Fermionization of the 2-D Ising model: The method of Schultz, Mattis and Lieb

F. A da Costa, R. T. G. de Oliveira, G. M. Viswanathan

This blog post was written in co-authorship with my physics department colleague Professor Francisco “Xico” Alexandre da Costa and Professor Roberto Teodoro Gurgel de Oliveira, of the UFRN mathematics department. Xico obtained his doctorate under Professor Sílvio Salinas at the University of São Paulo. Roberto was a student of Xico many years ago, but left physics to study mathematics at IMPA in Rio de Janeiro in 2010. During 2006–2007, Roberto and Xico had written up a short text in Portuguese that included the exact solution of the Ising model on the infinite square lattice using the method of fermion operators developed by Schultz, Mattis and Lieb. With the aim of learning this method, I adapted their text and expanded many of the calculations for my own convenience. I decided to post it on this blog since others might also find it interesting. I have previously written an introduction to the 2-D Ising model here, where I review a combinatorial method of solution.

1. Introduction

The spins in the Ising model can only take on two values, ${\pm 1}$. This behavior is not unlike how the occupation number ${n}$ for some single particle state for fermions can only take on two values, ${n=0,1}$. It thus makes sense to try to solve the Ising model via fermionization. This is what Schultz, Mattis and Lieb accomplished in their well-known paper of 1964. In turn, their method of solution is a simplified version of Bruria Kaufman’s spinor analysis method, which is in turn a simplification of Onsager’s original method.

We will proceed as follows. First we will set up the transfer matrix. Next we will reformulate it in terms of Pauli’s spin matrices for spin-${\tfrac 1 2}$ particles. Recall that in quantum field theory boson creation and annihilation operators satisfy the well-known commutation relations of the quantum harmonic oscillator, whereas fermion operators satisfy analogous anticommutation relations. The spin annihilation and creation operators ${\sigma_j^\pm }$ do not anticommute at distinct sites ${j}$ but instead commute, whereas fermion operators must anticommute at different sites. This problem of mixed commutation and anticommutation relations can be solved using a method known as the Jordan-Wigner transformation. This step completes the fermionic reformulation of the 2-D Ising model. To obtain the partition function in the thermodynamic limit, which is the largest eigenvalue of the transfer matrix, one diagonalizes the fermionized transfer matrix using appropriate canonical transformations.

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Are science and religion compatible?

This blog post explores whether or not science and religion are compatible. I use the term religion in the usual sense, to mean a system of faith, worship and sacred rituals or duties. Religions typically consist of an organized code or collection of beliefs related to the origins and purpose of humanity (or subgroups thereof), together with a set of practices  based on those beliefs. Can such belief systems be compatible with science?

Since this topic is controversial, I only reluctantly decided to write about it.  Being a physics professor and a research scientist, I decided not to flee debate on this issue (which is like the third rail of science). Instead, here I detail my thoughts in writing.

Actually, I spent decades trying to reconcile science and (organized) religion, however I made little or no significant progress. Eventually, after much hesitation and discomfort, I was forced to conclude that full reconciliation between science and organized religion may not be possible, even in principle.   Although this realization was initially surprising (and unpleasant) to me, I soon discovered new and more fulfilling ways of approaching issues such as ethics, morals and the purpose or meaning of life, which religion has traditionally monopolized.

1. Short answer: science and religion are incompatible

Religion is a culture of faith and science is a culture of doubt.’  This statement is usually attributed to Richard Feynman.   Faith and doubt are indeed antagonistic, like water and fire. How can it be possible to fully reconcile religious views, which are based on faith, with the systematic doubt and the skeptical questioning that are intrinsic to the scientific method? Like many scientists, I too have concluded that full reconciliation of science and religion is not possible.

One caveat: obviously, if one removes the element of dogma and faith from religion, then reconciliation might be possible. But religion without dogma is more like a social club than a traditional religion. What would become of Christianity without faith in Jesus Christ? Can you imagine Islam without faith in the Koran?  So, by religion I always mean organized religion, with a set of teachings or dogmas.

Below I explore these issues in some detail.

2. Dirac and Feynman on religion

In the list of the all-time greatest physicists, Newton and Einstein invariably take the top positions. Paul A. M. Dirac, of Dirac equation fame, is considered to be an intellectual giant, ranking just a few notches below Einstein or Newton. And Feynman, who usually ranks just below or comparable to Dirac, has rock star status in the physics community.

Feynman had the following to say about religion:

It doesn’t seem to me that this fantastically marvelous universe, this tremendous range of time and space and different kinds of animals, and all the different planets, and all these atoms with all their motions, and so on, all this complicated thing can merely be a stage so that God can watch human beings struggle for good and evil — which is the view that religion has. The stage is too big for the drama.

Dirac had the following to say about religion:

If we are honest — and scientists have to be — we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination. It is quite understandable why primitive people, who were so much more exposed to the overpowering forces of nature than we are today, should have personified these forces in fear and trembling. But nowadays, when we understand so many natural processes, we have no need for such solutions. I can’t for the life of me see how the postulate of an Almighty God helps us in any way. What I do see is that this assumption leads to such unproductive questions as why God allows so much misery and injustice, the exploitation of the poor by the rich and all the other horrors He might have prevented. If religion is still being taught, it is by no means because its ideas still convince us, but simply because some of us want to keep the lower classes quiet. Quiet people are much easier to govern than clamorous and dissatisfied ones. They are also much easier to exploit. Religion is a kind of opium that allows a nation to lull itself into wishful dreams and so forget the injustices that are being perpetrated against the people. Hence the close alliance between those two great political forces, the State and the Church. Both need the illusion that a kindly God rewards — in heaven if not on earth — all those who have not risen up against injustice, who have done their duty quietly and uncomplainingly. That is precisely why the honest assertion that God is a mere product of the human imagination is branded as the worst of all mortal sins.

