In three research articles we have studied the critical properties of effective lattice models for strongly correlated electron systems by Monte Carlo simulations. A similar model is used in a fourth article for investigating thermal fluctuations of vortices in a rotating Bose–Einstein condensate. In the first part of this thesis we review the necessary background and introduce the models one by one. The last part is a collection of the papers.\n\nPaper I [1]: We consider the scaling of the mean square dipole moment in a plasma with logarithmic interactions in a two- and three-dimensional system. In both cases, we establish the existence of a low-temperature regime where the mean square dipole moment does not scale with system size and a hightemperature regime does scale with system size. Thus, there is a nonanalytic change in the polarizability of the system as a function of temperature, and hence a metal-insulator transition in both cases. The relevance of this transition in three dimensions to quantum phase transitions in 2 + 1-dimensional systems is briefly discussed.\n\nPaper II [2]: The existence of a discontinuity in the inverse dielectric constant of the two-dimensional Coulomb gas is demonstrated on purely numerical grounds. This is done by expanding the free energy in an applied twist and performing a finite-size scaling analysis of the coefficients of higher-order terms. The phase transition, driven by unbinding of dipoles, corresponds to the Kosterlitz-Thouless transition in the 2D XY model. The method developed is also used for investigating the possibility of a Kosterlitz-Thouless phase transition in a threedimensional system of point charges interacting with a logarithmic pair-potential, a system related to effective theories of low-dimensional strongly correlated systems. We also contrast the finite-size scaling of the fluctuations of the dipole moments of the two-dimensional Coulomb gas and the three-dimensional logarithmic system to those of the three-dimensional Coulomb gas.\n\nPaper III [3]: We perform large-scale Monte Carlo simulations on an effective gauge theory for an easy plane quantum anti-ferromagnet, including a Berry phase term that projects out the S = 1/2 sector. Without a Berry phase term, the model exhibits a phase transition in the 3DXY universality class associated with proliferation of gauge-charge neutral U(1) vortices. The instantons that eliminate the phase transition in the gauge-charged sector are cancelled by the Berry phases. The result is a first order phase transition. This gauge theory therefore does not exhibit deconfined criticality.\n\nPaper IV [4]: We perform Monte Carlo studies of vortices in three dimensions in a cylindrical confinement, with uniform and nonuniform density. The former is relevant to rotating 4He, the latter is relevant to a rotating trapped Bose condensate. In the former case we find dominant angular thermal vortex fluctuations close to the cylinder wall. For the latter case, a novel effect is that at low temperatures the vortex solid close to the center of the trap crosses directly over to a tension-less vortex tangle near the edge of the trap. At higher temperatures an intermediate tensionful vortex liquid located between the vortex solid and the vortex tangle, may exist.
In three research articles we have studied the critical properties of effective lattice models for strongly correlated electron systems by Monte Carlo simulations. A similar model is used in a fourth article for investigating thermal fluctuations of vortices in a rotating Bose–Einstein condensate. In the first part of this thesis we review the necessary background and introduce the models one by one. The last part is a collection of the papers.\n\nPaper I [1]: We consider the scaling of the mean square dipole moment in a plasma with logarithmic interactions in a two- and three-dimensional system. In both cases, we establish the existence of a low-temperature regime where the mean square dipole moment does not scale with system size and a hightemperature regime does scale with system size. Thus, there is a nonanalytic change in the polarizability of the system as a function of temperature, and hence a metal-insulator transition in both cases. The relevance of this transition in three dimensions to quantum phase transitions in 2 + 1-dimensional systems is briefly discussed.\n\nPaper II [2]: The existence of a discontinuity in the inverse dielectric constant of the two-dimensional Coulomb gas is demonstrated on purely numerical grounds. This is done by expanding the free energy in an applied twist and performing a finite-size scaling analysis of the coefficients of higher-order terms. The phase transition, driven by unbinding of dipoles, corresponds to the Kosterlitz-Thouless transition in the 2D XY model. The method developed is also used for investigating the possibility of a Kosterlitz-Thouless phase transition in a threedimensional system of point charges interacting with a logarithmic pair-potential, a system related to effective theories of low-dimensional strongly correlated systems. We also contrast the finite-size scaling of the fluctuations of the dipole moments of the two-dimensional Coulomb gas and the three-dimensional logarithmic system to those of the three-dimensional Coulomb gas.