6.3: Lee-Yang Theory

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\)
\( \newcommand\Vpi

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\)
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\)
\( \newcommand\Vrho

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\)
\( \newcommand\Vvarrho

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\( \newcommand\Vtau

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\)
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\)
\( \newcommand\VTheta{\vec\Theta}\)
\( \newcommand\VLambda{\vec\Lambda}\)
\( \newcommand\VXi{\vec\Xi}\)
\( \newcommand\VPi{\vec\Pi}\)
\( \newcommand\VSigma{\vec\Sigma}\)
\( \newcommand\VUps{\vec\Upsilon}\)
\( \newcommand\VPhi{\vec\Phi}\)
\( \newcommand\VPsi{\vec\Psi}\)
\( \newcommand\VOmega{\vec\Omega}\)
\( \newcommand\BA{\mib A}\)
\( \newcommand\BB{\mib B}\)
\( \newcommand\BC{\mib C}\)
\( \newcommand\BD{\mib D}\)
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\( \newcommand\BG{\mib G}\)
\( \newcommand\BH{\mib H}\)
\( \newcommand\BI{\mib I}}\)
\( \newcommand\BJ{\mib J}\)
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\( \newcommand\Bups{\mib\upsilon}\)
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\( \newcommand\BDelta{\mib\Delta}\)
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\( \newcommand\Bxhi{\raise.35ex\hbox{$\Bchi$}}\)
\( \newcommand\RGamma{ \Gamma}\)
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\( \newcommand\RN{ N}\)
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\( \newcommand\Ra{ a}\)
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\( \newcommand\Rf{ f}\)
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\( \newcommand { m}\)
\( \newcommand\Rn{ n}\)
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\( \newcommand\BBA{\boldsymbol\RA}\)
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\( \newcommand\Tra{\mathop{\textsf{Tr}}\,}\)
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\( \def\tket#1{|  #1 \rangle}\)
\( \def\tbra#1{\langle #1|}\)
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\( \def\texpect#1#2#3{\langle #1 |   #2  |  #3 \rangle}\)
\( \def\sket#1{|  \, #1 \,  \rangle}\)
\( \def\sbra#1{\langle \,  #1 \, |}\)
\( \def\sbraket#1#2{\langle \, #1  \, |  \, #2 \,  \rangle}\)
\( \def\sexpect#1#2#3{\langle \, #1 \, | \,  #2  \, | \, #3 \, \rangle}\)
\(\def\ket#1{\big| \, #1\, \big\rangle}\)
\( \def\bra#1{\big\langle \, #1 \, \big|}\)
\( \def\braket#1#2{\big\langle \, #1\, \big| \,#2 \,\big\rangle}\)
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\( \newcommand\pz{\partial}\)
\( \newcommand\pzb{\bar{\partial}}\)
\( \newcommand\svph{\vphantom{\int}}\)
\( \newcommand\vph{\vphantom{\sum_i}}\)
\( \newcommand\bvph{\vphantom{\sum_N^N}}\)
\( \newcommand\nd{^{\vphantom{\dagger}}}\)
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\( \newcommand\gtau{g_\tau}\)
\( \newcommand\Htil{\tilde H}\)
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\( \newcommand\oc{\omega_\Rc}\)

\(\newcommand \gtwid{\approx}\)

\( \newcommand\index{\textsf{ind}}\)
\( \newcommand\csch{\,{ csch\,}}\)
\( \newcommand\ctnh{\,{ ctnh\,}}\)
\( \newcommand\ctn{\,{ ctn\,}}\)
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\( \def\tmapright#1{\xrightarrow \limits^{#1}}\)
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\( \newcommand\hfb{\hfill\break}\)
\( \newcommand\Rep{\textsf{Re}\,}\)
\( \newcommand\Imp{\textsf{Im}\,}\)
\( \newcommand\ncdot{\!\cdot\!}\)
\( \def\tmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits^{#1}}\ }\)
\( \def\bmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits_{#1}}\ }\)
\( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)

