A holographic approach to strongly coupling magnetism Run-Qiu Yang Institute of Theoretical Physics, Chinese Academy of Sciences

Content Magnetism in strong coupling electrons system; How to build holographic models; What we have done; Conclusion.

What is magnetism? Magnetism or magnetic force, as one part of electromagnetic interaction, has a very long history in human society. However, the reason why some materials show strong magnetism but some materials do not is understood only when the quantum theory about the materials had been built.

Typical magnetic state of magnetic materials Paramagnetic Ferromagnetic Antiferromagnetic Magnetic disordered state and have very weak response to external magnetic field. Magnetic ordered state and show lots of fascinating phenomenon in the condensed matter physics and real applications.

Why we need consider spontaneous magnetism In fact, in condensed matter theory about materials, there are two central properties attractting attention for a long time in strongly correlated system. One is the electronic transport and the other one is magnetic response properties. For the former one, we have made abundant of work in holographic framework to understand relevant phenomenon, such as superconducting, Fermi/non- Fermi liquid and so on.

Why we need consider spontaneous magnetism Though there are some papers which have discussed the magnetic properties in holographic superconducting models and other problem, the magnetic field is only a supporting player rather than the central role. In fact there are many important phenomenon in strong correllated electron systems which are controlled by the magnetic properties of materials

Why we need consider spontaneous magnetism Colossal magnetic resistance (CMR) in manganate superconducting ferromagnetic state in heavy fermion system Antiferromagnetic quantum phase transition … In all these phenomenons, strong coupling and magnetism play important role, which involve some deep understanding about physics.

CMR effect Colossal magnetic resistance effect, or just named CMR effect, was discoveried in 1995 in manganate, nearly 20 years ago. It is still a very active field about strongly correlated electron system.

CMR effect The main features of this effect can be shown in this figure: There is metallic/insulating phase transition at Curie temperature; Near the Curie point, the magnetic resistance is very sensitive to external magnetic field; This effect is found in a very large class of materials and shows universal properties.

CMR effect To show how this effect is popular in condensed matter community, I just show some results from arXive. From 1995 to today, there are more than 31 and 30 papers appeared in PRL and Science.

Now let’s come to the theme of the meeting. holographyCondensed matter physics Spontaneous magnetization CMR AFM QPT Superconducting ferromagnestim Kondo effecs …… In order to build a holographic framework to describe them, we first need to clarify how describe spontaneously magnetic ordered state in holography.

A well-known example for critical phenomenon involving the magnetic properties is paramagnetism/ferromagnetism phase transition. One may naturally wonder whether there exists a dual gravitational description of such a phase transition. If it exists, the gravitational description is of great interest and can be regarded as the starting point to understand the more complicated phenomenons controlled by magnetic properties in strongly correlated electron system.

How to build a holographic model? The answer is what we want to obtain.

From Spontaneous symmetry broken Break the time reversal symmetry spontaneously in low temperature; If spatial dimension is more than 2, it also breaks spatial rotation symmetry; Without internal symmetry broken.

From properties of covariance Magnetic properties of material relate to the response to Maxwell field strength rather than its gauge potential, gauge invariant needs the field coupling with the field strength; From the theoretical point, magnetic field is not a vector. In fact, magnetic field is the component of a SO(1,3) tensor F , Even in non-relativistic case, the magnetic field is not a vector but a pseudo-vector

Magnetic moment should also be the spatial components of an antisymmetric tensor field. Time components then give the polarization of electric field.

From the origin of magnetic moment As we know, magnetism of material comes from two parts. One is the induced electronic current, which is classical effect and can be neglected in magnetic materials. The other is the angular momentum of valance electrons, which is the origin of ferromagnetism and antiferromagnetism.

From the origin of magnetic moment The magnetic moment of valance electron is just proportional to total angular momentum. A free electron’s Lagrangian can be written as We can see that magnetic moment is the spatial components of an antisymmetric tensor field. This antisymmetric tensor field is proportional to spin generator of electron field.

In general, the valance electrons have also orbital angular momentum, which couple with spin. The total angular momentum is just the spatial components of generator of Lorentz transformation. This tells us that the magnetic moment of magnetic materials in fact is the spatial components of an antisymmetric tensor operator. From the origin of magnetic moment

How to built a holographic model? An effective field to describe magnetic moment in the boundary field in a covariant manner needs an antisymmetric tensor; Its spatial components correspond to the magnetic moment. We need an antisymmetric real tensor field in bulk theory!

Holographic model Add an antisymmetric effective polarization field M  in bulk with action as, V describes the self-interaction of the polarization tensor, We will discuss its physical meaning latter Phys. Rev. D 90, (R) (2014) arXiv:

Ansatz and magnetic moment We consider a self-consistent ansatz for the antisymmetric field as, In RN background and probe limit, we prove that the magnetic moment density is expressed by following integration Some details of mathematics, such as equations of motion, numerical methods and so on, will not be shown here. One can find them in our papers. Phys. Rev. D 90, (R) (2014) arXiv:

Results In the case of zero external magnetic field, the model realizes the paramagnetism- ferromagnetism phase transition. The critical exponents agree with the ones from mean field theory. In the case of nonzero magnetic field, the model realizes the hysteresis loop of single magnetic domain and the magnetic susceptibility satisfies the Curie- Weiss law.

