Physical review. B, Condensed matter

Phase structure and dual properties of ZN -symmetric models on a triangular lattice

Robert S. Savit

Published Oct 1, 1980• 4 citations

Abstract

Motivated by the search for physical two-dimensional systems with the unusual phase structure of the vector Potts (clock) model, we study two ${Z}_{N}$-symmetric periodic Gaussian models (PGM) on a triangular lattice. The models differ from each other only in their vortex structure, and should be in the same universality class, along with the $N$-state clock model. One of the models is self-dual. We also argue that the critical inverse temperature of the Gaussian approximation to the $\mathrm{XY}$ model on a square lattice, ${\ensuremath{\beta}}_{\ensuremath{\square}}^{*}$, should be related to ${\ensuremath{\beta}}_{\ensuremath{\Delta}}^{*}$, the critical inverse temperature of a similar model on a triangular lattice, by ${\ensuremath{\beta}}_{\ensuremath{\Delta}}^{*}\sqrt{3}={\ensuremath{\beta}}_{\ensuremath{\square}}^{*}$. Using this result and the self-dual property of one of our ${Z}_{N}$ PGM's, we find that the equation determining ${N}_{c}$, the value of $N$ above which there must be at least two phase transitions, is identical for the self-dual ${Z}_{N}$ PGM's on a square and triangular lattice.

DOI: 10.1103/physrevb.22.3443

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Physical review. B, Condensed matter

Phase structure and dual properties of ZN -symmetric models on a triangular lattice

Robert S. Savit

Published Oct 1, 1980• 4 citations

Publisher

Abstract

Motivated by the search for physical two-dimensional systems with the unusual phase structure of the vector Potts (clock) model, we study two

{Z}_{N}

-symmetric periodic Gaussian models (PGM) on a triangular lattice. The models differ from each other only in their vortex structure, and should be in the same universality class, along with the

N

-state clock model. One of the models is self-dual. We also argue that the critical inverse temperature of the Gaussian approximation to the

\mathrm{XY}

model on a square lattice,

{\ensuremath{\beta}}_{\ensuremath{\square}}^{*}

, should be related to

{\ensuremath{\beta}}_{\ensuremath{\Delta}}^{*}

, the critical inverse temperature of a similar model on a triangular lattice, by

{\ensuremath{\beta}}_{\ensuremath{\Delta}}^{*}\sqrt{3}={\ensuremath{\beta}}_{\ensuremath{\square}}^{*}

. Using this result and the self-dual property of one of our

{Z}_{N}

PGM's, we find that the equation determining

{N}_{c}

, the value of

N

above which there must be at least two phase transitions, is identical for the self-dual

{Z}_{N}

PGM's on a square and triangular lattice.

DOI: 10.1103/physrevb.22.3443

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Hao Chen, Pengcheng Hou, Sheng Fang +1 more

The two-dimensional $q$ -state clock model for $q\ensuremath{\ge}5$ undergoes two Berezinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square-lattice clock model for $q=5$ --9 in a pair of flow representations, from high- and low-temperature expansions, respectively. By finite-size scaling analysis of susceptibilitylike quantities, we determine the critical points with a precision improving over the existing results. Due to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are ${\ensuremath{\eta}}_{1}=1/4$ and ${\ensuremath{\eta}}_{2}=4/{q}^{2}$ at the two BKT transitions. Further, the approximate self-dual points ${\ensuremath{\beta}}_{\mathrm{sd}}(L)$, defined by the stringent condition that the susceptibilitylike quantities in both flow representations are identical, are found to be nearly independent of system size $L$ and behave as ${\ensuremath{\beta}}_{\mathrm{sd}}\ensuremath{\simeq}q/2\ensuremath{\pi}$ asymptotically at the large- $q$ limit. The exponent $\ensuremath{\eta}$ at ${\ensuremath{\beta}}_{\mathrm{sd}}$ is consistent with $1/q$ within statistical error as long as $q\ensuremath{\ge}5$. Based on this, we further conjecture that $\ensuremath{\eta}({\ensuremath{\beta}}_{\mathrm{sd}})=1/q$ holds exactly and is universal for systems in the $q$ -state clock universality class. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.The two-dimensional $q$-state clock model for $q\ensuremath{\ge}5$ undergoes two Berezinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square-lattice clock model for $q=5$--9 in a pair of flow representations, from high- and low-temperature expansions, respectively. By finite-size scaling analysis of susceptibilitylike quantities, we determine the critical points with a precision improving over the existing results. Due to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are ${\ensuremath{\eta}}_{1}=1/4$ and ${\ensuremath{\eta}}_{2}=4/{q}^{2}$ at the two BKT transitions. Further, the approximate self-dual points ${\ensuremath{\beta}}_{\mathrm{sd}}(L)$, defined by the stringent condition that the susceptibilitylike quantities in both flow representations are identical, are found to be nearly independent of system size $L$ and behave as ${\ensuremath{\beta}}_{\mathrm{sd}}\ensuremath{\simeq}q/2\ensuremath{\pi}$ asymptotically at the large-$q$ limit. The exponent $\ensuremath{\eta}$ at ${\ensuremath{\beta}}_{\mathrm{sd}}$ is consistent with $1/q$ within statistical error as long as $q\ensuremath{\ge}5$. Based on this, we further conjecture that $\ensuremath{\eta}({\ensuremath{\beta}}_{\mathrm{sd}})=1/q$ holds exactly and is universal for systems in the $q$-state clock universality class. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.

