This thesis presents computational studies of three different models of many-body physics with direct or indirect constraints. The presence of constraints in complex many-body systems calls for non-trivial numerical algorithms to study them. The first two models which have a direct form of local constraint are the Rokhsar-Kivelson Quantum Dimer model (QDM) and a classical statistical mechanics model of non-intersecting loops with attractive interactions, both on the square lattice. The investigations of such constrained models have found a recent resurgence with their direct realizations on Rydberg atom arrays quantum simulators. The study of the classical model uses a Monte Carlo directed loop algorithm while the QDM calls for a novel Quantum Monte Carlo scheme based on the framework of Stochastic Series Expansions called the Sweeping Cluster Algorithm (SCA). We present a modification of the SCA in order to render simulations fully ergodic at finite temperature. For both models, our numerical studies show the existence of a critical phase separated by a phase transition at finite temperature to an ordered phase of dimers or loops which spontaneously breaks certain lattice symmetries. We show that for the case where the interaction is attractive this phase transition is of Kosterlitz-Thouless type and can be understood by constructing a coarse-grained field theory through a height mapping. The finite temperature phase diagram of the QDM presents an unusual re-entrance behavior in the critical phase. The final part of this thesis deals with the role of non-abelian symmetries in the thermalization process of quantum many-body systems. We study the high-energy eigenstates of a SU(3) symmetric spin chain in presence of disorder. While the model does not directly have constraints, we perform exact diagonalization in a constrained basis of Young tableau making use of the full SU(3) symmetry of the model. By looking at the commonly used probes for thermalization (spectral statistics, distribution of local observables and scaling of entanglement entropy), we show that the model exhibits a non-ergodic regime over a broad range of system sizes for strong enough disorder, contrasting with the rapid thermalization observed at weak disorder.
This thesis presents computational studies of three different models of many-body physics with direct or indirect constraints. The presence of constraints in complex many-body systems calls for non-trivial numerical algorithms to study them. The first two models which have a direct form of local constraint are the Rokhsar-Kivelson Quantum Dimer model (QDM) and a classical statistical mechanics model of non-intersecting loops with attractive interactions, both on the square lattice. The investigations of such constrained models have found a recent resurgence with their direct realizations on Rydberg atom arrays quantum simulators. The study of the classical model uses a Monte Carlo directed loop algorithm while the QDM calls for a novel Quantum Monte Carlo scheme based on the framework of Stochastic Series Expansions called the Sweeping Cluster Algorithm (SCA). We present a modification of the SCA in order to render simulations fully ergodic at finite temperature. For both models, our numerical studies show the existence of a critical phase separated by a phase transition at finite temperature to an ordered phase of dimers or loops which spontaneously breaks certain lattice symmetries. We show that for the case where the interaction is attractive this phase transition is of Kosterlitz-Thouless type and can be understood by constructing a coarse-grained field theory through a height mapping. The finite temperature phase diagram of the QDM presents an unusual re-entrance behavior in the critical phase. The final part of this thesis deals with the role of non-abelian symmetries in the thermalization process of quantum many-body systems. We study the high-energy eigenstates of a SU(3) symmetric spin chain in presence of disorder. While the model does not directly have constraints, we perform exact diagonalization in a constrained basis of Young tableau making use of the full SU(3) symmetry of the model. By looking at the commonly used probes for thermalization (spectral statistics, distribution of local observables and scaling of entanglement entropy), we show that the model exhibits a non-ergodic regime over a broad range of system sizes for strong enough disorder, contrasting with the rapid thermalization observed at weak disorder.
