Sung-Sik Lee (McMaster U and Perimeter Institute, Canada)
From renormalization group to emergent gravity: holographic description of quantum many-body systems
Blogged by Piers Coleman.
Sung-Sik begins talking about examples of scs, CM systems, high Tc, heavy fermion systems near a QCP, but alos QCD, the quark gluon plasma.
The long distance, low energy physics is described by a strongly interacting QFT. We need a new way of approaching these problems.
One new approach he suggests, is "Duality", the idea that one theory can be mapped onto another. For example, the 3D xy model can be mapped to a U(1) gauge theory w charged boson. EM duality in 4D gauge theories.
The hope: find a new way of mapping strongly coupled theories onto weakly coupled theories, using duality. This kind of thing can happen if
Original particles remain strongly coupled and have a short life time, yet organized into long-lived, weakly coupled collective excitations
Non-trivial dynamical info contained in the change of variable
Dual variable may carry new, sometimes fractional quantum numbers
It may live in a different space.
Q: when doesn't it work?
A: it doesn't work in MOST cases, but I will tell you about a case where it may! Holography.
Holography: AdS/CFT correspondence. Here the bold new idea, due to Maldacena, is that a D-dimensional AFT is dual ot a D+1 dimensional gravity theory.
- N=4 SU(N) gauge theory in 4D = 11B superstring theory in AdS5xS5
- Weak coupling description for strongly coupled QFT for a large N
- Believed to be a general framework for a large class of qFs
- The correspondence has been used to "reporoduce" many phenomena in CMT - from hydrodynamics to SC and NFL.
- There is no first principle derivation.
- Purely phenomenological, with a few exceptions.
Now SS introduces the "Dictionary" in Duality. Given below. There are a number of key ideas here, Quantities in D dimensions have their D+1 dimensional correspondence, eg T - energy momentum maps onto the metric tensor in D+1. There is also a mapping between the new dimension, z, and the RG flow, so that large z corresponds to low energy, small z to high energy. There is also a boundary condition, so that the field j in the D+1 theory at the boundary z=0 is equal to the source term in the lower dimensional CFT.
SS poses questions
Q- Can one explicitly construct hte gravitational theory form a a general QFT/quantum many body system?
A- Yes we can! One has to Quantize the beta function. (Blogger: This is amazing - "third quantization"?! )
Here we go.
Step 0 Write Z[J[x]] with sources Jn coupled to the fundamental ops On, such that any local operator allowed by symmetry can be written as a polynomail of the fundamental ops and their derivatives. He gives the prescription for Ising, Vector and Matrix fields.
Step 1 We need to do RG, keeping track of beta function for infinite many coupling constants. Let us simplify by removing the non-fundamental operators by introducing auxilliary fields. This is done with a delta function something linke
jn (pn-On) - Jn pn + Jnm pn pm +....
jn is the Lagrange multiplier that enforces the constrain while pn is the dynamical operator.
Step2 We integrate out the high energy modes of j.
Q: why not j and p?
A: you can this, but lets see what we get by just integrating out the he j modes.
If we do this, three things happen:
- By integrating out the quantum field, you generate energy that depends on the configuration of low energy j's and p's
- One obtains a quantum correction for the fundamental operators O.
- One also generates quadratic terms of the O's, which become p's.
Q: arent the fundamental fields the primary fields of CFT?
A: the fundamental fields are not the primary fields of CFT. For example, they do not include spin...
Step 3 replace O by p. (Really the same as step 1).
Step 4 repeat 2-3 again and again until one has got rid of the original field completely. One has now a theory in terms of a set of dynamical sources and dynamical operators. The interation parameter, has become (magically) an additional dimension, with a j(n) and p(n) at each stage.
Q: always local?
A: no - one needs arbitrary derivatives - locality is not guaranteed.
Q: you have two indices, m and n.... Whats the guarantee that the rank of the operators will not increase?
A: you had this infinite no of fields at the beginning. It only becomes useful if you can truncate in tersm of a finite no of variables.
Q: how are you guaranteed that the truncation works at all stages?
A: so far I have not truncated... yet....
So now the iteration variable has become a continuous extra dimension (idea of Verlindes...) and somehow, if it works, what is left is local with a truncated set of variables. He says that this is sofar, an exact change of variables. A d-dimensinoal Z cn be written as a d+1 functional integration for dynamical sources and operator fields. Its a general scheme, but for a general theory, the holographic description is not useful. When all sysmmetry allowed ops are large - some large N limit of the original theory - the holographic theory becomes classical. The holographic theory always includes gravity - the energy momentum tensor becomes a spin-2 source.
Q: whats the simplest CMT for which this can be carried out as an engineer?
A: I think its the O(N) model - vector model. I will show you.
Astonishing observation: Useful to consider z as time, one has p d_z j, as if p and j are conjugate to each other. So I can view the problem as a quantum problem in one higher dimension. The partition fn is a transition amplitude of a D-dimensional Quantum wavefunction. The Hamiltonian generates scale transformation for the dynamical couplings. The Heisenebrg equation, is a quantum beta function.
Conventional RG is in J_n space,
dJ_nm/Dl = beta(Jn,Jnm....).
This is a "classical RG trajectory". There are no fluctuations in the RG flow.
