Quantum Protectorate Research Articles

Symmetry breaking, quantum protectorate and quasiaverages in condensed matter physics

Substantial progress in the understanding of the spontaneously broken symmetry concept is connected with Bogoliubov’s fundamental ideas about quasiaverages. The development and applications of the method of quasiaverages, formulated by N. N. Bogoliubov, to quantum statistical mechanics and to quantum solid state theory and, in particular, to quantum theory of magnetism, are discussed. Some physical implications involved in a new concept, termed the quantum protectorate, are considered.

BOGOLIUBOV'S VISION: QUASIAVERAGES AND BROKEN SYMMETRY TO QUANTUM PROTECTORATE AND EMERGENCE

In the present interdisciplinary review, we focus on the applications of the symmetry principles to quantum and statistical physics in connection with some other branches of science. The profound and innovative idea of quasiaverages formulated by N. N. Bogoliubov, gives the so-called macro-objectivation of the degeneracy in the domain of quantum statistical mechanics, quantum field theory and quantum physics in general. We discuss the complementary unifying ideas of modern physics, namely: spontaneous symmetry breaking, quantum protectorate and emergence. The interrelation of the concepts of symmetry breaking, quasiaverages and quantum protectorate was analyzed in the context of quantum theory and statistical physics. The chief purposes of this paper were to demonstrate the connection and interrelation of these conceptual advances of the many-body physics and to try to show explicitly that those concepts, though different in details, have certain common features. Several problems in the field of statistical physics of complex materials and systems (e.g., the chirality of molecules) and the foundations of the microscopic theory of magnetism and superconductivity were discussed in relation to these ideas.

Statistical mechanics and the physics of many-particle model systems

The development of methods of quantum statistical mechanics is considered in light of their applications to quantum solid-state theory. We discuss fundamental problems of the physics of magnetic materials and the methods of the quantum theory of magnetism, including the method of two-time temperature Green's functions, which is widely used in various physical problems of many-particle systems with interaction. Quantum cooperative effects and quasiparticle dynamics in the basic microscopic models of quantum theory of magnetism: the Heisenberg model, the Hubbard model, the Anderson Model, and the spin-fermion model are considered in the framework of novel self-consistent-field approximation. We present a comparative analysis of these models; in particular, we compare their applicability for description of complex magnetic materials. The concepts of broken symmetry, quantum protectorate, and quasiaverages are analyzed in the context of quantum theory of magnetism and theory of superconductivity. The notion of broken symmetry is presented within the nonequilibrium statistical operator approach developed by D.N. Zubarev. In the framework of the latter approach we discuss the derivation of kinetic equations for a system in a thermal bath. Finally, the results of investigation of the dynamic behavior of a particle in an environment, taking into account dissipative effects, are presented.

Superconductivity emerging near U(1) topological critical point and the strange metal phase in cuprates

Topological excitations are examined to establish an effective theory of the superconducting state evolving from the Mott insulator in high- T c cuprates. We emphasise the Coulomb interaction U that is governed by the electromagnetic U(1) compact group. The results are obtained for the layered t− t′− t ⊥− U− J system of strongly correlated electrons relevant for cuprates. We exploit the U(1) instanton phase field configurations for the cuprate phase diagram which displays the “hidden” quantum critical point covered by the superconducting lobe in addition to a sharp crossover between a compressible normal “strange metal” state and a region characterized by a vanishing compressibility, which marks the Mott insulator. The existence of robust topological quantum numbers explains the stability against small perturbation of the system and attributes to the topological “quantum protectorate” as observed in strongly correlated systems.