\n\nPaper III [3]: We perform large-scale Monte Carlo simulations on an effective gauge theory for an easy plane quantum anti-ferromagnet, including a Berry phase term that projects out the S = 1/2 sector. Without a Berry phase term, the model exhibits a phase transition in the 3DXY universality class associated with proliferation of gauge-charge neutral U(1) vortices. The instantons that eliminate the phase transition in the gauge-charged sector are cancelled by the Berry phases. The result is a first order phase transition. This gauge theory therefore does not exhibit deconfined criticality.\n\nPaper IV [4]: We perform Monte Carlo studies of vortices in three dimensions in a cylindrical confinement, with uniform and nonuniform density. The former is relevant to rotating 4He, the latter is relevant to a rotating trapped Bose condensate. In the former case we find dominant angular thermal vortex fluctuations close to the cylinder wall. For the latter case, a novel effect is that at low temperatures the vortex solid close to the center of the trap crosses directly over to a tension-less vortex tangle near the edge of the trap. At higher temperatures an intermediate tensionful vortex liquid located between the vortex solid and the vortex tangle, may exist.
The existence of a discontinuity in the inverse dielectric constant of the two-dimensional (2D) Coulomb gas is demonstrated on purely numerical grounds. This is done by expanding the free energy in an applied twist and performing a finite-size scaling analysis of the coefficients of higher-order terms. The phase transition, driven by unbinding of dipoles, corresponds to the Kosterlitz-Thouless transition in the 2D XY model. The method developed is also used for investigating the possibility of a Kosterlitz-Thouless phase transition in a three-dimensional system of point charges interacting with a logarithmic pair-potential, a system related to effective theories of low-dimensional strongly correlated systems. We also contrast the finite-size scaling of the fluctuations of the dipole moments of the two-dimensional Coulomb gas and the three-dimensional logarithmic system to those of the three-dimensional Coulomb gas.
Phase transitions and their critical scaling properties, especially in systems with disorder, are important both for our theoretical understanding of our environment, but also for their practical use in applications and materials in our everyday life. This thesis presents results from finite size scaling analysis of critical phenomena in systems with disorder, using high-precision Monte Carlo simulations and state of the art numerical methods. Specifically, theoretical models suitable for simulations in the presence of uncorrelated or correlated disorder are studied. Uncorrelated strong disorder, as present in the two dimensional gauge glass model to study the vortex glass phase of high temperature superconductors in an applied magnetic field is shown to lack a finite temperature phase transition. Further, results from dynamic quantities, such as resistance and autocorrelation functions, indicate the existence of two distinct diverging correlation times, one associated with local relaxation and one associated with vortex phase slips. Correlated disorder is studied both in the superfluid transition of helium-4 and in the anisotropic critical scaling of a transverse Meissner-like transition in an experimental setup of a high temperature superconductor. For the superfluid helium transition, it is shown that the presence of fractally correlated disorder presumably alters the universality class of the pure model. Also, a comparison with experimental data suggests that the critical scaling theory describing the heat capacity of helium-4 may need to be modified in the presence of the disorder. In the case of superconductors, analyzing experimental data from resistance measurements in a system with columnar defects together with an anisotropy in the applied magnetic field, reveals a fully anisotropic scaling regime. Finally, a data analysis is presented from simulations of a charged particle gas system in three dimensions, where the normal Coulomb interaction between charges is changed into a logarithmic interaction. Previous work indicates the possibility of a transition similar to the Kosterlitz-Thouless transition in certain two dimensional systems. On the contrary, our simulations seem to favor a system whose critical scaling behavior is consistent with a transition occurring only at zero critical temperature. Overall, disorder in the model systems studied leads to important modifications of the critical scaling properties of pure systems, and thereby also to possible changes of the corresponding universality classes. This results in interesting predictions with experimentally relevant consequences.
