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How Robust are Entropy Plateaus? A case study of triangular Ising antiferromagnets

Owen Bradley, Chunhan Feng, Richard T. Scalettar, Rajiv R. P. Singh

Published Jan 24, 2019• 0 citations
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Abstract

Residual entropy is a key feature associated with emergence in many-body systems. From a variety of frustrated magnets to the onset of spin-charge separation in Hubbard models and fermion-$Z_2$-flux variables in Kitaev models, the freezing of one set of degrees of freedom and establishment of local constraints is marked by a plateau in entropy as a function of temperature. Yet, with the exception of rare-earth pyrochlore family of spin-ice materials, evidence for such plateaus is rarely seen in real materials, raising questions about their robustness. Following recent experimental findings of the absence of such plateaus in triangular-lattice Ising antiferromagnet (TIAF) TmMgGaO$_4$ by Li et al, we explore in detail the existence and rounding of entropy plateaus in TIAF. We use a transfer matrix method to numerically calculate the properties of the system at different temperatures and magnetic fields, with further neighbor interactions and disorder. We find that temperature windows of entropy plateaus exist only when second neighbor interactions are no more than a couple of percent of the nearest-neighbor ones, and are also easily destroyed by disorder in the nearest-neighbor exchange variable, thereby explaining the challenge in observing such effects.

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Robustness of entropy plateaus: A case study of triangular Ising antiferromagnets

Owen Bradley, Chunhan Feng, Richard T. Scalettar +1 more

Residual entropy is a key feature associated with emergence in many-body systems. From a variety of frustrated magnets to the onset of spin-charge separation in Hubbard models and fermion-${Z}_{2}$-flux variables in Kitaev models, the freezing of one set of degrees of freedom and the establishment of local constraints are marked by a plateau in entropy as a function of temperature. Yet, with the exception of the rare-earth pyrochlore family of spin-ice materials, evidence for such plateaus is rarely seen in real materials, raising questions about their robustness. Following recent experimental findings of the absence of such plateaus in the triangular-lattice Ising antiferromagnet (TIAF) ${\mathrm{TmMgGaO}}_{4}$ by Li et al., we explore in detail the existence and rounding of entropy plateaus in TIAF. We use a transfer-matrix method to numerically calculate the properties of the system at different temperatures and magnetic fields, with further neighbor interactions and disorder. We find that temperature windows of entropy plateaus exist only when second-neighbor interactions are no more than a couple of percent of the nearest-neighbor ones, and they are also easily destroyed by disorder in the nearest-neighbor exchange variable, thereby explaining the challenge in observing such effects.

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We study an Ising model with nearest-neighbor antiferromagnetic interactions. It is placed on a triangular lattice, where each site is occupied by a spin with $x$ probability. There is no applied magnetic field. Randomness and frustration, two essential ingredients of spin-glasses, are present in this model. We study its critical properties here. The entropy is obtained by a transfer-matrix calculation as a function of $x$ at low temperature ($T=\frac{0.3J}{{k}_{B}}$) for systems on a lattice of 10 \ifmmode\times\else\texttimes\fi{} 20 sites. A fairly shallow minimum appears near $x\ensuremath{\simeq}0.9$, which suggests that this case is the one most likely to show a transition into an ordered state at low temperature. We study the cases $x=1, 0.9, \mathrm{and} 0.74$, which is about half-way to the critical percolation. We simulate systems on lattices of 50 \ifmmode\times\else\texttimes\fi{} 50 sites and 30 \ifmmode\times\else\texttimes\fi{} 30 sites by the Monte Carlo method. The specific heat has a broad maximum at $T\ensuremath{\simeq}0.9$ for $x=0.74 \mathrm{and} 0.9$. ${\ensuremath{\chi}}_{\mathrm{SG}}$, defined by ${\ensuremath{\chi}}_{\mathrm{SG}}={N}^{\ensuremath{-}1}\ensuremath{\Sigma}{i,j}^{}{〈{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{j}〉}^{2}$, and the relaxation time ($\ensuremath{\tau}$) are obtained for $T\ensuremath{\ge}\frac{0.6J}{{k}_{B}}$. Both quantities, $\ensuremath{\tau}$ and ${\ensuremath{\chi}}_{\mathrm{SG}}$, turn out to be proportional to $\mathrm{exp}[\frac{A}{{(T\ensuremath{-}{T}_{0})}^{c}}]$ and $0\ensuremath{\le}{T}_{0}\ensuremath{\lesssim}0.4$; a fit with ${T}_{0}=0$ yields $c\ensuremath{\approx}1$ for $x=1$ but $c\ensuremath{\approx}2$ for $x=0.74 \mathrm{and} 0.9$.