Constrained models play a pivotal role in condensed matter physics by capturing essential features of strongly interacting many-body systems. These models impose local or global restrictions that mirror real physical constraints, enabling the study of complex emergent behaviors. For instance, hard-core dimer models reflect physical systems with strong local constraints due to repulsive interactions. Similarly, loop models, vertex models, and height representations provide constrained frameworks that reveal rich phase structures and critical phenomena. In quantum systems, constraints often emerge naturally from microscopic Hamiltonian and lead to phenomena such as fractionalization and quantum spin liquids. Constrained models act as fertile grounds for theoretical insight, enabling exact solutions, mapping to field theories, and exploring universal behaviors. They also bridge condensed matter with disciplines like statistical mechanics, combinatorics, and computational complexity, offering a unified language to study order, disorder, and transitions in systems ranging from magnets to artificial spin ices and cold atomic gases. This thesis investigates critical phenomena, strongly correlated models, and optimization problems on aperiodic lattices, with a particular focus on NP-hard problems, dimer models, and their classical and quantum variants. Moving beyond periodic lattices like square and honeycomb structures, this research delves into the less understood area of aperiodic systems, such as quasicrystals, which exhibit long-range order without translational symmetry. These structures challenge traditional paradigms of condensed matter physics and open avenues for discovering new critical behaviors. I present original contributions that combine analytical methods and numerical algorithms to study constrained systems on quasicrystalline tilings. These include the solution of NP-hard optimization problems like the Hamiltonian cycle on Ammann-Beenker tilings, an energy-based worm algorithm for simulating classical dimers on random graphs and modified Penrose tilings, and the analytical solution of the dimer partition function on the newly discovered aperiodic Spectre tiling. I further extend the study to quantum systems by analytically investigating the quantum dimer model (QDM) on the Spectre tiling and numerically analyzing it on the Ammann-Beenker tiling, revealing distinctive ground-state properties and quantum phase transitions that arise due to aperiodicity and local tiling constraints. This work builds interdisciplinary bridges between condensed matter physics, graph theory, and computational complexity. It also sheds light on the feasibility and limitations of transferring universal results from periodic systems to aperiodic ones.
This doctoral thesis studies low dimensional quantum systems that can be realized in recent cold atom experiments. From the viewpoint of quantum statistical mechanics, the main emphasis is on the detailed study of the different quantum and thermal phases and their transitions using numerical methods, such as quantum Monte Carlo and the Tensor Network Renormalization Group. The first part of this work deals with a lattice Boson model subject to strong three-body losses. In a quantum-Zeno like manner, a dynamical three-body constraint arises that restricts the evolution of the quantum system to the subspace with at most two particles per site i.e., three-body losses will be suppressed. Its low-energy description by a Feshbach model leads to the appearance of a pair condensate phase with intriguing quantum critical properties controlled by the Coleman-Weinberg mechanism. A comprehensive numerical study, presented in Chapter 3, identifies the pair phase and provides evidence for an unconventional Berezinksii-Kosterlitz-Thouless transition due to the unbinding of half-vortices. The study of the pair phase with worm-like algorithms turns out to be challenging and an extension of the directed loop algorithm is presented that overcomes the sampling limitations posed by the appearance of fat-tail distributions. In the following chapters, attention is drawn to lattice systems of polar molecules in optical lattices. Their intriguing dipolar interaction and sensitivity towards external fields make them ideal candidates to realize new states of matter. Particularly, it has been shown that one can drive these systems into a regime where the interaction is dominated by competing two- and three-body potentials. First, the triangular lattice supersolid is considered in Chapter 4. An extensive numerical study combined with a detailed symmetry analysis answers the question of the nature of the supersolid quantum nucleation transition. Following this discussion, Chapter 5 investigates whether the characteristic features of this model persist in the presence of interactions for a realistic system with polar molecules. The three-body interaction regime in a honeycomb Bose-Hubbard model turns out to give rise to a set of highly complex phases in terms of valence bond solids and leads to frustration. Chapter 6 studies the phase diagram and shows how the competition between two- and three-body interactions exhibiting a cascade of phases in the regime of intermediate couplings. In addition, it is demonstrated how the Tensor Network Renormalization Group can be applied to systems with multibody interactions and generic algorithmic limitations are described. Finally, preliminary results concerning the effect of quantum disorder on the triangular supersolid and recent calculations for the entropy of one-dimensional trapped SU(N) Hubbard models are discussed in Chapter 7.