I do not accept arguments from authority. But it is nevertheless interesting to read about what these eminent physicists had to say.

3. Scientists abandon God and religion

Most scientists are non-religious. Many are atheist. A Pew survey from 2009 found that while over 80% of Americans believed in God, less than 50% of scientists believed in God. The percentage was actually 33% in that particular survey. These numbers are typical. For instance, among the members of the US National Academy of Sciences, more than 60% of biological scientists had disbelief in God (i.e., were what most people call atheists’) according to a study from 1998. In the physical sciences, 79% had disbelief in God.

This issue is relevant in society because most politicians and people in leadership positions are, at least outwardly, sympathetic to religion if not actively religious. So there is at least this one important difference between the majority of scientists and the rest of society. Whereas most people are religious, most scientists are non-believers.

More worrying is that many politicians actively campaign against science and science education. We have all heard about attempts by the religious to eliminate (or water down) the teaching of Darwinian evolution in schools. At least in the West, these attempts have largely failed.

Fortunately, the voting population does not particularly crave a return to the dark ages. It  is easy to understand why. The experience of the last few centuries has shown that social and economic development is only possible when there is political support and commitment to science research and education. Science is responsible for the invention of the Internet, cell phones, radio, TV, cars, trains, airplanes, X-rays, MRIs, the eradication of smallpox, etc.   Rich and socially developed countries are precisely those in which science education and research are well funded. Economic pressures have thus led to investment in science and in science education.

At the same time, science has led to unintended consequences. The more a person is exposed to science, the less religious they are likely to become. (Possibly as a consequence, wealth is also negatively correlated with religiosity. In other words, on average the richer you are, the less religious you are likely to be.)

Especially among those with less science education, there is a fear that exposure to science and to “subversive” ideas such as Darwinian evolution will infect the minds of young people and turn them into “Godless infidels.”  In fact, fear is a constant theme in religion: fear of God, fear of divine punishment, fear of hell, fear of forbidden knowledge, etc.  Science education dispels such fears, and replaces it with the cultivation of curiosity, wonder, questioning, doubt and awe. Since fear is often used as an instrument of control and power, the loss of fear can be a setback for the power structures of organized religion. In this sense, science and science education sometimes directly threaten some religious movements.

Consider, as an example, suicide bombing as a form of jihad by Islamic militant organizations. It is perfectly fair for us to ask: is it even remotely plausible that these hapless suicide bombers correctly understood the scientific method? This is a rhetorical question, of course. A genuinely curious and scientifically literate potential candidate for suicide bombing would immediately ask questions, especially when faced with death by suicide. Is life after death a sure thing? Will Allah really reward a suicide bomber? How it is possible for the big-breasted and hot Houri girls and women to recover their sexual virginity every morning? The young man may then go on to ask: is there a remote possibility, perhaps, that such ridiculous claims are not a sign of pulling the wool over the eyes of the naïve young men in their sexual prime who crave sex and intimacy with women, but who are forbidden by religion to engage in casual sex? And why is recreational and free sex allowed in Paradise but not on earth? A scientifically literate young man would probably say `Thanks but no thanks, I’ll let you go first to set the example!’  Hence the fear and loathing of doubt, curiosity and questioning. Indeed, scientific illiteracy makes people gullible and easier to manipulate.

There is no denying the statistics: exposure to science is correlated with loss of religious faith. This raises two questions: (i) why does this happen and (ii) is this good or bad? I am mostly concerned here with question (i) and only briefly touch upon point (ii).

Brazil’s new science minister is a restaurant owner

I was dismayed to learn that the new science and technology minister of Brazil is, once again,  neither a scientist nor particularly qualified to be science minister.  Celso Pansera is a federal representative in Brazil’s lower house of congress (Chamber of Deputies) and a member of the PMDB political party that President Dilma Rousseff now depends on to keep her government afloat.  He also owns a restaurant. As far as I know, he has no connection at all with science and technology.

President Dilma Rousseff could have at least appointed a professional career scientist.  For comparison, former President Lula nominated an internationally known scientist to the science ministry. Similarly, US President Obama chose a Nobel prize winning physicist to be his energy secretary.

President Dilma Rousseff is sending the wrong message to Brazilians about the importance of science and technology.

Onsager’s solution of the 2-D Ising model: The combinatorial method

1. Introduction

In this blog post, I derive, step by step, the exact partition function for the ferromagnetic Ising model on the square lattice. The result is celebrated as “Onsager’s solution” of the 2-D Ising model. It was originally derived by Lars Onsager in 1942 and published in 1944 in Physical Review [1]. That paper revolutionized the study of phase transitions and what we now call critical phenomena [2].