Critical charge instability on the verge of the Mott transition and the origin of quantum protection in high-Tccuprates

The concept of topological excitations and the related ground state degeneracy are employed to establish an effective theory of the superconducting state evolving from the Mott insulator for high-${T}_{c}$ cuprates. The theory includes the effects of the relevant energy scales with the emphasis on the Coulomb interaction $U$ governed by the electromagnetic U(1) compact group. The results are obtained for the layered $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}_{\ensuremath{\perp}}\ensuremath{-}U\ensuremath{-}J$ system of strongly correlated electrons relevant for cuprates. Casting the Coulomb interaction in terms of composite-fermions via the gauge flux attachment facility, we show that instanton events in the Matsubara ``imaginary time,'' labeled by topological winding numbers, are essential configurations of the phase field dual to the charge. This provides a nonperturbative concept of the topological quantization and displays the significance of discrete topological sectors in the theory governed by the global characteristics of the phase field. We show that for topologically ordered states these quantum numbers play the role of an order parameter in a way similar to the Landau order parameter for conventionally ordered states. In analogy to the usual phase transition that is characterized by a sudden change of the symmetry, the topological phase transitions are governed by a discontinuous change of the topological numbers signaled by the divergence of the zero-temperature topological susceptibility. This defines a quantum criticality ruled by topologically conserved numbers rather than the Landau principle of the symmetry breaking. We show that in the limit of strong correlations topological charge is linked to the average electronic filling number and the topological susceptibility to the electronic compressibility of the system. We exploit the impact of these nontrivial U(1) instanton phase field configurations for the cuprate phase diagram which displays the ``hidden'' quantum critical point covered by the superconducting lobe in addition to a sharp crossover between a compressible normal ``strange metal'' state and a region characterized by a vanishing compressibility, which marks the Mott insulator. Finally, we argue that the existence of robust quantum numbers explains the stability against small perturbation of the system and attributes to the topological ``quantum protectorate'' as observed in strongly correlated systems.

2D superconductivity: Classification of universality classes by infinite symmetry

I consider superconducting condensates which become incompressible in the infinite gap limit. Classical 2D incompressible fluids possess the dynamical symmetry of area-preserving diffeomorphisms. I show that the corresponding infinite dynamical symmetry of 2D superconducting fluids is the coset W 1 + ∞ ⊗ W ¯ 1 + ∞ U ( 1 ) diagonal , with W 1 + ∞ the chiral algebra of quantum area-preserving diffeomorphisms and I derive its minimal models. These define a discrete set of 2D superconductivity universality classes which fall into two main categories: conventional superconductors with their vortex excitations and unconventional superconductors. These are characterized by a broken U ( 1 ) vector ⊗ U ( 1 ) axial symmetry and are labeled by an integer level m. They possess neutral spinon excitations of fractional spin and statistics S = θ 2 π = m − 1 2 m , which carry also an SU ( m ) isospin quantum number; this hidden SU ( m ) symmetry implies that these anyon excitations are non-Abelian. The simplest unconventional superconductor is realized for m = 2 : in this case the spinon excitations are semions (half-fermions). My results show that spin–charge separation in 2D superconductivity is a universal consequence of the infinite symmetry of the ground state. This infinite symmetry and its superselection rules realize a quantum protectorate in which the neutral spinons can survive even as soft modes on a rigid, spinless charge condensate.

Superconductivity inNaxCoO2·yH2O: Protection of ad1+id2state by spin-charge separation

Superconductivity in ${\text{Na}}_{x}{\text{CoO}}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}y{\mathrm{H}}_{2}\mathrm{O}$ is likely to be a $p$ or $d$ wave; however, experiments are unable to pinpoint the symmetry. A simple estimate of pair breaking effects from an unavoidable ``${\text{Na}}^{+}$ vacancy disorder'' in an ordered ${\text{Na}}^{+}$ lattice, at an optimal ${x}_{\text{opt}}\ensuremath{\approx}0.30$ is shown to destroy a Fermi liquid-based $p$- or $d$-wave superconductivity. However, a robustness of superconducting and normal states, seen in experiments, is pointed out and argued to imply the presence of a ``quantum protectorate,'' possibly a spin-charge decoupling that protects a ${d}_{1}+i{d}_{2}$ and not a $p$ state. A calculation of Knight shift and $1∕{T}_{1}$ in the framework of resonating valence bond mean-field theory and a fit to the data of Kobayashi et al. (e-print, cond-mat/0305649; cond-mat/0306264) is made.

