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Register a new userBerry Phase in Quantum Field Theory: Diabolical Points and Boundary Phenomena
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We study aspects of Berry phase in gapped many-body quantum systems by means of effective field theory. Once the parameters are promoted to spacetime-dependent background fields, such adiabatic phases are described by Wess-Zumino-Witten (WZW) and similar terms. In the presence of symmetries, there are also quantized invariants capturing generalized Thouless pumps. Consideration of these terms provides constraints on the phase diagram of many-body systems, implying the existence of gapless points in the phase diagram which are stable for topological reasons. We describe such diabolical points, realized by free fermions and gauge theories in various dimensions, which act as sources of higher Berry curvature and are protected by the quantization of the corresponding WZW terms or Thouless pump terms. These are analogous to Weyl nodes in a semimetal band structure. We argue that in the presence of a boundary, there are boundary diabolical points---parameter values where the boundary gap closes---which occupy arcs ending at the bulk diabolical points. Thus the boundary has an anomaly in the space of couplings in the sense of Cordova et al. Consideration of the topological effective action for the parameters also provides some new checks on conjectured infrared dualities and deconfined quantum criticality in 2+1d.
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We extend the formalism of entanglement renormalization to the study of boundary critical phenomena. The multi-scale entanglement renormalization ansatz (MERA), in its scale invariant version, offers a very compact approximation to quantum critical ground states. Here we show that, by adding a boundary to the scale invariant MERA, an accurate approximation to the critical ground state of an infinite chain with a boundary is obtained, from which one can extract boundary scaling operators and their scaling dimensions. Our construction, valid for arbitrary critical systems, produces an effective chain with explicit separation of energy scales that relates to Wilsons RG formulation of the Kondo problem. We test the approach by studying the quantum critical Ising model with free and fixed boundary conditions.
We study the relation between Chern numbers and Quantum Phase Transitions (QPT) in the XY spin-chain model. By coupling the spin chain to a single spin, it is possible to study topological invariants associated to the coupling Hamiltonian. These invariants contain global information, in addition to the usual one (obtained by integrating the Berry connection around a closed loop). We compute these invariants (Chern numbers) and discuss their relation to QPT. In particular we show that Chern numbers can be used to label regions corresponding to different phases.
Composite Fermi liquid metals arise at certain special filling fractions in the quantum Hall regime and play an important role as parent states of gapped states with quantized Hall response. They have been successfully described by the Halperin-Lee-Read (HLR) theory of a Fermi surface of composite fermions coupled to a $U(1)$ gauge field with a Chern-Simons term. However, the validity of the HLR description when the microscopic system is restricted to a single Landau has not been clear. Here for the specific case of bosons at filling $ u = 1$, we build on earlier work from the 1990s to formulate a low energy description that takes the form of a {em non-commutative} field theory. This theory has a Fermi surface of composite fermions coupled to a $U(1)$ gauge field with no Chern-Simons term but with the feature that all fields are defined in a non-commutative spacetime. An approximate mapping of the long wavelength, small amplitude gauge fluctuations yields a commutative effective field theory which, remarkably, takes the HLR form but with microscopic parameters correctly determined by the interaction strength. Extensions to some other composite fermi liquids, and to other related states of matter are discussed.
We study a quantum quench of the mass and the interaction in the Sinh-Gordon model starting from a large initial mass and zero initial coupling. Our focus is on the determination of the expansion of the initial state in terms of post-quench excitations. We argue that the large energy profile of the involved excitations can be relevant for the late time behaviour of the system and common regularization schemes are unreliable. We therefore proceed in determining the initial state by first principles expanding it in a systematic and controllable fashion on the basis of the asymptotic states. Our results show that, for the special limit of pre-quench parameters we consider, it assumes a squeezed state form that has been shown to evolve so as to exhibit the equilibrium behaviour predicted by the Generalized Gibbs Ensemble.
The quantum analog of Lyapunov exponent has been discussed in the Sachdev-Ye-Kitaev (SYK) model and its various generalizations. Here we investigate possible quantum analog of Kolmogorov-Arnold-Moser (KAM) theorem in the $ U(1)/Z_2 $ Dicke model which contains both the rotating wave (RW) term $ g $ and the counter-RW term $ g ^{prime} $ at a finite $ N $. We first study its energy spectrum by the analytical $ 1/J $ expansion, supplemented by the non-perturbative instanton method.Then we evaluate its energy level statistic (ELS) at a given parity sector by Exact diagonization (ED) at any $ 0 < beta= g ^{prime}/g < 1 $. We establish an intimate relation between the KAM theorem and the evolution of the scattering states and the emergence of bound states as the ratio $ beta $ increases. We stress the important roles played by the Berry phase and instantons in the establishment of the quantum analogue of the KAM theorem to the $ U(1)/Z_2 $ Dicke model.Experimental implications in cavity QED systems such as cold atoms inside an optical cavity or superconducting qubits in side a microwave cavity are also discussed.
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