Chiral properties of two-flavour QCD at zero and non-zero temperature

Lattice Quantum Chromodynamics (LQCD) is the preferred tool for obtaining non-perturbative results from QCD in the low-energy regime. It has by nowrnentered the era in which high precision calculations for a number of phenomenologically relevant observables at the physical point, with dynamical quark degrees of freedom and controlled systematics, become feasible. Despite these successes there are still quantities where control of systematic effects is insufficient. The subject of this thesis is the exploration of the potential of todays state-of-the-art simulation algorithms for non-perturbativelyrn$\mathcal{O}(a)$-improved Wilson fermions to produce reliable results in thernchiral regime and at the physical point both for zero and non-zero temperature. Important in this context is the control over the chiral extrapolation. Thisrnthesis is concerned with two particular topics, namely the computation of hadronic form factors at zero temperature, and the properties of the phaserntransition in the chiral limit of two-flavour QCD.rnrnThe electromagnetic iso-vector form factor of the pion provides a platform to study systematic effects and the chiral extrapolation for observables connected to the structure of mesons (and baryons). Mesonic form factors are computationally simpler than their baryonic counterparts but share most of the systematic effects. This thesis contains a comprehensive study of the form factor in the regime of low momentum transfer $q^2$, where the form factor is connected to the charge radius of the pion. A particular emphasis is on the region very close to $q^2=0$ which has not been explored so far, neither in experiment nor in LQCD. The results for the form factor close the gap between the smallest spacelike $q^2$-value available so far and $q^2=0$, and reach an unprecedented accuracy at full control over the main systematic effects. This enables the model-independent extraction of the pion charge radius. The results for the form factor and the charge radius are used to test chiral perturbation theory ($\chi$PT) and are thereby extrapolated to the physical point and the continuum. The final result in units of the hadronic radius $r_0$ is rn$$ \left\langle r_\pi^2 \right\rangle^{\rm phys}/r_0^2 = 1.87 \: \left(^{+12}_{-10}\right)\left(^{+\:4}_{-15}\right) \quad \textnormal{or} \quad \left\langle r_\pi^2 \right\rangle^{\rm phys} = 0.473 \: \left(^{+30}_{-26}\right)\left(^{+10}_{-38}\right)(10) \: \textnormal{fm} \;, $$rn which agrees well with the results from other measurements in LQCD and experiment. Note, that this is the first continuum extrapolated result for the charge radius from LQCD which has been extracted from measurements of the form factor in the region of small $q^2$.rnrnThe order of the phase transition in the chiral limit of two-flavour QCD and the associated transition temperature are the last unkown features of the phase diagram at zero chemical potential. The two possible scenarios are a second order transition in the $O(4)$-universality class or a first order transition. Since direct simulations in the chiral limit are not possible the transition can only be investigated by simulating at non-zero quark mass with a subsequent chiral extrapolation, guided by the universal scaling in the vicinity of the critical point. The thesis presents the setup and first results from a study on this topic. The study provides the ideal platform to test the potential and limits of todays simulation algorithms at finite temperature. The results from a first scan at a constant zero-temperature pion mass of about 290~MeV are promising, and it appears that simulations down to physical quark masses are feasible. Of particular relevance for the order of the chiral transition is the strength of the anomalous breaking of the $U_A(1)$ symmetry at the transition point. It can be studied by looking at the degeneracies of the correlation functions in scalar and pseudoscalar channels. For the temperature scan reported in this thesis the breaking is still pronounced in the transition region and the symmetry becomes effectively restored only above $1.16\:T_C$. The thesis also provides an extensive outline of research perspectives and includes a generalisation of the standard multi-histogram method to explicitly $\beta$-dependent fermion actions.

O(N) Models with Topological Lattice Actions

A variety of lattice discretisations of continuum actions has been considered, usually requiring the correct classical continuum limit. Here we discuss “weird” lattice formulations without that property, namely lattice actions that are invariant under most continuous deformations of the field configuration, in one version even without any coupling constants. It turns out that universality is powerful enough to still provide the correct quantum continuum limit, despite the absence of a classical limit, or a perturbative expansion. We demonstrate this for a set of O(N) models (or non-linear σ-models). Amazingly, such “weird” lattice actions are not only in the right universality class, but some of them even have practical benefits, in particular an excellent scaling behaviour.