Somewhat ironically, I first heard about the Ising model when I was working in industry. I was 20 and held a summer job at what was then known as British Telecom Research Labs (BTRL), near Ipswich in the UK. This was before I had ever seen a cell phone or heard of the Internet (although I knew about BITNET and JANET). I worked there in the summer of 1990 and again for a month or so around April 1991. My job at BT involved writing C implementations of multilayer perceptrons and Hopfield neural nets. In those days, BT was interested in implementing hardware neural networks and my boss mentioned casually to me that certain kinds of neural nets are basically just special cases of the Ising model. (Indeed, the Hopfield network is closely related to the Ising spin glass.) Thus began my fascination with the Ising model. Later, in 1994 in Boston, I took a course given by Bill Klein at BU on statistical mechanics, where we went through the solution of the 1-D ferromagnetic Ising model. Still, I never had the chance to study properly the 2-D Ising model. As a PhD student, I would almost daily pass by a poster with a background photo of Lars Onsager (with a cigarette in his hand), hung near the office door of my advisor Gene Stanley, so I was regularly reminded of the 2-D Ising model. I kept telling myself that one day I would eventually learn how Onsager managed to do what seemed to me, at the time, an “impossible calculation.” That was 1994 and I am writing this in 2015!

In what follows, I solve the Ising model on the infinite square lattice, but I do not actually follow Onsager’s original argument. There are in fact several different ways of arriving at Onsager’s expression [3–9]. The method I use below is known as the combinatorial method and was developed by van der Waerden, Kac and Ward among others and relies essentially on counting certain kinds of closed graphs (see refs. [3,10–13]). I more or less follow Feynman [3] and I have also relied on the initial portions of ref. [13].

2. The 2-D Ising model

Consider a two dimensional lattice ${\Bbb Z^2}$ where at each point of the lattice is located a (somewhat idealized) spin-${\tfrac{1}{2}}$ particle. Consider a finite subset of this lattice, of size ${L\times L}$ and let ${i,j=1,2,\ldots L^2}$. Let ${\cal N}$ denote the set of pairs of integers ${(i,j)}$ such that spins ${\sigma_i}$ and ${\sigma_j}$ are nearest neighbors. In the ferromagnetic 2-D Ising model with nearest neighbor interactions, spins ${i}$ and ${j}$ interact if and only if ${(i,j)\in \cal N}$. Each spin ${\sigma_i}$ can assume only 2 values: ${\sigma=\pm 1}$.

Consider a system of ${N=L^2}$ spins. The Hamiltonian for a spin configuration ${\sigma=(\sigma_1,\sigma_2,\ldots \sigma_N)}$ is given by

$\displaystyle H(\sigma)= -J \displaystyle\sum_{i,j\in \cal N} \sigma_i \sigma_j ~. \ \ \ \ \ (1)$

The sum over the nearest neighbors should avoid double counting, so that ${(i,j)}$ and ${(j,i)}$ are not counted separately. Without loss of generality, we will assume ${J=1}$ for simplicity.

3. The canonical partition function

In the theory of equilibrium statistical mechanics, the canonical partition function contains all the information needed to recover the thermodynamic properties of a system with fixed number of particles, immersed in a heat bath, details of which can be found in any textbook on statistical mechanics [3-6,14].

I prefer to define the partition function as the two-sided Laplace transform of the degeneracy ${\Omega(E)}$ of the energy level ${E}$. But traditionally, the partition function is defined as a sum or integral over all possible states of the system:

$\displaystyle Z(\beta) = \displaystyle\sum _{\sigma} e^{-\beta H(\sigma)} ~. \ \ \ \ \ (2)$

The two ways of thinking are equivalent. The Laplace transform variable ${\beta}$ is related to the thermodynamic temperature ${T}$ via ${\beta =1/k_B T}$, where ${k_B}$ is the Boltzmann constant. What follows is the exact calculation of ${Z(\beta)}$.

Domain coloring for visualizing complex functions

The above figure shows a domain color plot for the function ${z\mapsto {1/(1 - i z )} -(1 + i z)}$ .

I have been relearning complex analysis and decided to have some fun plotting complex functions. It is easy to visualize graphically a real function ${f:\Bbb R\rightarrow \Bbb R}$ of a real variable. To visualize ${f(x)}$, we can plot ${x}$ on the horizontal axis and ${y=f(x)}$ on the vertical axis, as we learn in high school mathematics. It is more difficult, however, to visualize the function ${g:\Bbb C\rightarrow \Bbb C}$, because ${g}$  is a  complex function of a complex variable. A complex number ${z}$ can be expressed in terms of real numbers $x, ~y$ as ${z=x+ i y}$. To visualize the complex domain of a function, one thus requires 2 real dimensions. The complex function ${g(z)}$ must then be plotted on the remaining dimensions. If we lived in 4 dimensional space, it would be easy to plot a complex function of a complex variable. The problem is that we live in three dimensions, not four. Domain coloring is a method to overcome this limitation. The basic idea is to use colors and shades etc. as extra dimensions for visualizing functions.

I first heard about domain coloring when I came across Hans Lundmark’s complex analysis pages. For readers unfamiliar with domain coloring, I recommend reading up on it first. There is also a wikipedia article on the subject.

Introduction

The pictures below were generated using Wolfram’s Mathematica software (many thanks to my department colleague Professor Marcio Assolin). I adapted the code from the discussion on stackexchange here, and also here.

