quantum critical properties of the theory near the QCP. Therefore, we are led to consider the Lagrangian,

.

In terms of this theory, the Neel state corresponds to a phase where the spinons are condensed. This is the Higgs phase. In the VBS phase, away from the QCP, the instantons are more relevant. In this case, the spinons are confined and we have a non-zero VBS order parameter. The spinons are deconfined at the QCP. For this reason, the scenario described here is called deconfined quantum criticality. A schematic phase diagram is shown in the figure.

The conserved current of this theory is

.

In my recent paper, I was mainly interested in the current correlation function of this theory and how close the universal aspects of this current to the predictions of the gauge-gravity duality are. The results I obtained are actually very simple. Let us discuss the main results of the paper.

As we know from textbooks, the renormalized gauge coupling is given simply by multiplying the bare coupling by the wavefunction renormalization of the gauge field , i.e., . Recall that is directly calculated from the vacuum polarization . Indeed, the gauge field propagator is written in Euclidean space as

Thus, we have,

.

The current correlation function in momentum space is

,

where the function is related to the vacuum polarization by

.

From the above equations we immediately see that

.

On the other hand, by Fourier transforming Eq. (2), we obtain

,

and therefore we can write an exact formula for the renormalized gauge coupling in terms of the central charge :

.

By defining the dimensionless gauge coupling as , we can find the value of at the QCP by taking the limit . This QCP is the sameas the fixed point value of the RG function for . Note that we can determine the exact value of in terms of , although we cannot determine the exact function. This is because is the renormalized gauge coupling already at the QCP, since we have calculated it using the quantum critical current correlation function. Therefore, we obtain the following interesting formula for :

. (7)

There is a somewhat similar formula for a theory having a gravitational dual, except that in this case , which is the central charge for the non-gravitational theory living on the boundary, is related to the dimensionless coupling of the gravitational theory in the bulk. In fact, Freedman and collaborators [D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Nucl. Phys. B 546, 96 (1999);
http://arxiv.org/abs/hep-th/9804058] have derived the result:

,

where is the dimensionless gauge coupling of the theory in -dimensional anti-de Sitter spacetime. For both results for agree, except that is not the coupling constant in the bulk. Indeed, we have for the former, and for the latter. So, we see that both results look very similar.

Now, Kovtun and Ritz were able to derive also a formula for in terms of by using gauge-gravity duality arguments. They have shown that

,

Unfortunately, I was not able to derive a similar formula for for a deconfined QCP. Desirable would be a formula such that , where is some constant. Then it would be of course very nice if in such a formula we get for the same value as predicted in the gauge-gravity duality. But such a comparison will have to wait because at the moment I still don’t know how to derive . However, I was able to calculate the value of for a free theory featuring complex scalar fields. This actually corresponds to the same result as obtained from the interacting theory at the large . Since we are interested in comparing the ratio with the prediction (5), let us first consider in the large limit. After calculating in large limit and using Eq. (7), that

,

which is also the value of for a free theory with complex scalar fields.

From the calculation of the charge susceptibility we obtain

.

The ratio at agrees with the exact result for that case, as we would expect. Interestingly, for the same value as for is obtained.

Note that the large result holds also in the absence of the Maxwell term and by strictly assuming the constraint. It would be interesting to consider a finite result, like the case of interest , in which case the model is equivalent to the non-linear -model. There is actually a conjecture by Klebanov and Polyakov [Phys. Lett. B 550, 213 (2002); http://arxiv.org/abs/hep-th/0210114] where they argue that there should be a duality between the critical model at and a spin gauge theory defined in AdS. For this is simply . Unfortunately, it is not an easy task to consider finite theories within the framework of the gauge-gravity duality. While such an analysis would be of little interest for particle physics phenomenology, it would be a very important achievement in a condensed matter physics context as discussed here. I hope that with the time more condensed matter physicists will get interested on the AdS/CFT correspondence. At the moment most of the papers making condensed matter applications were written by string theorists and/or nuclear physicists, with only a few exceptions like Subir Sachdev, Markus Muller, and Lars Fritz. But I believe that the number of condensed matter physicists working on the subject is likely to increase considerably by the end of 2009.