Session Y60: Computational Methods for Statistical Mechanics I

Efficiency of the minimally entangled typical thermal state algorithm in quantum critical chains

Minimally entangled typical thermal states (METTS) are pure states that decompose the Gibbs state. By randomly sampling these states, one can compute finite temperature properties of quantum many-body systems. It is empirically known that METTS has only a small amount of entanglement and can be efficiently represented using a matrix product state. However, there has been a lack of analysis of the computational cost of the METTS algorithm, and its superiority over other simulation methods has not been clarified. In particular, in the case of quantum critical chains, the entanglement entropy of the ground state diverges, and thus, estimates of the computational cost in terms of the entropy break down. We study the computational efficiency of the METTS algorithm in quantum critical chains. We show that the entanglement entropy of generic METTS obeys the area law and grows logarithmically slowly with the inverse temperature. Furthermore, we find that it exhibits a universal behavior characterized only by the central charge of the conformal field theory that describes the critical point. Based on these results, we argue that the computational cost of constructing METTS is parametrically smaller than that of the purification of the Gibbs state in the low-temperature limit. Finally, we demonstrate that the METTS algorithm provides a significant speedup compared to the purification method when analyzing low-temperature thermal equilibrium states in quantum critical chains.   

Real-space renormalization group in 3D with well-controlled approximations

Real-space renormalization group (RG) often has uncontrolled approximations. Agreement of the estimated critical exponents with other methods can serve as a posterior justification for a scheme, without explaining why it works.

Inspired by quantum-information concepts, tensor-network RG (TNRG) is a type of real-space RG with definite approximation errors and free of Monte-Carlo sign problem. In 2D criticality, like Ising model, those errors decay exponentially while the dimensionality of RG flows increases algebraically. This provides a promising framework for an accurate 3D real-space RG. However, block-spin idea using tensor-network method in 3D has RG errors more than 20% near Ising critical fixed point. Estimations of critical exponents from linearized RG are unstable with respect to RG steps, with errors over 10%.

We design an entanglement-filtering (EF) scheme for eliminating lattice-scale physics in 3D TNRG. Reducing RG errors to about 5%, the EF-enhanced TNRG leads to more stable tensor RG flows near the Ising criticality. With a few minutes on a desktop computer, the new scheme gives stable estimations of thermal and spin exponents with errors 1% and 5%. This is an important step towards a systematically-improvable 3D real-space RG method.   

Quantum Monte Carlo Simulations of Rydberg Atom Arrays

Arrays of Rydberg atoms are a powerful platform to realize strongly-interacting quantum many-body systems. Through the use of optical tweezers ⁸⁷Rb atoms are arranged into a programmable 2D lattice and excited into a high principal quantum number state (called a "Rydberg" state) by lasers. It is then possible to encode a physical qubit into the Hilbert space spanned by an atom’s groundstate and excited state. The atoms interact with each other through a distance dependent potential which penalizes simultaneous excitation. A common Hamiltonian implemented on such arrays takes a form similar to a Transverse-Field Ising Model and is free of the sign problem, meaning its equilibrium properties are amenable to efficient simulation via quantum Monte Carlo (QMC).

Classically simulating such systems is useful both for verifying experimental results, and for probing the physics of larger systems which may not yet be available experimentally. While the stoquasticity of the Hamiltonian does in principle allow exact QMC simulations, devising an efficient simulation scheme is not always trivial. We discuss our recent QMC algorithm for exact simulations of Rydberg arrays in which we devise updates that allow for the efficient estimation of physical observables for typical experimental parameters, and show that the algorithm can reproduce experimental results of groundstates of large Rydberg arrays in two dimensions.   

A Monte Carlo algorithm for calculating the overlap between ground states

Gaining insight into the overlap between the respective ground states of two given Hamiltonians has become increasingly important in recent years This need arises in various contexts, including the detection of quantum phase transitions, the study of the Anderson orthogonality catastrophe, and the exploration of critical systems. Additionally, this concept plays a pivotal role in the realm of quantum quenches involving impurity systems. I will discuss a novel Monte Carlo algorithm for calculating the overlap between the ground states of two input quantum Hamiltonians and share some results demonstrating the applicability and scope of the technique.   

Detecting the Energy Spectrum of Bound States with Random Walks

In the study of complex systems, unexpected macroscopic behaviors often arise from microscopic mechanics. We introduce a simple model consisting of two types of corpuscular particles, 'walkers' and 'antiwalkers', undergoing an unbiased, homogeneous random walk. Walker-antiwalker pairs annihilate when locally brought into contact, and are subject to a survival test, the survival probability of which is determined by the potential energy at their location, following a negative exponential relationship. Analyzing the spatial distribution of survivors (walkers and antiwalkers) and renormalizing it, our system converges to a stationary state resembling the fundamental eigenfunction of the related Hamiltonian. Moreover, when subtracting the common component from a random initial condition with the first (N-1) eigenfunctions of the Hamiltonian, the system reaches a stable state corresponding to the N-th eigenfunction. This discovery enables the development of an eigenfunction detection algorithm that can be linked to Imaginary Time Propagation (ITP). We present analytical and numerical results, including the relationship between random walk variance and result precision. Finally, our model also offers a corpuscular interpretation of ITP.   