Problems in this model However, this model has some problems in theory. Because here we use a tensor field, so the problem in high spin theory such as ghost and causality violation may appear. To overcome these problems, a modified model was proposed in arXiv:

Modified model To overcome these problems, we modified this model by adding a divergence term, Then in order to give the correct degree of freedom, the value of c is not arbitrary. We find c=-1/2. We prove that this modified model is equivalent to a massive 2-form field with self-interaction.

Modified model We begin just from the theory in condensed matter theory to construct a self-consistent model, and then we reach at the p-form field in Dp-brane theory. This equivalent form gives us a manner to explain how this massive ATF field is generated from String/M theory. Surprising result!

Modified model This modified theory keeps all the results in our previous works and can be treated as a better framework to describe spontaneous magnetization. More details about this model can be found in arXiv:

Meaning of potential term This can be done if we can obtain the partition function of the system. However, the full consideration is too complicated. But if we only consider the probe limit, the thing is not too bad. Here we need a few of mathematics.

Meaning of potential term By holographic principle, partition function of dual boundary is obtained by bulk theory. At the classical level and in probe limit, free energy for magnetic part is this, There we not assume  is the solution of EoMs. Finding the solution of  for EoMs is just equivalent to find the function of  to minimize this integration. (arXiv: )

Meaning of potential term The near the critical temperature, we prove that the free energy can be written as this form, Then we see if J=0, there is not N^4 term. So the dual boundary theory is a free field theory and no phase transition will happen. The potential term not only describes the self- interaction of 2-form field in the bulk but also describes the self-interaction of magnetic moment in dual boundary theory.

Antiferromagnetic mdoel Antiferromagnetic material has not net magnetism, but it is magnetic ordered. The simplest antiferromagnetic materials have two magnetic sub-lattices. The magnetic moment in these two sub lattices just offset each other when external magnetic field is zero.

Magnetic susceptibility A peak at the Neel temperarture

Antiferromagnetic order parameter Let M A and M B stand for the two magnetic moments, then the order parameter of antiferromagnetic phase is The total magnetic moment is Based on this physical picture, we can add two 2- form fields in Lagrange to describe these two sub lattices.

Is it necessary to add two fields? It seems too complicated to use 2 tensor field to describe antiferromagnetism. Can we use only one field to describe antiferromagnetic materials? It may be yes if you don’t care about the response of antiferromagnetic materials to the external magnetic field. But, if there is external magnetic field, the answer is no!

Why we need to tensor fields Because, to describe antiferromagnetic order we need the value of M A -M B, to describe the response to external magnetic field, we need the total magnetic moment M A +M B. So a full description for antiferromagnetic materials needs at least two fields.

Holographic antiferromagnetic model Take all these into account, we proposed following model for antiferromagnetism, It contains two 2-form field and the interaction between them.

By this model, we can realize, The magnetic moments condense spontaneously in an antiparallel manner with the same magnitude below a critical temperature T N. In the case with the weak external magnetic field, the magnetic susceptibility density has a peak at the critical temperature and satisfies the Curie-Weiss law.

By this model, we can realize, When we open external magnetic field, the antiferromagnetic transition temperature is suppressed by magnetic field. There is a critical magnetic field Bc in the antiferromagnetic phase: when the magnetic field reaches Bc, the system will return into the paramagnetic phase. This is a very interesting result !

Our holographic model can not only give this quantum critical point and the phase boundary but also give some quantitative results which can be tested in experiments. For example, our model predicate the energy of antiferromagnetic excitation over the B-Bc is just near 5.0. The results from Er2-2xY2xTi2O7 show it is 4.2. Though they are different, it is still a surprising result! More details discussions can be found in these two papers: arXiv: , Er 2-2x Y 2x Ti 2 O 7

CMR effects Now I want to make a brief introduction about our recent work about CMR effect. This work has appeared in arXiv in the Tuesday of this week. ( ) As far as I know, this is the first paper to discuss CMR effect in holographic model.

Main results The computation shows DC resistivity has a peak and an insulator/metal phase transition happens at Curie temperature. A remarkable magnetic field- sensitive resistance peak emerges naturally for temperatures near the magnetic phase transition. We see that from two figures, our holographic model may be a good model for this effect.

Conclusion we introduce our recent works to build to framework to describe spontaneous magnetic ordered state and some relevant problem in strongly correlated system. The key point is that we need a 2-form field coupled with Maxwell strength field in an asymptotic AdS space-time. Maybe the models in our paper are not the best, but I have a strong feeling that there are lots of things we can do in future. They are calling more clever peoples to proposed new frameworks, new models and new methods.