Physical review. B, Condensed matter

Commensurate-incommensurate transitions and the Lifshitz point in the quantum asymmetric clock model

Steven F. Howes

Published Feb 1, 1983• 55 citations

Abstract

The asymmetric three-state clock model is studied in the context of a one-dimensional quantum Hamiltonian. Series expansions are used to investigate the commensurate-incommensurate transition and the incommensurate-liquid Kosterlitz-Thouless transition. Evidence is presented for a Lifshitz point at which the critical indices $\ensuremath{\alpha}$, $\ensuremath{\beta}$, and $\ensuremath{\gamma}$ are Potts-type, while the mass-gap index appears to take its Ising value, $\ensuremath{\nu}=1$.

DOI: 10.1103/physrevb.27.1762

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Physical review. B, Condensed matter

Commensurate-incommensurate transitions and the Lifshitz point in the quantum asymmetric clock model

Steven F. Howes

Published Feb 1, 1983• 55 citations

Publisher

Abstract

\ensuremath{\alpha}

\ensuremath{\beta}

, and

\ensuremath{\gamma}

are Potts-type, while the mass-gap index appears to take its Ising value,

\ensuremath{\nu}=1

DOI: 10.1103/physrevb.27.1762

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Nearest-neighbor Ising model with a uniaxial incommensurate phase and a Lifshitz point

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The anisotropic triangular nearest-neighbor Ising model, with antiferromagnetic interactions, is studied. The phase diagram, as a function of temperature and field, has a complex structure. Two commensurate ordered phases are found. The transition to one is Ising-type. The second is reached from the disordered phase in one of two ways: either via an intermediate uniaxial incommensurate phase---which is reached at high temperatures by a Kosterlitz-Thouless transition and at low temperatures undergoes an incommensurate-commensurate transition---or by a single continuous transition. The latter transition is in the same universality class as the chiral three-state Potts model, with thermal exponent within 4% of that of the three-state Potts model. These two transition regimes are found to be separated by a Lifshitz point. The phase diagram and critical behavior were determined by analytical (symmetry analysis and free-fermion approximation) as well as numerical (phenomenological renormalization-group and Monte Carlo) methods.The anisotropic triangular nearest-neighbor Ising model, with antiferromagnetic interactions, is studied. The phase diagram, as a function of temperature and field, has a complex structure. Two commensurate ordered phases are found. The transition to one is Ising-type. The second is reached from the disordered phase in one of two ways: either via an intermediate uniaxial incommensurate phase---which is reached at high temperatures by a Kosterlitz-Thouless transition and at low temperatures undergoes an incommensurate-commensurate transition---or by a single continuous transition. The latter transition is in the same universality class as the chiral three-state Potts model, with thermal exponent within 4% of that of the three-state Potts model. These two transition regimes are found to be separated by a Lifshitz point. The phase diagram and critical behavior were determined by analytical (symmetry analysis and free-fermion approximation) as well as numerical (phenomenological renormalization-group and Monte Carlo) methods.

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Entanglement entropy and massless phase in the antiferromagnetic three-state quantum chiral clock model

