Abstract Symmetry-breaking phases in many-fermion systems are characterized by anomalous functions that describe transient processes in which certain properties of free particles, such as spin or charge, are not conserved. Connecting the low-temperature symmetry-breaking phase with the high-temperature phase within the Baym-Kadanoff scheme, beyond the static mean-field approximation, remains a long-standing challenge. We identify the reason why approximations that include critical dynamical fluctuations in the Schwinger-Dyson equation yield mismatched transition temperatures between the high- and low-temperature phases. We find a solution to this generic problem and show that the mismatch in closed systems arises from the inappropriate inclusion of anomalous contributions to the free electron-hole propagator that fail to maintain the balance between the incoming and outgoing excited states. We illustrate this behavior with an antiferromagnetic state calculated in the FLEX approximation. We find a two-gap structure in the spectral function in the antiferromagnetic phase when strong electron correlations are present.
Abstract Symmetry-breaking phases in many-fermion systems are characterized by anomalous functions that describe transient processes in which certain properties of free particles, such as spin or charge, are not conserved. Connecting the low-temperature symmetry-breaking phase with the high-temperature phase within the Baym-Kadanoff scheme, beyond the static mean-field approximation, remains a long-standing challenge. We identify the reason why approximations that include critical dynamical fluctuations in the Schwinger-Dyson equation yield mismatched transition temperatures between the high- and low-temperature phases. We find a solution to this generic problem and show that the mismatch in closed systems arises from the inappropriate inclusion of anomalous contributions to the free electron-hole propagator that fail to maintain the balance between the incoming and outgoing excited states. We illustrate this behavior with an antiferromagnetic state calculated in the FLEX approximation. We find a two-gap structure in the spectral function in the antiferromagnetic phase when strong electron correlations are present.
Large N - slave fermion techniques have been established as a powerful computational tool to study quantum many-body systems. They have been applied successfully to the study of various systems of strongly correlated electrons in condensed matter physics, particularly to problems where a non-perturbative treatment is required. In the first part of this thesis we use this approach to investigate the Frustrated Quantum Antiferromagnet. We study the Chiral Spin Liquid state of the antiferromagnet at zerotemperature, which has a spin-gap generated by the spontaneous breakdown of time-reversalsymmetry. This state is known as the Chiral Spin State. We determine conditions for the restoration of time-reversal-symmetry in a bilayer antiferromagnet with an interlayer exchange interaction. We show the energetically favored ground state has Chiral Spin Liquids with opposite chiralities on each layer. This constitutes a dynamic mechanism which prevents the observation of broken time-reversal-symmetry in a bilayer system, even though this breakdown may be present on each layer independently. In the second part of this thesis we use the slave-fermion technique to study the physics of magnetic impurities both in Quantum Antiferromagnets (in a flux phase) and in d-wave superconductors. This is a system consisting of a magnetic impurity coupled to either the low lying excitations of a flux phase, or to the normal quasi-particles close to the Fermi surface of a superconductor whose gap function exhibits nodes. We show that both systems display a zero temperature quantum phase transition in the large N approximation. The transition takes place between a weak coupling regime, in which the magnetic impurity is effectively decoupled from the quasi-particles, to a strong coupling phase which exhibits a Ill Kondo effect and the magnetic impurities are screened. The ground state is a singlet. If the system has particle-hole symmetry the impurity appears to be overscreened. This effect is not due to a multichannel character in the system. The physical origin of the quantum phase transition is directly related to the fact that this is a non-marginal Kondo system. This follows from having a quasi-particle band with a density of states that vanishes linearly with the energy in the proximity of the Fermi surface. We show that, due to the presence of nodes in the gap function of a d-wave superconductor, there is a striking analogy between the problem of a Kondo impurity coupled to a flux phase and a magnetic impurity coupled to the superconductor. We consider a clean dx2-y2 superconductor in the presence of a fully quantum mechanical magnetic impurity. It is assumed that the temperature is sufficiently low so as to neglect both phase and amplitude fluctuations of the order parameter. We study the zero temperature behavior and the low temperature and low magnetic field regimes in the vicinity of the quantum phase transition. In the strong coupling phase, we obtain scaling behavior for the chemical potential of the impurity f.J and for the amplitude of the ground state singlet ~- At zero temperature we show that they present different scaling behavior with the distance to critical coupling: EJ scales linearly while ~ scales with a vanishing exponent. Both magnitudes present logarithmic corrections to scaling suggesting that the theory may be at an upper critical dimension. At finite field (zero temperature) and/or at finite temperature (zero field), at the critical coupling the magnetic field or the temperature respectively, play a similar role as the distance to the critical coupling in the scaling formulas. The susceptibility and the specific heat are investigated both at zero temperature, zero field limit and at low temperature, low field regimes. The results are consistent with the existence of perfect screening and a singlet ground state in the strong coupling phase. When particle-hole is an exact symmetry of the system, the impurity appears to be overscreened with a vanishing paramagnetic susceptibility.