The first figure below shows the identity function ${f(z)=z}$:

The horizontal axis is the real axis and the vertical the imaginary axis of the domain ${z}$. The 29×29 grid is not deformed in this case because the identity function ${f(z)=z}$ does “nothing.” But I decided to include this example to illustrate how the colors and shades code information. First of all, notice that as we go around the origin ${z=0}$, the colors go from red on the positive real line, to green on the positive imaginary axis, to cyan on the negative real axis, to purple on the negative imaginary axis and then back to red again.

The color represents the argument of the function. In this case, since we can write ${z=r e^{i\theta}}$ the argument ${\theta}$ wraps back around every time we turn ${2\pi}$ radians around the origin in the complex plane.

Note that in addition to color, there is shading. Notice that at ${|z|=1/2}$, ${|z|=1}$ and ${|z|=2}$ there are discontinuities in shading. As we increase the absolute value of ${z}$, the shading gets darker and discontinuously gets brighter and this process repeats itself. Indeed, the shading has been used to plot the absolute value of the function. Every time the absolute value doubles, the shading goes through one cycle, becoming discontinuously brighter.

Monomials

Let us now look at more complicated functions. The pictures below show the functions ${z\mapsto z^2}$ and ${z\mapsto z^3}$.

Now, things look more interesting! The first thing to note is that the 29×29 grid is distorted. The grid shown is the inverse image of the grid of the identity map. So for the function defined by ${f(z)=z^2}$, the procedure is as follows. Take the points of the grid and put them into some set ${S}$. Then calculate the inverse image ${f^{-1}(S)}$, i.e. the set of points that map to ${S}$. This inverse image is the grid shown above.

Notice how the distorted grid seems to preserve right angles, except at the origin. Indeed, holomorphic functions are typically also conformal (i.e., angle preserving) in many instances, and we will return to this topic further below. At the origin, the monomial functions above are clearly not conformal.

Take a look at the colors. Instead of cyclying through the rainbow colors going once around the origin, the colors cycle 2 and 3 times, respectively, for ${z^2}$ and ${z^3}$. This is easy to understand if we write ${z=re^{i \theta}}$ (for real ${r}$ and ${\theta}$), so that ${z^2=r^2 e^{2i\theta}}$ and ${z^3=r^3 e^{3i\theta}}$. So ${z^2}$ and ${z^3}$ circle the origin 2 and 3 times every time ${z}$ goes around the origin once.

Finally, notice that the shading cycles through more quickly. Indeed, ${|z^2|}$ and ${|z^3|}$ double more quickly than ${|z|}$.

Poles

The monomial functions above have zeroes, but no poles. What do poles look like? The image below shows the function ${z\mapsto 1/z}$.

Notice how the colors cycle around “backwards.” Indeed, the argument of ${z}$ and ${z^{-1}}$ are negative of each other. Notice also how the shading now has discontinuities in the “opposite” sense (compare with ${z\mapsto z}$ in the first image above).

It is also worthwhile to look at the grid. The original grid of lines has transformed into a patchwork of circles. To show this more clearly, I modifed the image to show only a few grid lines, corresponding to real and imaginary lines at ${\pm 1,2,3,4,5}$:

In the image above the grid lines are now clearly seen to have mapped into circles. In higher order poles, the colors cycle around (backwards) a number equal to the order of the pole. Here are poles of order 2 and 3, shown with only some grid lines for greater clarity:

In addition to poles, there are other kinds of singularities. Removable singularities are not too interesting, because basically a “point” is missing. If we “manually add” the point, the singularity is “removed” — hence the name.

In addition to removable singularities and poles, there are also what are known as essential singularities. Essential singularities can be thought of, loosely speaking, as poles of infinite order. Further below, we will take a look at essential singularities.

Now consider a function with a zero as well as a pole:

The function shown above is ${z\mapsto (z-1)+ 1/(z+1)}$, which has a zero at ${z=0}$ and simple pole at ${z=-1}$.

Non-holomorphic functions

Having looked at examples of holomorphic and meromorphic functions, let us look at more complicated non-holomorphic function. The figure below shows ${|z|}$, the absolute value of ${z}$.

The color is red because ${|z|}$ is always non-negative real. The grid is gone and we have circles instead: we do not have conformality. The Cauchy-Riemann equations are impossible to satisfy because ${|z|}$ is always real, the imaginary part being identically zero everywhere.

The complex conjugate function ${z\mapsto \bar z}$ also is not analytic. Here is ${z\mapsto \bar z}$:

Notice that it looks just like the identity map, but reflected along the real axis. The imaginary axis is “backwards”.  Still, angles are preserved, so why is this function not holomorphic? The answer is that the angle orientations are reversed, i.e. the function is antiholomorphic rather than holomorphic. The colors cycle around “backwards”  in this case because the complex conjugate of  $r e^{i \theta}$ is $r e ^{- i \theta}.$  Conformal maps preserve oriented angles, rather than just angles. Indeed, the complex conjugate function is neither holomorphic nor conformal.

The exponential function and its Taylor polynomials

Let us now look at a transcendental holomorphic function: the exponential function ${z\mapsto \exp(z)}$. The images below show the exponential function at two scales.

Notice that the colors now cycle through going up and down vertically. The reason for this is as follows. If we write ${z=x + i y }$, then

$\exp[z]=e^{x+i y } = e^x e^{i y} ~.\ \ \ \ \$

So ${y}$, which is the imaginary part of ${z}$, determines the argument, hence the color. The zoomed out version makes it clear that the argument is periodic with period ${2 \pi}$ in the imaginary direction.

