# Density of states, Potts zeros, and Fisher zeros of the Q

## Abstract

The Q-state Potts model can be extended to noninteger and even complex Q by expressing the partition function in the Fortuin-Kasteleyn (F-K) representation. In the F-K representation the partition function Z(Q,a) is a polynomial in Q and v=a{minus}1 (a=e{sup {beta}J}) and the coefficients of this polynomial, {Phi}(b,c), are the number of graphs on the lattice consisting of b bonds and c connected clusters. We introduce the random-cluster transfer matrix to compute {Phi}(b,c) exactly on finite square lattices with several types of boundary conditions. Given the F-K representation of the partition function we begin by studying the critical Potts model Z{sub CP}=Z(Q,a{sub c}(Q)), where a{sub c}(Q)=1+{radical}Q. We find a set of zeros in the complex w={radical}Q plane that map to (or close to) the Beraha numbers for real positive Q. We also identify {tilde Q}{sub c}(L), the value of Q for a lattice of width L above which the locus of zeros in the complex p=v/{radical}Q plane lies on the unit circle. By finite-size scaling we find that 1/{tilde Q}{sub c}(L){r_arrow}0 as L{r_arrow}{infinity}. We then study zeros of the antiferromagnetic (AF) Potts model in the complex Q plane and determine Q{sub c}(a), the largest value of Q for a fixed valuemore »

- Authors:

- Publication Date:

- Sponsoring Org.:
- (US)

- OSTI Identifier:
- 40203283

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review E

- Additional Journal Information:
- Journal Volume: 63; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevE.63.066107; Othernumber: PLEEE8000063000006066107000001; 024105PRE; PBD: Jun 2001; Journal ID: ISSN 1063-651X

- Publisher:
- The American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUNDARY CONDITIONS; PARTITION FUNCTIONS; POLYNOMIALS; SINGULARITY

### Citation Formats

```
Kim, Seung-Yeon, and Creswick, Richard J.
```*Density of states, Potts zeros, and Fisher zeros of the Q*. United States: N. p., 2001.
Web. doi:10.1103/PhysRevE.63.066107.

```
Kim, Seung-Yeon, & Creswick, Richard J.
```*Density of states, Potts zeros, and Fisher zeros of the Q*. United States. https://doi.org/10.1103/PhysRevE.63.066107

```
Kim, Seung-Yeon, and Creswick, Richard J. Fri .
"Density of states, Potts zeros, and Fisher zeros of the Q". United States. https://doi.org/10.1103/PhysRevE.63.066107.
```

```
@article{osti_40203283,
```

title = {Density of states, Potts zeros, and Fisher zeros of the Q},

author = {Kim, Seung-Yeon and Creswick, Richard J},

abstractNote = {The Q-state Potts model can be extended to noninteger and even complex Q by expressing the partition function in the Fortuin-Kasteleyn (F-K) representation. In the F-K representation the partition function Z(Q,a) is a polynomial in Q and v=a{minus}1 (a=e{sup {beta}J}) and the coefficients of this polynomial, {Phi}(b,c), are the number of graphs on the lattice consisting of b bonds and c connected clusters. We introduce the random-cluster transfer matrix to compute {Phi}(b,c) exactly on finite square lattices with several types of boundary conditions. Given the F-K representation of the partition function we begin by studying the critical Potts model Z{sub CP}=Z(Q,a{sub c}(Q)), where a{sub c}(Q)=1+{radical}Q. We find a set of zeros in the complex w={radical}Q plane that map to (or close to) the Beraha numbers for real positive Q. We also identify {tilde Q}{sub c}(L), the value of Q for a lattice of width L above which the locus of zeros in the complex p=v/{radical}Q plane lies on the unit circle. By finite-size scaling we find that 1/{tilde Q}{sub c}(L){r_arrow}0 as L{r_arrow}{infinity}. We then study zeros of the antiferromagnetic (AF) Potts model in the complex Q plane and determine Q{sub c}(a), the largest value of Q for a fixed value of a below which there is AF order. We find excellent agreement with Baxter{close_quote}s conjecture Q{sub c}{sup AF}(a)=(1{minus}a)(a+3). We also investigate the locus of zeros of the ferromagnetic Potts model in the complex Q plane and confirm that Q{sub c}{sup FM}(a)=(a{minus}1){sup 2}. We show that the edge singularity in the complex Q plane approaches Q{sub c} as Q{sub c}(L){similar_to}Q{sub c}+AL{sup {minus}y{sub q}}, and determine the scaling exponent y{sub q} for several values of Q. Finally, by finite-size scaling of the Fisher zeros near the antiferromagnetic critical point we determine the thermal exponent y{sub t} as a function of Q in the range 2{le}Q{le}3. Using data for lattices of size 3{le}L{le}8 we find that y{sub t} is a smooth function of Q and is well fitted by y{sub t}=(1+Au+Bu{sup 2})/(C+Du) where u={minus}(2/{pi})cos{sup {minus}1}({radical}Q/2). For Q=3 we find y{sub t}{approx_equal}0.6; however if we include lattices up to L=12 we find y{sub t}{approx_equal}0.50(8) in rough agreement with a recent result of Ferreira and Sokal [J. Stat. Phys. >96, 461 (1999)].},

doi = {10.1103/PhysRevE.63.066107},

url = {https://www.osti.gov/biblio/40203283},
journal = {Physical Review E},

issn = {1063-651X},

number = 6,

volume = 63,

place = {United States},

year = {2001},

month = {6}

}