\( \def\pabc#1#2#3{\left({\pz #1\over\pz #2}\right)\ns_{\!\!#3}}\)
\( \def\spabc#1#2#3{\big({\pz #1\over\pz #2}\big)\ns_{\!#3}}\)
\( \def\qabc#1#2#3{\pz^2\! #1\over\pz #2\,\pz #3}\)
\( \def\rabc#1#2#3#4{(\pz #1,\pz #2)\over (\pz #3,\pz #4)}\)
\( \newcommand\subA{\ns_\ssr{A}}\)
\( \newcommand\subB{\ns_\ssr{B}}\)
\( \newcommand\subC{\ns_\ssr{C}}\)
\( \newcommand\subD{\ns_\ssr{D}}\)
\( \newcommand\subAB{\ns_\ssr{AB}}\)
\( \newcommand\subBC{\ns_\ssr{BC}}\)
\( \newcommand\subCD{\ns_\ssr{CD}}\)
\( \newcommand\subDA{\ns_\ssr{DA}}\)
\( \def\lmapright#1{\ \ \smash{\mathop{\hbox to 55pt{\rightarrowfill}}\limits^{#1}}\ \ }\)
\( \def\enth#1{\RDelta {\textsf H}^0_\Rf[{ #1}]}\)
\( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr  \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
\( \newcommand\longrightharpoonup{\relbar\joinrel\rightharpoonup}\)
\( \newcommand\longleftharpoondown{\leftharpoondown\joinrel\relbar}\)
\( \newcommand\cds{\,\bullet\,}\)
\( \newcommand\ccs{\,\circ\,}\)
\( \newcommand\nsub{_{\vphantom{\dagger}}}\)
\( \newcommand\rhohat{\hat\rho}\)
\( \newcommand\vrhhat{\hat\vrh}\)
\( \newcommand\impi{\int\limits_{-\infty}^\infty\!\!\!}\)
\( \newcommand\brangle{\big\rangle}\)
\( \newcommand\blangle{\big\langle}\)
\( \newcommand\vet{\tilde\ve}\)
\( \newcommand\zbar{\bar z}\)
\( \newcommand\ftil{\tilde f}\)
\( \newcommand\XBE{\RXi\ns_\ssr{BE}}\)
\( \newcommand\XFD{\RXi\ns_\ssr{FD}}\)
\( \newcommand\OBE{\Omega\ns_\ssr{BE}}\)
\( \newcommand\OFD{\Omega\ns_\ssr{FD}}\)
\( \newcommand\veF{\ve\ns_\RF}\)
\( \newcommand\kF{k\ns_\RF}\)
\( \newcommand\kFu{k\ns_{\RF\uar}}\)
\( \newcommand\SZ{\textsf Z}}\) \( \newcommand\kFd{k\ns_{\RF\dar}\)
\( \newcommand\muB{\mu\ns_\ssr{B}}\)
\( \newcommand\mutB{\tilde\mu}\ns_\ssr{B}\)
\( \newcommand\xoN{\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N}\)
\( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
\( \newcommand\xhihOZ

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\)
\( \newcommand\jhz{\HJ(0)}\)
\( \newcommand\nda{\nd_\alpha}\)
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Analytic Properties of the Partition Function

How can statistical mechanics describe phase transitions? This question was addressed in some beautiful mathematical analysis by Lee and Yang⁷. Consider the grand partition function \(\Xi\),

\[\Xi(T,V,z)=\sum_{N=0}^\infty z^N\,Q\ns_N(T,V)\,\lambda_T^{-dN}\ ,\]

where

\[Q\ns_N(T,V)={1\over N!}\int\!\!d^dx\ns_1\cdots\int\!\!d^dx\ns_N\>e^{-U(\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N)/\kT}\]

is the contribution to the \(N\)-particle partition function from the potential energy \(U\) (assuming no momentum-dependent potentials). For two-body central potentials, we have

\[U(\Bx\ns_1,\ldots,\Bx\ns_N)=\sum_{i<j}v\big(|\Bx\ns_i-\Bx\ns_j|\big).\]

Suppose further that these classical particles have hard cores. Then for any finite volume, there must be some maximum number \(N\ns_V\) such that \(Q\ns_N(T,V)\) vanishes for \(N>N\ns_V\). This is because if \(N>N\ns_V\) at least two spheres must overlap, in which case the potential energy is infinite. The theoretical maximum packing density for hard spheres is achieved for a hexagonal close packed (HCP) lattice⁸, for which \(f\nd_{\ssr{HCP}}=\frac{\pi}{3\sqrt{2}}=0.74048\). If the spheres have radius \(r\ns_0\), then \(N\ns_V=V/4\sqrt{2}r_0^3\) is the maximum particle number.