While on the topic of the exponential function, let us take a look at Talyor polynomial expansions. The figure below shows the Talor polynomial of degree 5.

Notice the 5 zeroes, which lie on an arc like the letter “C” slightly to the left of the origin. The exponential function does not have zeroes, of course. We know that if we take the infinite degree Talyor polynomial, i.e. the infinite Taylor series expansion, then we recover the exponential function. We can already see that for positive real part and small imaginary part of ${z}$, the Taylor polymial above is starting to behave qualitatively like the exponential function.

Essential singularities

Having seen the exponential function, we can now look at essential singularities. Observe that the Laurent expansion of ${\exp(1/z)}$ around the origin in ${z}$ has an infinite number of terms of negative power in ${z}$. The singularity at the origin is thus stronger than a pole of any finite order. The figure below shows ${z\mapsto \exp(1/z)}$, shown at three different scales.  The third figure is a zoom of the second, which is a zoom of the first.

The software is apparently having some trouble near the origin, in the last figure! The reason for this is the Great Picard’s theorem, which says, loosely speaking, that an analytic function near an essential singularity takes all possible complex values, with at most 1 exception. In the case of ${z\mapsto \exp(1/z)}$, the function cannot become zero, which is the exceptional value. As we approach the origin, the argument (i.e. color) changes, cycles around, etc., increasingly quickly.

Let us now look at some trigonometric functions:

${\sin(z)}$:

${\tan(z)}$

We can clearly see the zeroes in ${\sin(z)}$ and the poles and zeroes of ${\tan(z)}$. Moreover, it is clear that these are periodic functions.

Compare the above trigonometric functions with their inverses:

${\arcsin(z)}$

${\arctan(z)}$

Notice that on the real line, for ${|z|>1}$ there is a discontinuity in color for arcsine. Similarly, for arctan there is a color discontinuity on the imaginary axis. To understand this jump in color, recall that ${|\sin(x)|\leq 1}$, which means that the inverse function ${\sin^{-1}(x)=\arcsin(x) }$ is not defined for ${x}$ outside the interval ${ -1\leq x\leq 1}$. Recall also that ${\sin(x)=\sin(x+2\pi)}$, so that the inverse function ${\sin^{-1}}$ must be multivalued. What is being shown above is the principal branch.

It helps to switch over to the logarithmic form. Recall that

$\displaystyle \sin(z)= {e^{iz} - e^{-iz} \over 2i } ~. \ \ \ \ \$

If we write ${z=\arcsin(w)}$, then ${\sin(z)=w}$. Substituting, we get

$\displaystyle 2iw ={e^{iz} - e^{-iz} }~. \ \ \ \ \$

To simplify the algebra, let ${e^{iz}=Z}$, so that

$\displaystyle 2iw = {Z - Z^{-1} } ~, \ \ \ \ \$

which gives us a quadratic equation:

$\displaystyle Z^2 -2iwz -1 =0~, \ \ \ \ \$

whose roots are

$\displaystyle Z= iw \pm \sqrt{1-w^2} ~. \ \ \ \ \$

So we finally get

$\displaystyle z=\arcsin(w)=-i \log Z= -i \log\left( iw \pm \sqrt{1-w^2} \right) ~. \ \ \ \ \$

So the branch cut in the ${\arcsin}$ function is due to the 2 possible values of the square root. There is a branch point at ${w=\pm 1}$ and there is actually another branch point at infinity.

Branch points and cuts

Let us look at branch points more closely. As we know, the square root is multivalued, and the figure below shows the two branches, with the principal branch at the bottom.

Note how in each branch alone the colors do not cycle all the way through the rainbow colors. The missing colors of one branch are on the other branch. To see both branches, one would need to visualize the Riemann surface for the square root, a topic beyong what I wish to cover here.

The figure below shows the 3 branches of the cube root function, with the principal branch at the top:

There is more than 1 type of branch point. Algebraic branch points are those that arise from taking square roots, cubic roots, and ${n}$-th roots (for positive integer ${n}$). In general there will be ${n}$ well defined branches.

What happens if one takes ${n}$ to be a positive irrational number? Here is a hint:

$z^{\alpha} = e^{\alpha \ln z} ~.$

If we choose ${\alpha}$ rational, say ${\alpha=p/q}$, then putting ${z=r e^{i\theta + 2n\pi i}}$ we get ${\ln z= \ln r + i\theta + 2n\pi i}$, so that

$\displaystyle z^\alpha = r^{p/q} e^{i \alpha \theta} e^{ 2\pi i (np /q)} ~.$

But ${n p/q}$ is rational, and so there can be at most ${q}$ branches. But if ${\alpha}$ is irrational, this argument does not work. Instead of ${np/q}$ we get ${n\alpha}$, which can never equal an integer. So the branches never cycle through and the number of branches is infinite. Indeed, the logarithm above leads to a infinite branching, as we will see below.

There is a more complicated type of branch point in the function ${z\mapsto \exp(1/z^{1/n})}$ for integer ${n}$. If we loop around the origin ${n}$ times, the function returns to the original starting point (i.e. there is finite monodromy). However, there is an essential singularity at the origin. In other words, there is the unhappy coincidence of the algebraic branch point of ${1/z^{1/n}}$ sitting exactly on top of an essential singularity. Such branch points are known as transcendental branch points.

