Riccardo Hertel
email: riccardo.hertel@ipcms.unistra.fr

Rajgowrav Cheenikundil

Defect-sensitive High-frequency Modes in a Three-Dimensional Arti cial Magnetic Crystal

Modern three-dimensional nanofabrication methods make it possible to generate arbitrarily shaped nanomagnets, including periodic networks of interconnected magnetic nanowires. Structurally similar to optical or acoustic metamaterials, these arrays could represent magnetic variants of such artificial materials. Using micromagnetic simulations, we investigate a three-dimensional array of interconnected magnetic nanowires with intersection points corresponding to atomic positions of a diamond lattice. The high-frequency excitation spectrum of this artificial magnetic crystal (AMC) is governed by its microstructure and, to a lesser extent, by the magnetic configuration. The magnetic system displays characteristics of three-dimensional artificial spin ice. It can contain Dirac-type magnetic defect structures, which modify the magnonic spectrum of the AMC similarly as defect sites in a natural diamond crystal influence optical absorption spectra. Our study opens new perspectives for applying such materials in high-density magnonic devices and shows that AMCs represent a promising category of magnonic materials with tunable properties.

The desire to design metamaterials-synthetically prepared material systems with properties that do not occur naturally-has long inspired condensed-matter physicists [START_REF] Kadic | [END_REF][2][3]. The unusual effective material parameters of metamaterials arise from their microstructure, consisting of microscopic building blocks repeated periodically in space [4,5]. This periodic arrangement results in an artificial crystal structure, and it affects the material's photonic, phononic, or electronic response, causing unusual optical [6], acoustic [7], or electromagnetic [8] properties. Research on this topic has recently experienced a significant boost owing to progress in three-dimensional (3D) fabrication techniques [START_REF] Kadic | [END_REF].

Similarly, the advent of three-dimensional nanomagnetic fabrication techniques has sparked considerable interest in the magnetism community [9][10][11][12][13][14]. Still, these techniques are relatively new, and they have not yet had a strong impact on the design of magnetic metamaterials. By harnessing the possibilities provided by FEBID (focussed electron beam induced deposition) it was recently shown that virtually any three-dimensional geometry can be generated [15,16]. The technique allows for a layer-by-layer deposition of magnetic materials with geometries of complex shape and nanoscale feature size. Using this method, it is possible to prepare periodic three-dimensional magnetic arrays of interconnected nanowires [17,18] so that their vertex positions correspond to atomic sites of molecules or crystals, with the nanowires connecting the nearest-neighbor sites [13]. The resulting artificial magnetic crystals (AMCs) are a largely unexplored category of magnetic nanostructures, especially their high-frequency magnetic properties [19].

Such periodic 3D arrays of interconnected nanowires could constitute magnetic metamaterials, with physical properties imprinted by their geometry and different from those of the material in its natural form. In analogy to optical metamaterials [4], whose electromagnetic spectrum is governed by the microstructure, AMCs may be designed so that they represent a medium in which magnons, the fundamental excitations in magnetism, display similarly unusual and microstructure-driven properties. If successful, the development of AMCs with tunable magnonic properties could conduce to a departure from the commonly used approach of designing magnonic crystals [20,21]. In their traditional form, magnonic crystals are arrays of spatially separated micromagnets arranged on a periodic grid, whichthrough their dipolar coupling-display collective features of the high-frequency magnetic excitations. Three-dimensional AMCs formed by interconnected unit cells could represent an advanced type of magnonic crystal in the form of bulk media with nanoscale microstructure and tunable magnonic properties.

The interconnected nanowire arrays forming an AMC constitute three-dimensional versions of artificial spin ice (ASI) systems in terms of their micromagnetic properties [22][23][24]. In the past decade, ASI lattices have advanced to an essential research topic in magnetism [25][26][27][28][29]. In their traditional form, these systems consist of sub-micron-sized patterned magnetic elements with single-domain magnetization arranged in a regular grid. The magnetic state of the ASI is characterized by the magnetic configurations at the crossing points of the array (the vertex sites) [30]. Vertices in which the surrounding magnetic configuration results in a non-zero net magnetic flux toward or away from them are magnetically frustrated defect sites with a monopole-type emergent charge [31][32][33][34]. Numerous magnetic defects, which appear in different forms [33], can be distributed onto the vertex sites throughout the ASI, resulting in a quasi-continuum of possible magnetic configurations [35,36]. From a fundamental perspective, research on ASI systems is driven by the interest to explore their rich and complex features. In addition, from an application point of view, ASIs are promising candidates for future components in low-energy programmable magnonic devices [37,38],

given that their high-frequency oscillations can be manipulated by changing the array's magnetic configuration [20,[39][40][41][42][43][44].

While ASIs have mainly been studied in the form of separated magnetic elements [28,29,45], recent studies have extended the concept to interconnected nanowire arrays [41,[46][47][48].

Moreover, the aforementioned three-dimensional nanomagnetic fabrication techniques have made it possible to study three-dimensional arrays of interconnected magnetic nanowires [11,19,24]. Using micromagnetic finite-element simulations, we investigate the magnetic structure as well as the high-frequency magnetization dynamics in a mesoscopic 3D array of nanowires forming a diamond-type artificial crystal. Our study shows that, similar to the effect of color centers in an actual diamond [49][50][51], magnetic defect structures in 3D AMCs generate pronounced peaks in the spectrum of magnetic modes, thereby leading to an effective "coloring" of the magnetic high-frequency spectrum. Besides the appealing physical analogy between the impact of these magnetic defects on the AMCs' magnonic properties and those of real defects on the optical properties of natural crystals, these findings could have implications for the development of magnonic metamaterials.

The dependence of the AMC's spectrum on the density and type of magnetic defect states discussed here opens up additional possibilities-unavailable in more traditional metamate-rial types-of manipulating the system's high-frequency properties. A control of this type is in agreement with the general goal of research in metamaterials, which is to tailor the effective material parameters of synthetic crystals by controlling the distribution of only a few elementary ingredients within the building blocks. In their recent review, Kadic et al. [START_REF] Kadic | [END_REF] have compared this approach to the ability of a graphical printer to generate a vast multitude of colors from just three color cartridges. In the context of magnetic metamaterials, the insertion of a defect site in the AMC can be interpreted as the replacement of one type of unit cell with another. Leveraging on the detailed knowledge acquired in the magnetism community on the properties of ASI and magnetic defect structures, our study provides new perspectives on the development of magnetic metamaterials with potential applications in devices for storage, magnonic operations, and neuromorphic computations.

I. MODEL SYSTEM

Our model of the AMC structure, illustrated in Fig. 1, consists of cylindrical nanowires connecting nearest-neighbor sites of a diamond lattice. Each nanowire has a length of l = 70 nm and a diameter d = 14 nm), resulting in a diamond-type AMC structure with a cubic lattice constant of about 162 nm.

The arrangement of the magnetization forming in the artificial crystal has the characteristics of a three-dimensional ASI, since each wire is magnetized homogeneously along its axis. This combination of two branches of research in magnetism-3D nanomagnetism and ASI-was recently discussed on the example of a buckyball-type geometry [40]. It is well known that the magnetization state of ASIs is not unique [35,52]. The lowest-energy arrangement of the magnetization is a defect-free magnetic configuration that satisfies the so-called ice rule at every vertex [33], but ASI lattices can also contain several magnetic defect structures carrying monopole-type magnetostatic charges [25,30,52].

II. VERTEX CHARGES AND STATIC CONFIGURATION

Before addressing the magnetic high-frequency modes of the three-dimensional array, it is instructive to analyze the possible micromagnetic configurations at the vertex sites, since-as we shall see-they have a decisive impact on the array's dynamic properties. The 4 FIG. 1: Three-dimensional nanowire array with a diamond-type structure. The array contains 238 nanowires and 155 vertices. Each wire has a length of 70 nm and a diameter of 12 nm. We assume ideally soft-magnetic material with properties corresponding to those of typical FEBID-deposited nanoarchitectures [11].

diamond-type AMC has a coordination number of four: each vertex inside the 3D lattice is the intersection point of four nanowires. Regarding the coordination number, this type of lattice is therefore comparable to the well-known two-dimensional square ASI lattice [28,39,53,54], which can be considered to be its two-dimensional (2D) counterpart. The main constituents of an ASI-in this case the cylindrical nanowires-are usually of Ising type, meaning that their magnetic structure is homogeneous and aligned along the particle axis.

The assembly of such Ising-type nanomagnets into a regular array can generate structures of varying complexity and magnetic frustration at the vertices. Vertex configurations in ASIs can be characterized by their magnetic charge [40,52]. The definition of the vertex charge is based on the fact that-assuming Ising-type magnetization in each nanowire-each nanowire carries magnetic flux either toward the vertex or away from it. Any imbalance in the number of wires with in-and out-flowing magnetic flux results in a net magnetic charge at the vertex, akin to a magnetic monopole [31,33,34].

