Transverse field Ising model on the checkerboard lattice: a plaquette-operator approach

M. Sadrzadeh, A. Langari

Published May 6, 2014• 0 citations

Abstract

We study the effect of quantum fluctuations by means of a transverse magnetic field ($\Gamma$) on the antiferromagnetic $J_1-J_2$ Ising model on the checkerboard lattice, the two dimensional version of pyrochlore lattice. A plaquette operator formalism has been implemented to bosonise the model, in which a single boson is associated to each eigenstate of a plaquette and the inter-plaquette interactions define an effective Hamiltonian. The excitations of a plaquette would represent an-harmonic fluctuations of the model, which lead not only to lower the excitation energy compared with a single-spin flip but also to lift the extensive degeneracy in favor of a plaquette ordered state, which breaks lattice rotational symmetry for $J_2>J_1$ (the collinear phase). The bosonic excitation gap vanishes at the critical boundary to N\'{e}el ($J_2 J_1$) phases, respectively. At the homogeneous coupling ($J_1=J_2$) a (canted) resonating plaquette state (RPS) is established from an-harmonic fluctuations, which lasts for low fields, $\Gamma/J_1\lesssim 0.4$, followed by a crossover to the fully polarized phase. We will show that our results predict an order-by-disorder as a results of transverse field at the critical point ($J_2=0.5 J_1$) of the antiferromagnetic $J_1-J_2$ Ising square lattice.

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Quantum phase diagram of the two-dimensional transverse-field Ising model: Unconstrained tree tensor network and mapping analysis

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Transverse field Ising model on the checkerboard lattice: a plaquette-operator approach

M. Sadrzadeh, A. Langari

Published May 6, 2014• 0 citations

OpenAlex

Abstract

We study the effect of quantum fluctuations by means of a transverse magnetic field (

\Gamma

) on the antiferromagnetic

J_1-J_2

Ising model on the checkerboard lattice, the two dimensional version of pyrochlore lattice. A plaquette operator formalism has been implemented to bosonise the model, in which a single boson is associated to each eigenstate of a plaquette and the inter-plaquette interactions define an effective Hamiltonian. The excitations of a plaquette would represent an-harmonic fluctuations of the model, which lead not only to lower the excitation energy compared with a single-spin flip but also to lift the extensive degeneracy in favor of a plaquette ordered state, which breaks lattice rotational symmetry for

J_2>J_1

(the collinear phase). The bosonic excitation gap vanishes at the critical boundary to N'{e}el (

J_2 J_1

) phases, respectively. At the homogeneous coupling (

J_1=J_2

) a (canted) resonating plaquette state (RPS) is established from an-harmonic fluctuations, which lasts for low fields,

\Gamma/J_1\lesssim 0.4

, followed by a crossover to the fully polarized phase. We will show that our results predict an order-by-disorder as a results of transverse field at the critical point (

J_2=0.5 J_1

) of the antiferromagnetic

J_1-J_2

Ising square lattice.

Related papers

Quantum phase diagram of the two-dimensional transverse-field Ising model: Unconstrained tree tensor network and mapping analysis

