Information geometry in quantum field theory: lessons from simple examples

Motivated by the increasing connections between information theory and high-energy physics, particularly in the context of the AdS/CFT correspondence, we explore the information geometry associated to a variety of simple systems. By studying their Fisher metrics, we derive some general lessons that may have important implications for the application of information geometry in holography. We begin by demonstrating that the symmetries of the physical theory under study play a strong role in the resulting geometry, and that the appearance of an AdS metric is a relatively general feature. We then investigate what information the Fisher metric retains about the physics of the underlying theory by studying the geometry for both the classical 2d Ising model and the corresponding 1d free fermion theory, and find that the curvature diverges precisely at the phase transition on both sides. We discuss the differences that result from placing a metric on the space of theories vs. states, using the example of coherent free fermion states. We compare the latter to the metric on the space of coherent free boson states and show that in both cases the metric is determined by the symmetries of the corresponding density matrix. We also clarify some misconceptions in the literature pertaining to different notions of flatness associated to metric and non-metric connections, with implications for how one interprets the curvature of the geometry. Our results indicate that in general, caution is needed when connecting the AdS geometry arising from certain models with the AdS/CFT correspondence, and seek to provide a useful collection of guidelines for future progress in this exciting area.Comment: 36 pages, 2 figures; added new section and appendix, miscellaneous improvement

Alternative routes and solvents in polymer chemistry : microwave irradiation and ionic liquids

