Download We need an antisymmetric real tensor field in bulk theory!

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/nonFermi 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.
holography
Condensed matter physics
AFM QPT
CMR
Superconducting
ferromagnestim
Spontaneous
magnetization
……
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 Fmn,
Even in non-relativistic case, the magnetic field is not
a vector but a pseudo-vector

0
E
x
Fmn  
Ey

 Ez


 Ex
 Ey
0
 Bz
By
Bz
0
 Bx
 Ez 
 By 
Bx 

0 
M mn
0
P
x

 Py

 Pz
 Px
 Py
0
 Mz
My
Mz
0
 Mx
 Pz 
 M y 
Mx 

0 
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.
s i   i jk jk

 mn  [ m ,  n ]
This antisymmetric tensor field is proportional to spin
generator of electron field.
From the origin of magnetic moment

In general, the valance electrons have also orbital
angular momentum, which couple with spin.
J


mn

mn

mn
J 
i
i
jk
J
jk
   
M J SL
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.
M i   i jk M jk
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
Mmn 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, 081901(R) (2014) arXiv: 1404.2856
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, 081901(R) (2014) arXiv: 1404.2856
Results

In the case of zero external
magnetic field, the model realizes
the paramagnetismferromagnetism 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 CurieWeiss 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: 1507.00546
Modified model

To overcome these problems, we modified this model
by adding a divergence term,
c m
LM  LM  ( M mn ) M v
2

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
Surprising
result!

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.
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: 1507.00546
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, (arXiv: 1507.00546)
There we not assume r is the solution of EoMs.
Finding the solution of r for EoMs is just equivalent
to find the function of r to minimize this integration.
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 selfinteraction 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 MA and MB stand for the two magnetic moments,
then the order parameter of antiferromagnetic phase is
M   MA  MB

The total magnetic moment is
MT  M A  M B

Based on this physical picture, we can add two 2form 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 MA-MB, to describe the response to
external magnetic field, we need the total magnetic
moment MA+MB.

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 TN.
 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:1501.04481, 1505.03405
Er2-2xY2xTi2O7
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. (1507.03105)

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 fieldsensitive 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.