QUANTUM PROTECTORATE AND MICROSCOPIC MODELS OF MAGNETISM

Some physical implications involved in a new concept, termed the "quantum protectorate" (QP), are developed and discussed. This is done by considering the idea of quantum protectorate in the context of quantum theory of magnetism. It is suggested that the difficulties in the formulation of quantum theory of magnetism at the microscopic level, that are related to the choice of relevant models, can be understood better in the light of the QP concept. We argue that the difficulties in the formulation of adequate microscopic models of electron and magnetic properties of materials are intimately related to dual, itinerant and localized behaviour of electrons. We formulate a criterion of what basic picture describes best this dual behaviour. The main suggestion is that quasi-particle excitation spectra might provide distinctive signatures and good criteria for the appropriate choice of the relevant model.

Fermion-condensation quantum phase transition in high temperature superconductors

The effect of a quantum phase transition associated with the appearance of the fermion condensate in an electron liquid on the properties of superconductors is considered. It is shown that the electron system in both superconducting and normal states exhibits characteristic features of a quantum protectorate after the point of this fermion-condensation quantum phase transition. The single-particle spectrum of a superconductor can be represented by two straight lines corresponding to two effective masses MFC∗ and ML∗. The MFC∗ mass characterizes the spectrum up to the binding energy E0 and ML∗ determines the spectrum at higher binding energies. Both effective masses are retained in the normal state. These results are used to explain the line shape of single-particle excitations and other properties of high-Tc superconductors and are in a good agreement with recent experimental data.

Quasiparticle picture of high-temperature superconductors in the frame of a Fermi liquid with the fermion condensate

A model of a Fermi liquid with the fermion condensate (FC) is applied to the consideration of quasiparticle excitations in high temperature superconductors, in their superconducting and normal states. Within our model the appearance of the fermion condensate presents a quantum phase transition, that separates the regions of normal and strongly correlated electron liquids. Beyond the phase transition point the quasiparticle system is divided into two subsystems, one containing normal quasiparticles and the other --- fermion condensate localized at the Fermi surface and characterized by almost dispersionless single-particle excitations. In the superconducting state the quasiparticle dispersion in systems with FC can be presented by two straight lines, characterized by effective masses $M^*_{FC}$ and $M^*_L$, respectively, and intersecting near the binding energy which is of the order of the superconducting gap. This same quasiparticle picture persists in the normal state, thus manifesting itself over a wide range of temperatures as new energy scales. Arguments are presented that fermion systems with FC have features of a quantum protectorate.

Fermi-condensate quantum phase transition in high-T c superconductors

The effect of a quantum phase transition associated with the appearance of the fermion condensate in an electron liquid on the properties of superconductors is considered. It is shown that the electron system in both superconducting and normal states exhibits characteristic features of a quantum protectorate after the point of this fermion-condensation quantum phase transition. The single-particle spectrum of a superconductor can be represented by two straight lines corresponding to two effective masses $M^*_{FC}$ and $M^*_{L}$. The $M^*_{FC}$ mass characterizes the spectrum up to the binding energy $E_0$, which is of the order of the superconducting gap in magnitude, and $M^*_{L}$ determines the spectrum at higher binding energies. Both effective masses are retained in the normal state; however, $E_0=4T$. These results are used to explain the lineshape of single-particle excitations and other remarkable properties of high-$T_c$ superconductors and are in a good agreement with recent experimental data.

Spin-charge separation is the key to the high Tc cuprates

The most striking fact about the high-T c cuprates is that in none of the relevant regions of the phase diagram is there any evidence of the usual effects of phonon or impurity scattering. This is strong evidence that these states are in a “quantum protectorate” to borrow Laughlin's terms. (Two striking experimental facts which demonstrate this are the absence of phonon self-energy in ARPES measurements, demonstrated recently by Johnson in BSCCO [1], and the scattering-independent T c in YBCO, even though the superconductivity is d-wave.) A quantum protectorate is a state in which the many-body correlations are so strong that the dynamics can no longer be described in terms of individual particles, and therefore perturbations which scatter individual particles are not effective.

Sources of quantum protection in high-T(c) superconductivity

The layer-structure cuprates with high superconducting transition temperatures T(c) exhibit a number of anomalous electronic properties in both superconducting and normal states. These anomalies are ascribed to the existence of independent spectra of excitations for charge and for spin, signaling a collective state, a "quantum protectorate."