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Antiferromagnetic Ising Model on a Triangular Lattice

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We study the critical properties of an Ising model with nearest neighbor antiferromagnetic interactions, placed on a triangular lattice, where each site is occupied by a spin, only with x probability. Randomness and frustration, two essential ingredients of spin-glasses, are present in this model. The entropy is obtained, by a transfer matrix calculation, as a function of x at low temperature (T=0.3 J/kB), for systems on a lattice of 10×19 sites. A fairly shallow minimum in the entropy appears near x=0.9, which suggests that this case is the most likely to show a transition into an ordered state at low temperature. We therefore simulate systems on lattices of 50×50 sites for x=0.9 and x=1 by the Monte Carlo method. The specific heat has a broad maximum at T=0.9 J/kB (1.2) for x=0.9 (1). The relaxation time (τ) obtained for T≥0.6 J/kB, turns out to be proportional to exp[A/(T−T0)c], and 0<T0≲0.4 J/kB; a fit with T0=0 yields c≂1 for x=1 but c≂2 for x=0.9.

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Residual entropy, which reflects the degrees of freedom in a system at absolute zero temperature, is crucial for understanding quantum and classical ground states. Despite its key role in explaining low-temperature phenomena and ground-state degeneracy, accurately measuring residual entropy remains a difficult task owing to computational limitations. In this Letter, we introduce the temperature incremental Monte Carlo (TIMC) method, our approach to overcoming these challenges. The TIMC method incrementally calculates the partition function ratio of neighboring temperatures within Monte Carlo simulations, enabling precise entropy calculations and revealing many other temperature-dependent properties in a single computational sweep of temperatures. We have rogorously tested TIMC on several complex systems, including the frustrated antiferromagnetic Ising model on both C60 and 2D triangular lattices, the Newman-Moore glassy model, and a 2D quantum transverse field Ising model. Notably, our method overcomes the difficulties encountered in partition function measurements when mapping $d$-dimensional quantum models to ($d+1$)-dimensional classical counterparts. These challenges arise from singular interactions that emerge in the small ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\tau}}$ limit during the quantum-to-classical mapping procedure. The TIMC method enables precise entropy calculations across the entire temperature range, as demonstrated in our studies of frustrated spin models, glassy phases, and phases exhibiting spontaneous symmetry breaking. This method's capability to calculate residual entropy could provide insights when applied to systems lacking analytical solutions.

