Thermal AND Quantum Fluctuations?

[maverick_starstrider]

Hi all,

I know how if a statistical partition function is written as a path integral in imaginary time (Wick's rotation) the fluctuations around the mean-field represent thermal fluctuations. If the path integral is instead done in real time then fluctuations from the mean-field/classical minima represent quantum fluctuations.

My question is: is it possible to describe in one equation for the partition function BOTH thermal and quantum fluctuations? I assume in terms of a path integral.

Thanks in advance.

 

[bhobba]

I know how if a statistical partition function is written as a path integral in imaginary time (Wick's rotation) the fluctuations around the mean-field represent thermal fluctuations.

It represents a Wiener process in imaginary time. It doesn't have to be thermal in nature - although you can interpret it that way - but the 'heat' source in that view is an issue, although some have interpreted it as being in a constant thermal bath of some sort.

If you are asking if it can be interpreted as a Wiener process and taking all paths simultaneously then the answer is no. To get a Wiener process you must go to imaginary time, to get the sum over all paths its real time with an imaginary exponent - that's why the majority of paths cancel and you get the Principle Of least Action.

In this context you may find the following of interest:
http://arxiv.org/pdf/1204.0653.pdf

Thanks
Bill

 

[maverick_starstrider]

That's what statistical field theory is. If, say, I have a system with a continuous degree of freedom at every site of a lattice of N spins (let's call the site variables S_i) then I can write the partition function as

$$Z = \frac{1}{N}\int^{\infty}_{-\infty} \ldots \int^{\infty}_{-\infty} dS_0 dS_1 \ldots dS_{N-2} dS_{N-1} e^{- \beta H(\hat{S}_i)}$$

there is then the "average" or minimizing field arrangement H_min and then thermal fluctuations about it. This is identical to a quantum description of the same system

$$Z = \frac{1}{N}\int^{\infty}_{-\infty} \ldots \int^{\infty}_{-\infty} dS_0 dS_1 \ldots dS_{N-2} dS_{N-1} e^{ i t S(\hat{S}_i)}$$

except now the S's are operators. Am I not correct?

 

[bhobba]

Am I not correct?

Yes that is correct.

The issue is interpretation. It does not have to be thermal in nature - its simply a Wiener process.

To put it another way you assumed thermal fluctuations then did a Wick rotation to get the quantum description. But now start with the quantum description and rotate back - you get the equation, but the fluctuations do not need to be interpreted thermally.

And you can't get it to display both at the same time ie you can't do a sum over histories in real time and have a Wiener process in imaginary time - you view time differently in either.

Thanks
Bill

 

[maverick_starstrider]

I'm specifically concerned with quantum critical regions at finite temperatures ( http://en.wikipedia.org/wiki/Quantum_phase_transition#mediaviewer/File:QuantumPhaseTransition.png ). At T=0 quantum fluctuations are the driving force of exotic behaviour. At T>0 you have, necessarily, thermal fluctuations but the quantum behaviour still exists in the distinctive fan region. Is it possible to encapsulate this notion in a single path integral formulation?

EDIT: Just to clarify, I don't actually care if I can SOLVE said path integral. I'm just wondering if I can write it down schematically.

 

[bhobba]

I am not sure exactly what you are getting at. I may have to leave it to someone else.

Thanks
Bill

 

[maverick_starstrider]

That's alright. I appreciate the effort regardless.

 

[stevendaryl]

Hi all,

I know how if a statistical partition function is written as a path integral in imaginary time (Wick's rotation) the fluctuations around the mean-field represent thermal fluctuations. If the path integral is instead done in real time then fluctuations from the mean-field/classical minima represent quantum fluctuations.

My question is: is it possible to describe in one equation for the partition function BOTH thermal and quantum fluctuations? I assume in terms of a path integral.

Thanks in advance.

I'm not 100% positive, since it's been a while since I've studied this topic, but I believe that quantum statistical mechanics takes into account both quantum and thermal effects.