Thus, if \(V\) itself is finite, then \(\Xi(T,V,z)\) is a finite degree polynomial in \(z\), and may be factorized as

\[\Xi(T,V,z)=\sum_{N=0}^{N\ns_V} z^N\,Q\ns_N(T,V)\,\lambda_T^{-dN} = \prod_{k=1}^{N\ns_V} \bigg(1-{z\over z\ns_k}\bigg)\ ,\]

where \(z\ns_k(T,V)\) is one of the \(N\ns_V\) zeros of the grand partition function. Note that the \(\CO(z^0)\) term is fixed to be unity. Note also that since the configuration integrals \(Q\ns_N(T,V)\) are all positive, \(\Xi(z)\) is an increasing function along the positive real \(z\) axis. In addition, since the coefficients of \(z^N\) in the polynomial \(\Xi(z)\) are all real, then \(\Xi(z)=0\) implies \(\overline{\Xi(z)}= \Xi({\bar z})=0\), so the zeros of \(\Xi(z)\) are either real and negative or else come in complex conjugate pairs.

Figure \(\PageIndex{1}\): In the thermodynamic limit, the grand partition function can develop a singularity at positive real fugacity \(z\). The set of discrete zeros fuses into a branch cut.

For finite \(N\ns_V\), the situation is roughly as depicted in the left panel of Figure \(\PageIndex{1}\), with a set of \(N\ns_V\) zeros arranged in complex conjugate pairs (or negative real values). The zeros aren’t necessarily distributed along a circle as shown in the figure, though. They could be anywhere, so long as they are symmetrically distributed about the \({ Re}(z)\) axis, and no zeros occur for \(z\) real and nonnegative.

Lee and Yang proved the existence of the limits

\[\begin{split} {p\over \kT}&=\lim_{V\to\infty} {1\over V}\,\ln\Xi(T,V,z)\\ n&=\lim_{V\to\infty} z\,{\pz\over\pz z}\bigg[{1\over V}\,\ln\Xi(T,V,z)\bigg]\ , \end{split}\]

and notably the result

\[n=z\,{\pz\over\pz z}\bigg({p\over\kT}\bigg)\ ,\]

which amounts to the commutativity of the thermodynamic limit \(V\to\infty\) with the differential operator \(z\,{\pz\over\pz z}\). In particular, \(p(T,z)\) is a smooth function of \(z\) in regions free of roots. If the roots do coalesce and pinch the positive real axis, then then density \(n\) can be discontinuous, as in a first order phase transition, or a higher derivative \(\pz^j p/\pz n^j\) can be discontinuous or divergent, as in a second order phase transition.

Electrostatic Analogy

There is a beautiful analogy to the theory of two-dimensional electrostatics. We write

\[\begin{split} {p\over\kT}&={1\over V}\sum_{k=1}^{N\ns_V}\ln\!\bigg(1-{z\over z\ns_k}\bigg)\\ &=-\sum_{k=1}^{N\ns_V}\Big[\phi(z-z\ns_k)-\phi(0-z\ns_k)\Big]\ , \end{split}\]

where

\[\phi(z)=-{1\over V}\,\ln (z)\]

is the complex potential due to a line charge of linear density \(\lambda=V^{-1}\) located at origin. The number density is then

\[n=z\,{\pz\over\pz z }\bigg({p\over\kT}\bigg)=-z\,{\pz\over\pz z}\,\sum_{k=1}^{N\ns_V} \phi(z-z\ns_k)\ ,\]

to be evaluated for physical values of \(z\), \(z\in{\mathbb R}^+\). Since \(\phi(z)\) is analytic,

\[{\pz\phi\over\pz {\bar z}}={1\over 2}\,{\pz\phi\over\pz x} + {i\over 2}\,{\pz\phi\over\pz y}=0\ .\]

If we decompose the complex potential \(\phi=\phi\ns_1+i\phi\ns_2\) into real and imaginary parts, the condition of analyticity is recast as the Cauchy-Riemann equations,

\[{\pz\phi\ns_1\over\pz x}={\pz\phi\ns_2\over\pz y}\qquad,\qquad {\pz\phi\ns_1\over\pz y}=-{\pz\phi\ns_2\over\pz x}\ .\]