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Residual entropy is a key feature associated with emergence in many-body systems. From a variety of frustrated magnets to the onset of spin-charge separation in Hubbard models and fermion-Z2Z_2Z2​-flux variables in Kitaev models, the freezing of one set of degrees of freedom and establishment of local constraints is marked by a plateau in entropy as a function of temperature. Yet, with the exception of rare-earth pyrochlore family of spin-ice materials, evidence for such plateaus is rarely seen in real materials, raising questions about their robustness. Following recent experimental findings of the absence of such plateaus in triangular-lattice Ising antiferromagnet (TIAF) TmMgGaO4_44​ by Li et al, we explore in detail the existence and rounding of entropy plateaus in TIAF. We use a transfer matrix method to numerically calculate the properties of the system at different temperatures and magnetic fields, with further neighbor interactions and disorder. We find that temperature windows of entropy plateaus exist only when second neighbor interactions are no more than a couple of percent of the nearest-neighbor ones, and are also easily destroyed by disorder in the nearest-neighbor exchange variable, thereby explaining the challenge in observing such effects.
Residual entropy is a key feature associated with emergence in many-body systems. From a variety of frustrated magnets to the onset of spin-charge separation in Hubbard models and fermion-Z2{Z}_{2}Z2​-flux variables in Kitaev models, the freezing of one set of degrees of freedom and the establishment of local constraints are marked by a plateau in entropy as a function of temperature. Yet, with the exception of the rare-earth pyrochlore family of spin-ice materials, evidence for such plateaus is rarely seen in real materials, raising questions about their robustness. Following recent experimental findings of the absence of such plateaus in the triangular-lattice Ising antiferromagnet (TIAF) TmMgGaO4{\mathrm{TmMgGaO}}_{4}TmMgGaO4​ by Li et al., we explore in detail the existence and rounding of entropy plateaus in TIAF. We use a transfer-matrix method to numerically calculate the properties of the system at different temperatures and magnetic fields, with further neighbor interactions and disorder. We find that temperature windows of entropy plateaus exist only when second-neighbor interactions are no more than a couple of percent of the nearest-neighbor ones, and they are also easily destroyed by disorder in the nearest-neighbor exchange variable, thereby explaining the challenge in observing such effects.
We study an Ising model with nearest-neighbor antiferromagnetic interactions. It is placed on a triangular lattice, where each site is occupied by a spin with xxx probability. There is no applied magnetic field. Randomness and frustration, two essential ingredients of spin-glasses, are present in this model. We study its critical properties here. The entropy is obtained by a transfer-matrix calculation as a function of xxx at low temperature (T=0.3JkBT=\frac{0.3J}{{k}_{B}}T=kB​0.3J​) for systems on a lattice of 10 \ifmmode\times\else\texttimes\fi{} 20 sites. A fairly shallow minimum appears near $x\ensuremath{\simeq}0.9$, which suggests that this case is the one most likely to show a transition into an ordered state at low temperature. We study the cases x=1,0.9,and0.74x=1, 0.9, \mathrm{and} 0.74x=1,0.9,and0.74, which is about half-way to the critical percolation. We simulate systems on lattices of 50 \ifmmode\times\else\texttimes\fi{} 50 sites and 30 \ifmmode\times\else\texttimes\fi{} 30 sites by the Monte Carlo method. The specific heat has a broad maximum at $T\ensuremath{\simeq}0.9$ for x=0.74and0.9x=0.74 \mathrm{and} 0.9x=0.74and0.9. ${\ensuremath{\chi}}_{\mathrm{SG}}$, defined by ${\ensuremath{\chi}}_{\mathrm{SG}}={N}^{\ensuremath{-}1}\ensuremath{\Sigma}{i,j}^{}{〈{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{j}〉}^{2}$, and the relaxation time ($\ensuremath{\tau}$) are obtained for $T\ensuremath{\ge}\frac{0.6J}{{k}_{B}}$. Both quantities, $\ensuremath{\tau}$ and ${\ensuremath{\chi}}_{\mathrm{SG}}$, turn out to be proportional to $\mathrm{exp}[\frac{A}{{(T\ensuremath{-}{T}_{0})}^{c}}]$ and $0\ensuremath{\le}{T}_{0}\ensuremath{\lesssim}0.4$; a fit with T0=0{T}_{0}=0T0​=0 yields $c\ensuremath{\approx}1$ for x=1x=1x=1 but $c\ensuremath{\approx}2$ for x=0.74and0.9x=0.74 \mathrm{and} 0.9x=0.74and0.9.