Talking about "quantum fluctuations" as something analogous to thermal fluctuations is a little misleading. If you express the state of the system as a superposition (in statistical mechanics, you would use mixtures, rather than superpositions) of energy eigenstates, then there are no (quantum) fluctuations. The appearance of fluctuations results when you consider operators (such as position or momentum) that don't commute with the Hamiltonian.

If you describe things in terms of a complete set of energy eigenstates, then the effects of both quantum mechanics and thermal fluctuations are described by the partition function.

 

[king vitamin]

I would say that stephendaryl is correct. Quantum stat mech completely accounts for quantum and thermal fluctuations at the same time. Once you form some mixed density matrix representing your ensemble, it's impossible to distinguish which parts of the ensemble come from classical and quantum uncertainties. From Landau and Lifshitz's stat mech textbook:

The averaging by means of the statisitcal matrix ... has a twofold nature. It comprises both the averaging due to the probalistic nature of the quantum description (even when as complete as possible) and the statistical averaging necessiated by the incompleteness of our information concerning the object considered... It must be borne in mind, however, that these constituents cannot be separated; the whole averaging procedure is carried out as a single operation, and cannot be represented as the result of succesive averagings, one purely quantum-mechanical and the other purely statistical.

The imaginary-time path integral describes both thermal and quantum fluctuations, though of course you need to make some analytic continuation later on to get relate it to the quantum partition function. For applications to quantum phase transitions, see Sachdev's textbook, which makes extensive use of path integrals and has considerable discussion on the "fan region."

 

[maverick_starstrider]

Thinking about it a bit more I believe the state of affairs is probably as follows:

Thermal fluctuations for a classical Hamiltonian:

$$Z = \int DS_i e^{- \beta H (S_i)} \rightarrow \int DS_i e^{- \beta H_{saddle pt.} + fluctuations}$$

Quantum fluctuations for a T=0 system

$$Z = \int DS_i e^{i t H (S_i)} \rightarrow \int DS_i e^{i t H_{saddle pt.} + fluctuations}$$

The general case is:

$$Z=tr(e^{-\beta \hat{H}})$$

where the crucial thing is that H has a hat (i.e. it is a description in non-commuting quantum variables). This can also be written as:

$$Z = \int DS_i e^{- (\beta + it) H (S_i)} \rightarrow \int DS_i e^{- (\beta + it) H_{saddle pt.} + fluctuations}$$

Does this make sense?

 

[king vitamin]

I'm afraid I don't follow past your first equation.

For the T = 0 quantum system, you have the following:

$$Z = \langle 0 | e^{-iH(\phi_n)\tau} | 0 \rangle = \int \mathcal{D}\phi_n e^{i \int_0^{\tau} dt L(\phi_n(t))}$$

In what follows, we consider long times $\tau \rightarrow \infty$. This is the vacuum amplitude at long times; notice that the saddle point (the extrema of $S[\phi]$) is just the classical equations of motion, and the fluctuations are the quantum corrections. I assume you're familiar with the derivation of the above equation?

Now let's turn on temperature, adding thermal fluctuations. Since we've rewritten the vacuum amplitude above, there's actually an analytic continuations allowing us to do the same procedure as above.

$$Z = tr(e^{-\beta \hat{H}}) = \sum_n \langle n | e^{-\beta \hat{H}} | n \rangle$$

but from above

$$\sum_n \langle n | e^{-i\hat{H}\tau} | n \rangle \xrightarrow{\tau \rightarrow -i\beta} \sum_n \langle n | e^{-\beta\hat{H}} | n \rangle = \int_{\phi(t=0) = \phi(t=\beta)} \mathcal{D}\phi_n e^{- \int_0^{\beta} dt L_E(\phi_n(t))}$$

where the sum over n means you include all $\phi$ with periodic boundary conditions along the imaginary time axis, and by performing the analytic continuation (check this, I probably have sign errors), $L_E = H$, the Hamiltonian.

Now, your saddle point again gives you classical configurations (minimizing the Hamiltonian), and the expansion is in fluctuations. You can't really say whether the fluctuations are quantum or thermal without further info, such as the relative magnitudes of temperatures and relevant energy scale (like an energy gap).