With this premise, the charge of a vertex with coordination number four can take one of five possible values: +4q ("four-in"), +2q ("three-in/one-out"), 0, ("two-in/two-out"), -2q

("one-in/three-out"), and -4q ("four-out"). The value of the magnetic charge q can be estimated to q = r 2 πM s , where r is the nanowire radius and M s is the saturation magnetization; a material-specific constant. Different vertex configurations and their resulting charges have been discussed in detail for 2D square ASI lattices [36,55], and the same type of chargebased classification can also be made for vertex configurations in 3D diamond-type lattices.

There is, however, a difference regarding the multiplicity of uncharged magnetic structures at the vertices in the 2D as compared to the 3D case. While in 2D lattices the zero-charge configuration exists in two different flavors [55], it only appears in one form in the case of 3D diamond-type lattices [68], see Fig. 2. This reduction of complexity in the 3D system -compared to its 2D analogue-simplifies the interpretation of the magnetic configuration in the diamond-type structure as it allows to identify the vertex configurations uniquely by means of their charge, without the need to distinguish different flavors of zero-charge vertices.

Inside the 3D diamond-type ASI, each nanowire is connected to two vertices with even charges (±4q, ±2q, or 0). In addition to the vertices within the volume, a different type of charge exists at the lattice boundary ("surface"), where the wires are connected only to one vertex. The dangling ends of these outer nanowires can be considered as vertices with coordination number one and charge ±1q, where the sign indicates whether the magnetization points inwards or outwards. In total, the 3D diamond-type ASI lattice with Ising-type nanowires thus contains vertices of charges ±4q, ±2q, ±1q, and 0. Owing to time-inversion symmetry, the magnetic configurations of positive and negative charged vertices are equivalent. It is therefore sufficient to distinguish the vertices only by their absolute charge, which effectively reduces the number of vertex configurations to four. In the diamond-type ASI lattice that we have studied, the diversity of vertex structures is further diminished by the fact that vertices with quadruple charges were never observed in the simulations, which indicates that these configurations are unstable. The magnetic state of the diamond-type ASI lattice is thus effectively characterized by the spatial distribution of double-charged defects of the type "three-in/one-out" and "one-in/three-out". A neutrality condition applies to the sum of vertex charges in the lattice, but not to the individual types: the presence of a double-charged vertex does not imply the formation of an additional double-charged vertex The double-charge vertex in (b) connects two nanowires at the crystal surface whose dangling ends carry an effective charge of ±1. In both images, the color code displays the divergence of the magnetization. With this representation, the single charges appear as a weak orange and blue contrast near the ends of the wires at the top left and the top right, respectively. For better visibility, only a small subset of the computed magnetic moments is displayed with arrows.

of opposite sign, so long as the sum of all vertex charges (including those at the surface) is zero.

The ice-rule-obeying zero-charge vertices are energetically optimal magnetic arrangements within the lattice, whereas charged vertex configurations are localized magnetic defect structures increasing both the exchange and the magnetostatic energy. Defect charges within the AMC increase the magnetic disorder, which can be quantified by the density of charged vertices. In spite of their heightened energy compared to the defect-free state, magnetic structures containing several charged vertices can develop as metastable states, and they may persist even at sizeable external fields. Such multiplicity of metastable magnetization states is well-known [25,36] from 2D ASI systems-a behavior that is found also in our 3D diamond-type lattice. Removing defect configurations from the system requires external fields strong enough to magnetically saturate the sample, which is only achieved at fields exceeding about 100 mT.

III. MAGNONIC SPECTRUM

To study the magnonic spectrum of the diamond-type AMC and how it is impacted by magnetic defects, we consider two different remanent magnetic configurations: a defect-free state, which will serve as reference state, and a meta-stable state with random disorder. In the defect-free state, all vertices within the volume have the same two-in/two-out-type zerocharge configuration. Since the magnetic structure of the defect-free AMC is identical for each unit cell, the magnetic state shares the same periodicity of the crystal structure. The disordered state, on the other hand, contains several vertices with double charge configurations (three-in/one-out or one-in/three-out), and its magnetic structure does not display any spatial periodicity. Double-charge vertices of different sign are randomly scattered within the AMC. In our case, the disordered state contains 35 double-charge vertices and 48 zerocharge vertices. This disordered state is only one realization out of a quasi-continuum of possible configurations [35,56,57].

Let us first analyze the high-frequency modes developing in the ordered configuration.

The magnonic spectrum of the defect-free configuration, shown in green (line a) in Fig. 4, displays five clearly defined peaks corresponding to distinct magnetic modes within the ASI lattice. Fourier analysis methods, through which the spatial distribution of the local oscillation amplitude (i.e., the mode profile) can be extracted for each frequency, reveal the central role played by the geometry in each of the peaks in the spectrum. Each of the prominent modes can be attributed to oscillations localized at different geometric constituents of the AMC: the vertices in the volume, the cylindrical nanowires, and the wire ends at the crystal surface.

The lowest-frequency mode at 9.9 GHz, denoted with the symbol 1 in Fig. 4, is a surface contribution arising from the oscillations of the single-charge vertices on the outer shell. The color-code through which the mode profile is represented in Fig. 5(a) and (b) shows inactive regions in purple color, while the yellow color indicates regions of maximum oscillation amplitude. Only the free ends of the single-connected wires partake in the magnetic oscillation at this frequency. This is the only surface contribution in the magnonic spectrum of the AMC. The relative height of the peak 1 with respect to the other modes therefore depends on the crystal size and the resulting surface-to-volume ratio. The oscillations at the wire ends appear to be decoupled from one another. No clear correlation is observed among the phases of this oscillation at different sites (see Movie 1 in the supplemental material).

The most pronounced peak in the spectrum develops at 14.1 GHz (see label 2 in Fig. 4).

The corresponding mode is an oscillation of the magnetization at the zero-charge vertices within the array. The profile of this mode, shown in Fig. 5c) and d), displays a clear localization at the center of the tetrapod-shaped vertices. Only the vertices within the volume are active; the free ends do not oscillate at this frequency. Contrary to the mode localized at the free ends at the surface, the oscillation phases of the individual vertices is correlated and almost synchronous throughout the volume of the lattice (see Movie 2 in the supplemental material). Three distinct high-frequency peaks can be identified in the spectrum of Fig. 4, labeled 3 , 4 , and 5 . At these frequencies, standing waves develop in the nanowires. Fig. 5e)

shows the mode at 28.2 GHz, in which the magnetic oscillation within each wire contains four nodes and three anti-nodes. The profiles of this category of modes are displayed in Fig. 5f). In that image, the wires have been isolated graphically from the main crystal for visualization purposes, but they are magnetically and structurally embedded into the AMC.

In these standing wave modes, remarkably, the wire ends (i.e., the vertices) neither act as free nor as fixed ends of the oscillation. At 16.7 GHz and 28.2 GHz, the vertices remain inactive, while at 19.1 GHz the magnetization also oscillates at the vertices. Although there appears to be a certain phase correlation of the standing-wave modes in different wires within the crystal, it is not always as clear as in the case of the vertex mode 2 . The modes can be viewed in the movies 3, 4, and 5 of the supplemental material. A clear macroscopic correlation can be observed only for mode 4 , which is the only one at which the volume vertices participate in the oscillation. This suggests that long-range dynamic correlations primarily result from the dynamic interaction between the volume vertices.

IV. MODES IN THE MAGNETICALLY DISORDERED STATE

Up to this point, we have investigated the magnetic high-frequency modes of an AMC in which the structure is magnetically ordered, such that all vertices in the volume have a zerocharge magnetic configuration. Although the magnetic structure is inhomogeneous (there are six different magnetization directions in the nanowires, according to their orientation in space), the magnetic modes do not appear to be influenced by the magnetic structure.

The modes discussed so far appear to be tightly connected to the structural or geometric elements of the AMC: vertices, nanowires, and surface; not to the magnetization.

A strong influence of the magnetic structure on the magnetic modes, however, is revealed by analyzing the magnetically disordered AMC. The excitation spectrum of the disordered AMC, printed in red in Fig. 4 (line b), displays significant differences with respect to the ordered AMC (line a). The most striking change is the appearance of an additional lowfrequency mode 6 at 7.9 GHz. This mode describes the oscillation of the double-charge vertices in the volume (three-in/one-out and one-in/three-out configurations), which are absent in the ordered state. The appearance of this mode is an example of a high-frequency feature of the AMC that is directly related to its magnetic configuration. The zero-charge vertices 2 , which are still present in the disordered AMC, oscillate at a frequency markedly different from the oscillation of the double-charge vertices, even though they are localized at identical geometrical sites.