Yan-Wei Dai, Sam Young Cho, Murray T. Batchelor +1 more

The von Neumann entanglement entropy is used to estimate the critical point ${h}_{c}/J\ensuremath{\simeq}0.143(3)$ of the mixed ferro-antiferromagnetic three-state quantum Potts model $H={\ensuremath{\sum}}_{i}[J({X}_{i}{X}_{i+1}^{\phantom{\rule{0.16em}{0ex}}2}+{X}_{i}^{\phantom{\rule{0.16em}{0ex}}2}{X}_{i+1})\ensuremath{-}h\phantom{\rule{0.16em}{0ex}}{R}_{i}]$, where ${X}_{i}$ and ${R}_{i}$ are standard three-state Potts spin operators and $J>0$ is the antiferromagnetic coupling parameter. This critical point value gives improved estimates for two Kosterlitz-Thouless transition points in the antiferromagnetic ($\ensuremath{\beta}<0$) region of the $\mathrm{\ensuremath{\Delta}}\ensuremath{-}\ensuremath{\beta}$ phase diagram of the three-state quantum chiral clock model, where $\mathrm{\ensuremath{\Delta}}$ and $\ensuremath{\beta}$ are, respectively, the chirality and coupling parameters in the clock model. These are the transition points ${\ensuremath{\beta}}_{c}\ensuremath{\simeq}\ensuremath{-}0.143(3)$ at $\mathrm{\ensuremath{\Delta}}=\frac{1}{2}$ between incommensurate and commensurate phases and ${\ensuremath{\beta}}_{c}\ensuremath{\simeq}\ensuremath{-}7.0(1)$ at $\mathrm{\ensuremath{\Delta}}=0$ between disordered and incommensurate phases. The von Neumann entropy is also used to calculate the central charge $c$ of the underlying conformal field theory in the massless phase $h\ensuremath{\le}{h}_{c}$. The estimate $c\ensuremath{\simeq}1$ in this phase is consistent with the known exact value at the particular point $h/J=\ensuremath{-}1$ corresponding to the purely antiferromagnetic three-state quantum Potts model. The algebraic decay of the Potts spin-spin correlation in the massless phase is used to estimate the continuously varying critical exponent $\ensuremath{\eta}$.The von Neumann entanglement entropy is used to estimate the critical point ${h}_{c}/J\ensuremath{\simeq}0.143(3)$ of the mixed ferro-antiferromagnetic three-state quantum Potts model $H={\ensuremath{\sum}}_{i}[J({X}_{i}{X}_{i+1}^{\phantom{\rule{0.16em}{0ex}}2}+{X}_{i}^{\phantom{\rule{0.16em}{0ex}}2}{X}_{i+1})\ensuremath{-}h\phantom{\rule{0.16em}{0ex}}{R}_{i}]$, where ${X}_{i}$ and ${R}_{i}$ are standard three-state Potts spin operators and $J>0$ is the antiferromagnetic coupling parameter. This critical point value gives improved estimates for two Kosterlitz-Thouless transition points in the antiferromagnetic ($\ensuremath{\beta}<0$) region of the $\mathrm{\ensuremath{\Delta}}\ensuremath{-}\ensuremath{\beta}$ phase diagram of the three-state quantum chiral clock model, where $\mathrm{\ensuremath{\Delta}}$ and $\ensuremath{\beta}$ are, respectively, the chirality and coupling parameters in the clock model. These are the transition points ${\ensuremath{\beta}}_{c}\ensuremath{\simeq}\ensuremath{-}0.143(3)$ at $\mathrm{\ensuremath{\Delta}}=\frac{1}{2}$ between incommensurate and commensurate phases and ${\ensuremath{\beta}}_{c}\ensuremath{\simeq}\ensuremath{-}7.0(1)$ at $\mathrm{\ensuremath{\Delta}}=0$ between disordered and incommensurate phases. The von Neumann entropy is also used to calculate the central charge $c$ of the underlying conformal field theory in the massless phase $h\ensuremath{\le}{h}_{c}$. The estimate $c\ensuremath{\simeq}1$ in this phase is consistent with the known exact value at the particular point $h/J=\ensuremath{-}1$ corresponding to the purely antiferromagnetic three-state quantum Potts model. The algebraic decay of the Potts spin-spin correlation in the massless phase is used to estimate the continuously varying critical exponent $\ensuremath{\eta}$.

arXiv:1608.04960Physical review. B./Physical review. BJan 18, 20179 citations

\ensuremath{\alpha}

\ensuremath{\beta}

, and

\ensuremath{\gamma}

are Potts-type, while the mass-gap index appears to take its Ising value,

\ensuremath{\nu}=1

p

. In particular, for

p=3

(the "chiral Potts" case), we conjecture that the phase diagram contains no Lifshitz point, in contrast to previous authors; for

p=1

, dislocations remove the original

\mathrm{CI}

transition completely.

{X}_{i}

and

{R}_{i}

are standard three-state Potts spin operators and

J>0

is the antiferromagnetic coupling parameter. This critical point value gives improved estimates for two Kosterlitz-Thouless transition points in the antiferromagnetic ($\ensuremath{\beta}<0$) region of the $\mathrm{\ensuremath{\Delta}}\ensuremath{-}\ensuremath{\beta}$ phase diagram of the three-state quantum chiral clock model, where $\mathrm{\ensuremath{\Delta}}$ and $\ensuremath{\beta}$ are, respectively, the chirality and coupling parameters in the clock model. These are the transition points ${\ensuremath{\beta}}_{c}\ensuremath{\simeq}\ensuremath{-}0.143(3)$ at $\mathrm{\ensuremath{\Delta}}=\frac{1}{2}$ between incommensurate and commensurate phases and ${\ensuremath{\beta}}_{c}\ensuremath{\simeq}\ensuremath{-}7.0(1)$ at $\mathrm{\ensuremath{\Delta}}=0$ between disordered and incommensurate phases. The von Neumann entropy is also used to calculate the central charge

c

of the underlying conformal field theory in the massless phase $h\ensuremath{\le}{h}_{c}$. The estimate $c\ensuremath{\simeq}1$ in this phase is consistent with the known exact value at the particular point $h/J=\ensuremath{-}1$ corresponding to the purely antiferromagnetic three-state quantum Potts model. The algebraic decay of the Potts spin-spin correlation in the massless phase is used to estimate the continuously varying critical exponent $\ensuremath{\eta}$.