The figure above shows ${z\mapsto \exp(1/z^{1/2} )}$. The essential singularity coincides with the branch point.

Finally, as we saw above, there are branch points where the number of branches is infinite. Consider the complex logarithm. If we again write ${z=r e^{i \theta}}$ then ${\log z= \log r + i \theta}$. Since ${\theta}$ and ${\theta \pm 2\pi}$ give same value of ${z}$, the logarithm is multivalued. The branch cut is usually taken at ${\theta=\pm \pi}$. Here is the logarithm:

The zero at ${z=1}$ is clearly visible, as is the branch cut along the negative real axis.

The complex logarithm, like other multivalued functions, could instead be visualized as Riemann surfaces. Here is an example of the Riemann surface of the complex logarithm.

Euler’s reflection formula

Let us next consider another topic related to Weierstrass’s beautiful factorization theorem, which states that every entire function can be expressed as an infinite product. Among the best known infinite products are the one used by Euler to solve the Basel problem, and the product formula for Euler’s gamma function. Indeed, the gamma function is, in a sense, one half of the sine function (or cosecant). Put differently, the sine function is the product of two gamma functions:

$\Gamma(z)\Gamma(1-z) = {\displaystyle \pi \over \displaystyle \sin (\pi z) }$ .

Proofs can be found in textbooks. Here I wish to focus on the zeroes and poles. Take a look at these figures:

The figures above show ${1/\Gamma(z)}$ and ${1/ [\Gamma(z)\Gamma(1-z) ]}$. Note the zeroes at the non-positive integers in the first image, correponding to the poles of the gamma function. If we multiply two gamma functions, so that there are zeroes at all integers, we basically get ${\sin(2\pi z)}$ upto a constant! Indeed, the last figure above is identical, up to scale, to that of the sine function seen earlier.

When are holomorphic functions conformal?

Finally, let us take a closer look at conformality, i.e. the angle-ṕreserving property found in many holomorphic functions. The examples above of monomials of degree greater than 1 and poles shows clearly that conformality can break down at zeroes and poles.

Consider again the function ${z \mapsto z^2}$ below:

Conformality indeed breaks down at the zero at the origin, as expected.

But if a function is holomorphic with no zeroes in a region, is it necessarily conformal in that region? The answer is NO, as seen from the following counter-example:

${z \mapsto z^2+1}$

There is no zero at the origin, yet conformality breaks down!

To understand why, recall that to preserve angles, the map must locally be a scaled rotation (upto a translation). So the Jacobian determinant of the conformal map must be some positive constant. It is easy to show that the Cauchy-Riemman equations lead to a scaled rotation, provided the derivative is not zero. If the derivative is zero, however, the function need no longer be a scaled rotation, so angles need not be preserved.

In the example above, the function is holomorphic at the origin, but the derivative is zero at the origin, in other words, the origin is a critical point. Moreover, as ${z}$ moves around the origin once, the function ${z^2+1}$ moves around 1 twice, hence the breakdown in conformality.

Conversely, if a holomorphic function has a critical point at the origin, then its Taylor series does not contain a term of degree 1. But all higher order monomials ${z^n}$ break the conformal property at the origin, so the function cannot be conformal at the critical point. By translating arbitrary functions such that a critical point is on the origin, we can understand the following well known result:

A holomorphic function is conformal if and only if there are no critical points in the region of interest.

NOTES:

[1]. Below is the Mathematica code I used for the plot of the identity map. The code has been adapted from the discussion on stackexchange here and also here.

f[z_] := z;

paint[z_] :=
Module[{x = Re[z], y = Im[z]},
color = Hue[Rescale[ArcTan[-x, -y], {-Pi, Pi}]];
shade = Mod[Log[2, Abs[x + I y]], 1];

ParametricPlot[{x, y}, {x, -3, 3}, {y, -3, 3},
ColorFunctionScaling -> False,
ColorFunction -> Function[{x, y}, paint[f[x + y I]]], Frame -> True,
MaxRecursion -> 1, PlotPoints -> 300, PlotRangePadding -> 0,
Axes -> False , Mesh -> 29,
MeshFunctions -> {(Re@f[#1 + I #2] &), (Im@f[#1 + I #2] &)},
PlotRangePadding -> 0, MeshStyle -> Opacity[0.3], ImageSize -> 400]

A independência do banco central vai impedir os avanços sociais?

O assunto da independência do Banco Central (BC) tornou-se polêmico nas últimas semanas devido à entrada da candidata do PSB, Marina Silva, na disputa presidencial. Economistas e gestores experientes, como o Henrique Meirelles, que foi presidente do BC durante o governo Lula, já se declararam a favor de maior independência  para o BC (veja aqui).

Mesmo assim, boatos e mentiras espalham medo e ansiedade sobre os efeitos supostamente malignos da independência do BC. Devido a esta campanha de desinformação, naturalmente há uma certa confusão sobre o assunto. Por exemplo, representantes de alguns partidos políticos estão fazendo propaganda, afirmando que a independência do BC vai impedir os avanços sociais.  Poderia isso ser verdade?  Caso afirmativo, por que pessoas reconhecidamente competentes, como Henrique Meirelles, manifestam-se a favor da independência do BC? Onde está a verdade?