The different oscillation profiles of the zero-charge and double-charge vertices are shown in Fig. 6 a) and b), respectively. These mode profiles are complementary in the sense that only double-charge vertices are active at 7.9 GHz, and only the zero-charge vertices participate in the mode at 14.2 GHz. In spite of this clear separation in terms of frequency and micromagnetic configuration, the oscillation amplitude of the different vertices is not Only the vertices with zero-charge type magnetic configuration partake in the oscillation at that frequency. The frequency as well as the local mode profile of the oscillating zero-charge vertices are the same as in the ordered state, shown in Figs. 5c,d). The vertices with double charge configuration are active at a significantly lower frequency (7.9 GHz). The mode profile of this oscillation, shown in frame (b), highlights the activity of a set of vertices which is complementary to that in the mode shown in frame (a).

identical within each of these modes. Some vertices display strong oscillations, whereas a few vertices show only little activity. We attribute this inhomogeneity in the amplitude to local magnetic fields within the array arising from the charged vertices. The frequency of the zero-charge vertices and the local mode profile of the active vertices remain mostly the same as in the ordered AMC, but the phase correlation between the vertices is lost in the disordered state (see Movie 6 of the supplemental material). Compared with the ordered case, the intensity of the zero-charge vertex mode 2 is reduced in the disordered state, in accordance with the lower density of this type of vertices in the disordered AMC.

A more significant difference in the spectra concerns the peaks 3 and 4 describing the first and the second standing-spin-wave modes in the nanowires, which significantly broaden and become almost indiscernible in the disordered AMC. Instead of the two well-defined peaks, a weakly defined quasi-continuum of overlapping modes appears in the frequency range between 15 and 19 GHz. A detailed Fourier analysis of the individual nanowires shows that these modes are still present in the disordered crystal. Their frequency, however, differs from one nanowire to another, resulting in a broadened and blurred spectrum in that frequency range. This effect, too, can be explained by spatial variations of local magnetic fields in the disordered state.

Charged vertices generate local magnetic fields within the disordered crystal that are absent in the ordered state. The magnetic divergences developing, for example, in a vertex with negative double charge (three-in/one-out configuration) and those in a positive double charge vertex (one-in/three-out) are monopole-type sources and sinks of the magnetic field H, respectively. Correspondingly, a longitudinal magnetic field develops in a nanowire forming a Dirac string that connects two such vertices of opposite charges [31,39]. Such longitudinal magnetic fields result in a shift of the standing-wave modes to different frequencies [58]. There are numerous possible combinations of vertices that can be pairwise connected by a nanowire in the disordered state since vertices can differ in both magnitude and sign of the magnetic charge they carry. Accordingly, although the homogeneous and axial magnetization structure is the same in all nanowires, their high-frequency magnetic dynamic properties differ as a result of a broad distribution of local fields throughout the crystal.

In addition to differences in magnitude, the magnetic field developing in wires connecting oppositely charged vertices can also differ in sign. The dipolar magnetic field generated by the vertex charges can be parallel or antiparallel to the wire magnetization. This leads to two qualitatively different cases, resulting in an additional lift of the degeneracy of nanowire modes. Compared to the modes 4 and 5 , the higher-frequency excitation 6 at 28 GHz remains largely unaffected by the magnetic disorder of the crystal. In general, high-frequency modes are dominated by the exchange interaction, which leads to local effective fields that are much larger than those due to dipolar fields. Correspondingly, changes in the dipolar field distribution have a weaker impact on exchange-dominated high-frequency modes.

V. IMPACT OF STRUCTURAL DEFECTS

Having established that charged vertices, which represent magnetic defects, strongly affect the high-frequency properties of the AMC, we now analyze how a structural defect changes the magnetic spectrum. We introduce a structural defect by removing a single nanowire inside the AMC, as shown in Fig. 7a). This removal results in two defect vertices with reduced coordination number as they represent the junction of three instead of four nanowires. To isolate the effect of this structural defect on the high-frequency properties of the AMC, it is introduced as a modification of the otherwise defect-free magnetic state, i.e., the ordered reference state. Despite obeying the spin-ice rule, both defect vertices have a charged magnetic structure. With their two-in/out-out and one-in/two-out type structures, the defects display a magnetic charge of type ±1q while preserving the overall charge neutrality of the AMC. Although they have a magnetostatic charge equal to that of surface vertices, the defect vertices are different in terms of their magnetic structure and charge type. The defect vertices carry volume-type magnetostatic charges, while the free ends have surface-type charges [START_REF] Brown | Magnetostatic Principles in Ferromagnetism[END_REF]. The magnonic spectrum of the AMC with a structural defect -represented as line (c) in Fig. 4-demonstrates the appearance of an additional, weakly pronounced mode (label 7 in the inset). As shown in Fig. 7b), this mode is localized at the defect vertices.

It has a low intensity in the spectrum because the signal originates from only two defect vertices out of a total of more than 100 surface and volume vertices in the AMC. Besides this additional low-amplitude peak, the introduction of a structural defect also leads to a clear shift of the volume modes 3 , 4 , and 5 toward lower frequencies. Adding more structural defects exacerbates this effect and leads to further changes in the spectrum. Upon increasing the density of structural defects, the extent to which the spectrum changes quickly reaches a degree at which it becomes impossible to consider the defects as mere perturbations or modifications of the defect-free state.

VI. CONCLUSION

We have identified a wide variety of high-frequency magnetic modes in an AMC consisting of a three-dimensional network of interconnected nanowires arranged in the form of a diamond-type artificial crystal structure. The magnetic modes of the AMC can be classified into two categories: the fundamental excitations, which are related to the geometrical constituents of the network (vertices, nanowires, surfaces), and the defect modes, which are linked to the magnetic structure of the AMC. Magnetically charged vertices within the crystal give rise to distinct additional modes and yield unmistakable fingerprints in the magnonic spectrum. These charged vertices, which represent magnetic defects in the terminology of artificial spin ice, thus lead to effects analogous to the changes that atomic defect sites induce in the optical spectrum of natural diamonds [49,50]. We anticipate that our findings, which demonstrate that high-frequency modes can be controlled through both the 3D geometry and the magnetic structure of an AMC, will further promote three-dimensional interconnected nanowire arrays as a new category of magnonic metamaterials [START_REF] Kadic | [END_REF] with promising potential for applications in devices for storage, magnonic operations, and neuromorphic computations [26,[START_REF] Grollier | [END_REF].

when the magnitude of the local torque M × H eff drops below a threshold value at every discretization point. In this type of LLG-based energy-minimization, the magnetization dynamics towards the relaxed state is unimportant and therefore, to accelerate the calculation, the damping parameter α can be set to an unrealistically large value. We typically use α = 0.5 in these cases.

The defect-free reference state of the AMC is obtained by starting from a homogeneous magnetization state and relaxing the system at zero applied field. The disordered state, on the other hand, is prepared by minimizing the system's total energy at zero field after starting from a random initial configuration.

C. Mode calculation

The high-frequency modes are generated by applying a small perturbation to the relaxed state-typically a low-amplitude picosecond magnetic field pulse-and analyzing the magnetization dynamics that it induces. It is important to choose a perturbation that is, on one hand, sufficient to generate a dynamic response while, on the other hand, avoiding magnetic switching processes or other persistent modifications of the zero-field configuration. When a suitable perturbation is applied, it triggers a characteristic "ringing" of the magnetization arising from the small-angle precession dynamics of the local magnetization around the equilibrium state. This ringing fades out after a few nanoseconds as the system asymptotically returns to the converged state. To extract the magnetic modes, the oscillations during this ringdown process are recorded and Fourier-analyzed [39,40,66]. The overall spectrum, as shown in Fig. 4, is the sum over all discretization points of the Fourier-transformed magnetic oscillation. A low value of the damping constant α must be chosen to ensure a sufficiently large number of oscillations and narrow line width in the Fourier spectrum; we have set α = 0.01 in these dynamic calculations.

The profiles of the individual modes are filtered from the Fourier-transformed magnetic oscillations by means of a windowed Fourier back-transform from the frequency to the time domain. This Fourier back-transformation is performed such as to only consider the oscillations occurring within a narrow frequency range near the frequency of interest [66,67]. In addition to this windowing in the frequency domain, we also sometimes restrict the analysis to certain spatial regions, for instance when we analyze the modes unfolding within

FIG. 2 :

 2 FIG. 2: Zero-charge vertex configuration in a tetrapod structure-the elementary building block of a diamond-type 3D ASI lattice (a). Only one type of zero-charge configuration exists in 3D, whereas different zero-charge vertex structures are known from 2D square ice systems. The flavors of zerocharge vertex configurations shown in panels (b) and (c) known from 2D square ice ASI lattices differ qualitatively in their micromagnetic structure, and they have different physical properties.Nevertheless, they can be interpreted as projections of a single 3D structure onto different planes.The color code represents the x-component of the magnetization.

FIG. 3 :

 3 FIG. 3: Different types of effective magnetic charges in the diamond-type 3D lattice. (a) Doublecharge vertex with a "three-in/one-out" configuration, leading to an effective charge of +2. The orange colored region in panel (b) shows an equivalent "one-in/three-out" vertex with charge -2.

FIG. 4 :

 4 FIG. 4: Magnetic excitation spectrum of the ordered magnetic crystal and defect-induced modifications. The AMC displays characteristic frequencies at which the magnetization oscillates. The spectrum (a) of the ordered state contains five distinct peaks, each corresponding to a different magnetic excitation. The magnetically disordered state yields a spectrum (b) with an additional low-frequency mode 6 . The spectrum (c) refers to an AMC containing a structural defect site.