arXiv:2205.02642Physical review. EAug 5, 202222 citations

Motivated by the search for physical two-dimensional systems with the unusual phase structure of the vector Potts (clock) model, we study two

{Z}_{N}

-symmetric periodic Gaussian models (PGM) on a triangular lattice. The models differ from each other only in their vortex structure, and should be in the same universality class, along with the

N

-state clock model. One of the models is self-dual. We also argue that the critical inverse temperature of the Gaussian approximation to the

\mathrm{XY}

model on a square lattice,

{\ensuremath{\beta}}_{\ensuremath{\square}}^{*}

, should be related to

{\ensuremath{\beta}}_{\ensuremath{\Delta}}^{*}

, the critical inverse temperature of a similar model on a triangular lattice, by

{\ensuremath{\beta}}_{\ensuremath{\Delta}}^{*}\sqrt{3}={\ensuremath{\beta}}_{\ensuremath{\square}}^{*}

. Using this result and the self-dual property of one of our

{Z}_{N}

PGM's, we find that the equation determining

{N}_{c}

, the value of

N

above which there must be at least two phase transitions, is identical for the self-dual

{Z}_{N}

PGM's on a square and triangular lattice.

We investigate the critical behavior and the duality property of the ferromagnetic

q

-state clock model on the square lattice based on the tensor-network formalism. From the entanglement spectra of local tensors defined in the original and dual lattices, we obtain the exact self-dual points for the model with $q \leq 5 $ and approximate self-dual points for

q \geq 6

. We calculate accurately the lower and upper critical temperatures for the six-state clock model from the fixed-point tensors determined using the higher-order tensor renormalization group method and compare with other numerical results.

Quantum phase transitions are studied in the nonchiral

p

-clock chain, and a new explicitly U(1)-symmetric clock model, by monitoring the ground-state fidelity susceptibility. For $p\ensuremath{\ge}5$, the self-dual

{\mathbb{Z}}_{p}

-symmetric chain displays a double-hump structure in the fidelity susceptibility with both peak positions and heights scaling logarithmically to their corresponding thermodynamic values. This scaling is precisely as expected for two Beresinskii-Kosterlitz-Thouless (BKT) transitions located symmetrically about the self-dual point, and so confirms numerically the theoretical scenario that sets

p=5

as the lowest

p

supporting BKT transitions in

{\mathbb{Z}}_{p}

-symmetric clock models. For our U(1)-symmetric, non-self-dual minimal modification of the

p

-clock model we find that the phase diagram depends strongly on the parity of

p

and only one BKT transition survives for $p\ensuremath{\ge}5$. Using asymptotic calculus we map the self-dual clock model exactly, in the large

p

limit, to the quantum

O(2)

rotor chain. Finally, using bond-algebraic dualities we estimate the critical BKT transition temperatures of the classical planar

p

-clock models defined on square lattices, in the limit of extreme spatial anisotropy. Our values agree remarkably well with those determined via classical Monte Carlo for isotropic lattices. This work highlights the power of the fidelity susceptibility as a tool for diagnosing the BKT transitions even when only discrete symmetries are present.

{k}_{B}

{T}_{1}

/J=0.68\ifmmode\pm\else\textpm\fi{}0.02 and

{k}_{B}

{T}_{2}

/J=0.92\ifmmode\pm\else\textpm\fi{}0.01 and are of the Kosterlitz-Thouless type. The susceptibility diverges in the intermediate phase and the exponent \ensuremath{\eta} varies between 0.100 at

{T}_{1}

and 0.275 at

{T}_{2}

The two-dimensional

q

-state clock model for $q\ensuremath{\ge}5$ undergoes two Berezinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square-lattice clock model for

q=5

--9 in a pair of flow representations, from high- and low-temperature expansions, respectively. By finite-size scaling analysis of susceptibilitylike quantities, we determine the critical points with a precision improving over the existing results. Due to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are ${\ensuremath{\eta}}_{1}=1/4$ and ${\ensuremath{\eta}}_{2}=4/{q}^{2}$ at the two BKT transitions. Further, the approximate self-dual points ${\ensuremath{\beta}}_{\mathrm{sd}}(L)$, defined by the stringent condition that the susceptibilitylike quantities in both flow representations are identical, are found to be nearly independent of system size

L

and behave as ${\ensuremath{\beta}}_{\mathrm{sd}}\ensuremath{\simeq}q/2\ensuremath{\pi}$ asymptotically at the large-

q

limit. The exponent $\ensuremath{\eta}$ at ${\ensuremath{\beta}}_{\mathrm{sd}}$ is consistent with

1/q

within statistical error as long as $q\ensuremath{\ge}5$. Based on this, we further conjecture that $\ensuremath{\eta}({\ensuremath{\beta}}_{\mathrm{sd}})=1/q$ holds exactly and is universal for systems in the

q

-state clock universality class. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.