M. Sadrzadeh, Reza Haghshenas, A. Langari

We investigate the ground-state phase diagram of the frustrated transverse-field Ising (TFI) model on the checkerboard lattice (CL), which consists of N\'eel, collinear, quantum paramagnet, and plaquette-valence bond solid (-VBS) phases. We implement a numerical simulation that is based on the recently developed unconstrained tree tensor network ansatz, which systematically improves the accuracy over the conventional methods as it exploits the internal gauge selections. At the highly frustrated region $({J}_{2}={J}_{1})$ , we observe a second-order phase transition from the plaquette-VBS state to the paramagnet phase at the critical magnetic-field ${\mathrm{\ensuremath{\Gamma}}}_{c}=0.28$ with the associated critical exponents $\ensuremath{\nu}=1$ and $\ensuremath{\gamma}\ensuremath{\simeq}0.4$, which are obtained within the finite-size scaling analysis on different lattice sizes $N=4\ifmmode\times\else\texttimes\fi{}4,6\ifmmode\times\else\texttimes\fi{}6,8\ifmmode\times\else\texttimes\fi{}8$. The stability of the plaquette-VBS phase at low magnetic fields is examined by a spin-spin correlation function, which verifies the presence of plaquette-VBS at ${J}_{2}={J}_{1}$ and rules out the existence of a N\'eel phase. In addition, our numerical results suggest that the transition from the N\'eel (for ${J}_{2}<{J}_{1})$ to the plaquette-VBS phase is a deconfined phase transition. Moreover, we introduce a mapping, which renders the low-energy effective theory of the TFI on the CL to be the same model on the ${J}_{1}\ensuremath{-}{J}_{2}$ square lattice (SL). We show that the plaquette-VBS phase of the highly frustrated point ${J}_{2}={J}_{1}$ on the CL is mapped to the emergent string-VBS phase on the SL at ${J}_{2}=0.5{J}_{1}$ .We investigate the ground-state phase diagram of the frustrated transverse-field Ising (TFI) model on the checkerboard lattice (CL), which consists of N\'eel, collinear, quantum paramagnet, and plaquette-valence bond solid (-VBS) phases. We implement a numerical simulation that is based on the recently developed unconstrained tree tensor network ansatz, which systematically improves the accuracy over the conventional methods as it exploits the internal gauge selections. At the highly frustrated region $({J}_{2}={J}_{1})$, we observe a second-order phase transition from the plaquette-VBS state to the paramagnet phase at the critical magnetic-field ${\mathrm{\ensuremath{\Gamma}}}_{c}=0.28$ with the associated critical exponents $\ensuremath{\nu}=1$ and $\ensuremath{\gamma}\ensuremath{\simeq}0.4$, which are obtained within the finite-size scaling analysis on different lattice sizes $N=4\ifmmode\times\else\texttimes\fi{}4,6\ifmmode\times\else\texttimes\fi{}6,8\ifmmode\times\else\texttimes\fi{}8$. The stability of the plaquette-VBS phase at low magnetic fields is examined by a spin-spin correlation function, which verifies the presence of plaquette-VBS at ${J}_{2}={J}_{1}$ and rules out the existence of a N\'eel phase. In addition, our numerical results suggest that the transition from the N\'eel (for ${J}_{2}<{J}_{1})$ to the plaquette-VBS phase is a deconfined phase transition. Moreover, we introduce a mapping, which renders the low-energy effective theory of the TFI on the CL to be the same model on the ${J}_{1}\ensuremath{-}{J}_{2}$ square lattice (SL). We show that the plaquette-VBS phase of the highly frustrated point ${J}_{2}={J}_{1}$ on the CL is mapped to the emergent string-VBS phase on the SL at ${J}_{2}=0.5{J}_{1}$.

arXiv:1901.00278Physical review. B./Physical review. BApr 16, 201911 citations

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Louis-Paul Henry, P. C. W. Holdsworth, Frédéric Mila +1 more

The ground-state properties of the $S=1/2$ transverse-field Ising model on the checkerboard lattice are studied using linear spin-wave theory. We consider the general case of different couplings between nearest neighbors ( ${J}_{1}$ ) and next-to-nearest neighbors ( ${J}_{2}$ ). In zero field, the system displays a large degeneracy of the ground state, which is exponential in the system size (for ${J}_{1}={J}_{2}$ ) or in the system's linear dimensions (for ${J}_{2}>{J}_{1}$ ). Quantum fluctuations induced by a transverse field are found to be unable to lift this degeneracy in favor of a classically ordered state at the harmonic level. This remarkable fact suggests that a quantum-disordered ground state can be instead promoted when nonlinear fluctuations are accounted for, in agreement with existing results for the isotropic case ${J}_{1}={J}_{2}$ . Moreover, spin-wave theory shows sizable regions of instability, which are further candidates for quantum-disordered behavior.The ground-state properties of the $S=1/2$ transverse-field Ising model on the checkerboard lattice are studied using linear spin-wave theory. We consider the general case of different couplings between nearest neighbors (${J}_{1}$) and next-to-nearest neighbors (${J}_{2}$). In zero field, the system displays a large degeneracy of the ground state, which is exponential in the system size (for ${J}_{1}={J}_{2}$) or in the system's linear dimensions (for ${J}_{2}>{J}_{1}$). Quantum fluctuations induced by a transverse field are found to be unable to lift this degeneracy in favor of a classically ordered state at the harmonic level. This remarkable fact suggests that a quantum-disordered ground state can be instead promoted when nonlinear fluctuations are accounted for, in agreement with existing results for the isotropic case ${J}_{1}={J}_{2}$. Moreover, spin-wave theory shows sizable regions of instability, which are further candidates for quantum-disordered behavior.

arXiv:1202.1462Physical Review BApr 16, 201223 citations

Plaquette valence-bond solid in the square-lattice J1 - J2 antiferromagnet Heisenberg model: A bond operator approach