From quantum to classical dynamics: The relativistic O ( N ) model in the framework of the real-time functional renormalization group

We investigate the transition from unitary to dissipative dynamics in the relativistic O(N) vector model with the λ(φ2)2 interaction using the nonperturbative functional renormalization group in the real-time formalism. In thermal equilibrium, the theory is characterized by two scales, the interaction range for coherent scattering of particles and the mean free path determined by the rate of incoherent collisions with excitations in the thermal medium. Their competition determines the renormalization group flow and the effective dynamics of the model. Here we quantify the dynamic properties of the model in terms of the scale-dependent dynamic critical exponent z in the limit of large temperatures and in 2≤d≤4 spatial dimensions. We contrast our results to the behavior expected at vanishing temperature and address the question of the appropriate dynamic universality class for the given microscopic theory.

Field theory in superfluid 3He: What are the lessons for particle physics, gravity, and high-temperature superconductivity?

There are several classes of homogeneous Fermi systems that are characterized by the topology of the energy spectrum of fermionic quasiparticles: (i) gapless systems with a Fermi surface, (ii) systems with a gap in their spectrum, (iii) gapless systems with topologically stable point nodes (Fermi points), and (iv) gapless systems with topologically unstable lines of nodes (Fermi lines). Superfluid 3He-A and electroweak vacuum belong to the universality class 3. The fermionic quasiparticles (particles) in this class are chiral: they are left-handed or right-handed. The collective bosonic modes of systems of class 3 are the effective gauge and gravitational fields. The great advantage of superfluid 3He-A is that we can perform experiments by using this condensed matter and thereby simulate many phenomena in high energy physics, including axial anomaly, baryoproduction, and magnetogenesis. 3He-A textures induce a nontrivial effective metrics of the space, where the free quasiparticles move along geodesics. With 3He-A one can simulate event horizons, Hawking radiation, rotating vacuum, etc. High-temperature superconductors are believed to belong to class 4. They have gapless fermionic quasiparticles with a “relativistic” spectrum close to gap nodes, which allows application of ideas developed for superfluid 3He-A.

Condensados de Bose-Einstein em redes óticas: a transição superfluido-isolante de Mott em redes hexagonais e a classe de universalidade superfluido-vidro de Bose em 3D

Estudamos transições de fases quânticas em gases bosônicos ultrafrios aprisionados em redes óticas. A física desses sistemas é capturada por um modelo do tipo Bose-Hubbard que, no caso de um sistema sem desordem, em que os átomos têm interação de curto alcance e o tunelamento é apenas entre sítios primeiros vizinhos, prevê a transição de fases quântica superfluido-isolante de Mott (SF-MI) quando a profundidade do potencial da rede ótica é variado. Num primeiro estudo, verificamos como o diagrama de fases dessa transição muda quando passamos de uma rede quadrada para uma hexagonal. Num segundo, investigamos como a desordem modifica essa transição. No estudo com rede hexagonal, apresentamos o diagrama de fases da transição SF-MI e uma estimativa para o ponto crítico do primeiro lobo de Mott. Esses resultados foram obtidos usando o algoritmo de Monte Carlo quântico denominado Worm. Comparamos nossos resultados com os obtidos a partir de uma aproximação de campo médio e com os de um sistema com uma rede ótica quadrada. Ao introduzir desordem no sistema, uma nova fase emerge no diagrama de fases do estado fundamental intermediando a fase superfluida e a isolante de Mott. Essa nova fase é conhecida como vidro de Bose (BG) e a transição de fases quântica SF-BG que ocorre nesse sistema gerou muitas controvérsias desde seus primeiros estudos iniciados no fim dos anos 80. Apesar dos avanços em direção ao entendimento completo desta transição, a caracterização básica das suas propriedades críticas ainda é debatida. O que motivou nosso estudo, foi a publicação de resultados experimentais e numéricos em sistemas tridimensionais [Yu et al. Nature 489, 379 (2012), Yu et al. PRB 86, 134421 (2012)] que violam a lei de escala $\\phi= u z$, em que $\\phi$ é o expoente da temperatura crítica, $z$ é o expoente crítico dinâmico e $ u$ é o expoente do comprimento de correlação. Abordamos essa controvérsia numericamente fazendo uma análise de escalonamento finito usando o algoritmo Worm nas suas versões quântica e clássica. Nossos resultados demonstram que trabalhos anteriores sobre a dependência da temperatura de transição superfluido-líquido normal com o potencial químico (ou campo magnético, em sistemas de spin), $T_c \\propto (\\mu-\\mu_c)^\\phi$, estavam equivocados na interpretação de um comportamento transiente na aproximação da região crítica genuína. Quando os parâmetros do modelo são modificados de maneira a ampliar a região crítica quântica, simulações com ambos os modelos clássico e quântico revelam que a lei de escala $\\phi= u z$ [com $\\phi=2.7(2)$, $z=3$ e $ u = 0.88(5)$] é válida. Também estimamos o expoente crítico do parâmetro de ordem, encontrando $\\beta=1.5(2)$.