Strongly correlated systems on geometrically frustrated lattices can stabilize a large number of interesting phases that includes a wide array of novel Mott insulators in both bosonic and electronic systems. Charge fluctuations in a Mott insulator are suppressed due to strong mutual interaction among the particles. The presence of frustration is of particular importance as the physics it offers is often rich, unexpectedly complicated, and continues to raise many open questions. The thesis elucidates some of these issues on a kagome lattice where strong interactions among the particles in the Mott phase impose non-trivial local constraints depending on the filling fraction on the lattice. These Mott insulators, in addition to featuring unusual magnetic and/or charge ordering, can also harbor topologically ordered states of quantum matter, e.g., resonating valence bond liquids realized in certain quantum dimer models on non-bipartite lattices. The dimer models can be regarded as low-energy effective theories for different types of bosonic models in the strong-coupling limit. Exploring this connection is a central theme of this thesis with the aim of realizing novel strongly correlated ground states. Past studies of these models have revealed the existence of various ordered and disordered phases with distinct signatures. Among these low-energy phases, the presence of a stable topological liquid at a particular point, known as Rokhsar-Kivelson point, in the phase diagram is notable. The classical versions of the dimer model are also known to have garnered a vast interest in various fields ranging from problems of pure mathematical origin to ones in physical chemistry as well as statistical physics. Pioneered by Kasteleyn, several analytical works came forward to exactly calculate the partition function of the problem from which other physical observables can be derived. Classical numerical methods are extensively applied to these models to verify the analytical predictions. We introduce a new classical algorithm here to compute the correlation functions of a classical dimer model on a square (bipartite) and a triangular (non-bipartite) lattice based on a tensor network construction. The method, called tensor network renormalization group, turns out to be a powerful tool for simulating short-ranged gapped systems as inferred from our results benchmarked against the classical Monte-Carlo technique and compared with past analytical studies. One should note that the quantum dimer model at the Rokhsar-Kivelson point can also be described as an infinite temperature canonical ensemble of classical dimers because of the particular structure of the ground state which is an equal weight superposition in the configuration manifold. The geometry of the lattice plays a pivotal role in deciding the nature of the phases that arise in the dimer models. Many physical properties of the dimer liquid phase can be extracted in the simple classical setting which certainly allows for a deep understanding of the classical models to be developed. The liquid phase is gapped on non-bipartite lattices and gapless on bipartite lattices, which is reflected in the decay of correlation functions with spatial distances. In general on non-bipartite lattices, the topological nature of the dimer liquid is characterized by a Z2 topological order which survives even when the model is perturbed away from the Rokhsar-Kivelson point. Stability of this liquid phase not only depends on the lattice geometries but notably on dimer concentrations also. In this context, we focus on a particular variant of the dimer model on a triangular lattice which is known as the quantum fully packed loop model. The model is composed of nonintersecting closed loops made of dimers and governed by the same Hamiltonian as the quantum dimer model. The loop model provides an effective low-energy description of a strongly correlated bosonic system at 1/3 filling on the kagome lattice. The corresponding Bose-Hubbard Hamiltonian consists of nearest-neighbor hopping and all possible repulsive interactions within a hexagonal plaquette. Conspicuous features of the zero-temperature phase diagram for this model include (i) presence of a stable Z2 liquid even without any Rokhsar-Kivelson potential term (in distinction to the standard quantum dimer model), and (ii) an unconventional phase transition from the liquid phase to a novel crystalline phase that has nematic order (dubbed lattice nematic). For a deeper understanding of the physics, a mapping to an Ising gauge theory is presented. The gauge theoretic description provides a useful way to predict the nature of the quantum phase transition to lie in the O(3) universality class. Finally a fermionic model at the same 1/3 filling is considered in which the ground state exhibits a number of exotic local orderings resulting from the spin-charge interplay of electrons. The Hamiltonian comprises nearest-neighbor hopping, strong on-site Coulomb interaction, and repulsive interaction terms only between nearest-neighbors. In the strong correlation limit, this fermionic problem maps to a two-color fully packed loop model – a model in which the loop segments carry an additional quantum number as color on a honeycomb lattice. The effective theory is governed by coherent three-particle ring exchanges and nearest-neighbor antiferromagnetic spin exchanges. The competition between these two leads to a phase diagram composed of a novel plaquette ordered state (known as the plaquette phase) that undergoes phase transition to a new kind of charge ordered state which we call a short loop phase. From our numerical analysis, we conclude that the plaquette phase features an unusual antiferromagnetic order with gapless spin excitations while the charge-ordered state is subjugated by spin fluctuations of localized electrons arranged in small hexagonal loops on the kagome lattice.