The dynamic conductivity \ensuremath{\sigma}(\ensuremath{\omega}) of strongly correlated electrons in a symmetry-broken phase is investigated in the present work. The model considered consists of spinless fermions with repulsive interaction on a simple cubic lattice. The investigated symmetry-broken phase is the charge density wave (CDW) with wave vector Q=(\ensuremath{\pi},\ensuremath{\pi},\ensuremath{\pi}${)}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$ which occurs at half-filling. The calculations are based on the high dimensional approach, i.e., an expansion in the inverse dimension 1/d is used. The finite dimensionality is accounted for by the inclusion of linear terms in 1/d and the true finite dimensional DOS. Special care is paid to the setup of a conserving approximation in the sense of Baym/Kadanoff without inconsistencies. The resulting Bethe-Salpeter equation is solved for the dynamic conductivity in the nonsymmetry-broken and in the symmetry-broken phase (AB-CDW). The dc-conductivity is reduced drastically in the CDW. Yet it does not vanish in the limit T\ensuremath{\rightarrow}0 due to a subtle cancellation of diverging mobility and vanishing DOS. In the dynamic conductivity \ensuremath{\sigma}(\ensuremath{\omega}) the energy gap induced by the symmetry breaking is clearly discernible. In addition, the vertex corrections of order 1/d lead to an excitonic resonance lying within the gap. \textcopyright{} 1996 The American Physical Society.
In the last decades, multifaceted manifestations of the breakdown of the self-consistent perturbation theory have been identified for the many-electron problem. Yet, the investigations have been so far mostly limited to paramagnetic states, where symmetry breaking is not allowed. Here, we extend the analysis to the spontaneously symmetry-broken antiferromagnetic (AF) phase of the repulsive Hubbard model. To this aim, we calculated two-particle quantities using dynamical mean-field theory for the AF-ordered Hubbard model and studied the possible occurrence of divergences of the irreducible vertex functions in the charge and spin sectors. Our calculations pinpoint the divergences in the AF phase diagram, showing that while the onset of AF order mitigates the breakdown of the perturbation expansion, it does not fully prevent it. Moreover, we have been able to link the changes in the dynamical structure of the corresponding generalized susceptibilities to the physical crossover from a weak-coupling (Slater) to a strong-coupling (Heisenberg) antiferromagnet, which takes place as the interaction strength is gradually increased. Finally, we discuss possible physical consequences of the irreducible vertex divergences in triggering phase-separation instabilities within the AF phase and elaborate on the implications of our findings for two-dimensional systems, where the onset of a long-range AF order is prevented by the Mermin-Wagner theorem.
The Landau theory of phase and phase transition, which describes the state of matter by the spontaneous symmetry breaking, and the Fermi liquid theory of interacting Fermi systems are the two corner stones of contemporary condensed matter physics. They act as our starting points to understand the structure, excitations and phase transitions of any condensed matter system. According to such a Landau paradigm, an interacting Fermi system will behave just as non-interacting Fermi gas system at sufficient low energy, if the interaction is not strong enough to induce a symmetry breaking phase transition in the system. In such a case, the perturbation expansion in the interaction strength is assumed to converge and the low energy excitation can be connected to that of a Fermi gas system adiabatically. In strongly correlated electron systems, such as the high temperature superconductors and quantum antiferromagnets, a pertubative picture on the interaction effect usually fails. The correlation effect between the electrons can induce new forms of order that are beyond the Landau symmetry breaking paradigm. Non-Fermi liquid behavior can emerge in the fully symmetric correlated phases. The Mott insulating phase and the quantum spin liquid phase are the most prominent example of such exotic state of matter. After about three decades of intensive studies, people find that the quantum non-locality effect is at the heart of such new forms of order. The quantum non-locality effect manifests itself in the emergence of gauge degree of freedom in the low energy physics of such strongly correlated system. Such emergent gauge filed is not only important for the characterization and classification of the ground state structure, but is also directly responsible for the non-Fermi liquid behavior of the system. The Landau paradigm should be extended to accommodate such new forms of order and the related new form of excitations and new universal class of phase transitions. Such an extension will elevate our understanding of the structure of matter to the level of quantum wave function. Unlike the conventional symmetry breaking order, which is characterized by the correlation functions between local physical observables, the new form of orders, which are commonly called quantum order (with the topological order as one of its typical example), can be better characterized by the entanglement property in the quantum wave function. From a mathematical point of view, the new form of order is closely related to the projective representation theory of the symmetry group, while the conventional symmetry breaking order is well described by the linear representation theory of the symmetry group.