Phase Transitions in Disordered Systems: The Example of the Random-Field Ising Model in Four Dimensions

By performing a high-statistics simulation of the D = 4 random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute to a high accuracy the complete set of critical exponents for this class, including the correction-to-scaling exponent. Our results indicate that in four dimensions (i) dimensional reduction as predicted by the perturbative renormalization group does not hold and (ii) three independent critical exponents are needed to describe the transition.

Fast simulation of a quantum phase transition in an ion-trap realizable unitary map

We show that a specific implementation of a unitary map on multiple qubits in an ion trap is physically equivalent to a Hamiltonian evolution that belongs to the same universality class as the transverse Ising Hamiltonian. We suggest experimental signatures, and present numerical simulations for the case of four qubits.

Memory effects in a non-equilibrium growth model

We study memory effects in a kinetic roughening model. For d=1, a different dynamic scaling is uncovered in the memory dominated phases; the Kardar-Parisi-Zhang scaling is restored in the absence of noise. dc=2 represents the critical dimension where memory is shown to smoothen the roughening front (a=0). Studies on a discrete atomistic model in the same universality class reconfirm the analytical results in the large time limit, while a different scaling behavior shows up for t

Stochastically driven instability in rotating shear flows

Origin of hydrodynamic turbulence in rotating shear flows is investigated. The particular emphasis is on flows whose angular velocities decrease but specific angular momenta increase with increasing radial coordinate. Such flows are Rayleigh stable, but must be turbulent in order to explain observed data. Such a mismatch between the linear theory and observations/experiments is more severe when any hydromagnetic/magnetohydrodynamic instability and the corresponding turbulence therein is ruled out. The present work explores the effect of stochastic noise on such hydrodynamic flows. We focus on a small section of such a flow which is essentially a plane shear flow supplemented by the Coriolis effect. This also mimics a small section of an astrophysical accretion disk. It is found that such stochastically driven flows exhibit large temporal and spatial correlations of perturbation velocities, and hence large energy dissipations, that presumably generate instability. A range of angular velocity profiles (for the steady flow), starting with the constant angular momentum to that of the constant circular velocity are explored. It is shown that the growth and roughness exponents calculated from the contour (envelope) of the perturbed flows are all identical, revealing a unique universality class for the stochastically forced hydrodynamics of rotating shear flows. This work, to the best of our knowledge, is the first attempt to understand origin of instability and turbulence in the three-dimensional Rayleigh stable rotating shear flows by introducing additive stochastic noise to the underlying linearized governing equations. This has important implications in resolving the turbulence problem in astrophysical hydrodynamic flows such as accretion disks.

Stochastically driven instability in rotating shear flows

Origin of hydrodynamic turbulence in rotating shear flows is investigated. The particular emphasis is on flows whose angular velocities decrease but specific angular momenta increase with increasing radial coordinate. Such flows are Rayleigh stable, but must be turbulent in order to explain observed data. Such a mismatch between the linear theory and observations/experiments is more severe when any hydromagnetic/magnetohydrodynamic instability and the corresponding turbulence therein is ruled out. The present work explores the effect of stochastic noise on such hydrodynamic flows. We focus on a small section of such a flow which is essentially a plane shear flow supplemented by the Coriolis effect. This also mimics a small section of an astrophysical accretion disk. It is found that such stochastically driven flows exhibit large temporal and spatial correlations of perturbation velocities, and hence large energy dissipations, that presumably generate instability. A range of angular velocity profiles (for the steady flow), starting with the constant angular momentum to that of the constant circular velocity are explored. It is shown that the growth and roughness exponents calculated from the contour (envelope) of the perturbed flows are all identical, revealing a unique universality class for the stochastically forced hydrodynamics of rotating shear flows. This work, to the best of our knowledge, is the first attempt to understand origin of instability and turbulence in the three-dimensional Rayleigh stable rotating shear flows by introducing additive stochastic noise to the underlying linearized governing equations. This has important implications in resolving the turbulence problem in astrophysical hydrodynamic flows such as accretion disks.