Phase transitions and their critical scaling properties, especially in systems with disorder, are important both for our theoretical understanding of our environment, but also for their practical use in applications and materials in our everyday life. This thesis presents results from finite size scaling analysis of critical phenomena in systems with disorder, using high-precision Monte Carlo simulations and state of the art numerical methods. Specifically, theoretical models suitable for simulations in the presence of uncorrelated or correlated disorder are studied. Uncorrelated strong disorder, as present in the two dimensional gauge glass model to study the vortex glass phase of high temperature superconductors in an applied magnetic field is shown to lack a finite temperature phase transition. Further, results from dynamic quantities, such as resistance and autocorrelation functions, indicate the existence of two distinct diverging correlation times, one associated with local relaxation and one associated with vortex phase slips. Correlated disorder is studied both in the superfluid transition of helium-4 and in the anisotropic critical scaling of a transverse Meissner-like transition in an experimental setup of a high temperature superconductor. For the superfluid helium transition, it is shown that the presence of fractally correlated disorder presumably alters the universality class of the pure model. Also, a comparison with experimental data suggests that the critical scaling theory describing the heat capacity of helium-4 may need to be modified in the presence of the disorder. In the case of superconductors, analyzing experimental data from resistance measurements in a system with columnar defects together with an anisotropy in the applied magnetic field, reveals a fully anisotropic scaling regime. Finally, a data analysis is presented from simulations of a charged particle gas system in three dimensions, where the normal Coulomb interaction between charges is changed into a logarithmic interaction. Previous work indicates the possibility of a transition similar to the Kosterlitz-Thouless transition in certain two dimensional systems. On the contrary, our simulations seem to favor a system whose critical scaling behavior is consistent with a transition occurring only at zero critical temperature. Overall, disorder in the model systems studied leads to important modifications of the critical scaling properties of pure systems, and thereby also to possible changes of the corresponding universality classes. This results in interesting predictions with experimentally relevant consequences.
We have studied the critical properties of three-dimensional U(1)-symmetric lattice gauge theories. The models apply to various physical systems such as insulating phases of strongly correlated electron systems as well as superconducting and superfluid states of liquid metallic hydrogen under extreme pressures. This thesis contains an introductory part and a collection of research papers of which seven are published works and one is submitted for publication.\n\nPaper I: Critical properties of the 2+1-dimensional compact abelian Higgs model with gauge charge q=2 are studied. We introduce a novel method of computing the third moment M3 of the action which allows us to extract correlation length and specific heat critical exponents ν and α without invoking hyperscaling. Finite-size scaling analysis of M3 yields the ratio (1+α)/ν and 1/ν separately. We find that α and ν vary along the critical line of the theory, which however exhibits a remarkable resilience of Z2 criticality. We conclude that the model is a fixed-line theory, which we propose to characterize the zero temperature quantum phase transition from a Mott-Hubbard insulator to a charge fractionalized insulator in two spatial dimensions.\n\nPaper II: Large scale Monte Carlo simulations are employed to study phase transitions in the three-dimensional compact abelian Higgs model in adjoint representations of the matter field, labeled by an integer q, for q=2,3,4,5. We also study various limiting cases of the model, such as the Zq lattice gauge theory, dual to the 3DZq spin model, and the 3D xy spin model which is dual to the Zq lattice gauge theory in the limit q → ∞. In addition, for benchmark purposes, we study the 2D square lattice 8-vertex model, which is exactly solvable and features non-universal critical exponents. The critical exponents α and ν are calculated from finite size scaling of the third moment of the action, and the method is tested thoroughly on models with known values for these exponents. We have found that for q=3, the three-dimensional compact abelian Higgs model exhibits a second order phase transition line which joins a first order phase transition line at a tricritical point. The results for q=2 in Paper I are reported with a higher lever of detail.\n\nPaper III: This paper is based on a talk by F. S. Nogueira in the Aachen HEP 2003 conference where a review of the results for the compact abelian Higgs model from Paper I and Paper II was presented, as well as the results for the q=1 case studied by F. S. Nogueira, H. Kleinert and A. Sudbø.