Using the Hubbard model and the corresponding t-J model, we study the properties of correlated states with valence-bond structures. Mixed states with such structures and with antiferromagnetic spin ordering can be constructed by means of local unitary transformations of uncorrelated states. The latter turn out to have lower energy than mean-field antiferromagnetic solutions. Spin correlations for various degrees of doping δ = n −1 are in good agreement with the results of exact calculations for finite systems. In contrast to mean-field solutions, allowance for valence-bond correlations leads to a reasonable value of the critical δ , at which long-range antiferromagnetic order disappears. A calculation of the spectral functions that describe photoemission reveals typical behavior in two bands of effective hole (and electron) excitations, and energy transport in bands as the quasimomentum varies from (0,0) to ( π,π ), consistent with calculations in finite systems. We construct a homogeneous correlated state of fluctuating valence bonds (the band-model analog of states of fluctuating valence bonds), and demonstrate that its energy is lower than that of valence-bond alternant structures.
Exotic phases and associated phase transitions in low
dimensions have been a fascinating frontier and a driving force in
modern condensed matter physics since the 80s. Due to strong
correlation effect, they are beyond the description of mean-field
theory based on a single-particle picture and Landau's
symmetry-breaking theory of phase transitions. These new phases of
matter require new physical quantities to characterize them and new
languages to describe them. This thesis is devoted to the study on
exotic phases of correlated electrons in two spatial dimensions. We
present the following efforts in understanding two-dimensional
exotic phases: (1) Using Zn vertex algebra, we give a complete
classification and characterization of different one-component
fractional quantum Hall (FQH) states, including their ground state
properties and quasiparticles. (2) In terms of a non-unitary
transformation, we obtain the exact form of statistical
interactions between composite fermions in the lowest Landau level
(LLL) with v=1/(2m), m=1,2... By studying the pairing instability
of composite fermions we theoretically explains recently observed
FQHE in LLL with v=1/2,1/4. (3) We classify different Z2 spin
liquids (SLs) on kagome lattice in Schwinger-fermion representation
using projective symmetry group (PSG). We propose one most
promising candidate for the numerically discovered SL state in
nearest-neighbor Heisenberg model on kagome lattice}. (4) By
analyzing different Z2 spin liquids on honeycomb lattice within PSG
classification, we find out the nature of the gapped SL phase in
honeycomb lattice Hubbard model, labeled sublattice pairing state
(SPS) in Schwinger-fermion representation. We also identify the
neighboring magnetic phase of SPS as a chiral-antiferromagnetic
(CAF) phase and analyze the continuous phase transition between SPS
and CAF phase. For the first time we identify a SL called 0-flux
state in Schwinger-boson representation with one (SPS) in
Schwinger-fermion representation by a duality transformation. (5)
We show that when certain non-collinear magnetic order coexists in
a singlet nodal superconductor, there will be Majorana bound states
in vortex cores/on the edges of the superconductor. This proposal
opens a window for discovering Majorana fermions in strongly
correlated electrons. (6) Motivated by recent numerical discovery
of fractionalized phases in topological flat bands, we construct
wavefunctions for spin-polarized fractional Chern insulators (FCI)
and time reversal symmetric fractional topological insulators (FTI)
by parton approach. We show that lattice symmetries give rise to
different FCI/FTI states even with the same filling fraction. For
the first time we construct FTI wavefunctions in the absence of
spin conservation which preserve all lattice symmetries. The
constructed wavefunctions also set up the framework for future
variational Monte Carlo simulations.