Statistical mechanics and thermodynamics of texts from various authors in the English language

Just as pairwise energetic interactions (e.g., Lennard-Jones) can inform model Hamiltonians from which the statistical mechanics and thermodynamics of physical systems can be studied, it is possible to study analogous quantities in the English language by considering the specific sequences of letters in words to be the result of ``energetic’’ interactions between pairs of these letters. Previous work has used the empirical frequencies of letters in words, in combination with Jaynes’ principle of maximum entropy, to obtain the interaction potentials which maximize the entropy of the resulting Boltzmann distribution of the energy of words [1,2]. In this work, we obtain these interaction potentials for various authors and examine differences in the resulting ``thermodynamic’’ properties. We find that signals in the quantity analogous to the heat capacity indicate changes in an author’s word use (both type and composition) and, moreover, that this quantity can be used to distinguish between authors.

[1] G. J. Stephens and W. Bialek, Physical Review E 81, 066119 (2010) 

[2] A. Corral and M. G. del Muro, Entropy 22, 179 (2020)    

Interfacial Adsorption and Wetting Transition in the Classical q-State Potts Model on Archimedean Lattices

We explore the phenomena of interfacial adsorption and critical wetting transition in the classical 2-dimensional q-state Potts model for several Archimedean lattices, including the triangular, square, honeycomb, and Kagome lattices. Our choice of lattice tilings is motivated by the availability of various 2-dimensional materials exhibiting these lattice structures. Using standard Monte Carlo simulations coupled with a cluster Graphics Processing Units (GPU) algorithm, we simulate lattices with up to 10⁹ sites. Our results indicate the emergence of a critical wetting point in each of the lattice systems, suggesting the possibility of critical wetting transitions in these lattice systems. Furthermore, we investigate the phenomenon of interfacial adsorption in these lattices and demonstrate the possibility and emergence of adsorption in these systems.   

Yang-Lee Zeros of Certain Antiferromagnetic Models

We revisit the somewhat less studied problem of Yang-Lee zeros of the Ising antiferromagnet. For this purpose, we study two models, the nearest-neighbor model on a square lattice, and the more tractable mean-field model corresponding to infinite-ranged coupling. In the high-temperature limit, we show that the logarithm of the Yang-Lee zeros can be written as a series in half odd integer powers of the inverse temperature, k, with the leading term∼k^1/2. This result is true in any dimension and for arbitrary lattices. We also show that the coefficients of the expansion satisfy simple identities for the nearest-neighbor case. These new identities are verified numerically by computing the exact partition function for a square lattice of size 16×16. For the mean-field model, we write down the partition function for the ferromagnetic (FM) and antiferromagnetic (AFM) cases. We analytically show that at high temperatures the zeros of the AFM mean-field polynomial scale as ∼k^1/2 as well. Using a simple numerical method, we find the roots lie on certain curves (root curves), in the thermodynamic limit for the AFM case as well as for the FM one. Our results show a new root curve, that was not found earlier. Our results also clearly illustrate the phase transition expected for the FM and AFM cases, in the language of Yang-Lee zeros. Moreover, for the AFM case, we observe that the root curves separate two distinct phases of zero and non-zero complex staggered magnetization, and thus depict a complex phase boundary.   

Brownian Bridges for Stochastic Containment using a Self-Adjoint Formulation of the Backward Fokker-Planck Equation

We show that continuous random walks, processes which model a wide variety of chemical and physical systems, can be conditioned to efficiently generate rare events. Specifically, we use a Brownian bridge to examine contained trajectories, i.e. processes which stay within a given region of state-space for a set period time T, or in other words, paths whose survival time within that region is greater than T. The bridge acts as a conditioned process so that we generate only the subset of sample paths which meet this containment criteria. Bridges are constructed via the solution of the Backwards Fokker-Planck (BFP) equation. We derive a method which reformulates the BFP into a self-adjoint representation, effectively reducing its complexity. This representation shows that in the asymptotic limit, T>>1, the bridge is time-independent and is a function of only the dominant eigenfunction of the self-adjoint BFP operator. In this limit, we show that the subset of paths contained within a specified region is equivalent to the set of paths sampled from a modified potential energy landscape.  We demonstrate that this idea is accurate for many systems, even for times T ~ O(1).  Lastly, we discuss how this idea could be scaled to higher dimensions, using existing numerical techniques to approximate dominant eigenfunctions.   

Efficient Sampling of Equilibrium Distributions with Energy-Informed, Scalable Diffusion Models

Many problems in statistical mechanics involve inference in high-dimensional spaces, where sampling techniques such as Markov Chain Monte Carlo (MCMC) frequently suffer from slow decorrelation and computational bottlenecks. To enhance sampling efficiency,MCMC can be augmented with deep generative models, a class of machine learning models that can learn and generate samples from desired probability distributions. We demonstrate the potential of integrating force field information into a specific type of deep generative model known as score-based diffusion models, with practical applications demonstrated in the generation of Lennard-Jones liquid configurations. This approach also gives rise to the possibiltiy of conditional generation, allowing particles to be generated based on their chemical environment. We aim to achieve scalability and generalizability in our sampling methods through the use of physics-informed generative models.   

Symmetry Breaking in the Grasshopper Problem

A grasshopper lands at a random point on a planar lawn of area one. It then makes one jump of fixed distance d in a random direction. What shape should the lawn be to maximize the chance that the grasshopper remains on the lawn after jumping? This easily stated yet hard-to-solve problem has intriguing connections to quantum information and statistical physics. A discrete version can be modeled by a spin system, representing a new class of statistical models with fixed-range interactions where the range d can be large. Surprisingly, there is no d > 0 for which a disc shaped lawn is optimal. For jump distances smaller than the radius of a unit disc, the optimal lawn shapes resemble cogwheels that break rotational symmetry. When the problem is generalized to higher dimensions, rotational symmetry is restored in the optimal lawn shapes. We will discuss numerical results as well as an analysis of why and how rotational symmetry is broken.