Strongly correlated low-dimensional quantum spin models provide a well-established frame- \nwork to study magnetic properties of insulators, and are of great theoretical interest and experimental relevance in condensed-matter physics. In this thesis, I use quantum Monte Carlo methods to numerically study quantum critical behavior in low-dimensional quantum spin models and wavefunctions. \n \nFirst, I study spinons &ndash emergent spin-1/2 bosonic excitations &ndash at certain one- and two-dimensional quantum phase transitions (QPTs) in spin models, by characterizing their size and confinement length quantitatively. In particular, I focus on the QPT from an antiferromagnetic (AFM) phase into a valence-bond solid (VBS) phase, which is an example of a violation of the standard Landau-Ginzburg-Wilson paradigm for phase transitions. This transition in two dimensions (2D) is instead likely described by a novel theory called "deconfined quantum criticality" (DQC). According to the theory, spinons should be deconfined. The degree of deconfinement is quantified in my calculations. \n \nSecond, I present a comprehensive study of so-called short-bond resonating-valence-bond (RVB) spin liquids in 2D, which have been suggested as a good starting point for understanding the spin physics of high-temperature cuprates. I find that these RVB states can also be classified as quantum-critical VBS states, which indicates that RVB is less disordered than expected. This work suggests a possible mapping from the quantum RVB states to classical dimer models via a classical continuum field theory--the height model. This map explicitly bridges well-established classical results to future quantum studies. \n \nThird, I consider 1D amplitude product (AP) states, which are generalized versions of RVB states, with different wavefunction weightings of bonds according to their lengths. AP states constitute a good ansatz for certain Hamiltonians and are of broad interest in quantum magnetism. I study phase transitions from AFM-VBS phases in AP states by tuning their amplitudes, and obtain continuously varying critical exponents. In addition, I classify the 1D AP states through entanglement entropy calculations of the central charge in (1+1)D conformal field theory. This new classification could serve as guide for AP states as trial wavefunctions to search for ground states of corresponding quantum spin models.
Strongly correlated low-dimensional quantum spin models provide a well-established frame- \nwork to study magnetic properties of insulators, and are of great theoretical interest and experimental relevance in condensed-matter physics. In this thesis, I use quantum Monte Carlo methods to numerically study quantum critical behavior in low-dimensional quantum spin models and wavefunctions. \n \nFirst, I study spinons &ndash emergent spin-1/2 bosonic excitations &ndash at certain one- and two-dimensional quantum phase transitions (QPTs) in spin models, by characterizing their size and confinement length quantitatively. In particular, I focus on the QPT from an antiferromagnetic (AFM) phase into a valence-bond solid (VBS) phase, which is an example of a violation of the standard Landau-Ginzburg-Wilson paradigm for phase transitions. This transition in two dimensions (2D) is instead likely described by a novel theory called "deconfined quantum criticality" (DQC). According to the theory, spinons should be deconfined. The degree of deconfinement is quantified in my calculations. \n \nSecond, I present a comprehensive study of so-called short-bond resonating-valence-bond (RVB) spin liquids in 2D, which have been suggested as a good starting point for understanding the spin physics of high-temperature cuprates. I find that these RVB states can also be classified as quantum-critical VBS states, which indicates that RVB is less disordered than expected. This work suggests a possible mapping from the quantum RVB states to classical dimer models via a classical continuum field theory--the height model. This map explicitly bridges well-established classical results to future quantum studies. \n \nThird, I consider 1D amplitude product (AP) states, which are generalized versions of RVB states, with different wavefunction weightings of bonds according to their lengths. AP states constitute a good ansatz for certain Hamiltonians and are of broad interest in quantum magnetism. I study phase transitions from AFM-VBS phases in AP states by tuning their amplitudes, and obtain continuously varying critical exponents. In addition, I classify the 1D AP states through entanglement entropy calculations of the central charge in (1+1)D conformal field theory. This new classification could serve as guide for AP states as trial wavefunctions to search for ground states of corresponding quantum spin models.
