Walter Metzner

Correlated Lattice Fermions in High Dimensions

A new approach to correlated Fermi systems such as the Hubbard model, the periodic Anderson model etc. is discussed, which makes use of the limit of high spatial dimensions. This limit — which is wellknown in the case of classical as well as localized quantum spin models — is found to be very helpful also in the case of quantum mechanical models with itinerant degrees of freedom. Many investigations, which are prohibitively difficult in lower dimensions, become tractable in this limit. In particular, essential features of systems in d = 3, and even lower dimensions, are very well described by the results in d = ∞ or expansions around this limit. A brief review of the state-of-the-art is presented.

Foraging behaviour and echolocation in the rufous horseshoe bat (Rhinolophus rouxi) of Sri Lanka

SummaryIn October 1984 foraging areas and foraging behaviour of the rufous horseshoe bat, Rhinolophus rouxi, were studied around a nursery colony on the hill slopes of Sri Lanka. The bats only foraged in dense forest and were not found in open woodlands (Fig. 1). This strongly supports the hypothesis that detection of fluttering prey is by pure tone echolocation within or close to echo-cluttering foliage. During a first activity period after sunset for about 30–60 min, the bats mainly caught insects on the wing. This was followed by a period of inactivity for another 60–120 min. Thereafter the bats resumed foraging throughout the night. They mainly alighted on specific twigs and foraged in flycatcher style. Individual bats maintained individual foraging areas of about 20x20 m. They stayed in this area throughout the night and returned to the same area on subsequent nights. Within this area the bats generally alighted on twigs at the same spots. Foraging areas were not defended against intruders. The bats echolocated throughout the night at an average repetition rate of 9.6±1.4 sounds/s. While hanging on twigs they scanned the surrounding area for flying prey by turning their bodies continuously around their legs. On average they performed one brief catching flight every 2 min and immediately returned to one of their favourite vantage points. Echolocation sounds may consist of up to three parts, a brief initial frequency-modulated (FM) component, a long constant frequency (CF) part lasting for about 40–50 ms, and a final FM part again (Fig. 4b, c). Adult males and females emitted pure tone frequencies in separate bands, the males from 73.5–77 kHz and the females from 76.5–79 kHz (Fig. 5). During scanning for prey from vantage points, the bats mostly emitted pure tones without any FM component (Fig. 4a). The last few pure tones emitted before take-off were prolonged to about 60 ms duration. The final FM part was therefore not an obligatory component of the echolocation signals in horseshoe bats. During flight and especially during emergence from the cave, most sounds consisted of a pure tone and loud initial and final FM sweeps. We therefore suggest that the initial FM part might also be relevant for echolocation. From our observations we conclude that the FM components are especially important during obstacle avoidance. In most sounds emitted in the field a fainter first harmonic was present. It was usually up to 30 dB fainter than the second harmonic, but in some instances it was as loud or even distinctly louder than the second one (Fig. 6a). Even within one sound the intensity relationship between the two harmonics may be reversed. We therefore suggest that the first harmonic is an integral part of the signal and relevant for information analysis in echolocation.

d-wave superconductivity and pomeranchuk instability in the two-dimensional hubbard model

We present a systematic stability analysis for the two-dimensional Hubbard model, which is based on a new renormalization group method for interacting Fermi systems. The flow of effective interactions and susceptibilities confirms the expected existence of a d-wave pairing instability driven by antiferromagnetic spin fluctuations. More unexpectedly, we find that strong forward scattering interactions develop which may lead to a Pomeranchuk instability breaking the tetragonal symmetry of the Fermi surface.

Functional renormalization group approach to correlated fermion systems

Numerous correlated electron systems exhibit a strongly scale-dependent behavior. Upon lowering the energy scale, collective phenomena, bound states, and new effective degrees of freedom emerge. Typical examples include (i) competing magnetic, charge, and pairing instabilities in two-dimensional electron systems; (ii) the interplay of electronic excitations and order parameter fluctuations near thermal and quantum phase transitions in metals; and (iii) correlation effects such as Luttinger liquid behavior and the Kondo effect showing up in linear and nonequilibrium transport through quantum wires and quantum dots. The functional renormalization group is a flexible and unbiased tool for dealing with such scale-dependent behavior. Its starting point is an exact functional flow equation, which yields the gradual evolution from a microscopic model action to the final effective action as a function of a continuously decreasing energy scale. Expanding in powers of the fields one obtains an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes. This review is a comprehensive introduction to the functional renormalization group method for interacting Fermi systems. A self-contained derivation of the exact flow equations is presented and frequently used truncation schemes are described. Reviewing selected applications it is shown how approximations based on the functional renormalization group can be fruitfully used to improve our understanding of correlated fermion systems.

