NEQNET: Non-equilibrium Phenomena

43. Schwinger-Keldysh: Martin-Siggia-Rose diagrammatics (non-equilibrium diagrammatic methods 2)

Posted by Dmitry

at May 14, 2008

This post is again from the series devoted to the discussion of non-equilibrium diagrammatic methods. Last time we have found trivial saddle points of the Schwinger-Keldysh action and discovered that quasi-classical dynamics of a quantum system is actually governed by the Langevin-type (and therefore Fokker-Planck) equations as it should be. I also briefly mentioned in the end that the opposite procedure is in fact possible: one can start from a Langevin/Fokker-Planck equation (or, generally speaking, with any equation describing diffusive dynamics) and construct a classical limit of the correponding Schwinger-Keldysh diagrammatic technique (note that derivation of Langevin, Fokker-Planck and kinetic equations typically includes coarse-graining, so it is impossible to recover the full Schwinger-Keldysh action just, say, knowing Fokker-Planck; it means that different quantum theories can give similar classical dynamics).

The procedure was first introduced by Martin, Siggia and Rose (and independently by DeDominics).
Martin-Siggia-Rose diagrammatic methods are used a lot in applications to the theory of turbulence and various problems involving stochasticity (so, yes, in market studies, too :-))

Starting with the Fokker-Planck equation

$\frac{\partial P}{\partial\tau}=\hat{H}P=D\triangle P+\delta\hat{H}P$ ,

one writes down the corresponding “generating functional” as a functional $\delta$ -function

$Z=\int{\cal D}P\delta\left(\frac{\partial P}{\partial t}-\hat{H}P\right),$

which, after introducing the auxiliary field $\bar{P}$ , acquires the form

$Z=\int{\cal D}P{\cal D}\bar{P}\exp\left(i\int dtd\vec{n}\,\bar{P}\left(\frac{\partial P}{\partial t}-\hat{H}P\right)\right)$ .(1)

There exists a remarkable similarity between this generating functional and the Schwinger-Keldysh generating functional describing the quasiclassical approximation of the quantum non-equilibrium dynamics - for example, in the Martin-Siggia-Rose diagrammatic technique the field $\bar{P}$ plays the same role as the quatum field $\chi_{q}$ plays in the Schwinger-Keldysh technique.

Of course, as it is, the diagrammatic expansion of the generating functional (1) is not of much use, since we are not interested in correlation functions of or $\bar{P}$ - is itself the probability measure for a random walk/diffusion process governed by the corresponding Fokker-Planck equation.

(There are exceptions for this rule though - for example, in eternal inflation such diagrammatic technique is useful as it is - we will discuss it later).

Observables in the problem under consideration (such as the mean square displacement $\langle\vec{n}^{2}(\tau)\rangle$ ) are related to the physics of this random walk. However, the correlation functions of $\vec{n}$ can be determined from the correlation functions in the momentum representation generated by (1) by differentiating over momenta. For example, one has

$\langle\vec{n}^{2}(\tau)\rangle=-\frac{d}{dq^{\alpha}}\frac{d}{dq^{\alpha}}\int d\omega e^{-i\omega\tau}G(\omega,\vec{q})|_{\vec{q}=0},$

where $G(\omega,\vec{q})$ is the Fourier component of the Green function $\langle PP\rangle$ . This is due to the correspondence between the Langevin equation describing the dynamics of the observable $\vec{n}$ and the Fokker-Planck equation describing the dynamics of probability distribution for the diffusion process.

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Condensed Matter, Fellow Craft, Non-equilibrium diagrammatic methods, Quantum field theory

33. Schwinger-Keldysh: information carried by Keldysh Green’s functions (nonequilibrium diagrammatic methods 1)

31. Schwinger-Keldysh: brief review (Nonequilibrium diagrammatic methods 1)

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42. Recalling a couple of facts about 2D and 3D Ising models

Posted by Dmitry

at May 12, 2008

In reply to my comment about string theory applicability for the solution of 3D Ising model, Thomas Larsson writes on the Peter Woit’s blog:

I am well aware that the 3D Ising model is Kramers-Wannier dual to the 3D Ising gauge model and admits a random surface representation. Even better, one can write down a funny or lattice gauge model on a 3D brick lattice, with a log action, that can be exactly mapped onto a model of self-avoiding random surfaces; the weight of each graph equals $N^\chi u^A$ , where $\chi$ is the Euler characteristic, is related to the coupling constant, and the surface area.

Does this construction solve gauge theory, or make me into a string theorist? Hardly. Just because I have mapped one untractable model into another does not mean that I have solved it. If anything, this shows that perturbative string theory is inadequate for this problem because steric repulsion is crucial here (self-avoiding surfaces do not self-intersect!).