This thesis is focused on many-body localization (MBL) and the development of algorithms using the tensor networks representation for many-body localized systems. Chapter 1 is a detailed discussion on the MBL phenomenon. Chapter 2 gives an introduction to the numerical techniques that are frequently used in studying many-body quantum systems. Chapter 3 studies the shift-and-invert matrix product state method for obtaining highly excited states of MBL systems. Chapter 4 studies an efficient full diagonalization method using Wegner unitary gates. Chapter 5 studies the distribution of the entanglement entropy in the MBL-ergodic transition region. Chapter 6 studies the the influence of pseudo-spin SU (2) symmetry on the MBL phase. Chapter 7 is a side project that studies the influence of the duality-twisted boundary conditions on the critical twist defect chains. The MBL-ergodic transition is a dynamical phase transition which happens at finite energy density for a disordered and isolated many-body interacting system. It is a natural extension of Anderson localization where the localized phase survives the inter-particle interaction. In Chapter 1, we describe its relation to quantum thermalization, typical lattice models that harbor a MBL phase, MBL system’s phenomenological description, the characteristics of the MBL phase, experimental studies of MBL phenomenon in 1D and higher dimensions, the latest developments and the open questions on the nature of the MBL-ergodic transition. In Chapter 2, we provide a short but self-contained discussion of basic numerical methods that are frequently used in studying MBL physics. We mainly focus on the exact diagonalization methods and the matrix product state (MPS) representation, and explain their advantages and disadvantages. In Chapter 3, because obtaining the interior eigenstates of MBL Hamiltonians is a key step to studying the MBL phases and the transitions, we develop algorithms that can capture individual excited states to high fidelity. Using the fact that eigenstates of many-body localized systems have area-law entanglement and can therefore be efficiently represented as a matrix product state (MPS), we designed two algorithms that can generate excited states in the MPS representation, and use them to test the basic properties of MBL in the regime of large one-dimensional random field Heisenberg chains that were previously inaccessible due to the limitations of exact diagonalization (ED). In Chapter 4, we investigate an efficient way to diagonlize a fully many-body localized (FMBL) Hamiltonian. Using the Wegner flow technique for subsystems, we constructively build unitary gates that can greedily transform a local part of the FMBL Hamiltonian with the aim of lowering its average energy variance. Applying the Wegner unitary successively can make the system “flow” towards the diagonalized form. We find that the performance of this method is mainly controlled by the length of each individual Wegner unitary gate, and gets better with increasing length of the gate. We compare the performance of this constructive method with previous methods based on quasi-Newtonian optimization methods, and find that they have similar performance, while our constructive method is far more efficient. We also comment the possibility of constructing a tensor network state that resembles a multi-scale entanglement renormalization ansatz (MERA) using this constructive method. In Chapter 5, using strong subadditivity (SSA) theorem, we develop two order parameters – cut-averaged entanglement entropy (CAEE) and its slope (SCAEE), and show that they can be used to directly identify the volume or area law scaling in single eigenstates. We study the distribution of the SCAEE over disorder realizations. This distribution appears Gaussian at weak disorder, while on the MBL side, the distribution of the SCAEE is peaked at zero slope and has an exponential tail. In the critical regime we find that the distribution of the SCAEE is bimodal both over multiple disorder realizations as well as for single disorder realizations. The variance of the SCAEE distribution in the transition region seems to grow with system size when considered over disorder realizations. Our system sizes are too small to pin down its maximal value, but they are consistent with (among other possibilities) the maximal variance possible which would lead to half the states having zero SCAEE and half having maximal SCAEE. This scenario would lead to an entanglement entropy at the transition which scales as a volume law with half its thermal value. In Chapter 6, we consider the strongly disordered one-dimensional Hubbard model with spin disorder, which, as we argue, cannot be categorized as a conventional MBL phase. Though the spin rotation SU (2) symmetry is broken by the spin disorder, this model still preserves the pseudo-spin SU (2) symmetry under periodic boundary conditions. On the theoretical side, using the pseudo-spin algebra we show that a significant number of the excited states at any disorder strength have logarithmic correction to their von Neumann entanglement entropy, which violates the area-law entanglement for a typical MBL phase. A common feature of this group of excited states is that they usually have far more double occupancies than single occupancies. On the numerical side, we studied the time evolution after quantum quench from two extreme cases of product states – all single occupancies at quarter filling, and all double occupancies at half filling. We find in the former case entanglement entropy behaves as in a typical MBL system, while the latter is clearly delocalized, which suggests the existence of incomplete set of local integrals of motion. With the above evidence, it is convincing that this model provides a playground for studying a non-ergodic, non-MBL phase, and its relation to continuous SU (2) symmetry. Chapter 7 deviates from the common scene of MBL physics. In this chapter, we consider a series of one-dimensional critical twist defect models with various boundary conditions. These twist defects are embedded in the background of a generalization of Kitaev Z 2 toric code model, and can also be understood as a discrete Z k gauge theory in its deconfined phase. We analytically construct the duality transformation of these critical models, and use numerical methods to extract information on the central charges, sound velocities, and conformal dimensions. With the numerical data, we show the relation between the CFT contents of the critical chains with periodic boundary conditions and the critical chains with duality-twisted boundary conditions.