An antiferromagnetic Ising system on a triangular lattice is studied by means of a transfer matrix and the phenomenological scaling method. System consists of a semi-infinite strip and helical boundary conditions are imposed at the end of each row. At external magnetic fields h =0 and 6 J , where J is a nearest neighbor coupling constant, a unite residual entropy is obtained at the zero temperature. With a small field, specific heat is found to have two peaks; a high-temperature peak corresponding to the enhanced short-range order, and a low-temperature peak to the appearance of a long-range order. A phase diagram in h - T space is obtained by means of the phenomenological scaling, and agrees fairly well with the one obtained by Monte Carlo simulation. The main deficiency is the finite transition temperature at h =0, which may be due to a crossover to the XY-like phase transition.
Geometric frustration in quantum systems can stabilize unconventional phases of matter that avoid traditional magnetic ordering at low temperatures. Here, we observe this phenomenon while mapping out the finite temperature phase diagram of the spin-1/2 Heisenberg antiferromagnet on the ruby lattice with next-nearest-neighbor interactions. Using an infinite tensor network state (iTNS) optimized and measured with belief propagation (BP) and corrections to BP, we observe the low temperature formation of stable magnetic plateaus at various magnetic field strengths. We find these plateaus host a novel `simplex liquid state' -- a disordered phase involving strongly paired spin simplices that retains non-zero residual entropy due to an exponentially large subspace of crystalline configurations. We accurately quantify the energy gap associated with these states and show that, as the temperature of the system is lowered, it does not go through a phase transition to reach them: the heat capacity remains finite and continuous at all observed temperatures. Our work demonstrates how BP-based tensor network techniques provide a powerful route to understanding frustrated quantum magnets at finite temperature.
We study the critical properties of an Ising model with nearest neighbor antiferromagnetic interactions, placed on a triangular lattice, where each site is occupied by a spin, only with x probability. Randomness and frustration, two essential ingredients of spin-glasses, are present in this model. The entropy is obtained, by a transfer matrix calculation, as a function of x at low temperature (T=0.3 J/kB), for systems on a lattice of 10×19 sites. A fairly shallow minimum in the entropy appears near x=0.9, which suggests that this case is the most likely to show a transition into an ordered state at low temperature. We therefore simulate systems on lattices of 50×50 sites for x=0.9 and x=1 by the Monte Carlo method. The specific heat has a broad maximum at T=0.9 J/kB (1.2) for x=0.9 (1). The relaxation time (τ) obtained for T≥0.6 J/kB, turns out to be proportional to exp[A/(T−T0)c], and 0<T0≲0.4 J/kB; a fit with T0=0 yields c≂1 for x=1 but c≂2 for x=0.9.
Residual entropy, which reflects the degrees of freedom in a system at absolute zero temperature, is crucial for understanding quantum and classical ground states. Despite its key role in explaining low-temperature phenomena and ground-state degeneracy, accurately measuring residual entropy remains a difficult task owing to computational limitations. In this Letter, we introduce the temperature incremental Monte Carlo (TIMC) method, our approach to overcoming these challenges. The TIMC method incrementally calculates the partition function ratio of neighboring temperatures within Monte Carlo simulations, enabling precise entropy calculations and revealing many other temperature-dependent properties in a single computational sweep of temperatures. We have rogorously tested TIMC on several complex systems, including the frustrated antiferromagnetic Ising model on both C60 and 2D triangular lattices, the Newman-Moore glassy model, and a 2D quantum transverse field Ising model. Notably, our method overcomes the difficulties encountered in partition function measurements when mapping ddd-dimensional quantum models to (d+1d+1d+1)-dimensional classical counterparts. These challenges arise from singular interactions that emerge in the small ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\tau}}$ limit during the quantum-to-classical mapping procedure. The TIMC method enables precise entropy calculations across the entire temperature range, as demonstrated in our studies of frustrated spin models, glassy phases, and phases exhibiting spontaneous symmetry breaking. This method's capability to calculate residual entropy could provide insights when applied to systems lacking analytical solutions.