 

[maverick_starstrider]

Yes thank you, I see. So a quantum partition function or statistical field theory partition function includes both thermal and quantum fluctuations provided the Hamiltonian is made of non-commuting operators (i.e. quantum).

Is it then correct to say the thermal partition function of a, say, classical kinetic theory can be described as saddle point + thermal fluctuations and that a quantum path intregral can be described as saddle point + quantum fluctuations but that a quantum partition function integrated from 0 to beta necessarily contains both types of fluctuations.

Furthermore, if one considers high-temperatures (beta goes to zero) and the requirement that all paths start and end at the same point then that terms in the Lagrangian that related to DERIVATIVES in time of the field are quantum fluctuations which are heavily penalized in this high-temperature system leaving only a path integral over the POTENTIAL term with the path integral essentially being the thermal fluctuations?

In essence can we not separate the two types of fluctuations in that quantum fluctuations are "time" fluctuations and thermal fluctuations are in the path integral sum-over-histories at constant time?

Does this make sense (sorry for the nitpickyness, I'm trying to figure out a precise statement for my PhD thesis.)

Thanks a lot for the feedback so far. It's been very illuminating.

 

[atyy]

In the case where "time" is involved that would be non-equilibrium. One formalism that handles the equilibrium case is the Matsubara formalism, which king vitamin outlined above. A non-equilibrium formalism is the Schwinger-Keldysh formalism.

 

[maverick_starstrider]

An equilibrium system still has time dynamics for it has terms with derivatives in real or imaginary time. This is not the same as saying the system is not in equilibrium. A typical action would be something like

$$S = \overline{\psi} \partial_{t} \psi + H$$

where the first term is the time dynamics.

 

[atyy]

One argument that thermal and quantum fluctuations cannot be distinguished is to consider the thermal density matrix as a possible reduced density matrix. In the latter case, the fluctuations are entirely quantum.

In an isolated quantum system, since the true dynamics is quantum, although a particular reduced density matrix may be almost thermal, it isn't obvious that Schroedinger's equation applied to the whole system will lead to a reduced density matrix that is almost thermal, and remains so. Some discussion about when that happens is found in papers like http://arxiv.org/abs/1007.3957.

 

[maverick_starstrider]

Again, nothing I`m talking about is out-of-equilibrium, nothing is time dependent. I`m talking about separating quantum and thermal fluctuations in something like a spin system in equilibrium (i.e Mott insulator with strong on-site repulsions at low temperatures).

 

[king vitamin]

Yes thank you, I see. So a quantum partition function or statistical field theory partition function includes both thermal and quantum fluctuations provided the Hamiltonian is made of non-commuting operators (i.e. quantum).

Is it then correct to say the thermal partition function of a, say, classical kinetic theory can be described as saddle point + thermal fluctuations and that a quantum path intregral can be described as saddle point + quantum fluctuations but that a quantum partition function integrated from 0 to beta necessarily contains both types of fluctuations.

This sounds right to me.

Furthermore, if one considers high-temperatures (beta goes to zero) and the requirement that all paths start and end at the same point then that terms in the Lagrangian that related to DERIVATIVES in time of the field are quantum fluctuations which are heavily penalized in this high-temperature system leaving only a path integral over the POTENTIAL term with the path integral essentially being the thermal fluctuations?

Yeah, this is right. There's a nice way to think about it if you're familiar with thermal Feynman diagrams: When you develop the perturbation theory of your finite temperature path integral, you develop Feynman diagrams in direct relation to standard QFT, except your periodic imaginary time boundary conditions forces you to develop a Fourier series in time instead of a Fourier transform. Your Fourier frequencies must take discrete values, $\omega_n = 2\pi n/\beta$, where $n$ is an integer. Now, when $T \rightarrow \infty, \beta \rightarrow 0$, only the $n = 0$ frequency will contribute. This is basically what you're saying, but in frequency space.

In essence can we not separate the two types of fluctuations in that quantum fluctuations are "time" fluctuations and thermal fluctuations are in the path integral sum-over-histories at constant time?