Many strongly correlated quantum systems are difficult to study numerically because of the exponential growth of the Hilbert space combined with the failure of numerical methods in important parameter regimes. This includes the long-standing problems of high-temperature superconductivity and frustrated magnetism. A promising solution to this dilemma are quantum simulators, well-controlled (synthetic) quantum systems, emulating model Hamiltonians not only of condensed matter systems, where parameters can be tuned at will and model complexity can be added step by step. This dissertation presents numerical simulation studies that are motivated by quantum simulators. Using large-scale quantum Monte Carlo computer simulations and mean-field theory the semiquantitative ground state phase diagram of a Penning ion trap quantum simulator, realizing a two-dimensional model of quantum magnetism with long-range interactions, is obtained. We study also thermal and quantum phase transitions in one-dimensional transverse-field Ising models with long-range interactions, which exhibit a Kosterlitz-Thouless transition and universality classes with continuously varying critical exponents. The second part of this thesis is concerned with the Fermi-Hubbard model, which has been realized recently with ultracold atoms in optical lattices, both for repulsive and attractive interactions. Equipped with a quantum gas microscope, which allows single-site single-atom detection, these experimental setups can collect histograms of the particle number from repeated projective measurements. This dissertation presents a method to compute such full counting statistics for quadratic operators such as particle number or magnetization in the framework of determinantal quantum Monte Carlo. From an analysis of the full counting statistics of the particle number for different subsystem sizes the size of preformed pairs or Cooper pairs in the attractive Hubbard model is inferred. Furthermore, excellent agreement of the numerically computed probability distribution of the staggered magnetization with a recent quantum gas experiment demonstrates that current experiments are capable of resolving the differences between the Hubbard model and its limiting Heisenberg model. An extensive method section reviews the stochastic series expansion quantum Monte Carlo method for Ising models in a transverse field, which is free of the sign problem for arbitrary interactions, with special emphasis on the efficient simulation of long-range interactions. Also a self-contained introduction to the determinantal quantum Monte Carlo method is provided.
The physics of strongly interacting particles is often extraordinarily rich, leading to the \textit{emergence} of fundamentally new phases and phenomena that cannot exist in the absence of correlations. This thesis is devoted to the study of different classes of quantum systems with such strong correlations. Programmable quantum simulators based on Rydberg atom arrays have recently emerged as versatile platforms to probe quantum many-body phases and their dynamics. In Chapter 2, we start with theories of quantum phase transitions (QPTs) in one-dimensional chains of Rydberg atoms. We demonstrate the existence of a novel QPT in the universality class of the Z3 chiral clock model, calculate its critical exponents, and discuss their measurement with the quantum Kibble-Zurek mechanism. Proceeding to two dimensions, Chapter 3 describes the zero-temperature phase diagram of a square-lattice Rydberg atom array, revealing several experimentally realizable density-wave-ordered phases and QPTs. For a different geometry, namely, the kagome lattice studied in Chapter 4, we identify a regime that constitutes a likely candidate for hosting a phase with long-range quantum entanglement and topological order. Based on mappings to dimer models and gauge theories, we argue for the existence of a quantum spin liquid phase that has proved elusive in solid-state materials. In the next part of this thesis, we focus on similarly strongly correlated phases in the realm of quantum materials. Motivated by recent observations of an enhanced thermal Hall response in the high-temperature cuprate superconductors, in Chapter 5, we study the thermal Hall conductivity, κxy, of chiral spin-liquid ans\"{a}tze on the square lattice, focusing on the contribution from neutral (bosonic or fermionic) spinons. Thereafter, going beyond mean-field theory, we consider the corrections due to emergent gauge fields, which yield a response of opposite sign to that of the fermionic matter. We additionally consider the contribution of phonons to κxy and establish its relation to the spinon Hall viscosity. Thermal Hall measurements also leave intriguing fingerprints in Kitaev materials like α-RuCl3, and in this context, Chapter 6 investigates the unquantized κxy obtained in a gapless spinon Fermi surface state.