A holographic approach to strongly coupling magnetism

1
A holographic approach to strongly coupling
magnetism

• Run-Qiu Yang

Institute of Theoretical Physics, Chinese Academy
of Sciences
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Content

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

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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.

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Typical magnetic state of magnetic materials

• Paramagnetic
• Ferromagnetic
• Antiferromagnetic

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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.

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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

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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.
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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.

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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.

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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.

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• Now lets come to the theme of the meeting.

holography
Condensed matter physics
CMR
AFM QPT
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.
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• 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.

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

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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

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

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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.

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From the origin of magnetic moment

• The magnetic moment of valance electron is just
  proportional to total angular momentum.
• A free electrons 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.

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From the origin of magnetic moment

• 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.

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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!
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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
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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
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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.

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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

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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.

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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.

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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

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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.

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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 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.

(arXiv 1507.00546)
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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 J0, there is not N4 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.

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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.

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Magnetic susceptibility
A peak at the Neel temperarture
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Antiferromagnetic order parameter

• Let MA and MB 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.

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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 dont care about the
  response of antiferromagnetic materials to the
  external magnetic field.
• But, if there is external magnetic field, the
  answer is no!

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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 MAMB.
• So a full description for antiferromagnetic
  materials needs at least two fields.

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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.

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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.

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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.

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• 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 arXiv1501.04481, 1505.03405

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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.

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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.

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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.

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Thank you