EDIT: After writing up this whole post, I realized that I don't really need the next two paragraphs. atyy's point that a finite temperature density matrix is equivalent to a zero temperature reduced density matrix (actually, is this always true?) gets rid of any need to establish a quantum-classical correspondence, which may be ruined by a badly defined Wick rotation. I'll keep them here in case it's new/interesting info to anyone.

Maybe you could elaborate? The problem is that we can find equivalent quantum statistical and thermal statistical ensembles with identical path integrals. This can be seen by reinterpreting the imaginary time path integral for a quantum stat mech system in d spatial dimensions as a classical partition function in d+1 spatial dimensions where the last spatial dimension is periodic with length $\beta$. If you now take the extreme quantum case, $\beta \rightarrow \infty$, you find that a quantum statistical system in d spatial dimensions at zero temperature is equivalent to some infinite classical system at finite temperature in d+1 spatial dimensions. (There are some caveats here - for some quantum actions with topological/Berry phase terms the corresponding classical Hamiltonian contains imaginary terms and can't really be properly interpreted).

For example: the zero temperature quantum Ising model in one (spatial) dimension in a transverse field is completely equivalent to Onsager's famous 2D classical Ising model at finite temperature (with some correspondence between the 1D Ising coupling/transverse field and the 2D Ising coupling/temperature). This is discussed in both Sachdev and Fradkin's textbooks: the quantum phase transition in the former case even has identical critical exponents as the classical finite temperature transition in the latter case. In one system we can only have quantum fluctuations, in the other system we can only have thermal fluctuations, but the same universal phase transition.

Ok, now having said all that, you might ask why we care about quantum phase transitions at all, since they're (usually) just equivalent to some classical phase transition, maybe with some funky periodic dimension. This has to do with the major technical problem in the imaginary time path integral: you need to eventually analytically continue your results to real time if you want dynamic correlation functions or really any real-time transport properties. These are dominated by some new time scale related to the phase coherence time, or equivalently, some energy scale ($\tau \sim \hbar/E$). This relates to what I mentioned in the last sentence of my last post. The "fan structure" that you mentioned earlier (which I suspect you're hinting at in your questions) is determined by different regimes, where either some relevant quantum energy scale/energy gap or the temperature is the scale determining the phase coherence time and the transport properties. The fan occurs because the quantum energy scale must go to zero at zero temperature for a second order phase transition, so you can define different regimes depending on whether temperature dominates over the energy scale. So really, you can talk about the relative thermal and quantum fluctuation effects on your model, but this heavily depends on the specific model you're considering.

atyy is totally correct that there exist reduced density matrices equivalent to thermal density matrices, so I suspect you could come up with some equivalent model to the above which could generate the "temperature" fluctuations via entanglement at zero temperature. I just wanted to explain the "fan" since you mentioned it earlier, since its physical realizations do involve competing quantum and thermal fluctuations.

 

[atyy]

atyy's point that a finite temperature density matrix is equivalent to a zero temperature reduced density matrix (actually, is this always true?) gets rid of any need to establish a quantum-classical correspondence, which may be ruined by a badly defined Wick rotation.

Would something like "the Church of the larger Hilbert space" http://www.quantiki.org/wiki/The_Church_of_the_larger_Hilbert_space work? I think it's only for finite dimensional Hilbert spaces, but they do say at the end that every mixed state has a purification.

 

[maverick_starstrider]

Alright, I think I've got the section nailed down. Thanks all for the great help!

 

[king vitamin]

Would something like "the Church of the larger Hilbert space" http://www.quantiki.org/wiki/The_Church_of_the_larger_Hilbert_space work? I think it's only for finite dimensional Hilbert spaces, but they do say at the end that every mixed state has a purification.

I'm not a mathematician, but that looks fine to me! The only possible problem I can conceive of is taking the thermodynamic limit, and if that's needed it seems like you could do so on the system and the "ancilla" at once.

In a way it's conceptually simple; the same way classical stat mech comes from throwing away individual degrees of freedom, an exact quantum description of the universe is described by the pure density matrix of the universe. Then any subsystem is just the reduced matrix, which now has no reference to temperature (this should also apply to closed systems which fail to thermalize).