Thus,

\[\begin{split} -{\pz\phi\over\pz z}&=-{1\over 2}\,{\pz\phi\over\pz x}+{i\over 2}\,{\pz\phi\over\pz y}\\ &=-{1\over 2}\bigg({\pz\phi\ns_1\over\pz x} + {\pz\phi\ns_2\over\pz y}\bigg) + {i\over 2}\bigg({\pz\phi\ns_1\over\pz y} - {\pz\phi\ns_2\over\pz x}\bigg)\\ &=-{\pz\phi\ns_1\over\pz x}+i\,{\pz\phi\ns_1\over\pz y}=E\ns_x-i E\ns_y\ , \end{split}\]

where \(\BE=-\bnabla\phi\ns_1\) is the electric field. Suppose, then, that as \(V\to\infty\) a continuous charge distribution develops, which crosses the positive real \(z\) axis at a point \(x\in{\mathbb R}^+\). Then

\[{n\ns_+-n\ns_-\over x}=E\ns_x(x^+)-E\ns_x(x^-)=4\pi\sigma(x)\ ,\]

where \(\sigma\) is the linear charge density (assuming logarithmic two-dimensional potentials), or the two-dimensional charge density (if we extend the distribution along a third axis).

Example

As an example, consider the function

\[\begin{split} \Xi(z)&={(1+z)^M\,(1-z^M)\over 1-z} \\ &=(1+z)^M\,\big(1+z+z^2+\ldots+z^{M-1}\big)\ . \end{split}\]

The \((2M-1)\) degree polynomial has an \(M^{\ssr{th}}\) order zero at \(z=-1\) and \((M-1)\) simple zeros at \(z=e^{2\pi i k/M}\), where \(k\in\{1,\ldots,M\!-\!1\}\). Since \(M\) serves as the maximum particle number \(N\ns_V\), we may assume that \(V=M v\ns_0\), and the \(V\to\infty\) limit may be taken as \(M\to\infty\). We then have

\[\begin{split} {p\over\kT}&=\lim_{V\to\infty} {1\over V}\,\ln\Xi(z)\\ &={1\over v\ns_0}\,\lim_{M\to\infty}{1\over M}\ln\Xi(z)\\ &={1\over v\ns_0}\,\lim_{M\to\infty}{1\over M}\bigg[ M\ln(1+z) + \ln\!\big(1-z^M\big) - \ln(1-z)\bigg]\ . \end{split}\]

The limit depends on whether \(|z|>1\) or \(|z|<1\), and we obtain

\[{p\,v\ns_0\over\kT}=\begin{cases} \ln(1+z) & \hbox{ if $|z|<1$} \\ &\\ \Big[\ln(1+z)+\ln z\Big] & \hbox{ if $|z|>1$}\ . \end{cases}\]

Figure \(\PageIndex{2}\): Fugacity \(z\) and \(pv_0/k_{\ssr{B}}T\) versus dimensionless specific volume \(v/v_0\) for the example problem discussed in the text.

Thus,

\[n=z\,{\pz\over\pz z}\bigg({p\over\kT}\bigg)=\begin{cases} {1\over v\ns_0}\cdot{z\over 1+z} & \hbox{ if $|z|<1$} \\ &\\ {1\over v\ns_0}\cdot\Big[{z\over 1+z}+1\Big] & \hbox{ if $|z|>1$}\ . \end{cases}\]

If we solve for \(z(v)\), where \(v=n^{-1}\), we find

\[z=\begin{cases} {v\ns_0\over v-v\ns_0} & \hbox{ if $v>2v\ns_0$} \\ &\\ {v\ns_0-v\over 2v-v\ns_0} & \hbox{ if ${1\over 2}v\ns_0 < v < {2\over 3} v\ns_0$} \ . \end{cases}\]

We then obtain the equation of state,

\[{p\, v\ns_0\over\kT}=\begin{cases} \ln\!\Big({v\over v-v\ns_0}\Big) & \hbox{ if $v>2v\ns_0$} \\ &\\ \ln 2 & \hbox{ if ${2\over 3} v\ns_0 <v<2v\ns_0$} \\ &\\ \ln\!\Big({v(v\ns_0-v)\over (2v-v\ns_0)^2}\Big) & \hbox{ if ${1\over 2}v\ns_0 < v < {2\over 3} v\ns_0$}\ . \end{cases}\]