We study a number of quantum phase transitions, which are exotic in their nature and separates non-trivial phases of matter. Since quantum fluctuations, which drive these phase transitions, are stronger in low-dimensions, we concentrate on low-dimensional systems. We consider two different two-dimensional systems in this thesis and study their phase transition. \n\nFirst, we investigate a phase transition in graphene, one of the most famous two-dimensional systems in condensed matter. For a suspended bilayer graphene in ν = 0 quantum Hall regime, the conductivity data and mean-field analysis suggests a phase transition from an antiferromagnetic (AF) state to a valence bond solid (VBS) state, when perpendicular electric field is increased. This AF to VBS phase transition is reminiscent of deconfined criticality, which is a novel phase transition that cannot be explained by Landau’s theory of symmetry breaking. We show that in the strong coupling regime of bilayer graphene, the AF state is destabilized by the transverse electric field, likely resulting in a VBS state. We also consider monolayer and bilayer graphene in the large cyclotron gap limit and show that the effective action for the AF and VBS order parameters have a topological Wess-Zumino-Witten term, supporting that the phase transition observed in experiments is in the deconfined criticality class. \n\nSecond, we study the model systems of cuprate superconductor, which is effectively a two-dimensionalal system in the CuO_2 plane. The proposal that the pseudogap metal is a fractionalized Fermi liquid described by a quantum dimer model is extended using the density matrix renormalization group. Measuring the Friedel oscillations in the open boundaries reveals that the fermionic dimers have dispersion minima near (π/2,π/2), which is compatible with the Fermi arcs in photoemission. Moreover, investigating the entanglement entropy suggests that the dimer model with low fermion density is similar to the free fermion system above the Lifshitz transition. We also study the phase transition from a metal with SU(2) spin symmetry to an AF metal. By applying the functional renormalization group to the two-band spin-fermion model, we establish the existence of a strongly coupled fixed point and calculate critical exponents of the fixed point.
Observing constituent particles with fractional quantum numbers in confined and deconfined states is an interesting and challenging problem in quantum many-body physics. Here, we further explore a computational scheme [Y. Tang and A. W. Sandvik, Phys. Rev. Lett. 107, 157201 (2011)] based on valence-bond quantum Monte Carlo simulations of quantum spin systems. Using several different one-dimensional models, we characterize S=21 spinon excitations using the intrinsic spinon size $\ensuremath{\lambda}$ and confinement length $\mathrm{\ensuremath{\Lambda}}$ (the size of a bound state). The spinons have finite size in valence-bond-solid states, infinite size in the critical region (with overlaps characterized by power laws), and become ill defined (completely unlocalizable) in the N\'eel state (which we stabilize in one dimension by introducing long-range interactions). We also verify that pairs of spinons are deconfined in uniform spin chains but become confined upon introducing a pattern of alternating coupling strengths (dimerization) or coupling two chains (forming a ladder). In the dimerized system, an individual spinon can be small when the confinement length is large; this is the case when the imposed dimerization is weak but the ground state of the corresponding uniform chain is a spontaneously formed valence-bond solid (where the spinons are deconfined). Based on our numerical results, we argue that a system with $\ensuremath{\lambda}\ensuremath{\ll}\mathrm{\ensuremath{\Lambda}}$ is associated with weak repulsive short-range spinon-spinon interactions. In principle, both the length scales $\ensuremath{\lambda}$ and $\mathrm{\ensuremath{\Lambda}}$ can still be individually tuned from small to infinite (with $\ensuremath{\lambda}\ensuremath{\le}\mathrm{\ensuremath{\Lambda}}$) by varying model parameters. In contrast, in the ladder system the two lengths are always similar, and this is the case also in the weakly dimerized systems when the corresponding uniform chain is in the critical phase. In these systems, the effective spinon-spinon interactions are purely attractive and there is only a single large length scale close to criticality, which is reflected in the standard spin correlations as well as in the spinon characteristics.
The interplay between lattice gauge theories and fermionic matter accounts for fundamental physical phenomena ranging from the deconfinement of quarks in particle physics to quantum spin liquid with fractionalized anyons and emergent gauge structures in condensed matter physics.However, except for certain limits (for instance, a large number of flavors of matter fields), analytical methods can provide few concrete results.Here we show that the problem of compact Uð1Þ lattice gauge theory coupled to fermionic matter in ð2 þ 1ÞD is possible to access via sign-problem-free quantum Monte Carlo simulations.One can hence map out the phase diagram as a function of fermion flavors and the strength of gauge fluctuations.By increasing the coupling constant of the gauge field, gauge confinement in the form of various spontaneous-symmetrybreaking phases such as the valence-bond solid (VBS) and Néel antiferromagnet emerge.Deconfined phases with algebraic spin and VBS correlation functions are also observed.Such deconfined phases are incarnations of exotic states of matter, i.e., the algebraic spin liquid, which is generally viewed as the parent state of various quantum phases.The phase transitions between the deconfined and confined phases, as well as that between the different confined phases provide various manifestations of deconfined quantum criticality.In particular, for four flavors N f ¼ 4, our data suggest a continuous quantum phase transition between the VBS and Néel order.We also provide preliminary theoretical analysis for these quantum phase transitions.