\n\nPaper IV: We study the effects of a Chern-Simons (CS) term in the phase structure of two different abelian gauge theories in three dimensions. By duality transformations we show how the compact U(1) gauge theory with a CS term for certain values of the CS coupling can be written as a gas of vortex loops interacting through steric repulsion. This theory is known to exhibit a phase transition governed by proliferation of vortex loops. We also employ Monte Carlo simulations to study the non-compact U(1) abelian Higgs model with a CS term. Finite size scaling of the third moment of the action yields critical exponents α and ν that vary continuously with the strength of the CS term, and a comparison with available analytical results is made.\n\nPaper V: The critical properties of N-component Ginzburg-Landau theory are studied in d=2+1 dimensions. The model is dualized to a theory of N vortex fields interacting through a Coulomb and a screened potential. The model with N=2 shows two anomalies in the specific heat. From Monte Carlo simulations we calculate the critical exponents α and ν and the mass of the gauge field. We conclude that one anomaly corresponds to an inverted 3D xy fixed point, while the other corresponds to a 3D xy fixed point. There are N fixed points, namely one corresponding to an inverted 3D xy fixed point, and N-1corresponding to neutral 3D xy fixed points. Applications are briefly discussed.\n\nPaper VI: The phase diagram and critical properties of the N-component London superconductor are studied both analytically and through large-scale Monte-Carlo simulations in d=2+1 dimensions. The model with different bare phase stiffnesses for each flavor is a model of superconductivity which should arise out of metallic phases of light atoms under extreme pressure. A projected mixture of electronic and protonic condensates in liquid metallic hydrogen under extreme pressure is the simplest example, corresponding to N=2 with individually conserved matter fields. We compute critical exponents α and ν for N=2 and N=3. The results from Paper V are presented at a higher level of detail. For the arbitrary N case, there are N fixed points,namely one charged inverted 3D xy fixed point, and N-1 neutral 3D xy fixed points. We explicitly identify one charged vortex mode and N-1 neutral vortex modes. The model for N=2 and equal bare phase stiffnesses corresponds to a field theoretical description of an easy-plane quantum antiferromagnet. In this case, the critical exponents are computed and found to be non 3D xy values. Furthermore, we study the model in an external magnetic field, and find a novel feature, namely N-1 superfluid phases arising out of N charged condensates. In particular, for N=2 we point out the possibility of two novel types of field-induced phase transitions in ordered quantum fluids: i) A phase transition from a superconductor to a superfluid or vice versa, driven by tuning an external magnetic field. This identifies the superconducting phase of liquid metallic hydrogen as a novel quantum fluid. ii) A phase transition corresponding to a quantum fluid analogue of sublattice melting, where a composite field-induced Abrikosov vortex lattice is decomposed and disorders the phases of the constituent condensate with lowest bare phase stiffness. Both transitions belong to the 3D xy universality class.\n\nPaper VII: We consider the vortex superconductor with two individually conserved condensates in a finite magnetic field. The ground state is a lattice of cocentered vortices in both order parameters. We find two novel phase transitions when temperature is increased at fixed magnetic field. i) A "vortex sublattice melting" transition where vortices in the field with lowest phase stiffness ("light vortices") loose cocentricity with the vortices with large phase stiffness ("heavy vortices"), entering a liquid state (the structure factor of the light vortex sublattice vanishes continuously.) This transition is in the 3D xy universality class. ii) A first order melting transition of the lattice of heavy vortices in a liquid of light vortices.\n\nPaper VIII: We report on large-scale Monte Carlo simulations of a novel type of a vortex matter phase transition which should take place in a three dimensional two-component superconductor. We identify the regime where first, at a certain temperature a field-induced lattice of co-centered vortices of both order parameters melts, causing the system to loose superconductivity. In this state the two-gap system retains a broken composite symmetry and we observe that at a higher temperature it undergoes an extra phase transition where the disordered composite one-flux-quantum vortex lines are "ionized" into a "plasma" of constituent fractional flux vortex lines in individual order parameters. This is the hallmark of the superconductor-to-superfluid-to-normal fluid phase transitions projected to occur in e.g. liquid metallic hydrogen.