Sendo cientista, tendo a abordar qualquer questão sobre o mundo observável do ponto de vista empírico e experimental. Sempre me pergunto: Para onde os dados apontam? No fundo no fundo, o método científico nada mais é do que a acareação entre teorias e dados experimentais. Muitas teorias já foram derrubadas no confronto com os dados empíricos. Neste caso, os dados sugerem que a independência de um banco central não prejudica os avanços sociais. Muito pelo contrário.

Mesmo sem entrar em detalhes, o leitor pode chegar a uma conclusão independentemente, simplesmente olhando para os polos extremos no espectro de políticas econômicas adotadas em países diferentes. Por exemplo, podemos comparar a qualidade de vida, IDH, etc. nos países com e sem bancos centrais independentes. Então, vamos considerar exemplos extremos: (1) Países com bancos centrais quase totalmente independentes e (2) países com bancos centrais marionetes.

A União Europeia, os EUA, o Reino Unido e o Japão têm bancos centrais independentes. Esses bancos centrais agem sem interferência do governo e são imunes à pressão por grupos com conflito de interesses. Esses países foram pioneiros em conquistar a independência de seus bancos centrais. Sem exceção, são países ricos e com baixas taxas de pobreza, analfabetismo, mortalidade infantil, etc.

Podemos também olhar para os países onde os bancos centrais são marionetes do governo. Venezuela é exemplo de um país onde o banco central vem perdendo autonomia: O governo manda no banco central e faz o que bem quer. Zimbabwe é outro país onde o governo interferiu na atuação do banco central. Em ambos países, o governo tem amplos poderes para ditar as regras e passar o rolo compressor por cima de qualquer questionamento técnico dos economistas que trabalham na instituição. A situação é bem diferente nos países que tem um banco central independente. Por exemplo, nem mesmo o presidente dos EUA, Barack Obama, tem o poder para ajustar as taxas de juros do banco central americano  (Federal Reserve).

Temos então os exemplos extremos de países com e sem bancos centrais. Milhares de brasileiros e pessoas de outras nacionalidades tentam imigrar todo ano para os EUA, para a Europa, e até mesmo para o Japão. Mas quantos brasileiros sonham em imigrar para Venezuela ou para Zimbabwe?

Se o leitor está convencido, então por que os políticos resistem à ideia de um banco central independente? Será que os políticos dos partidos que são contra a independência do BC estão tão enganados? Eles não conseguem distinguir entre uma Venezuela e um Japão, ou entre Zimbabwe e Europa? O leitor deve estar pensando que, certamente, deve haver alguma razão boa e bem pensada contra a independência do BC. Tantos políticos não poderiam estar todos errados! A final de contas, quem esse tal de Henrique Meirelles acha que ele é? O leitor pode estar pensando:  Se estudarmos com mais cuidado os detalhes de como funciona um banco central, certamente iremos encontrar boas razoes para manter o status quo ante.

Mas a resposta é bem mais simples.  A verdade é que todos esses políticos estão enganados. Ou pior, agem por interesses diferentes do interesse público. Havia uma época quando quase todo mundo acreditava que o sol girava em torno da terra. Nem por isso era verdade. Da mesma forma, acreditar que um BC independente vai impedir os avanços sociais não faz com que isso seja uma verdade.  Opinião popular não é um bom indício de veracidade.  A única fonte de conhecimento sobre o mundo observável é o método científico.   Para saber a verdade, temos que estudar os dados.  Os dados, diferentemente dos políticos, não mentem.

Trabalhei por mais de uma década na área de pesquisa que ficou conhecida como Econofísica, estudando fenômenos econômicos do ponto de vista da Física de processos estocásticos. Meu interesse principal tem sido as propriedades estatísticas de retornos financeiros.  Minhas publicações sobre o assunto podem ser encontradas aqui. Uma taxa de juro, no final das contas, nada mais é do que uma taxa de retorno financeiro. Considero-me, portanto minimamente equipado para discursar e opinar qualificadamente sobre essa questão da independência do BC.

A seguir, vou resumir, de forma extremamente condensada, o principal argumento a favor da independência do BC.  Na maioria dos países com bancos centrais independentes, o governo fixa a meta de inflação e o banco central administra as taxas de juros para tentar atingir essa meta de inflação. O governo não interfere nessa atuação. Os critérios que norteiam as decisões sobre as taxas de juros são relativamente transparentes. Os agentes econômicos, portanto, têm o luxo de poder planejar para o futuro, sabendo o quanto esperar da taxa de inflação anual. Essa previsibilidade gera confiança na meta de inflação, o que concede credibilidade ao BC e logo também ao governo.

O efeito colateral mais importante desse aumento da credibilidade  e da confiança é a redução da taxa de juros do BC. A razão é simples: Os investidores, o empresariado, os especuladores financeiros, os trabalhadores e todos os outros agentes econômicos perdem o incentivo para apostar suas fichas numa inflação diferente da meta de inflação. Assim, as taxas de juros (como por exemplo a taxa SELIC no Brasil) necessários para garantir a meta de inflação acabam sendo significativamente menor em países que têm bancos centrais independentes, comparados aos países sem bancos centrais independentes.