FIG. 5 :

 5 FIG. 5: Fundamental modes of the defect-free diamond-type AMC. The low-frequency modes, corresponding to the peaks labeled 1 and 2 in Fig. 4, are oscillations localized at the surface and volume vertices, respectively.(a) The mode at 9.9 GHz refers to oscillations of the free ends, i.e, to the outermost vertices at the crystal surface, as shown in the magnified view in frame (b). The strongest excitation in the spectrum, displayed in frame (c), is due to vertices within the volume which become active at 14.2 GHz. The magnified view (d) shows the localization of the mode at the vertex sites. (e) The high-frequency modes at 16.7 GHz, 19.1 GHz, and 28.2 GHz (corresponding to the peaks 3 , 4 , and 5 , respectively) are oscillations describing longitudinal standing spin waves in the nanowires. Frame (f) shows the computed mode profile along the nanowires for these standing-wave modes, with frequency increasing from the top to the bottom. In all panels, the color code ranging from purple to yellow describes regions of minimum and maximum oscillation amplitude, respectively.

FIG. 6 :

 6 FIG. 6: Vertex modes in the disordered state. Frame (a) shows the mode profile at 14.2 GHz.

FIG. 7 :

 7 FIG. 7: Structural defect within the AMC giving rise to an additional mode localized at the defect site. Frame (a) shows the modified AMC in which a single nanowire has been removed. The removal of a nanowire introduces two vertices with reduced coordination number which oscillate at 11.7 GHz. The spatial distribution of this mode is localized at the defect site (b). This defect mode appears in the spectrum as a small peak labeled 7 in the inset of Fig. 4 .

Acknowledgments This work was funded by the LabEx NIE (ANR-11-LABX-0058 NIE) in the framework of the Interdisciplinary Thematic Institute QMat (ANR-17-EURE-0024), as part of the ITI 2021-2028 program supported by the IdEx Unistra (ANR-10-IDEX-0002-002) and SFRI STRATUS (ANR-20-SFRI-0012) through the French Programme d'Investissement d'Avenir. The authors acknowledge the High Performance Computing center of the University of Strasbourg for supporting this work by providing access to computing resources.

Data availability

The raw data on the static magnetization structures of the reference state and the disordered state is available from the authors upon reasonable request.

Code availability

Our proprietary finite-element micromagnetic code tetmag is closed-source and not publicly available.

VII. METHODS

A. Micromagnetic model

The magnetic structure and the dynamic modes are calculated using our customdeveloped micromagnetic finite-element software tetmag, [61]. Given an initial magnetic configuration M ( x), the algorithm integrates the Landau-Lifshitz-Gilbert equation [62,63]

in time and thereby yields information on the structure of the magnetization and its dynamics. The effective field H eff is defined by the variational derivative of the micromagnetic energy E with respect to the magnetization, µ 0 H eff = -δE( M )/δ M , where

and the magnetostatic energy E mag = (µ 0 /2) M • ∇u dV , where A is the ferromagnetic exchange constant, H ext is an externally applied field, and u is the magnetostatic potential [START_REF] Brown | Magnetostatic Principles in Ferromagnetism[END_REF].

In equation ( 1), α is a dimensionless damping constant and γ 0 /2π = 28.02 GHz T -1 is the gyromagnetic ratio, and M s = M is a material-specific constant; the saturation magnetization. The magnetic material of this study is described by the parameters of FEBID-deposited cobalt: µ 0 M s = 1.2 T, zero magnetocrystalline anisotropy, and A = 1.5 × 10 11 J m -1 [64].

The numerical calculation uses a discretized representation of the magnetization in which the vector field M is defined at the nodes of a tetrahedral finite element mesh. We use netgen [65] to generate the finite-element meshes and made sure that the element size remained everywhere below the magnetostatic exchange length l s = 2µ 0 A/M 2 s = 5.1 nm. In this study, the choice of the maximum cell size was primarily imposed by geometrical aspects rather than by the intrinsic material parameters. To accurately reproduce the nanometric rounded features of the modeled shape, the discretization size was set to 2.5 nm -significantly smaller than l s . Both authors prepared the figures and contributed to the discussions and analyses of the data.

Competing interests

The authors declare no competing interests.

Supplementary Files

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Alexandre Ferrere
email: ferrere.alexandre@hotmail.fr

Creative Environments: The Geo-Poetics of Allen Ginsberg

Keywords: Allen Ginsberg, Beat Generation, poetry, poetics, memory, Guy Debord, psychogeography, landscape, ecocriticism

As was the case for other writers from the Beat Generation, geography is more than simply a setting for Allen Ginsberg's work, as his poetry also bears the imprint of the influence of the landscapes through which he traveled in his mind and poetic practice. In the 1950s, the same decade which saw the composition of Ginsberg's Howl, Guy Debord and his followers developed the concept of "psychogeography" and "dérive" to analyze the influence of landscapes on one's mind. The Debordian concept of psychogeography implies then that an objective world can have unknown and subjective consequences. Inspired by Debord's theories and through the analysis of key poems, this paper argues that a psychogeographical focus can shed new light on ecocritical studies of Ginsberg's poetry. It can indeed unveil the complex construction of the poet's own space-time poetics, from hauntological aspects to his specific composition process.

Introduction

In his introduction to Geocriticism: Real and Fictional Spaces, Bertrand Westphal writes: "The perception of space and the representation of space do not involve the same thing" (Westphal 2011, p. 1). Passing from an objective notion of "space" to a subjective concept of "place" is to take a step towards psychogeography, as it involves both the perception and the representation of one's surroundings. In order to understand and to study the influence of geography on emotion, Guy Debord and the Situationist International developed the concept of psychogeography as well as an experimental method they named the "dérive"-to wander in a landscape so as to re-discover it. Besides reinforcing the position of the individual in the world, psychogeography is, in the Debordian system, a way for citizens to reclaim their territories. But the Debordian system also advocates for a complete revolution of urbanism and architecture in creating new zones according to psychogeographical reports-as written in Potlatch #5: "Poetry is written in the shape of cities. We will thus build some overwhelming ones" (Conord 1954c. My translation).

In the wider theoretical "environment" of ecocriticism, psychogeography can be a useful tool to analyze literature, especially if we consider written texts as creative psychogeographical accounts. From this perspective, Allen Ginsberg's work is particularly striking, as the number of poems involving his experience of different places can reveal (as well as his journals, sometimes written in very specific places, like his Indian Journals or his South American Journals). Indeed, his Odyssean adventures in more than sixty countries around the globe have inevitably shaped his poetry, not only in terms of redefinitions of space but also in his vision of society as well as in his very methods of composition. And while in political terms psychogeography could be seen as a defense of freedom of movement, it could be considered poetically as an emotional interpretation of the world. The self is at the center of psychogeography, as it can only be experienced by a subject, it is therefore not surprising that Ginsberg, often labeled a "confessional" poet, wrote many poems about his relationship to his close environment.

The deep involvement of the Beats in environmental (re)actions is well-known; Michael McClure even considered the "poetry of the Beats" as "the first literary wing of the environmental movement" (Meltzer 2001, p. 185). Nevertheless, a part of Ginsberg's poetry is not only about having to cope with an environmental crisis. Sometimes, it is also the poet's own crisis, lamentations, or visions that are explored in the larger frame of an environmental and urban downfall. In other words, it seems that ever-changing landscapes are not always observed for their essence nor for the impact mankind can have on them; it is also a way to express what these landscapes can symbolize in the poet's life and how they can represent the inner tensions between Ginsberg and the outside world in a social and political way.

Historically, analyzing Ginsberg's poetry through the lens of Debord's theories could appear as a counterintuitive experiment. Indeed, Debord and the Situationists were rather hostile to Beat writers-they are described as "mystical cretins" in the first issue of Internationale Situationniste (Debord 1958a). Furthermore, Greil Marcus remarks that Alexander Trocchi, who was very close to Debord, was even indirectly excluded from the Situationists because of his relationship with Allen Ginsberg (Marcus 2011, p. 359). Nonetheless, both Ginsberg and Debord's followers were advocating for a new vision of landscapes through direct and subjective experiences of them, in a larger countercultural and intellectual renewal. Their works were also often linked to political criticisms of capitalism and urbanism. While the Beats were writing "both a critique of urban-industrial civilization and a search to recover authentic human identity" (Stephenson 2009, p. 177), Debord wrote, "urbanism […] is capitalism's method for taking over the natural and human environment" (Debord 2005, p. 95). Their ideological points of contention are in that sense not antinomic when it comes to environment, ecology, and urbanism. Rooted in Debordian theories, this study, therefore, considers psychogeography as a critical tool that can bring a new understanding to how the objectivity of the world has been subjectively and poetically transcribed by the poet. In this paper, we will explore Allen Ginsberg's subtle and heterogeneous constructions of his own development of space-time poetics. From hauntological aspects to eco-linguistical approaches and genetic criticism of the poet's methods of composition, Allen Ginsberg's creative process will be analyzed with regard to what Guy Debord called "psychogeographical variations", as applied to the "field" of poetry.