In this thesis, we investigate several topics pertaining to emergent collective quantum phenomena in the domain of correlated fermions, using the quantum Monte Carlo method. They display exotic low temperature phases as well as phase transitions which are beyond the Landau–Ginzburg theory. The interplay between three key points is crucial for us: fermion statistics, many body effects and topology. We highlight the following several achievements: 1. Successful modeling of continuum field theories with lattice Hamiltonians, 2. their sign-problem-free Monte Carlo simulations of these models, 3. and numerical results beyond mean field descriptions. First, we consider a model of Dirac fermions with a spin rotational invariant inter- action term that dynamically generates a quantum spin Hall insulator. Surprisingly, an s-wave superconducting phase emerges due to the condensation of topological de- fects of the spin Hall order parameter. When particle-hole symmetry is present, the phase transition between the topological insulator and the superconducting phase is an example of a deconfined quantum critical point(DQCP). Although its low energy effec- tive field theory is purely bosonic, the exact conservation law of the skyrmion number operator rules out the possibility of realizing this critical point in lattice boson models. This work is published in Ref. [1]. Second, we dope the dynamically generated quantum spin Hall insulator mentioned above. Hence it is described by a field theory without Lorentz invariance due to the lack of particle-hole symmetry. This sheds light on the extremely hot topic of twisted bilayergraphene: Why is superconductivity generated when the repulsive Coulomb interaction is much stronger than the electron-phonon coupling energy scale? In our case, Cooper pairs come from the topological skyrmion defects of the spin current order parameter, which are charged. Remarkably, the nature of the phase transition is highly non-mean-field-like: one is not allowed to simply view pairs of electrons as single bosons in a superfluid-Mott insulator transition, since the spin-current order parameter can not be ignored. Again, due to the aforementioned skyrmions, the two order parameters are intertwined: One phase transition occurs between the two symmetry breaking states. This work is summarized in Ref. [2]. Third, we investigate the 2 + 1 dimensional O(5) nonlinear sigma model with a topological Wess-Zumino-Witten term. Remarkably, we are able to perform Monte Carlo calculations with a UV cutoff given by the Dirac Landau level quantization. It is a successful example of simulating a continuous field theory without lattice regularization which leads to an additional symmetry breaking. The Dirac background and the five anti-commuting Dirac mass terms naturally introduce the picture of a non-trivial Berry phase contribution in the parameter space of the five component order parameter. Using the finite size scaling method given by the flux quantization, we find a stable critical phase in the low stiffness region of the sigma model. This is a candidate ground state of DQCP when the O(5) symmetry breaking terms are irrelevant at the critical point. Again, it has a bosonic low energy field theory which is seemingly unable to be realized in pure boson Hamiltonians. This work is summarized in Ref. [3].
This thesis presents results from large scale Monte Carlo simulations of systems subject to a superfluid phase transition in the presence of disorder. The simulations are performed by state-of-the-art, collective Monte Carlo algorithms treating phase degrees of freedom in effective models with amplitude fluctuations integrated out. In Paper I a model system for the possible solid to supersolid transition in 4He is presented.The Wolff cluster algorithm is used to study how the presence of linearly correlated random defects is able to alter the universality class of the 3-dimensional XY-model. In the pure case the superfluid density and heat capacity have singular onsets, which are not seen in the supersolid experiments where instead a smooth onset is obtained. Using finite size scaling of Monte Carlo data, we find a similar smooth onset in our simulations, governed by exponents ν=1 for the superfluid density and α=-1 for the heat capacity. These results are in qualitative agreement with experiments for the observed transition in solid 4He. In Paper II a systematic investigation of the scaling result z=d for the dynamic critical exponentat the Bose glass to superfluid quantum phase transition is performed. The result z=d has been believed to be exact for about 20 years, but although it has been questioned lately no accurate estimate of z has been available. An effective link current model of quantum bosons at T=0 with disorder in 2D is simulated using highly effective worm Monte Carlo simulations.The data analysis is based on a finite size scaling approach todetermine the quantum correlation time from simulationdata for boson world lines without any a priori assumption on the critical parameters. The resulting critical exponents are z=1.8 \pm 0.05, ν=1.15 \pm 0.03, and η=-0.3 \pm 0.1. This suggests that z=d is not satisfied.