Análisis de cianuro

The sign problem is actually a problem that can be described quite simply in a few lines, but the simply described problem has vast consequences when one attempts to perform calculations using probability distributions. This problem has been the root cause for many road blocks of computing in condensed matter physics, QCD and the like. It is because of this, in fact, that many physically interesting regimes have been completely unattainable (e.g. repulsive fermions). Likewise there have been a plethora of projects whose sole purpose has been to find a way to handle this exact problem, like the one we will discuss shortly, the Complex Langevin method.

So without further ado, what exactly is the sign problem? To answer this question let’s examine eq. [3.20].

P[σ] ˆO[σ] =|P[σ]|[sgn(P[σ]) ˆO[σ]]

We have briefly discussed the sign problem without directly mentioning it. That is, we asked the question, what happens if the measure is not positive definite? If the measure does carry some portion that is negative, then it cannot be considered a valid probability density. To handle this, we remove the overall sign of the measure, and carry it with the operator. The problem with this is that we have shifted the fluctuation in sign to the observable, which is good since we have assured a positive definite measure, but these fluctuations can be quite severe (19).

To examine this further, let’s address an example regarding fermions. Considering the probability defined using the Grand Partition function, eq. [2.69]

sgn(P[σ]) = P[σ] |P[σ]| = e−βH[σ] e−βH[σ] = e −βH[σ] e−βHB[σ] =e −β(H[σ]−HB[σ])_{. (3.28)}

WhereHB denotes the same type of Hamiltonian asH in the numerator (same kinetic, and interaction

terms), but with the fermions replaced with Bosons. In the limit of low temperature (β → ∞) the average sign can become extremely small, meaning that the expectation of our observable,hOi, is given by the ratio of two exponentially small numbers, determined stochastically. Furthermore, since the average sign is in fact the average of±1, then the overall variance becomes quite large. Finally, when the error bars become proportional to the average sign, massive fluctuations in the computation ofhOioccur.

In this instance the sign problem was caused by fermion statistics, but in general this problem comes about in many scenarios for both QMC and HMC. For instance, in HMC, when the action becomes complex, S[σ, π] = SR[σ, π] +iSI[σ, π], one immediately encounters a non-positive definite probability distribution,

and therefor a sign problem (19).

The next question would then be, how does one handle the sign problem? Sadly, the answer to that question is largely that one does not have a good handle on this problem. One would have to find some way of implementing a classical approximation, or avoid the problem all together. We have utilized a method, however, that attempts to handle the sign problem in regards to HMC. That is, it attempts to make use of the imaginary portion of the action directly. We will now discuss that method,Complex Langevin.

3.3.1: Complex Langevin Method

The idea of stochastic quantization is that a random process that obeys the Langevin equation, ∂σ(θ)

∂θ =− δS[σ]

δσ(θ)+η, (3.29)

will yield configurations σ that are distributed according to e−S[σ]_{. Here, θ is Langevin time and η is}

gaussian random noise. We generally assume that our Hubbard-Stratonovich fieldσ is real, but whenS[σ] is complex, however, we have no way of dealing with the complex part of the action as eq [3.29] stands. Complex Langevin on the other hand is a means of stochastic quantization, using a complexified auxiliary field that helps to surpass the sign problem. We want to use the Langevin equation to update our field given a complex action. So what we do is convert the real auxiliary field into a real and imaginary component.

σ=σR+iσI (3.30)

Thus, stochastic quantization requires complexified fields and the (spacetime local) Langevin equation now has both real and imaginary parts,

σR(n+ 1) = σR(n) +KR(n) +√η(n), (3.31)

σI(n+ 1) = σI(n) +KI(n), (3.32)

where the Langevin time has been discretized asθ=n,is the time step, andKR andKI are the real and imaginary drift terms respectively. Again,η, is the real Gaussian noise such thathη(n)i= 0. We may now

specify the drift terms as, K_a,xR = −Re " δS δσ _σ →σR+iσI # , (3.33) K_a,xI = −Im " δS δσ _σ →σR_+iσI # . (3.34)

These terms can be computed using various updating techniques such as the Euler and Runga Kutta methods. Again this allows us to specifically treat the imaginary part of the action with its own force, and with the hope that the imaginary component will average to zero if you expect a real result.

CHAPTER 4: Friedel Oscillations

Motivated by the realization of hard-wall boundary conditions in experiments with ultracold atoms, in this chapter we investigate the ground-state properties of spin-1/2 fermions with attractive interactions in a one-dimensional box. We use lattice Monte Carlo methods, namely hybrid Monte Carlo, explained in the previous chapter, to determine essential quantities like the energy, which we compute as a function of coupling strength and particle number in the regime from few to many particles. These many-fermion systems bound by hard walls display non-trivial density profiles characterized by so-called Friedel oscillations (which are similar to those observed in harmonic traps). In non-interacting systems, the characteristic length scale of the oscillations is set by (2k_F)−1_{, where k}

F is the Fermi momentum, while repulsive interactions tend

to generate Wigner-crystal oscillations of period (4k_F)−1_{. Based on the non-interacting result, we find a}

simple parametrization of the density profiles of the attractively interacting case, which we generalize to the one-body density matrix. In addition, we determine the spatially varying on-site density-density correlation, which in turn yields Tan’s contact density.