R. L. Doretto

We study the plaquette valence-bond solid phase of the spin- $1/2$ ${J}_{1}$ - ${J}_{2}$ antiferromagnet Heisenberg model on the square lattice within the bond-operator theory. We start by considering four $S=1/2$ spins on a single plaquette and determine the bond operator representation for the spin operators in terms of singlet, triplet, and quintet boson operators. The formalism is then applied to the ${J}_{1}$ - ${J}_{2}$ model and an effective interacting boson model in terms of singlets and triplets is derived. The effective model is analyzed within the harmonic approximation and the previous results of Zhitomirsky and Ueda [Phys. Rev. B 54, 9007 (1996)] are recovered. By perturbatively including cubic (triplet-triplet-triplet and singlet-triplet-triplet) and quartic interactions, we find that the plaquette valence-bond solid phase is stable within the parameter region $0.34<{J}_{2}/{J}_{1}<0.59$ , which is narrower than the harmonic one. Differently from the harmonic approximation, the excitation gap vanishes at both critical couplings ${J}_{2}=0.34{J}_{1}$ and ${J}_{2}=0.59{J}_{1}$ . Interestingly, for ${J}_{2}<0.48{J}_{1}$ , the excitation gap corresponds to a singlet-triplet excitation at the $\ensuremath{\Gamma}$ point while, for ${J}_{2}>0.48{J}_{1}$ , it is related to a singlet-singlet excitation at the $\mathbf{X}=(\ensuremath{\pi}/2,0)$ point of the tetramerized Brillouin zone.We study the plaquette valence-bond solid phase of the spin-$1/2$${J}_{1}$-${J}_{2}$ antiferromagnet Heisenberg model on the square lattice within the bond-operator theory. We start by considering four $S=1/2$ spins on a single plaquette and determine the bond operator representation for the spin operators in terms of singlet, triplet, and quintet boson operators. The formalism is then applied to the ${J}_{1}$-${J}_{2}$ model and an effective interacting boson model in terms of singlets and triplets is derived. The effective model is analyzed within the harmonic approximation and the previous results of Zhitomirsky and Ueda [Phys. Rev. B 54, 9007 (1996)] are recovered. By perturbatively including cubic (triplet-triplet-triplet and singlet-triplet-triplet) and quartic interactions, we find that the plaquette valence-bond solid phase is stable within the parameter region $0.34<{J}_{2}/{J}_{1}<0.59$, which is narrower than the harmonic one. Differently from the harmonic approximation, the excitation gap vanishes at both critical couplings ${J}_{2}=0.34{J}_{1}$ and ${J}_{2}=0.59{J}_{1}$. Interestingly, for ${J}_{2}<0.48{J}_{1}$, the excitation gap corresponds to a singlet-triplet excitation at the $\ensuremath{\Gamma}$ point while, for ${J}_{2}>0.48{J}_{1}$, it is related to a singlet-singlet excitation at the $\mathbf{X}=(\ensuremath{\pi}/2,0)$ point of the tetramerized Brillouin zone.

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Phase diagram of the frustrated J 1 − J 2 transverse field Ising model on the square lattice

M. Sadrzadeh, A. Langari

We study the zero-temperature phase diagram of transverse field Ising model on the J 1 − J 2 square lattice. In zero magnetic field, the model has a classical Néel phase for J 2 / J 1 < 0.5 and an antiferromagnetic collinear phase for J 2 / J 1 > 0.5. We incorporate harmonic fluctuations by using linear spin wave theory (LSWT) with single spin flip excitations above a magnetic order background and obtain the phase diagram of the model in this approximation. We find that harmonic quantum fluctuations of LSWT fail to lift the large degeneracy at J 2 / J 1 = 0.5 and exhibit some inconsistent regions on the phase diagram. However, we show that anharmonic fluctuations of cluster operator approach (COA) resolve the inconsistency of the LSWT, which reveals a string-valence bond solid ordered phase for the highly frustrated region.We study the zero-temperature phase diagram of transverse field Ising model on the J 1 − J 2 square lattice. In zero magnetic field, the model has a classical Néel phase for J 2 / J 1 < 0.5 and an antiferromagnetic collinear phase for J 2 / J 1 > 0.5. We incorporate harmonic fluctuations by using linear spin wave theory (LSWT) with single spin flip excitations above a magnetic order background and obtain the phase diagram of the model in this approximation. We find that harmonic quantum fluctuations of LSWT fail to lift the large degeneracy at J 2 / J 1 = 0.5 and exhibit some inconsistent regions on the phase diagram. However, we show that anharmonic fluctuations of cluster operator approach (COA) resolve the inconsistency of the LSWT, which reveals a string-valence bond solid ordered phase for the highly frustrated region.