Motor control of the jamming avoidance response of Apteronotus leptorhynchus: evolutionary changes of a behavior and its neuronal substrates

The two closely related gymnotiform fishes, Apteronotus and Eigenmannia, share many similar communication and electrolocation behaviors that require modulation of the frequency of their electric organ discharges. The premotor linkages between their electrosensory system and their medullary pacemaker nucleus, which controls the repetition rate of their electric organ discharges, appear to function differently, however. In the context of the jamming avoidance response, Eigenmannia can raise or lower its electric organ discharge frequency from its resting level. A normally quiescent input from the diencephalic prepacemaker nucleus can be recruited to raise the electric organ discharge frequency above the resting level. Another normally active input, from the sublemniscal prepacemaker nucleus, can be inhibited to lower the electric organ discharge frequency below the resting level (Metzner 1993). In contrast, during a jamming avoidance response, Apteronotus cannot lower its electric organ discharge frequency below the resting level. The sublemniscal prepacemaker is normally completely inhibited and release of this inhibition allows the electric organ discharge frequency to rise during the jamming avoidance response. Further inhibition of this nucleus cannot lower the electric organ discharge frequency below the resting level. Lesions of the diencephalic prepacemaker do not affect performance of the jamming avoidance response. Thus, in Apteronotus, the sublemniscal prepacemaker alone controls the change of the electric organ discharge frequency during the jamming avoidance response.

Renormalization-group analysis of the two-dimensional Hubbard model

Salmhofer [Commun. Math. Phys. 194, 249 (1998)] recently developed a new renormalization-group method for interacting Fermi systems, where the complete flow from the bare action of a microscopic model to the effective low-energy action, as a function of a continuously decreasing infrared cutoff, is given by a differential flow equation which is local in the flow parameter. We apply this approach to the repulsive two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping amplitudes. The flow equation for the effective interaction is evaluated numerically on a one-loop level. The effective interactions diverge at a finite-energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signaled by the diverging interactions we extend Salmhofers renormalization group for the calculation of susceptibilities. We compute the singlet superconducting susceptibilities for various pairing symmetries, and also charge- and spin-density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength, and band filling) we find commensurate and incommensurate antiferromagnetic instabilities or d-wave superconductivity as leading instability. We present the resulting phase diagram in the vicinity of half-filling, and also results for the density dependence of the critical energy scale.

The coding of signals in the electric communication of the gymnotiform fish Eigenmannia : from electroreceptors to neurons in the torus semicircularis of the midbrain

SummaryIn the context of aggression and courtship, Eigenmannia repeatedly interrupts its electric organ discharges (EODs) These interruptions (Fig. 1) contain low-frequency components as well as high-frequency transients and, therefore, stimulate ampullary and tuberous electroreceptors, respectively (Figs. 2, 3). Information provided by these two classes of receptors is relayed along separate pathways, via the electrosensory lateral line lobe (ELL) of the hindbrain, to the dorsal torus semicircularis (TSd) of the midbrain. Some neurons of the torus receive inputs from both types of receptors (Figs. 14, 15), and some respond predominantly to EOD interruptions while being rather insensitive to other forms of signal modulations (Figs. 12, 13). This high selectivity appears to result from convergence and gating of inputs from individually less selective neurons.

Foraging areas and foraging behavior in the notch-eared bat, Myotis emarginatus (Vespertilionidae)

SummaryField observations in a maternity colony of Myotis emarginatus (Vespertilionidae) were made during the summers of 1986 and 1987 in southern Germany. The nursery colony consisted of about 90 adult and 30 juvenile bats which roosted in a dimly lit and relatively cool church attic. Telemetry data from six adult M. emarginatus disclosed that some individuals also use secondary day roosts in trees or small buildings located close to their foraging areas. During the night, radiotagged individuals spent most of the time on the wing in forested areas (Fig. 2). Stationary bouts lasted no longer than 63 min. Individual bats returned to the same foraging areas on consecutive nights. All major foraging areas were situated in or at the fringes of forests, at distances as far as 10 km from the nursery roost. During commuting flights to the forests, M. emarginatus avoided open fields and preferred flight paths which offered cover such as orchards, hedges, overhanging foliage along creeks, etc. On the way to the forests, the bats started to forage within buildings, in open spaces where aggregations of insects were present, and around or within the foliage of various types of trees at the level of tree tops or the upper third of the foliage. At these transient foraging areas close to the maternity roost, M. emarginatus displayed flexible foraging strategies: (1) They gleaned prey (mainly flies and spiders) from the substrate, (2) seized insects in aerial pursuit, and (3) occasionally hovered in front of foliage and walls.Our observations confirm the conclusion from morphometric data on the wings that M. emarginatus is a predominantly gleaning bat and contradict the suggestion that it makes only brief flights of short distances. On the contrary, our field data suggest that M. emarginatus spends most of the night on the wing and commutes over distances of at least 10 km.

