Andrews, Baxter, and Forrester have recently solved two infinite sequences of models that are generalizations of the Ising and hard-square models and exhibit multicriticality. The nature of these new two-dimensional multicritical points is examined in order to determine what universality classes they may represent. Appropriate order parameters are defined and their critical exponents reported. One infinite sequence of multicritical points are continuous melting transitions of $p\ifmmode\times\else\texttimes\fi{}1$ commensurate ordered phases for all integers $p\ensuremath{\ge}2$. The other sequence appears to consist of "generic" multicritical points terminating lines of $n$ -phase coexistence, again for all integers $n\ensuremath{\ge}2$. The multicritical exponents in this latter sequence coincide with those found by Friedan, Qiu, and Shenker on the basis of assuming conformal invariance and unitarity.Andrews, Baxter, and Forrester have recently solved two infinite sequences of models that are generalizations of the Ising and hard-square models and exhibit multicriticality. The nature of these new two-dimensional multicritical points is examined in order to determine what universality classes they may represent. Appropriate order parameters are defined and their critical exponents reported. One infinite sequence of multicritical points are continuous melting transitions of $p\ifmmode\times\else\texttimes\fi{}1$ commensurate ordered phases for all integers $p\ensuremath{\ge}2$. The other sequence appears to consist of "generic" multicritical points terminating lines of $n$-phase coexistence, again for all integers $n\ensuremath{\ge}2$. The multicritical exponents in this latter sequence coincide with those found by Friedan, Qiu, and Shenker on the basis of assuming conformal invariance and unitarity.

Physical review. B, Condensed matterOct 1, 1984252 citations

Review on multicritical phase transitions

Ammon Aharony

Annals of the Israel Physical SocietyJan 1, 19780 citations

Coexistence of spontaneous dimerization and magnetic order in a transverse-field Ising ladder with four-spin interactions

J. C. Xavier, Rodrigo G. Pereira, Maria Eugênia Silva Nunes +1 more

The spin-1/2 transverse field two-leg Ising ladder with nearest-neighbor exchange and plaquette four-spin interaction ${J}_{4}$ is studied analytically and numerically with the density matrix renormalization group approach. The quantum phase diagram in the transverse field $B$ versus ${J}_{4}$ plane has been obtained. There are three different phases: a paramagnetic (PM) phase for high values of the transverse field and, for low values of $B$ , a ferromagnetic (FM) ordered phase for small ${J}_{4}$ , and a dimerized-rung (DR) phase for large negative values of ${J}_{4}$ . All phases are separated by quantum phase transition lines meeting at a multicritical point. The critical lines have been obtained by exploring the entanglement entropy. The results show that along the critical lines the central charge is $c=1/2$ , while at the multicritical point one has $c=1$ . The scaling dimension of the energy operator is ${X}_{\ensuremath{\epsilon}}=1$, in agreement with the universality class of the critical behavior of the quantum Ising chain. An effective field theory for the multicritical point is also discussed. The FM and the DR order parameters have also been computed and we found a region where the FM and the DR phases coexist.The spin-1/2 transverse field two-leg Ising ladder with nearest-neighbor exchange and plaquette four-spin interaction ${J}_{4}$ is studied analytically and numerically with the density matrix renormalization group approach. The quantum phase diagram in the transverse field $B$ versus ${J}_{4}$ plane has been obtained. There are three different phases: a paramagnetic (PM) phase for high values of the transverse field and, for low values of $B$, a ferromagnetic (FM) ordered phase for small ${J}_{4}$, and a dimerized-rung (DR) phase for large negative values of ${J}_{4}$. All phases are separated by quantum phase transition lines meeting at a multicritical point. The critical lines have been obtained by exploring the entanglement entropy. The results show that along the critical lines the central charge is $c=1/2$, while at the multicritical point one has $c=1$. The scaling dimension of the energy operator is ${X}_{\ensuremath{\epsilon}}=1$, in agreement with the universality class of the critical behavior of the quantum Ising chain. An effective field theory for the multicritical point is also discussed. The FM and the DR order parameters have also been computed and we found a region where the FM and the DR phases coexist.

arXiv:2111.11404Physical review. B./Physical review. BJan 31, 20226 citations

The second order phase transitions and critical phenomena

WORLD SCIENTIFIC eBooksOct 3, 20182 citations

Competing order parameters and a tricritical point with a difference

Arghya Dutta, Jayanta K. Bhattacharjee

arXiv:1011.6521Physica B Condensed MatterJun 5, 20121 citation

Phase diagrams and multicritical points in randomly mixed magnets. III. Competing spin-glass and magnetic ordering