Journal of Physics Conference SeriesMar 1, 20183 citations

Spin- 12 J1 - J2 Heisenberg antiferromagnet on a square lattice: A plaquette renormalized tensor network study

Ji-Feng Yu, Ying-Jer Kao

We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg ${J}_{1}$ - ${J}_{2}$ model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with $L$ $=$ 8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different ${J}_{2}/{J}_{1}$ values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations, and plaquette order parameters. In addition to the well-known N\'eel order and collinear order at low and high ${J}_{2}/{J}_{1}$ , we observe a plaquettelike order at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.5$. A continuous transition between the N\'eel order and the plaquettelike order near ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.40{J}_{1}$ is observed. The collinear order emerges at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.62{J}_{1}$ through a first-order phase transition.We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg ${J}_{1}$-${J}_{2}$ model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with $L$ $=$ 8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different ${J}_{2}/{J}_{1}$ values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations, and plaquette order parameters. In addition to the well-known N\'eel order and collinear order at low and high ${J}_{2}/{J}_{1}$, we observe a plaquettelike order at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.5$. A continuous transition between the N\'eel order and the plaquettelike order near ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.40{J}_{1}$ is observed. The collinear order emerges at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.62{J}_{1}$ through a first-order phase transition.

arXiv:1108.3393Physical Review BMar 5, 201270 citations

Order-by-disorder in the antiferromagnetic J 1 − J 2 − J 3 transverse-field Ising model on the ruby lattice

Antonia Duft, Jan Alexander Koziol, Patrick Adelhardt +2 more

We investigate the quantum phase diagram of the J 1 − J 2 − J 3 antiferromagnetic transverse-field Ising model on the ruby lattice. In the low-field limit we derive an effective quantum dimer model, analyzing how the extensive ground-state degeneracy at zero field is lifted by an order-by-disorder scenario. We support our analysis by studying the gap closing of the high-field phase using series expansions. For J 2 > J 3 , we find a columnar phase at low fields, followed by a clock-ordered phase stabilized by resonating plaquettes at intermediate field values, and an emergent three-dimensional (3D)-XY quantum phase transition to the polarized high-field phase. For J 3 > J 2 , an order-by-disorder mechanism stabilizes a distinct k = ( 0 , 0 ) order and a quantum phase transition in the 3D-Ising universality class is observed. Further, we discuss the possible implementation of the columnar- and clock-ordered phase in existing Rydberg atom quantum simulators. When taking into account the full algebraically decaying long-range interactions on the ruby lattice, we find that long-range interactions favor the same ground state as the quantum fluctuations induced by a transverse field, which could make the ruby lattice a promising candidate for the realization of a clock-ordered phase. Published by the American Physical Society 2024We investigate the quantum phase diagram of the J 1 − J 2 − J 3 antiferromagnetic transverse-field Ising model on the ruby lattice. In the low-field limit we derive an effective quantum dimer model, analyzing how the extensive ground-state degeneracy at zero field is lifted by an order-by-disorder scenario. We support our analysis by studying the gap closing of the high-field phase using series expansions. For J 2 > J 3 , we find a columnar phase at low fields, followed by a clock-ordered phase stabilized by resonating plaquettes at intermediate field values, and an emergent three-dimensional (3D)-XY quantum phase transition to the polarized high-field phase. For J 3 > J 2 , an order-by-disorder mechanism stabilizes a distinct k = ( 0 , 0 ) order and a quantum phase transition in the 3D-Ising universality class is observed. Further, we discuss the possible implementation of the columnar- and clock-ordered phase in existing Rydberg atom quantum simulators. When taking into account the full algebraically decaying long-range interactions on the ruby lattice, we find that long-range interactions favor the same ground state as the quantum fluctuations induced by a transverse field, which could make the ruby lattice a promising candidate for the realization of a clock-ordered phase. Published by the American Physical Society 2024

arXiv:2312.12941Physical Review ResearchSep 26, 20248 citations

We study the effect of quantum fluctuations by means of a transverse magnetic field (

\Gamma

) on the antiferromagnetic

J_1-J_2

J_2>J_1

(the collinear phase). The bosonic excitation gap vanishes at the critical boundary to N'{e}el (

J_2 J_1

) phases, respectively. At the homogeneous coupling (

J_1=J_2

) a (canted) resonating plaquette state (RPS) is established from an-harmonic fluctuations, which lasts for low fields,

\Gamma/J_1\lesssim 0.4

, followed by a crossover to the fully polarized phase. We will show that our results predict an order-by-disorder as a results of transverse field at the critical point (

J_2=0.5 J_1

) of the antiferromagnetic

J_1-J_2

Ising square lattice.