Structure and function of neurons in the complex of the nucleus electrosensorius of the gymnotiform fish Eigenmannia: Detection and processing of electric signals in social communication

SummaryThe complex of the diencephalic nucleus electrosensorius (nE) provides an interface between the electrosensory processing performed by the torus semicircularis and the control of specific behavioral responses. The rostral portion of the nE comprises two subdivisions that differ in the response properties and projection patterns of their neurons. First, the nEb (Fig. 1 B), which contains neurons that are driven almost exclusively by beat patterns generated by the interference of electric organ discharges (EODs) of similar frequencies. Second, the area medial to the nEb, comprising the lateral pretectum (PT) and the nE-acusticolateralis region (nEar, Fig. 1 B-D), which contains neurons excited predominantly by EOD interruptions, signals associated with aggression and courtship. Neurons in the second area commonly receive convergent inputs originating from ampullary and tuberous electroreceptors, which respond to the low-frequency and high-frequency components of EOD interruptions, respectively. Projections of these neurons to hypothalamic areas linked to the pituitary may mediate modulations of a fishs endocrine state that are caused by exposure to EOD interruptions of its mate.

The nuclei of the lateral lemniscus in the rufous horseshoe bat,Rhinolophus rouxi

Summary1.In the rufous horseshoe bat,Rhinolophus rouxi, responses to pure tones and sinusoidally frequency modulated (SFM) signals were recorded from 289 single units and 241 multiunit clusters located in the nuclei of the lateral lemniscus (NLL).2.The distribution of best frequencies (BFs) of units in all three nuclei of the lateral lemniscus showed an overrepresentation in the range corresponding to the constant-frequency (CF) part of the echolocation signal (‘filter frequency’ range): in the ventral nucleus of the lateral lemniscus (VNLL) ‘filter neurons’ represented 43% of all units encountered, in the intermediate nucleus (INLL) 33%, and in dorsal nucleus (DNLL) 29% (Fig. 2a). Neurons with best frequencies in the filter frequency range had highest Q10dB-values (maxima up to 400, Fig. 2c) and only in low-frequency units were values comparable to those found in other mammals. On the average, filter neurons in ventral nucleus had higher Q10dB-values (about 220) than did those in intermediate and dorsal nucleus (both about 160, Fig. 2d).3.Response patterns and tuning properties showed higher complexity in the dorsal and intermediate nucleus than in the ventral nucleus of the lateral lemniscus (Figs. 4 and 6). Multiple best frequencies were found in 12 neurons, nine of them with harmonically related excitation maxima (Fig. 5c, d). Best frequencies of six of these harmonically tuned units could not be correlated with any harmonic components of the echolocation signal. Half of all multiple tuned neurons were located in the caudal dorsal nucleus the other half in the caudal intermediate nucleus.4.Synchronization of responses to sinusoidally frequency modulated (SFM) signals occurred in VNLL-units in the average up to modulation frequencies of 515 Hz (maximum about 800 Hz) whereas in the intermediate and dorsal nucleus of the lateral lemniscus responses were synchronized in the average only up to modulation frequencies of about 300 Hz (maximum about 600 Hz) (Figs. 7 and 8).5.A tonotopic arrangement of units was found in the intermediate nucleus of the lateral lemniscus with units having high best frequencies located medially and those with low best frequencies laterally. In the dorsal nucleus the tonotopic distribution was found to be fairly similar to that in the intermediate nucleus but much less pronounced. In more rostral parts of the dorsal nucleus additionally higher best frequencies predominated whereas in caudal areas of that nucleus and also of the intermediate nucleus low BFs were found more regularly. The ventral nucleus of the lateral lemniscus was characterized by a remarkably high proportion of neurons having best frequencies in the filter frequency range. In fact, the entire lateral, central and dorsomedial parts were devoted to the representation of the filter frequency. Lower best frequencies were present only in the medial parts of the ventral nucleus of the lateral lemniscus (Fig. 9).