Simplificando um pouco o argumento, a independência do BC gera otimismo e segurança com relação à previsibilidade de inflação. Já a falta de independência gera desconfiança e insegurança pois ninguém sabe quanto será a inflação. Na verdade, nos países com bancos centrais marionetes, os agentes econômicos simplesmente não confiam nas projeções de inflação do governo. Quantos brasileiros já compraram dólares para fazer “hedge” contra a inflação? Às vezes, nem mesmo as estatísticas do governo são confiáveis. Na Venezuela, por exemplo, a inflação real hoje é bem maior do que a taxa de inflação oficial. O governo faz uma maquiagem para esconder a inflação. Em resumo, quando um banco central não pode agir independentemente do governo, o preço cobrado é maior para controlar a inflação.   Há situações em que nada garante que a inflação ficará sob controle, como por exemplo na Venezuela ou em Zimbabwe.

Um banco central independente é um dos vários ingredientes que fazem um país ser economicamente próspero e socialmente evoluído. Por que, então, o Brasil ainda não tem um BC independente? O custo de um banco central independente é mínimo. Os benefícios são enormes, pois a inflação é um mal que atinge principalmente as classes mais pobres. Um BC independente também ajuda a economia a crescer, da seguinte maneira: Com as expectativas da inflação futura sob controle, as taxas de juros do BC ficam reduzidas e juros baixos ajudam a economia a crescer.

Henrique Meirelles, durante os anos do governo Lula, teve total autonomia operacional no BC e consequentemente a inflação ficou sob controle, como deve se esperar. No entanto, sob seu próximo presidente  o BC teve uma atuação bem diferente inicialmente. Mesmo com a taxa de inflação acima do centro da meta de inflação de 4,5%, o BC baixou a taxa SELIC sem nenhuma explicação que fazia sentido.  Exatamente como a teoria prevê, a inflação subiu e quase estourou o teto da meta de inflação, de 6,5%. O que poderia ter levado o BC a agir desta forma tão irracional? A única explicação plausível é também a reposta mais óbvia:  A decisão veio de cima. Quando deu no que deu, o BC novamente elevou a taxa SELIC o hoje pagamos o troco em triplo: (1) inflação alta, (2) juros altos e (3) economia em recessão técnica.   O Brasil literalmente pagou caro para aprender essa lição importante sobre a necessidade da independência do BC.

Frente ao exposto, venho aplaudir a coragem e o acumen  de Henrique Meirelles e de Marina Silva.  Apoio 100% a proposta de um BC independente.

O Brasil merece.

Agradecimentos: Tiago Viera, Carlos Chesman e Claudionor Bezerra  contribuíram com comentários e correções.

Feynman on worthwhile problems

A colleague of mine, Luiz Felipe Pereira, brought to my attention this letter of Richard P. Feynman addressed to Koichi Mano, dated 3 Feb 1966. It conveys immediately a clarity about what are the problems that are worthwhile studying. I also found it moving.

I copy the entire text here.

Dear Koichi,

I was very happy to hear from you, and that you have such a position in the Research Laboratories.

Unfortunately your letter made me unhappy for you seem to be truly sad. It seems that the influence of your teacher has been to give you a false idea of what are worthwhile problems. The worthwhile problems are the ones you can really solve or help solve, the ones you can really contribute something to. A problem is grand in science if it lies before us unsolved and we see some way for us to make some headway into it. I would advise you to take even simpler, or as you say, humbler, problems until you find some you can really solve easily, no matter how trivial. You will get the pleasure of success, and of helping your fellow man, even if it is only to answer a question in the mind of a colleague less able than you. You must not take away from yourself these pleasures because you have some erroneous idea of what is worthwhile.

You met me at the peak of my career when I seemed to you to be concerned with problems close to the gods. But at the same time I had another Ph.D. Student (Albert Hibbs) whose thesis was on how it is that the winds build up waves blowing over water in the sea. I accepted him as a student because he came to me with the problem he wanted to solve. With you I made a mistake, I gave you the problem instead of letting you find your own; and left you with a wrong idea of what is interesting or pleasant or important to work on (namely those problems you see you may do something about). I am sorry, excuse me. I hope by this letter to correct it a little.

I have worked on innumerable problems that you would call humble, but which I enjoyed and felt very good about because I sometimes could partially succeed. For example, experiments on the coefficient of friction on highly polished surfaces, to try to learn something about how friction worked (failure). Or, how elastic properties of crystals depends on the forces between the atoms in them, or how to make electroplated metal stick to plastic objects (like radio knobs). Or, how neutrons diffuse out of Uranium. Or, the reflection of electromagnetic waves from films coating glass. The development of shock waves in explosions. The design of a neutron counter. Why some elements capture electrons from the L-orbits, but not the K-orbits. General theory of how to fold paper to make a certain type of child’s toy (called flexagons). The energy levels in the light nuclei. The theory of turbulence (I have spent several years on it without success). Plus all the “grander” problems of quantum theory.

No problem is too small or too trivial if we can really do something about it.

You say you are a nameless man. You are not to your wife and to your child. You will not long remain so to your immediate colleagues if you can answer their simple questions when they come into your office. You are not nameless to me. Do not remain nameless to yourself – it is too sad a way to be. Know your place in the world and evaluate yourself fairly, not in terms of your naïve ideals of your own youth, nor in terms of what you erroneously imagine your teacher’s ideals are.

Best of luck and happiness.

Sincerely,
Richard P. Feynman.

Source: “Perfectly Reasonable Deviations from the Beaten Track: The Letters of Richard P. Feynman” (Basic Books, 2008).