Psychogeography or the Art of Awareness

Almost every issue of the magazine Potlatch (1954)(1955)(1956)(1957), the "information bulletin of the French group of the Lettrist International" 1 in which Guy Debord actively participated, is infused with the notion of psychogeography-a concept directly inherited from Ivan Chtcheglov (also known as Gilles Ivain)2 and his essay Formulaire pour un urbanisme nouveau. The word psychogeography is first introduced along with the notion of a game in the article Psychogeographical Game of the Week, in Potlatch #1:

Depending on what you are after, choose an area, a more or less populous city, a more or less lively street. Build a house. Furnish it. Make the most of its decoration and surroundings. Choose the season and the time. Gather together the right people, the best records and drinks. Lighting and conversation must of course be appropriate, along with the weather and your memories.

If your calculations are correct, you should find the outcome satisfying. (Please inform the editors of the results.) (Conord 1954a) In this unattributed article, the term "psychogeography" is not defined, but the juxtaposition of words like "area", "city", "memories", and "outcome" suggests that "psychogeography" focuses on 1 It will be later subtitled the "information bulletin of the Lettrist International" until finally the Lettrist International becomes the Situationist International. what happens at the frontier of two worlds, the physical and the emotional. In the second issue of Potlatch, Guy Debord (who signed Guy-Ernest Debord) published an article entitled Exercise in Psychogeography, which emphasizes the interactive features of the still undefined term "psychogeography":

The postman Cheval is psychogeographical in architecture Arthur Cravan is psychogeographical in hurried drifting. Jacques Vaché is psychogeographical in dress. Louis II of Bavaria is psychogeographical in royalty. Jack the Ripper is probably psychogeographical in love. (Conord 1954b) Psychogeography is then, in its first two mentions in Potlatch, either a game to create a new "situation"-based on the "dialectic setting-behavior" ("Potlatch" 1957)-or a poetic exercise unveiling the confrontation of a subject and their environment (whether it be a natural environment or an intimate environment). But those are not satisfying in terms of intellectual definition and implication; is psychogeography a concept? A tool? A process? Subsequent issues of Potlatch bring no solid answer to those questions, even though the term is used several times. But psychogeography is part of a wider philosophy in which urbanism plays a key role. The aim of this avant-garde group of thinkers, who sought to meld arts and politics, is to "create situations" (Conord 1954d), to offer the unexpected, partly through what they called "influential urbanism" (Conord 1954d). It is important to underline that, even though urbanism could be thought of as in opposition to ecological questions, this is not the case for Debord's vision of it. Indeed, besides advocating against the "temporal fixation of cities" (and "fixation of people at certain points of a city"), the museumification of cities, and purely aesthetical architecture, Debord asks for "moving cities", "the overgrowth of tropical vegetation", and a "marriage with nature more audacious than anything attempted by Frank Lloyd Wright" (Debord 1959).

Despite the fact that psychogeographical reports (as well as maps) were published in Potlatch by members of the Lettrist group, in which they described as precisely as possible the influence of the environment on their feelings (and in that sense being what could be called ecocritical reports), Debord would define "psychogeography" only in 1955. Indeed, in an article entitled Introduction to a Critique of Urban Geography, published in the Belgian magazine Les Lèvres Nues 3 , Debord wrote: Psychogeography could set for itself the study of the precise laws and specific effects of the geographical environment, consciously organized or not, on the emotions and behavior of individuals. The adjective psychogeographical, retaining a rather pleasing vagueness, can thus be applied to the findings arrived at by this type of investigation, to their influence on human feelings, and even more generally to any situation or conduct that seems to reflect the same spirit of discovery. (Debord 1955) One interesting aspect of psychogeography as defined by Debord is, therefore, its protean nature, whether it be in its use or its object. Nevertheless, the "pleasing vagueness" of the word does not mean imprecise analysis, as at the heart of the word lays the idea that geography, landscapes, and environments actively trigger emotion and physical reactions. In a sense, Debord and the Situationists sought to analyze the mechanisms at the frontier between an open world and a secret mind-what is seen and what is felt. Thanks to this "vagueness"-and thus, to the capacity of psychogeography to be adapted to new situations and contexts-the Debordian definition of psychogeography helps to unveil certain characteristics of Allen Ginsberg's poetry, especially when used in an ecocritical perspective.

Intrinsically linked to the notion "psychogeography" and equally important for Debord's followers, the concept of "dérive" is also a key element when it comes to psychogeographical analysis. The "dérive", which Nathalie Caritoux defines as the "experimental method" of psychogeography ((Caritoux and Villard 2017, p. 8) My translation), is first described in the first issue of Internationale Situationniste as: 3

To which the Lettrist Internationale collaborated from issue #6 onward.

A mode of experimental behavior linked to the conditions of urban society: a technique of rapid passage through varied ambiances. The term also designates a specific uninterrupted period of dériving. (Debord 1958a) In other words, the "dérive" 4 is an exploratory process-cabs are in this regard the preferred vehicle (Dahou 1954a)-used by psychogeographers in order to produce reports and maps, as architecture is seen by the Situationists as "a means of knowledge and a means of action" (Debord 1958a). In Potlatch #14, the "dérive" is also defined as a "method of aimless displacement" founded on the "influence of scenery" (Dahou 1954b. My translation). This point will be important when we will study some poems written during Allen Ginsberg's own wanderings.

But a few years after offering the first definition of "psychogeography", Debord wrote in the first issue of Internationale Situationniste, in 1958:

A situation is also an integrated ensemble of behavior in time. It is composed of actions contained in a transitory decor. These actions are the product of the decor and of themselves, and they in their turn produce other decors and other actions. How can these forces be oriented? (Debord 1958a) In this passage, besides advocating for an active reshaping of urbanism, Debord adds a new feature to psychogeography that will be important when studying Allen Ginsberg's poetry. Psychogeographical reactions seem indeed to be circular and could be seen as active reciprocity between the landscape and observer. More than a mere mirroring effect, the landscape affects the observer who, in turn, affects the image and representation of the landscape following a phenomenological impulse. We will see later how this sensitive exchange triggered Ginsberg's compositions, but also what forms this circularity takes on in his poetry.

Towards a Geography of Memories

In Geocriticism: Real and Fictional Spaces, Bertrand Westphal wrote that "The concrete details of geography often relate to a spiritual hermeneutic rather than to immediate observation" (Westphal 2011, p. 2). Adapted to an ecocritical stance, what is implied here is strongly psychogeographical, even phenomeno-geographical, to the extent that landscapes convey meaning for the psychogeographer, the writer, or simply the conscious individual-in most cases, an internal representation almost always associated with feelings. In that sense, psychogeography can be generally considered, as implied above, a re-evaluation of urbanism, an invitation to build anew everyday scenery according to inner perceptions. This metaphysical sense of belonging to a territory that exists more through how it is emotionally perceived than objectively observed is rooted in ecocriticism and central to Ginsberg's poetry. His tribute poem entitled At Apollinaire's Grave, recalling the poet's visit to the grave of the famous French poet, reflects this experience of emotionally heightened response to the poet's immediate surroundings: the universe is a graveyard and I walk around alone in here knowing that Apollinaire was on the same street 50 years ago his madness is only around the corner and Genet is with us stealing books (Ginsberg 2006, p. 188) Ginsberg emphasizes here what is called the spirit of places: when a place is so loaded with a strong sense of history that it can transcend temporal structures. Time and space appear here as two sides of the same coin-the poem is somewhat balancing on its edge. The place becomes a personal book of history. The place is, as Nathalie Caritoux wrote, not "an empty frame to fill with behaviors" but "the cause, the source of behaviors" ((Caritoux and Villard 2017, p. 10) My translation). Therefore, this phenomenal impulse (in the phenomenological sense) of a past inhabiting a place could be seen as an extension of Walter Benjamin's idea of unconscious memories, to the extent that unintentional memories emerge from the junction of collective and personal memories (Benjamin 2000, p. 335). And 4The term is already used in Gilles Ivain's essay mentioned earlier.

it is also the persistence of place through memories that is at work in this poem, as symbolized in this passage where hands are part of the forces pushing the poet into the geography of the mind (Ginsberg 2006, p. 188):

Already our hands have vanished from that place my hand writes now in a Room in Paris Git-le-Coeur Between those lines, the reader understands that a place, when infused with poetics, can complicate the notion of time-not as in the cliché of overcoming death in writing 5 , but in transcending time through space (reversing thus the equation of space transcended by time). At the heart of this poem is not so much the temporal difference between two eras (a dividing force that is physically experienced), but the common ground that allows the melding of those eras into one another (the merging force geography gives birth to). Therefore, in the second part of the poem, Ginsberg describes himself at the Bateau Lavoir, in the company of artists like Jacob, Picasso, Tzara, Breton, Cendrars, Gide, etc. The reader easily understands that the conventional notion of time is abolished by the very fact that the poet wanders ("dérive") in the same place as these French artists did once. Another important element in this poem (as in other works involving "meeting" other deceased artists, like Whitman or Lorca in A Supermarket in California) is that Ginsberg uses his imagination to physically describe himself amongst those artists he admired. Once again, time is overcome through the place and those souvenirs are ones of situations that never happened.

