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arXiv:2402.17827 · Physical review. B./Physical review. B

Residual entropy from the temperature incremental Monte Carlo method

Z. Dai, Xiao Yan Xu

Published Feb 14, 2025• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.111.l081108

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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 ddd-dimensional quantum models to (d+1d+1d+1)-dimensional classical counterparts. These challenges arise from singular interactions that emerge in the small \ensuremathΔ\ensuremathτ{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\tau}}\ensuremathΔ\ensuremathτ​ 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.
Abstract We develop a nonequilibrium increment method in quantum Monte Carlo simulations to obtain the Rényi entanglement entropy of various quantum many-body systems with high efficiency and precision. To demonstrate its power, we show the results on a few important yet difficult (2 + 1)dquantum lattice models, ranging from the Heisenberg quantum antiferromagnet with spontaneous symmetry breaking, the quantum critical point with O(3) conformal field theory (CFT) to the toric code Z2{{\mathbb{Z}}}_{2}Z2​ Z2 topological ordered state and the Kagome Z2{{\mathbb{Z}}}_{2}Z2​ Z2 quantum spin liquid model with frustration and multi-spin interactions. In all these cases, our method either reveals the precise CFT data from the logarithmic correction or extracts the quantum dimension in topological order, from the dominant area law in finite-size scaling, with very large system sizes, controlled errorbars, and minimal computational costs. Our method, therefore, establishes a controlled and practical computation paradigm to obtain the difficult yet important universal properties in highly entangled quantum matter.
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.
In spin systems, geometrical frustration describes the impossibility of minimizing simultaneously all the interactions in a Hamiltonian, often giving rise to macroscopic ground-state degeneracies and emergent low-temperature physics. In this thesis, combining tensor network (TN) methods to Monte Carlo (MC) methods and ground-state energy lower bound approaches, we study two-dimensional frustrated classical Ising models. In particular, we focus on the determination of the residual entropy in the presence of farther-neighbor interactions in kagome lattice Ising antiferromagnets (KIAFM). In general, using MC to determine the residual entropy is a significant challenge requiring ad-hoc updates, a precise evaluation of the energy at all temperatures to allow for thermodynamic integration, and a good control of the finite-size scaling behavior. As an alternative, we turn to TNs; however, we argue that, in the presence of frustration and macroscopic ground-state degeneracy, standard algorithms fail to converge at low temperatures on the usual TN formulation of partition functions. Inspired by methods for constructing ground-state energy lower bounds, we propose a systematic way to find the ground-state local rule using linear programming. Characterizing the rules as tiles that can be tessellated to form ground states of the model gives rise to a natural contractible TN formulation of the partition function. This method provides a direct access to the ground-state properties of frustrated models and, in particular, allows an extremely precise determination of their residual entropy. We then study two models inspired by artificial spin systems on the kagome lattice with out-of-plane (OOP) anisotropy. The first model is motivated by experiments on an array of chirally coupled nanomagnets. We argue that the farther-neighbor to nearest-neighbor couplings ratios in this system are much smaller than in the dipolar case, J2/J1 being of the order of 2%. A comparison of the experimental correlations with the results of extensive TN and MC simulations shows that (1) the experimental second- and third-neighbor correlations are inverted as compared to those of a pure nearest-neighbor model at equilibrium (even with a magnetic field), and (2) second-neighbor couplings as small as 1% of the nearest-neighbor couplings will affect the spin-spin correlations even at fairly high temperatures. Motivated by dipolar coupled artificial spin systems, we turn to the progressive lifting of the ground-state degeneracy of the KIAFM. We provide a detailed study of the ground-state phases of this model with up to third neighbor interactions, for arbitrary J2, J3 such that J1 >> J2, J3, obtaining exact results for the ground-state energies. When all couplings are antiferromagnetic, we exhibit three macroscopically degenerate ground-state phases and establish their residual entropy using our TN approach. Furthermore, in the phase corresponding to the dipolar KIAFM truncated to third neighbors, we use the ground-state tiles to establish the existence of a mapping to the ground-state manifold of the triangular Ising antiferromagnet.
Quantum magic, or nonstabilizerness, provides a crucial characterization of quantum systems, regarding the classical simulability with stabilizer states. In this work, we propose an alternative and efficient algorithm for computing stabilizer R\'enyi entropy, one of the measures for quantum magic, in spin systems with sign-problem free Hamiltonians. This algorithm is based on the quantum Monte Carlo simulation of the path integral of the work between two partition function ensembles and it applies to all spatial dimensions and temperatures. We demonstrate this algorithm on the one- and two-dimensional transverse field Ising model at both finite and zero temperatures and show the quantitative agreements with tensor-network based algorithms. We analyze the computational cost and provide analytical and numerical evidences for it to be polynomial in system size. This work also suggests a unifying framework for calculating various types of entropy quantities including entanglement R\'enyi entropy and entanglement R\'enyi negativity.
We develop a technique for calculating three-dimensional classical partition functions using projected entangled-pair states (PEPS). Our method is based on variational PEPS optimization algorithms for two-dimensional quantum spin systems, and allows us to compute free energies directly in the thermodynamic limit. The main focus of this work is classical frustration in three-dimensional many-body systems leading to an extensive ground-state degeneracy. We provide high-accuracy results for the residual entropy of the dimer model on the cubic lattice, water ice Ih{I}_{h}Ih​, and water ice Ic{I}_{c}Ic​. In addition, we show that these systems are in a Coulomb phase by computing the dipolar form of the correlation functions. As a further benchmark of our methods, we calculate the critical temperature and exponents of the Ising model on the cubic lattice.