A method has only really contributed to our understanding of a model if it helps to extract some kind of quantitative information about it, not necessarily critical exponents and not necessarily exactly. The methods that Peter O. mentioned do that, as do high- and low-temperature expansions, real-space RG and computer simulations. However, AFAIK no quantitative results about 3D Ising have come out of string theory.

So, I decided to refresh my memory concerning the subject and the problem of self-avoiding. Sorry if the post will be too lengthy, but I write it mostly for myself…

(By the way, Lubos, 3D Ising model is “mine” in the same sense as it is “yours” :-), I am cosmologist, not string theorist or condensed matter expert, I just happen to vaguely remember this staff.)

Let me first recall what happens in 2D case (exactly solved by Onsager), start with descrete case

$Z=\sum_{\rm paths} e^{-\beta{}H(\sigma_x)}$ ,

$H=-\sum_{x,\delta}\sigma_{x}\sigma_{x+\delta}$ .

We are generally interested in behavior of correlators of the spin variable $\sigma$ but one can also consider correlation functions of disorder variables $\mu$ . Disorder variables are constructed in the following way:

1. We consider the point of the dual lattice x_d , i.e., the center of one of the cells of initial lattice.
2. We put a string (or, better say, path or dislocation line) going from x_d to infinity. This path can be arbitrary.
3. We change the sign of $\beta$ for all connections on the lattice our dislocation line intersects. As a result, partition function is changed (perturbed partition function depends on the dislocation line and initial point x_d ).

Disorder variable $\mu$ is defined through its correlation functions:

$\langle\mu(x_d)\rangle=Z_{\rm perturbed}/Z$ etc.

Equations for corr. functions of $\sigma$ are as bad as eqs. for corr. functions of $\mu$ . But let us construct the following object:

$\psi_a(x)=\sigma(x)\mu(x+e_a)$ ,

where vectors e_a , $a=1,\cdots,4$ connect the point and the centers of four adjacent plackets $x_{d,a}=x+e_a$ .

Equation for the corr. functions of $\psi$ can be found from the equality

$\psi_a(x)={\rm cosh}(2\beta)\psi_{a+1}(x)-{\rm sinh}(2\beta)\psi_{a+2}(x+\delta_{a+1}x)$ .(1)

Also, $\psi$ satisfies the following condition: if we rotate by angle $\pi/2$ and simultaneously change the index , $\psi$ will remain invariant:

$\psi_i(x)=\psi_{i+1}(e^{i\pi/2}x)$ ,

so we can say that $\psi$ is actually spin variable. In the continuous limit Eq. (1) acquires the form

$(\partial_0+i\partial_1)u_+=imu_-$
$(\partial_0-\partial_1)u_-=imu_+$

where $m\sim\frac{\beta-\beta_c}{\beta_c}$ - nothing else but the Dirac equation (the spinor $\psi$ is actually four-component, not 2-component, but I dropped other 2 comp.)

This is well known free fermion representation for 2D Ising model. Where in this discussion is the problem of self-avoiding Thomas mentioned? The answer is very simple: fermion does not want to self-intersect its path due to the Pauli principle. Does it preclude me from considering continuous limit of the 2D Ising model? No way. I will easily write down amplitude of the propagation for the free fermion - self-intersecting paths will give strongly oscillating contribution into the overall function integral defining the amplitude. In the end it will give me

$\langle{}R^2\rangle=T^2$

for the random Brownian motion of the fermionic particle instead of linear diffusion for bosonic particle

$\langle{}R^2\rangle=T$

but one has no doubts that the model of free fermions is tractable and describes 2D Ising near the critical point.

No, let us proceed to 3D Ising. Dislocation lines $\mu(x)$ now become dislocation surfaces $\mu(C)$ and instead of free fermion $\psi_a$ describing critical behavior of the 2D Ising we have to deal with the construct

$\psi_{a_1,\cdots,a_L}(C_L)=\mu(C_L)\prod_{k=1}^L\sigma(x_k+e_a)$ .

Equation for this construct is remarkably similar to (1) but very long - I will not write it here ;-)
Although it is unclear how to write its continuous limit (and this is the main difficulty today in this prolem as I understand), from its descrete form one can see that it describes free fermionic (NSR?) string (with free fermions - elementary segments).

I do not think that self-avoidance problem will important role here as well, since Pauli principle for the string segments will preclude it from self crossing (at the level of functional integral again one will get suppression of self-crossing paths as it was for the case of free fermion in 2D).

Thomas, due to the lack of my education, I do not quite understand why you mention U(N)/O(N) sigma models in connection with 3D Ising; the latter has just discrete Z_2 symmetry. If I am wrong and there is relation, what is corresponding to Ising model, are you talking about large expansion or something else?