Allen Ginsberg ends his three-part poem with a line that summarizes this syncretism of perspectives: "I am buried here and sit by my grave beneath a tree" (Ginsberg 2006, p. 190). This line does not only cloud spatial and temporal structures, but it also connects the collective and Ginsberg's own memories. This work balances between a tribute and a personal reflection, and the place is the first impulse of this movement. This poem, by essence, unveils the geographical tensions at work within the poet's mind: the poet shuttles between two different eras made from the same material, and what they have in common is the scenery. To use Debordian vocabulary, the result of the confrontation between the poet and his very surrounding is a psychogeographical poem in itselfand Ginsberg's reflexive "dérive" through the graves gives birth to a new, specific and actualizing "situation", as was the aim of the Situationists.

But the landscape in Allen Ginsberg's poetry is not lived through the senses only: imagination is here an essential but paradoxical condition 6 . In fact, Jean-Jacques Wunenburger distinguishes imagination, which he describes as "too close to fiction", from the imaginal which is made of "a group of images, loaded with affective values (positive or negative) […] which are the subject of a shared experience, of a vision of the world shared by the participants of a joint space" ((Caritoux and Villard 2017, p. 220) My translation). In this poem, Ginsberg composes then from both imagination and imaginal. Nathalie Caritoux adds to this notion of imaginal the idea of kaïros, which could be defined as the right moment from which a revelation can occur (Caritoux and Villard 2017, p. 21). These two elements, the imaginal of the place, which is large, and the kaïros, the moment of the visit, which is intimate, form the invisible frame of the poem. In Ginsberg's poem, the place has a subjective meaning rooted in an objective ground. But the place is as much a projection, through the emotional phenomenology of the poet (Caritoux and Villard 2017, pp. 220-21), as a projector actualizing the position of the poet's mind in a specific environment, which will lead to the poetic gesture. Here, the whole poem, because of its raw material, could be considered a projection, while traces of the place 5 On this topic, and in this poem, Ginsberg was rather clear, and he wrote lines like "my temporary hands" and "my temporary American Howl" (Ginsberg 2006, p. 188). 6 In Over Kansas, Ginsberg writes about the limit of imagination:

And down there's Hollywood, The starry world below […]. Not even the human Imagination satisfies The endless emptiness of the soul (Ginsberg 2006, p. 124).

acting as a projector on the poet are scattered throughout the poem-a line summarizes it: "Peter Orlovsky and I walked softly thru Père Lachaise we both knew we/would die" (Ginsberg 2006, p. 188). The place is an active muse that transforms the poet and his knowledge of his own identity and temporality; and the place, in return, is transformed by the poet in an effort of imagination. This dynamic exchange, therefore, changes the nature of the relation between the poet and the place.

This psychogeographical porosity is at work in many of Ginsberg's poems; one of the most striking examples is his famous Kaddish, written after the death of his mother. Many passages in the poem highlight the importance of places, which are always a poetic trigger for memories. For example, right from the beginning, the poet is taken by the city that seems to be the only solid ground on which Ginsberg could stay in his mourning, his thoughts being spread in place and time (Ginsberg 2006, p. 217):

Strange now to think of you, gone without corsets & eyes, while I walk on the sunny pavement of Greenwich Village. downtown Manhattan, clear winter noon, and I've been up all night, talking, talking, reading the Kaddish aloud, listening to Ray Charles blues shout blind on the phonograph As in At Apollinaire's Grave, the "dérive" of the poet motivates his later recollections. The genealogy of the poem is thus blown away in the landscape-born from a "situation" that is intrinsically lost because experienced. A few lines below, Ginsberg writes:

It leaps about me, as I go out and walk the street, look back over my shoulder, Seventh Avenue, the battlements of window office buildings shoulddering each other high, under a cloud, tall as the sky an instant-and the sky above-an old blue place. or down the Avenue to the south, to-as I walk toward the Lower East Side -where you walked 50 years ago, little girl-from Russia, eating the first poisonous tomatoes of America-frightened on the dockthen struggling in the crowds of Orchard Street toward what?-toward Newark- (Ginsberg 2006, p. 217).

Again, the poet walks and remembers his mother who died earlier, and like in At Apollinaire's Grave, the place impacts the poet and the poet impacts the place. The main difference though is that the poet is transported by a historical force, which is here not collective as in At Apollinaire's Grave but personal. In the entire reflection that is Kaddish the notion of place is confused, disturbed by actual memories, imaginative ghosts, and projections of the mind. The landscape blurs the notion of time and acts like a mirror reflecting the past for Ginsberg. The city becomes an intimate wound and this poem a report of eco-trauma, the simple fact of moving triggering metaphysical questions, each detail of the city forming a scar, like in this passage:

[…] Strange to have moved thru Paterson, and the West, and Europe and here again, with the cries of Spaniards now in the doorstoops doors and dark boys on the street, fire escapes old as you -Tho you're not old now, that's left here with me- (Ginsberg 2006, p. 218) Time and place are blended into each other, the poet making one with the universe ("Myself, anyhow, maybe as old as the universe"). The poet is here a psychogeographer of personal trauma, exploring his own memories in space throughout the poem.

Natural Echoes: Eco-Linguistical Readings

The very moving final part of the poem is also interesting to analyze in terms of ecocriticismthe last lines read: caw caw all years my birth a dream caw caw New York the bus the broken shoe the vast highschool caw caw all Visions of the Lord Lord Lord Lord caw caw caw Lord Lord Lord caw caw caw Lord (Ginsberg 2006, p. 235) This passage is almost an animistic ritual, merging religious ("Lord") with natural power ("caw"), echoing each other. Therefore, nature harmoniously embodies the mourning of the poet in a natural, elegiac form. This could be linked to a claim formulated by Mircea Eliade: "All over the world learning the language of animals, especially of birds, is equivalent to knowing the secret of nature" (Manes 1996, p. 19). In that sense, natural elements accompany the poet's sentiments throughout the absurdity of death: a secret pact, known only to the landscapes and the poet is formed. The intimacy of death is made part of the poet's surroundings and goes beyond the language spoken by mortals. Confronting death becomes, then, a metaphysical secret that can only be revealed in the mourning process, when the poet is made lonesome, his "naked" soul contemplating his surroundings. Death is reevaluated, debunked, and re-mystified through space and nature. The power of nature is invoked, and the crow is more than a mere symbol here-it is a vision of nature as an understanding entity. Christopher Manes' take on animism and language (from which I draw Eliade's quotation) is also insightful:

[…] the "animistic subject" [is] a shifting, autonomous, articulate identity that cuts across the human/nonhuman distinction. Here, human speech is not understood as some unique faculty, but as a subset of the speaking of the world. (Manes 1996, p. 19) This last part of Kaddish is a cry for meaning before the unreal feature of death and a desperate realization that it is all part of a natural cycle. The whole poem is not only about what is located within the Anthropocene but also it is a way to zoom out, to shift focus from the individual towards the gist of the universe, and to a wider sense of belonging. In other words, Allen Ginsberg's poetics could be seen as a counterpart of the Camusian idea of the absurdity of mankind before the universe. On the contrary, Ginsberg's poetic is one of union, transcendence, and humility-as Nathalie Caritoux wrote "The distinction between the subject and the world becomes ineffective" ((Caritoux and Villard 2017, p. 71) My translation).

But a short eco-linguistical survey of one term used in different Ginsberg's poems can also underline the extent to which Ginsberg's poetry is tentatively overcoming space through language and against time. One of the most interesting examples is his use of the word "Mannahatta" which appears mostly in his collection The Fall of America (which will be at the heart of the last part of this essay). The Ginsbergian use of the word "Mannahatta" reflects the influence of Walt Whitman on the Beat poet, "Mannahatta" being the name of one of Whitman's poems. This word refers to the "aboriginal" (Ginsberg 2006, p. 790) name of New York City, a term used by the Lenape. Ginsberg uses this word as an invocation (often near the mention of Whitman) like in "Bayonne Entering NYC" (Ginsberg 2006, p. 428 Ginsberg develops here an interesting linguistical conception of the name "Manhattan", going from "Man city" to "my city", emphasizing a sense of both objective and subjective belonging to the city. But the mention of Mannahatta also invokes historical forces, in an almost Rousseauist vision of a nature that has been lost and destroyed by modernity. This point recalls the beginning of Ginsberg's poem entitled "Ecologue" (a wordplay on "eclogue"), a Rousseauist vision of an apocalyptic America:

In a thousand years, if there's History America'll be remembered as a nasty little Country Full of Pricks, thorny hothouse rose Cultivated by the Yellow Gardeners.