Dynamics of cholesteric liquid crystals in the presence of a random magnetic field:stochastic dynamics of cholesteric liquid crystal

Based on dynamic renormalization group techniques, this letter analyzes the effects of external stochastic perturbations on the dynamical properties of cholesteric liquid crystals, studied in presence of a random magnetic field. Our analysis quantifies the nature of the temperature dependence of the dynamics; the results also highlight a hitherto unexplored regime in cholesteric liquid crystal dynamics. We show that stochastic fluctuations drive the system to a second-ordered Kosterlitz-Thouless phase transition point, eventually leading to a Kardar-Parisi-Zhang (KPZ) universality class. The results go beyond quasi-first order mean-field theories, and provides the first theoretical understanding of a KPZ phase in distorted nematic liquid crystal dynamics.

Numerical study of the enlarged O(5) symmetry of the 3D antiferromagnetic RP^(2) spin model

We investigate by means of Monte Carlo simulation and finite-size scaling analysis the critical properties of the three dimensional O (5) non-linear σ model and of the antiferromagnetic RP^(2) model, both of them regularized on a lattice. High accuracy estimates are obtained for the critical exponents, universal dimensionless quantities and critical couplings. It is concluded that both models belong to the same universality class, provided that rather non-standard identifications are made for the momentum-space propagator of the RP^(2) model. We have also investigated the phase diagram of the RP^(2) model extended by a second-neighbor interaction. A rich phase diagram is found, where most of the phase transitions are of the first order.

Dynamical generation of a gauge symmetry in the double-exchange model

It is shown that a bosonic formulation of the double-exchange model, one of the classical models for magnetism, generates dynamically a gauge-invariant phase in a finite region of the phase diagram. We use analytical methods, Monte Carlo simulations and finite-size scaling analysis. We study the transition line between that region and the paramagnetic phase. The numerical results show that this transition line belongs to the universality class of the antiferromagnetic RP^(2) model. The fact that one can define a universality class for the antiferromagnetic RP^(2) model, different from the one of the O(N) models, is puzzling and somehow contradicts naive expectations about universality.

Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

It was recently shown [Phys. Rev. Lett. 110, 227201 (2013)] that the critical behavior of the random-field Ising model in three dimensions is ruled by a single universality class. This conclusion was reached only after a proper taming of the large scaling corrections of the model by applying a combined approach of various techniques, coming from the zero-and positive-temperature toolboxes of statistical physics. In the present contribution we provide a detailed description of this combined scheme, explaining in detail the zero-temperature numerical scheme and developing the generalized fluctuation-dissipation formula that allowed us to compute connected and disconnected correlation functions of the model. We discuss the error evolution of our method and we illustrate the infinite limit-size extrapolation of several observables within phenomenological renormalization. We present an extension of the quotients method that allows us to obtain estimates of the critical exponent a of the specific heat of the model via the scaling of the bond energy and we discuss the self-averaging properties of the system and the algorithmic aspects of the maximum-flow algorithm used.

[N]pT ensemble and finite-size-scaling study of the critical isostructural transition in the generalized exponential model of index 4.

First-order transitions of system where both lattice site occupancy and lattice spacing fluctuate, such as cluster crystals, cannot be efficiently studied by traditional simulation methods, which necessarily fix one of these two degrees of freedom. The difficulty, however, can be surmounted by the generalized [N]pT ensemble [J. Chem. Phys. 136, 214106 (2012)]. Here we show that histogram reweighting and the [N]pT ensemble can be used to study an isostructural transition between cluster crystals of different occupancy in the generalized exponential model of index 4 (GEM-4). Extending this scheme to finite-size scaling studies also allows us to accurately determine the critical point parameters and to verify that it belongs to the Ising universality class.