({J}_{2}={J}_{1})

, we observe a second-order phase transition from the plaquette-VBS state to the paramagnet phase at the critical magnetic-field ${\mathrm{\ensuremath{\Gamma}}}_{c}=0.28$ with the associated critical exponents $\ensuremath{\nu}=1$ and $\ensuremath{\gamma}\ensuremath{\simeq}0.4$, which are obtained within the finite-size scaling analysis on different lattice sizes $N=4\ifmmode\times\else\texttimes\fi{}4,6\ifmmode\times\else\texttimes\fi{}6,8\ifmmode\times\else\texttimes\fi{}8$. The stability of the plaquette-VBS phase at low magnetic fields is examined by a spin-spin correlation function, which verifies the presence of plaquette-VBS at

{J}_{2}={J}_{1}

and rules out the existence of a N\'eel phase. In addition, our numerical results suggest that the transition from the N\'eel (for

{J}_{2}<{J}_{1})

to the plaquette-VBS phase is a deconfined phase transition. Moreover, we introduce a mapping, which renders the low-energy effective theory of the TFI on the CL to be the same model on the ${J}_{1}\ensuremath{-}{J}_{2}$ square lattice (SL). We show that the plaquette-VBS phase of the highly frustrated point

{J}_{2}={J}_{1}

on the CL is mapped to the emergent string-VBS phase on the SL at

{J}_{2}=0.5{J}_{1}

The ground-state properties of the

S=1/2

transverse-field Ising model on the checkerboard lattice are studied using linear spin-wave theory. We consider the general case of different couplings between nearest neighbors (

{J}_{1}

) and next-to-nearest neighbors (

{J}_{2}

). In zero field, the system displays a large degeneracy of the ground state, which is exponential in the system size (for

{J}_{1}={J}_{2}

) or in the system's linear dimensions (for

{J}_{2}>{J}_{1}

). Quantum fluctuations induced by a transverse field are found to be unable to lift this degeneracy in favor of a classically ordered state at the harmonic level. This remarkable fact suggests that a quantum-disordered ground state can be instead promoted when nonlinear fluctuations are accounted for, in agreement with existing results for the isotropic case

{J}_{1}={J}_{2}

. Moreover, spin-wave theory shows sizable regions of instability, which are further candidates for quantum-disordered behavior.

We study the plaquette valence-bond solid phase of the spin-

1/2

{J}_{1}

{J}_{2}

antiferromagnet Heisenberg model on the square lattice within the bond-operator theory. We start by considering four

S=1/2

spins on a single plaquette and determine the bond operator representation for the spin operators in terms of singlet, triplet, and quintet boson operators. The formalism is then applied to the

{J}_{1}

{J}_{2}

model and an effective interacting boson model in terms of singlets and triplets is derived. The effective model is analyzed within the harmonic approximation and the previous results of Zhitomirsky and Ueda [Phys. Rev. B 54, 9007 (1996)] are recovered. By perturbatively including cubic (triplet-triplet-triplet and singlet-triplet-triplet) and quartic interactions, we find that the plaquette valence-bond solid phase is stable within the parameter region

0.34<{J}_{2}/{J}_{1}<0.59

, which is narrower than the harmonic one. Differently from the harmonic approximation, the excitation gap vanishes at both critical couplings

{J}_{2}=0.34{J}_{1}

and

{J}_{2}=0.59{J}_{1}

. Interestingly, for

{J}_{2}<0.48{J}_{1}

, the excitation gap corresponds to a singlet-triplet excitation at the $\ensuremath{\Gamma}$ point while, for

{J}_{2}>0.48{J}_{1}

, it is related to a singlet-singlet excitation at the $\mathbf{X}=(\ensuremath{\pi}/2,0)$ point of the tetramerized Brillouin zone.

We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg

{J}_{1}

{J}_{2}

model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with

L

=

8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different

{J}_{2}/{J}_{1}

values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations, and plaquette order parameters. In addition to the well-known N\'eel order and collinear order at low and high

{J}_{2}/{J}_{1}

, we observe a plaquettelike order at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.5$. A continuous transition between the N\'eel order and the plaquettelike order near ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.40{J}_{1}$ is observed. The collinear order emerges at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.62{J}_{1}$ through a first-order phase transition.