Shmuel Fishman, Amnon Aharony A

The phase diagrams and critical properties of a quenched random alloy of a ferromagnet and an antiferromagnet (or of two antiferromagnets with different periodicities) are studied in the mean-field approximation and by renormalization-group techniques in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, using the $n\ensuremath{\rightarrow}0$ replica method. If only nearest-neighbor interactions are assumed, then one finds ferromagnetic ( $F$ ), antiferromagnetic (AF), and spin-glass (SG) phases. As the average strength of the next-nearest-neighbor interactions is increased (e.g., by an additional component in the alloy), the $F$ and the AF phases approach each other, and one may find regions in parameter space where a mixed $F$ -AF phase exists, as discussed in Paper II of this series. The phase diagram exhibits lines of multicritical points where the paramagnetic ( $P$ ), $F$ (or AF) and SG phases coexist ( $P$ -SG- $F$ or $P$ -SG-AF), or where the $P$ , $F$ , and AF phases coexist ($P\ensuremath{-}F$-AF). These lines meet at a new multicritical point, where all four phases coexist ( $P$ -SG- $F$ -AF). Recent renormalization-group analysis of $P$ -SG- $F$ , by Chen and Lubensky, yields complex exponents for the $\mathrm{XY}$ and Heisenberg cases. We show that the phase diagram predicted by that analysis is difficult to understand even for the Ising case. The same method of analysis yields similar difficulties for $P$ -SG- $F$ -AF. A modified way to take the limit $n\ensuremath{\rightarrow}0$, which resolves these difficulties, is presented. This modified way still lacks rigorous justification.The phase diagrams and critical properties of a quenched random alloy of a ferromagnet and an antiferromagnet (or of two antiferromagnets with different periodicities) are studied in the mean-field approximation and by renormalization-group techniques in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, using the $n\ensuremath{\rightarrow}0$ replica method. If only nearest-neighbor interactions are assumed, then one finds ferromagnetic ($F$), antiferromagnetic (AF), and spin-glass (SG) phases. As the average strength of the next-nearest-neighbor interactions is increased (e.g., by an additional component in the alloy), the $F$ and the AF phases approach each other, and one may find regions in parameter space where a mixed $F$-AF phase exists, as discussed in Paper II of this series. The phase diagram exhibits lines of multicritical points where the paramagnetic ($P$), $F$ (or AF) and SG phases coexist ($P$-SG-$F$ or $P$-SG-AF), or where the $P$, $F$, and AF phases coexist ($P\ensuremath{-}F$-AF). These lines meet at a new multicritical point, where all four phases coexist ($P$-SG-$F$-AF). Recent renormalization-group analysis of $P$-SG-$F$, by Chen and Lubensky, yields complex exponents for the $\mathrm{XY}$ and Heisenberg cases. We show that the phase diagram predicted by that analysis is difficult to understand even for the Ising case. The same method of analysis yields similar difficulties for $P$-SG-$F$-AF. A modified way to take the limit $n\ensuremath{\rightarrow}0$, which resolves these difficulties, is presented. This modified way still lacks rigorous justification.

Physical review. B, Condensed matterJan 1, 198056 citations

n

-phase coexistence, again for all integers $n\ensuremath{\ge}2$. The multicritical exponents in this latter sequence coincide with those found by Friedan, Qiu, and Shenker on the basis of assuming conformal invariance and unitarity.

The spin-1/2 transverse field two-leg Ising ladder with nearest-neighbor exchange and plaquette four-spin interaction

{J}_{4}

is studied analytically and numerically with the density matrix renormalization group approach. The quantum phase diagram in the transverse field

B

versus

{J}_{4}

plane has been obtained. There are three different phases: a paramagnetic (PM) phase for high values of the transverse field and, for low values of

B

, a ferromagnetic (FM) ordered phase for small

{J}_{4}

, and a dimerized-rung (DR) phase for large negative values of

{J}_{4}

. All phases are separated by quantum phase transition lines meeting at a multicritical point. The critical lines have been obtained by exploring the entanglement entropy. The results show that along the critical lines the central charge is

c=1/2

, while at the multicritical point one has

c=1

. The scaling dimension of the energy operator is ${X}_{\ensuremath{\epsilon}}=1$, in agreement with the universality class of the critical behavior of the quantum Ising chain. An effective field theory for the multicritical point is also discussed. The FM and the DR order parameters have also been computed and we found a region where the FM and the DR phases coexist.

F

), antiferromagnetic (AF), and spin-glass (SG) phases. As the average strength of the next-nearest-neighbor interactions is increased (e.g., by an additional component in the alloy), the

F

and the AF phases approach each other, and one may find regions in parameter space where a mixed

F

-AF phase exists, as discussed in Paper II of this series. The phase diagram exhibits lines of multicritical points where the paramagnetic (

P

F

(or AF) and SG phases coexist (

P

-SG-

F

P

-SG-AF), or where the

P

F

, and AF phases coexist ($P\ensuremath{-}F$-AF). These lines meet at a new multicritical point, where all four phases coexist (

P

-SG-

F

-AF). Recent renormalization-group analysis of

P

-SG-

F

, by Chen and Lubensky, yields complex exponents for the

\mathrm{XY}

and Heisenberg cases. We show that the phase diagram predicted by that analysis is difficult to understand even for the Ising case. The same method of analysis yields similar difficulties for

P

-SG-

F

-AF. A modified way to take the limit $n\ensuremath{\rightarrow}0$, which resolves these difficulties, is presented. This modified way still lacks rigorous justification.