Classical and quantum spin systems are widely used models in both experimental and theoretical condensed matter physics. In many materials, the electronic interactions can be difficult to model exactly. However, in some insulators (Mott insulators), the magnetic (spin) interactions can be captured well with spin-only models. Several models are studied in this thesis. First, I report the solution of a long-standing issue in a classical frustrated spin model (i.e., where the quantum effects are neglected and the complexity is due to competing interactions): the nature of the thermal phase transition to a stripe state in the two dimensional (2D) J1-J2 model. Here J1 and J2 are nearest- and next-nearest-neighbor couplings. Monte Carlo simulations with single-spin updates are used for the calculations, and an extended-ensemble method, a generalization of the Wang-Landau algorithm, is also developed and tested. I focus on the study of "weak universality" behavior (continuously varying critical exponents, with one of the exponents staying fixed), for which I show a correspondence with a known class of conformal field theories with charge c = 1. Next, moving to quantum spins, to shed light on magnetic systems studied experimentally and to investigate new types of quantum states of interest in developing theories of quantum magnetism, I study the S = 1/2 Heisenberg model on the 2D square lattice with added six-spin interactions (the so-called J-Q3 model) as well as a set of 3D quantum antiferromagnets on dimerized lattices. Here I use the stochastic series expansion quantum Monte Carlo method. In the study of the J-Q3 model, I report on a similar weak-universality behavior as in the classical J1-J2 model, but with a mapping to a different known class of c = 1 conformal field theories. The critical behavior of the system again shows continuously changing exponents, in a way which corresponds to a gradual weakening of Z4 symmetry-breaking to an emergent U(1) symmetry. In the study of dimerized antiferromagnets, I report on a universal behavior of the Néel temperature TN, which can be related to ground state parameters independently of the microscopic interaction details in several different models.
We perform a numerical study of a spin-1/2 model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry in one dimension which demonstrates an interesting similarity to the physics of two-dimensional deconfined quantum critical points (DQCP). Specifically, we investigate the quantum phase transition between Ising ferromagnetic and valence bond solid (VBS) symmetry-breaking phases. Working directly in the thermodynamic limit using uniform matrix product states, we find evidence for a direct continuous phase transition that lies outside of the Landau-Ginzburg-Wilson paradigm. In our model, the continuous transition is found everywhere on the phase boundary. We find that the magnetic and VBS correlations show very close power-law exponents, which is expected from the self-duality of the parton description of this DQCP. Critical exponents vary continuously along the phase boundary in a manner consistent with the predictions of the field theory for this transition. We also find a regime where the phase boundary splits, as suggested by the theory, introducing an intermediate phase of coexisting ferromagnetic and VBS order parameters. Interestingly, we discover a transition involving this coexistence phase which is similar to the DQCP, being also disallowed by the Landau-Ginzburg-Wilson symmetry-breaking theory.
We use Monte Carlo methods to study spinons in two-dimensional quantum spin systems, characterizing their intrinsic size $\ensuremath{\lambda}$ and confinement length $\ensuremath{\Lambda}$. We confirm that spinons are deconfined, $\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ and $\ensuremath{\lambda}$ finite, in a resonating valence-bond spin-liquid state. In a valence-bond solid, we find finite $\ensuremath{\lambda}$ and $\ensuremath{\Lambda}$, with $\ensuremath{\lambda}$ of a single spinon significantly larger than the bound state---the spinon is soft and shrinks as the bound state is formed. Both $\ensuremath{\lambda}$ and $\ensuremath{\Lambda}$ diverge upon approaching the critical point separating valence-bond solid and N\'eel ground states. We conclude that the spinon deconfinement is marginal in the lowest-energy state in the spin-1 sector, due to weak attractive spinon interactions. Deconfinement in the vicinity of the critical point should occur at higher energies.