[…] Earth rolling round, epics on archaic tongues Fishermen telling island tales-All autos rusted away, Trees everywhere. (Ginsberg 2006, p. 550) Later, in the same poem, Ginsberg also writes, "All landscapes have become Phantom" (Ginsberg 2006, p. 463). This dystopian or Rousseauist cartography of a future America-depending on the reader's point of view-echoes Deleuze and Guattari's claim that "writing has nothing to do with signifying. It has to do with surveying, mapping, even realms that are yet to come" (Deleuze and Guattari 1987, pp. 4-5). But in the passage of Bayonne Entering NYC mentioned above, the city is also associated with human features, like the "East River's rib"-this goes with the deep sense of belonging to a city that is growing, and here somewhat on the verge of dying. The term "Mannahatta", which also appears on the back cover of The Fall of America is also present in the long poem Iron Horse-again associated with organic terms (Ginsberg 2006, p. 464):

Mind wanders. Sleep, cough & sweat… Mannahattas tunnel-door cobbled for traffic, trucks into that mouth Ginsberg also writes in the same poem: "crowd iron cancer on the city's throat" (Ginsberg 2006, p. 457) recalling Mathieu Perrot's analysis of "the poetic of the poisoned city" which is here insightful: "just like cancer changes living matter into dead matter, industrial societies' inclination to inorganic matters is the very proof of their degeneration" (Perrot 2018, p. 43). In his pamphlet A Sick Planet Debord would argue that "it is not the symptoms but the illness itself that must be cured" (Debord 2008, p. 91). Therefore, Ginsberg's use of an aboriginal name for Mannahatta would be an act of resistance against this metaphorical degeneration. Occurrences of "Mannahatta" can be found in other poems like Memory Gardens, Friday the Thirteenth or Contest of Bards. Of course, this use of the word "Mannahatta" is part of a wider Ginsbergian vision inherited from Whitman, of a primary land, "an appreciation of the Amer-Indian vision of America as "Turtle Island", as he said in an interview (Geneson and Ginsberg 1975, p. 30).

This vision of an original lost land, of which traces can be perceived through the spirit of places, is also linked to-and contrasted by-a Ginsbergian and apocalyptical vision of the city, most famously represented in Howl. An eco-Marxist reading of Ginsberg's poems (as well as of other Beat works, such as Kerouac's On the Road, Diane di Prima's Revolutionary Letters, etc.), would stress the need to reclaim (or to save) the landscape from capitalistic views. This is what the Situationists were fighting against with their own idea of urbanism. They, for example, criticized Haussmann for modeling streets according to the size of cannons, Le Corbusier for wanting to "suppress the streets", ending thus their "chances of insurrections and meetings", and modern urbanism for following Police directives (Conord 1954c). Mainstream urbanism becomes a tool of police or capitalistic oppression against the citizen, as Ginsberg observes in the opening lines of the second part of Howl:

What sphinx of cement and aluminum bashed open their skulls and ate up their brains and imagination?

In Greek mythology, the sphinx is a cruel and heartless creature killing and eating those who are not able to answer its riddles, famously represented in the myth of Oedipus. The city is seen then as an unsolvable riddle, brutally crushing its crazed youth. And Ginsberg goes on to compare the city to Moloch:

Moloch whose eyes are a thousand blind windows! Moloch whose skyscrapers stand in the long streets like endless Jehovahs! Moloch whose factories dream and croak in the fog! Moloch whose smokestacks and antennae crown the cities! Biblically, the figure of Moloch is connected to the practice of child sacrifice, so then again to a youth that is being sacrificed to the city and capitalism-Caitlin Cater analyzes it as Ginsberg's fear of "the increasingly pervasive mass culture and [...] its deleterious impact on mankind" (Cater 2008, p. 15). And if we keep in mind Gilles Ivain's remark from his Formulaire pour un urbanisme nouveau that "Architecture is the simplest means of articulating time and space, to modulate reality, to make dream" (Ivain 1953), Ginsberg's poetry underlines thus that capitalism took over urbanism to make it a real nightmare. The whole poem recalls a quotation Marx Debord uses about the theory of the "dérive" in the second issue of Internationale Situationniste: "Men can see nothing around them that is not their own (Debord 1958b). Hopefully, and this might be the most important point in Ginsberg's work, the creative act can be, if not lifesaving, at least an impetus towards reawakening an awareness of our surroundings. This is precisely the mechanism at work in Ginsberg's "Sunflower Sutra", in which the ruins of an old locomotive (which could be interpreted as an industrial vision of America) form a big sunflower. Here, the poetic sensibility of the observers (Ginsberg and Kerouac) helps them to overcome the disenchantment of nature, while the poet advocates, on the contrary, for a re-enchantment of landscapes.

Composing on/from the Tongue: Recording Geography

But the psychogeographical aspect of Allen Ginsberg's poetry is also important in terms of methodology-as David Wills writes, "travel" gave Allen Ginsberg "opportunity to make significant poetic breakthroughs" (Wills 2019, p. 17). The most striking examples and tangible evidence of the impact of traveling on Ginsberg's methods of composition could be dated between December 1965 and March 1966. During that interval indeed, Allen Ginsberg read in many cities and was almost always on the road, in a car, or in a bus through Wichita, Kansas City, Bloomington, and more (sometimes in the company of Gary Snyder). Earlier that year, Bob Dylan gave Allen Ginsberg six hundred dollars so he could buy a brand-new tape-recorder. During that period, Allen Ginsberg recorded himself on his tape-recorder, improvising and creating spontaneous pieces, a vast majority of which is included in his book The Fall of America: Poems of These States. Allen Ginsberg's moving environment made it easier for him to record his thoughts rather than writing them. He himself called these poems a "kind of collage of the simultaneous data of the actual sensory situation" (Ginsberg 1980, p. 26). The poems published in The Fall of America are then born from movements, the poet being the active reader of his environment. These poems are therefore approaching and revealing the psychogeographical variations at work in Ginsberg's creative process in a very different way. This difference is a direct consequence of the time spent in a place and of the potential duration of geographical observations. In other words, the frontier between geography and the psyche is everchanging, inducing thus, a new poetic structure, both in form and content. Allen Ginsberg's taperecorder constitutes a precious archive tool, as the resulting tapes show that it was not only a way to record disconnected thoughts but a way to record lively lines and to include materials from his immediate environment.

Among the most relevant examples is a passage from his poem entitled "Bayonne Entering NYC": More Chimney fires than all Kansas in a mile, Sulphur chemical Humble gigantic viaducts networked by road side What smell burning rubber, oil "freshens your mouth" Railroad rust, deep marsh garbage-fume Nostril hornscity Announcer jabbering at City Motel (Ginsberg n.d.a) The word "horns" above is intentionally in bold character because it is in fact a fragment of the real, added in real-time into the poem. In the recording, Allen Ginsberg recorded those lines quite normally, when all of sudden, just after saying "nostril", one can hear horns in the background and Allen Ginsberg adding spontaneously the word "horns" in his recording, making those horns instantaneously and definitively part of the poem. In this example, the outside world of the city and the inside world of the poet are united, and the psychogeographical aspect of the poem is even more concrete. This incorporation of already existing elements his poetry, a kind of collage of the real in a way, underlines at the same time the porosity of the frontier between the outside and the inside, but also the referentiality of the poem, as Ginsberg decided to keep those fragments of raw reality in his poems. To that extent, Ginsberg's poems of that period are made from and born in the landscapes through which he traveled.

Wichita Vortex Sutra, perhaps the best-known poem from that period, was also first taperecorded-Allen Ginsberg told Michael Aldrich that this poem was "the transcription of only one day" (Ginsberg 1980, p. 26). Again, the poem is composed with elements taken from the flux of reality and those concrete elements were usually the impulse of Allen Ginsberg's recorded thoughts, both being mixed in the vortex of the poem. There is an interesting example of this creative vortex at the beginning of the poem: Thy sins are forgiven, Wichita! Thy lonesomeness annulled, O Kansas dear! as the western Twang prophesied thru banjo, when lone cowboy walked the railroad track past an empty station toward the sun sinking giant-bulbed orange down the box canyon-Music strung over his back and empty handed singing on this planet earth I'm a lonely Dog, O Mother! (Ginsberg n.d.b) Here again, the lines in bold characters are revealing, when compared to what could be considered Allen Ginsberg's first draft. One can hear how Ginsberg worked on this passage: he heard on the radio some country music, and he improvised over it, including the very song in his poem. The creative process, its imagery, and its structure are here inherent to the temporal and geographical structures perceived.

