6.7: Appendix I- Potts Model in One Dimension
( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand\Dalpha
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[1], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Dbeta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[2], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Dgamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[3], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Ddelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[4], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Depsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[5], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Dvarepsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[6], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Dzeta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[7], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Deta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[8], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Dtheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[9], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Dvartheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[10], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Diota
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[11], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Dkappa
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[12], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Dlambda
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[13], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Dvarpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[14], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\DGamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[15], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\DDelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\DTheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vmu
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[18], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vnu
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vxi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vom
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vrho
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarrho
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vsigma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarsigma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vtau
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vupsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[29], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vphi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[30], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vvarphi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[31], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vchi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[32], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vpsi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[33], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\Vomega
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[34], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\VGamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[35], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\VDelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[36], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\newcommand\BI{\mib I}}
\)
\newcommand { M}
\newcommand { m}
}
\( \newcommand\tcb{\textcolor{blue}\)
\( \newcommand\tcr{\textcolor{red}\)
1$#1_$
\newcommand\SZ{\textsf Z}} \( \newcommand\kFd{k\ns_{\RF\dar}\)
\newcommand\mutB{\tilde\mu}\ns_\ssr{B}
\( \newcommand\xhihOZ
Callstack: at (Template:MathJaxArovas), /content/body/div/span[1], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
\( \newcommand\labar
Callstack: at (Template:MathJaxArovas), /content/body/div/span[2], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.07:_Appendix_I-_Potts_Model_in_One_Dimension), /content/body/p/span, line 1, column 23
Definition
The Potts model is defined by the Hamiltonian
H=−J∑⟨ij⟩δ†σ∗i,σ∗j−h∑iδ†σ∗i,1 .
Here, the spin variables σ∗i take values in the set {1,2,…,q} on each site. The equivalent of an external magnetic field in the Ising case is a field h which prefers a particular value of σ (σ=1 in the above Hamiltonian). Once again, it is not possible to compute the partition function on general lattices, however in one dimension we may once again find Z using the transfer matrix method.
Transfer matrix
On a ring of N sites, we have
Z=Tre−βH=∑{σ∗n}eβhδσ∗1,1eβJδσ∗1,σ∗2⋯eβhδσ∗N,1eβJδσ∗N,σ∗1=Tr(RN) ,
where the q×q transfer matrix R is given by
R†σσ′=eβJδσσ′e12βhδσ,1e12βhδσ′,1={eβ(J+h) if σ=σ′=1eβJ if σ=σ′≠1eβh/2 if σ=1 and σ′≠1eβh/2 if σ≠1 and σ′=11 if σ≠1 and σ′≠1 and σ≠σ′ .
In matrix form,
R=(eβ(J+h)eβh/2eβh/2⋯eβh/2eβh/2eβJ1⋯1eβh/21eβJ⋯1⋮⋮⋮⋱⋮eβh/211⋯eβJ1eβh/211⋯1eβJ)
The matrix R has q eigenvalues λ∗j, with j=1,…,q. The partition function for the Potts chain is then
Z=q∑j=1λNj .
We can actually find the eigenvalues of R analytically. To this end, consider the vectors
ϕ=(10⋮0),ψ=(q−1+eβh)−1/2(eβh/21⋮1) .
Then R may be written as
R=(eβJ−1)I+(q−1+eβh)|ψ⟩⟨ψ|+(eβJ−1)(eβh−1)|ϕ⟩⟨ϕ| ,
where I is the q×q identity matrix. When h=0, we have a simpler form,
R=(eβJ−1)I+q|ψ⟩⟨ψ| .
From this we can read off the eigenvalues:
λ∗1=eβJ+q−1λ∗j=eβJ−1,j∈{2,…,q} ,
since |ψ⟩ is an eigenvector with eigenvalue λ=eβJ+q−1, and any vector orthogonal to |ψ⟩ has eigenvalue λ=eβJ−1. The partition function is then
Z=(eβJ+q−1)N+(q−1)(eβJ−1)N .
In the thermodynamic limit N→∞, only the λ†1 eigenvalue contributes, and we have
F(T,N,h=0)=−NkBTln(eJ/kBT+q−1) for N→∞ .
When h is nonzero, the calculation becomes somewhat more tedious, but still relatively easy. The problem is that |ψ⟩ and |ϕ⟩ are not orthogonal, so we define
|χ⟩=|ϕ⟩−|ψ⟩⟨ψ|ϕ⟩√1−⟨ϕ|ψ⟩2 ,
where
x≡⟨ϕ|ψ⟩=(eβhq−1+eβh)1/2 .
Now we have ⟨χ|ψ⟩=0, with ⟨χ|χ⟩=1 and ⟨ψ|ψ⟩=1, with
|ϕ⟩=√1−x2|χ⟩+x|ψ⟩ .
and the transfer matrix is then
R=(eβJ−1)I+(q−1+eβh)|ψ⟩⟨ψ|+(eβJ−1)(eβh−1)[(1−x2)|χ⟩⟨χ|+x2|ψ⟩⟨ψ|+x√1−x2(|χ⟩⟨ψ|+|ψ⟩⟨χ|)]=(eβJ−1)I+[(q−1+eβh)+(eβJ−1)(eβh−1)(eβhq−1+eβh)]|ψ⟩⟨ψ|+(eβJ−1)(eβh−1)(q−1q−1+eβh)|χ⟩⟨χ|+(eβJ−1)(eβh−1)((q−1)eβhq−1+eβh)1/2(|χ⟩⟨ψ|+|ψ⟩⟨χ|) ,
which in the two-dimensional subspace spanned by |χ⟩ and |ψ⟩ is of the form
R=(accb) .
Recall that for any 2×2 Hermitian matrix,
M=a†0I+a⋅τ=(a∗0+a∗3a∗1−ia∗2a∗1+ia∗2a∗0−a∗3) ,
the characteristic polynomial is
P(λ)=det(λI−M)=(λ−a∗0)2−a21−a22−a23 ,
and hence the eigenvalues are
놱=a∗0±√a21+a22+a23 .
For the transfer matrix of Equation ???, we obtain, after a little work,
λ†1,2=eβJ−1+12[q−1+eβh+(eβJ−1)(eβh−1)]±12√[q−1+eβh+(eβJ−1)(eβh−1)]2−4(q−1)(eβJ−1)(eβh−1) .
There are q−2 other eigenvalues, however, associated with the (q−2)-dimensional subspace orthogonal to |χ⟩ and |ψ⟩. Clearly all these eigenvalues are given by
λ∗j=eβJ−1,j∈{3,…,q} .
The partition function is then
Z=λN1+λN2+(q−2)λN3 ,
and in the thermodynamic limit N→∞ the maximum eigenvalue λ∗1 dominates. Note that we recover the correct limit as h→0.