Quantum RG is holography. In this, flow, only fundamental operators appear. The price you pay is that these operators are now dynamical, and one has to consider a sum over RG trajectories. The RG flow is governed by a quantum hamiltonian
dJn/dl = [H,Jn]
If the large N theory is classical, this becomes classical.
Local RG prescription, spacetime dependent coarse graining - Gamma(x)=Gamma Exp [-alpha[x]dz]
By construction, Z is independent of alpha(x,z). Choosing different RG schemes corresponds to a gauge choice.
Q: how will a fixed point arise in this quantum approach?
A: if you find a saddle point solution, then the sp soln will have scale invariance.
Q: how can you justify one of the phenomenological models in the 1/N expansion?
A: I hope, but it has not been worked out so far.
Q: Can you reproduce or disprove the recent result of Verlinde, that gravity can be understood as an entropy.
A: Yes - if you integrate high energy mode, you generate an action by a kind of order from disorder - gravity appears - non trivial action a kind of entropic, quantum source.
Blogger- what a beautiful talk. The idea of quantum RG as a route to holography may be a revolution. Lets hope so, because we need one.
Q- Can one explicitly construct hte gravitational theory form a a general QFT/quantum many body system?
A- Yes we can! One has to Quantize the beta function. (Blogger: This is amazing - "third quantization"?! )
Here we go.
jn (pn-On) - Jn pn + Jnm pn pm +....
jn is the Lagrange multiplier that enforces the constrain while pn is the dynamical operator.
Q: why not j and p?
A: you can this, but lets see what we get by just integrating out the he j modes.
If we do this, three things happen:
- By integrating out the quantum field, you generate energy that depends on the configuration of low energy j's and p's
- One obtains a quantum correction for the fundamental operators O.
- One also generates quadratic terms of the O's, which become p's.
Q: arent the fundamental fields the primary fields of CFT?
A: the fundamental fields are not the primary fields of CFT. For example, they do not include spin...
Step 3 replace O by p. (Really the same as step 1).
Step 4 repeat 2-3 again and again until one has got rid of the original field completely. One has now a theory in terms of a set of dynamical sources and dynamical operators. The interation parameter, has become (magically) an additional dimension, with a j(n) and p(n) at each stage.
Q: always local?
A: no - one needs arbitrary derivatives - locality is not guaranteed.
Q: you have two indices, m and n.... Whats the guarantee that the rank of the operators will not increase?
A: you had this infinite no of fields at the beginning. It only becomes useful if you can truncate in tersm of a finite no of variables.
Q: how are you guaranteed that the truncation works at all stages?
A: so far I have not truncated... yet....
So now the iteration variable has become a continuous extra dimension (idea of Verlindes...) and somehow, if it works, what is left is local with a truncated set of variables. He says that this is sofar, an exact change of variables. A d-dimensinoal Z cn be written as a d+1 functional integration for dynamical sources and operator fields. Its a general scheme, but for a general theory, the holographic description is not useful. When all sysmmetry allowed ops are large - some large N limit of the original theory - the holographic theory becomes classical. The holographic theory always includes gravity - the energy momentum tensor becomes a spin-2 source.
Q: whats the simplest CMT for which this can be carried out as an engineer?
A: I think its the O(N) model - vector model. I will show you.
Astonishing observation: Useful to consider z as time, one has p d_z j, as if p and j are conjugate to each other. So I can view the problem as a quantum problem in one higher dimension. The partition fn is a transition amplitude of a D-dimensional Quantum wavefunction. The Hamiltonian generates scale transformation for the dynamical couplings. The Heisenebrg equation, is a quantum beta function.
Conventional RG is in J_n space,
dJ_nm/Dl = beta(Jn,Jnm....).
This is a "classical RG trajectory". There are no fluctuations in the RG flow.
Quantum RG is holography. In this, flow, only fundamental operators appear. The price you pay is that these operators are now dynamical, and one has to consider a sum over RG trajectories. The RG flow is governed by a quantum hamiltonian
dJn/dl = [H,Jn]
If the large N theory is classical, this becomes classical.
Local RG prescription, spacetime dependent coarse graining - Gamma(x)=Gamma Exp [-alpha[x]dz]
By construction, Z is independent of alpha(x,z). Choosing different RG schemes corresponds to a gauge choice.
Q: how will a fixed point arise in this quantum approach?
A: if you find a saddle point solution, then the sp soln will have scale invariance.
Q: how can you justify one of the phenomenological models in the 1/N expansion?
A: I hope, but it has not been worked out so far.
Q: Can you reproduce or disprove the recent result of Verlinde, that gravity can be understood as an entropy.
A: Yes - if you integrate high energy mode, you generate an action by a kind of order from disorder - gravity appears - non trivial action a kind of entropic, quantum source.
Blogger- what a beautiful talk. The idea of quantum RG as a route to holography may be a revolution. Lets hope so, because we need one.
Q1: A basic one:
ReplyDeleteIn light of the $T_{\mu\nu}\leftrightarrow g_{\mu\nu}$ correspondence, how to tackle the problem of non-conserved total EM tensor?
Q2:Another one:
Why the set of fundamental operators $O_{\mu}$ need to be infinite? Is it necessary for having a complete set of operators in the corresponding Fock space?
Q3:
Does $\log\Lambda$ becomes an 'equivalent time'?