In another passage, from the poem rightly entitled "Hiway Poesy: L.A.-Albuquerque-Texas-Wichita", Allen Ginsberg literally quotes a real conversation:

Hitchhiking student supported by National Defense Fund with his black horn rimmed glasses, thin blond hair, "If your country calls you, would you go?" "If my country drafted me… then I would go." (Ginsberg n.d.b) Once again, those lines are taken from the vivid flux of life, as one can hear in the recording. This way of composing in real time with and from the outside world hooks and involves the reader into the vortex Ginsberg creates, where one cannot distinguish what is imagination and what is reality. This kind of collage also adds new shades of meaning by exploring words in a new context and therefore, could be understood in the vein of Gilles Deleuze and Felix Guattari's notions of deterritorialization and re-territorialization, in the sense that a fragment of the real is de-contextualized and re-contextualized-what was a trivial experience now becomes a poetical statement and shifted in category. Another example can be found in the line "Ham steak please waitress, in the warm café" in "Wichita Vortex Sutra", as Ginsberg tape-recorded himself ordering food in real-time in a café. For Deleuze and Guattari, the book (here the poem) is not "an image of the world" (Deleuze and Guattari 1987, p. 11) but "it assures the deterritorialization of the world" and "the world affects a reterritorialization of the book" (Deleuze and Guattari 1987, p. 11). To borrow the words of the two philosophers, Ginsberg's poems are an "assemblage, a "multiplicity" made of "lines of articulation" as well as "movements of deterritorialization and destratification" (Deleuze and Guattari 1987, p. 3). In other words, Ginsberg's lines are creative movements-a process of shifting territories-from a world that builds upon itself and with its own raw material. In fact, this also recalls some characteristics of Deleuze and Guattari's concept of a rhizome: "Any point of a rhizome can be connected to anything other, and must be" and "a rhizome ceaselessly establishes connections between semiotic chains […]. A semiotic chain is like a tuber agglomerating very diverse acts, not only linguistic but also perceptive, mimetic, gestural, and cognitive" (Deleuze and Guattari 1987, p. 7). Ginsberg's poems could be then considered as poetic rhizomes: the lines know more than one ramification and layer and connect between themselves and each reader's experience in more than one direction.

The poems from that specific period are psychogeographical experiences and the occurrences that structure these experiences for the reader are too numerous to be listed here. But one can come across news broadcasts 7 , radio signals 8 , Ginsberg reading from newspapers 9 , declarations of the President 10 , and even a Bob Dylan's song-completing thus the circle: Angelic Dylan singing across the nation "When all your children start to resent you Won't you come see me, Queen Jane?" His youthful voice making glad the brown endless meadows His tenderness penetrating aether, soft prayers on the airwaves, (Ginsberg n.d.b) In the recording of this passage, even though it is of poor quality, one can distinctly hear Bob Dylan singing in his unique voice-and once again, Dylan's lyrics are re-contextualized in a country at war. To paraphrase Deleuze and Guattari once again, "there is no difference between what a [poem] talks about and how it is made" (Deleuze and Félix 1987, p. 4).

In those psychogeographical poems, Allen Ginsberg also underlines the extraordinary surfeit of language overflowing our environment to the point of being overused, with usually a strong emphasis on the manipulative side of language in the Vietnam War era, like in these lines (Ginsberg 2006, p. 409):

The war is language language abused 7 (Ginsberg 2006, p. 402).

In advance of the Cold Wave

Snow is spreading eastward to the Great Lakes 8 (Ginsberg 2006, p. 407).

Quietness, quietness over this countryside except for unmistakable signals on the radio followed by the honkytonk tinkle of a city piano 9 (Ginsberg 2006, p. 416).

Continued from page one area after the Marines killed 256 Vietcong captured 31 ten day operation Harvest Moon last December 10 (Ginsberg 2006, p. 408). "We will negotiate anywhere anytime" said the giant President.

for advertisement, language used like magic for power on the planet Mathieu Perrot describes in that sense Ginsberg's "green poetry" and "ecological poetic" as a way to "reawaken consciousness put to sleep by the verbal pollution which hypnotizes the masses." (Perrot 2018, p. 46). But besides language, the place itself bears violence (Ginsberg 2006, p. 418):

Here fifty years ago, by her [Carry Nation] violence began a vortex of hatred that defoliated the Mekong Delta-Proud Wichita! vain Wichita cast the first stone!-

The place seems here to be morally poisoned by its inhabitants and instead of rippling throughout the land, this hauntological violence ripples through time.

This vortex of information, this juxtaposition of different materials unveils what is otherwise hidden in the landscape. In that sense, the reservoir of images present in the outside world is transformed into a Whitmanic scream by the poet so it can be heard by everyone, with shared materials that are accessible not only through imagination but also through a physical experience of the world. As he writes in Wichita Vortex Sutra (Ginsberg 2006, p. 414):

I search for the language that is also yours-Almost all our language has been taxed by war.

Moreover, in the works Allen Ginsberg composed during that period, changes in typographical treatment are also a creative process, as the experience of traveling influences not only the content but also the form of his poems. Indeed, Allen Ginsberg himself admitted that "When transcribing, [he] pai [d] attention to the clicking on and off of the machine", and described this process as writing in a "notebook", arranging the lines "according to their organic time-spacing as per the mind's coming up with the phrases and the mouth pronouncing them" (Ginsberg 1980, pp. 28-29). Here are lines from Wichita Vortex Sutra-the "clicking on and off" of the tape-recorder are represented here with an asterisk, according to the recording (Ginsberg n.d.b):

While the triangle-roofed Farmer's Grain Elevator sat quietly by the side of the road along the railroad track * American Eagle beating its wings over Asia * million dollar helicopters * a billion dollars worth of Marines * who loved Aunt Betty * Drawn from the shores and farms * shaking from the high schools to the landing barge * blowing the air thru their cheeks with fear * in Life on Television * Put it this way on the radio * Put it this way in television language * Use the words language, language: "A bad guess" * Most of the time, the line breaks coincide with the "clicking on and off" of the tape-recorder, making each line a unit of thought. The layout of the poem is thus, deeply influenced by Allen Ginsberg's physical actions on the tape-recorder, actions which are themselves influenced by auditive or visual experiences during his travel. His use of a tape-recorder is in that sense a tool with which he sculpted his poem on the page but also a way to archive the poetry available in the world.

As we have seen, the notions of places and landscapes have a complex and pluralistic influence on Allen Ginsberg's poetry, whether words are taken from an always evolving geographical environment or born from a collective spirit of place. What is at work in most of the passages studied is a dynamic exchange between what is retrievable through poetry and what is lost in time, what is composed in the poem and what is archived in the mind.

Conclusions

The psychogeographical poems of Ginsberg are the embodiment of an inner struggle, of a crisis that is deployed in a sensible field: the difficult process of transcribing the world and the poet's evanescent reactions to it into poetry, into a new vocabulary or context. Ginsberg's poetry is complex, subtle, but visionary-he welcomes the reader in his mind and in the vortex of the world itself. Introspective analysis of his position in the world is almost always a defining aspect of his poetry. Ginsberg, who described himself as "a permanent traveler" (Wills 2019, p. 129)-one could add, "a contemporary Ulysses"-is most of the time a psychogeographer without knowing it, someone who "wants to know and to create" ((Caritoux and Villard 2017, p. 72) My translation). He is not simply providing a passive description of his experience: he invites the reader to explore with him the liminal space between what is outside and what is inside, in the form of a creative and poetic study. In a way, Ginsberg's poetry transcends the notion of "dérive" as introduced by Guy Debord, as it is not a mere observation of psychogeographical variations or a poetic report that Ginsberg offers. It is, on the contrary, a deconstruction of horizontal observations, and a lively reconstruction of sceneries into a vertical work that allows the reader to oversee the environment.

Of course, the overlapping of the psyche and geography indirectly points towards the ambiguity of psychogeography: the observed environment is as influential as the observer's mindset. Poetry might be a solution to the impasse Guy Debord describes: "The sectors of the city are, at a certain level, decipherable. But the personal meaning they have for us is incommunicable" (Debord [1961] 2005, p. 129). In that sense, the poet is the medium between two worlds, as he is able, in his practice, to cross what Wunenburger called "the invisible line of scission" ((Caritoux and Villard 2017, p. 226) My translation). The psychogeographical poems of Allen Ginsberg are composed in a pendulum movement: from geographical triggers to poetic observations, from geographical observations to poetic triggers.

The Situationist vision of psychogeography, in the sense that it is a practice that invites the gathering of first-hand experiences of surroundings, and also to the extent that it is a non-mediated knowledge based on direct observations, is linked to what Guy Debord writes in his seminal book The Society of the Spectacle: "While eliminating geographical distance, this society produces a new internal distance in the form of spectacular separation." (Debord 2005, p. 2). In adding "psyche", that is a subjective element, to "geography", the work of the Situationist reintroduced the importance of the self, of relationship, of freedom. Ginsberg's poetry creates a poetic space in which the schizophrenic representation of the subject as against the outside world is negotiated, confronted, and finally abjured in the creative process. Landscapes, as poetic experiences, work in a reciprocal movement in Ginsberg's work. They are the poet's own tinted images and inclinations projected on an environment that, in return, swallows the poet in a union. Therefore, this union between the poet and the landscape appears more like a visionary harmony of a specific moment rather than a spontaneous poetic epiphany. Reading Ginsberg's poetry could be thus seen as an interiorized form of psychogeography, and as an examination of one's response to the environment. As for the mechanism at work for the layout on the page, this study scratched the surface of new territory to explore: a "text-ritory" to discover through poetico-geographical observations and genetic studies.

Translations of articles extracted from the magazines Internationale Situationniste and Potlatch are from The Situationist International Online unless stated otherwise.

Gilles Ivain will be excluded from the Internationale Lettrist in 1954, as mentioned in Potlatch #2.

Funding: This research received no external funding.

Conflicts of Interest:

The author declare no conflict of interest.