The concept of sustainable chemistry represents an area of innovation, which not only preserves resources, but also stands for a progress in the chemical industry. The principle of sustainable chemistry comprises important elements in areas like environment, economy and society, dealing with the whole life of intrinsic safe chemicals and products, including their production, processing, use and disposal. One of the largest amounts of auxiliary wastes in industry is produced by the usage of solvents. Therefore, alternative reaction media are investigated in order to reduce or replace organic solvents. The most widely used green solvents in, e.g., polymer research are ionic liquids, zupercritical CO2 and water. In addition, also alternative energy sources, such as photochemistry, microwave energy, electron beam and ultrasound, are investigated in order to replace conventional heat sources for, e.g., polymer processing. The main goal of utilizing alternative energy sources is to improve the efficiency of the process by reducing the polymerization time. Ionic liquids are considered to be ‘green’ solvents on account of their non-volatility and nonflammability – which are results of their negligible vapor pressure – as well as their reusability. On the basis of ecological concerns, ionic liquids seem to be an attractive alternative to conventional volatile organic solvents. Ionic liquids with a linear alkyl side chain can be synthesized in a fast and efficient way at elevated temperatures (170 °C) by using microwave irradiation. In case of the ionic liquids with branched alkyl side chains, the synthesis could be accelerated as well, but the equilibrium was shifting towards undesired side products compared to the synthesis at conventional conditions (80 °C). In this regard, several new branched ionic liquids, e.g. 1-(1-ethylpropyl)-3-methylimidazolium iodide and 1-(1-methylbenzyl)-3- methylimidazolium chloride and their tetrafluoroborate containing analogues, were synthesized applying two different synthetic approaches. The direct scaling for the ionic liquids with linear alkyl side chain was investigated from small scale (0.01 mol) to large scale (1.15 mol). In this case, comparable results were obtained for the direct up-scaling utilizing different microwave reactors under otherwise similar reaction conditions. The results of the continuous flow experiments indicated that 1-butyl-3-methylimidazolium chloride can be synthesized with short reaction times by using continuous flow microwave systems. However, direct scaling from the batch experiments was not possible. Even when employing a residence time of 16 min, a complete conversion could not be obtained. Nonetheless, for the first time, the synthesis of ionic liquids in continuous flow reactors was achieved. In case of [C2MIM][Et2PO4], higher conversions were achieved, since the reaction proceeds in a homogeneous phase, but unfortunately only strongly colored ionic liquids could be obtained with the applied conditions, while not showing any decomposition products in the 1H NMR spectrum. In order to elucidate first structure–property relationships, the synthesized ionic liquids, both linear and branched, were investigated by thermogravimetric analysis, differential scanning calorimetry, and water uptake measurements of selected ionic liquids. The results obtained for the decomposition temperature support a SN2 (alkyl) and SN1 (aryl) decomposition pathway for branched ionic liquids with alkyl and aryl side chain, respectively, containing chloride as counter ion. In case of tetrafluoroborate containing ionic liquids a decomposition mechanism initiated by the anion seems to take place. Moreover, tetrafluoroborate containing ionic liquids and ionic liquids with linear alkyl side chains revealed lower glass transition temperatures compared to the ionic liquids with chloride anion or branched alkyl side chains, respectively. In general, the ionic liquids with an aromatic group showed the highest Tg values of all the investigated ionic liquids. In addition, the water uptake of the ionic liquids was measured and revealed a systematic dependency on the length of the alkyl side chain and on the branching. It was found that the water absorption decreased with the length of the alkyl chain and that branched alkyl chains increased the water uptake as a result of their decreased ability to self-assembly. The described results provide a better insight into the structure-property relationship of ionic liquids, allowing the fine-tuning of the chemical and physical properties. Cellulose is the most abundant natural polymer in nature and its derivative products have many important applications. However, cellulose is insoluble in water and most common organic solvents, because of its fibril structure and the pronounced presence of inter- and intermolecular hydrogen bonding. In recent years, ionic liquids were found to dissolve cellulose, but the candidates known are still limited. In order to extend the range of suitable ionic liquids, we screened known but also new tailor-made ionic liquids. In particular, the influence of different alkyl chain lengths, branched alkyl side chains and the anion on the dissolution of cellulose was investigated. A strong odd-even effect of the alkyl chain length on the solubility of cellulose in the ionic liquid was observed for imidazolium based ionic liquids with linear and branched alkyl side chains bearing chloride as counter ion. Alkyl side chains with an odd number of CH2 repeating units showed in general good dissolving properties, whereas an even number of CH2 repeating units was not able to dissolve cellulose. The difference in solubility might be explained by a different range of conformations for odd and even alkyl chains. In general, only the ionic liquids with chloride, acetate and phosphate counter anions showed good dissolving properties for cellulose. Moreover, the microwave-assisted dissolution of cellulose was investigated and optimized. Selected ionic liquids were used as solvent in the tritylation reaction. It was found, that pyridine is required to capture hydrogen chloride and that the reaction time could be reduced from 48 h (reaction in DMA/LiCl) to 3 h ([C4MIM][Cl]) in order to reach the desired DS of nearly 1.0. Unfortunately, recycling of the ionic liquid could not be achieved when pyridine was used as a base. However, this was possible when triethylamine was used as a base. New 4,4-imidazolium ionenes were synthesized under microwave irradiation. The polymerization times could be decreased from 24 to 1 h as a result of elevated temperatures above the boiling points of the applied solvents. Different approaches, such as monomer imbalance and monofunctional reagents, were applied in order to control the molar mass of the polymers. Analytical ultracentrifugation measurements indicate the formation of macrocyclic rings to a large extend (82 to 93%). Furthermore, the properties of the synthesized 4,4-imidazolium ionenes, such as thermal behavior, solubility behavior and water uptake were investigated as well. It was found that the decomposition temperatures were comparable to the values reported in literature for ammonium ionenes, while the glass transition temperatures obtained were lower compared to values reported in literature. In addition, the 4,4-imidazolium ionenes showed a high water uptake. The ability to absorb water is mainly depending on the counter ions (chloride showed a higher water uptake than bromide). The combination of the thermal auto-initiated free radical polymerization of styrene and the precipitation polymerization were investigated in order to develop a fast and environmentally friendly approach to produce polystyrene. To achieve high reaction temperatures in a short time, microwave irradiation was utilized as heating source. Styrene was used without any purification, e.g. without distillation or column filtration. First experiments were carried out using nearcritical water (water in the temperature range of 250 to 350 °C) as solvent, because the polarity and hydrogen-bonding of water are highly temperature depending. Due to the auto-initiation of styrene at high temperatures no radical initiator was required. The polymerization of styrene in near-critical water always led to polymers with comparable molar masses although different styrene concentrations were applied. In case of ethanol as solvent, the obtained molar masses could be controlled by the ethanol-to-styrene ratio although the monomer conversions were rather low under the applied conditions (1 to 13%). In order to achieve a better control over the molar mass SG-1, a commercially available stable free nitroxide, was applied to mediate the polymerization. It was found that the molar masses can be controlled by different styrene:SG-1 ratios (from 10:1 to 400:1). In this case moderate polydispersity indices (PDI = 1.3 to 1.9) could be obtained. Finally, the developed polymerization processes only require a simple purification step due to the precipitation of the polystyrene in the reaction solvent. Another example of using near-critical water is the hydrolytic ring-opening polymerization of polyamide. In this thesis a polyamide 12 pre-polymer was synthesized under microwave irradiation at high temperatures and pressures, indicating that less side products are formed compared to the thermal polymerization. Since these are preliminary results, further experiments are required in order to investigate if the utilization of microwave irradiation can provide advantages over thermal heating. In general, it was shown that microwave irradiation and ionic liquids are interesting alternatives to conventional energy sources and solvents. In particular, a better understanding of the structure-property relationships of branched ionic liquids was achieved resulting in a superior knowledge about the influence of the alkyl side chains of ionic liquids on the cellulose dissolution process. New concepts (combination of thermal and precipitation polymerization, near-critical water) and polymers (imidazolium ionenes) were investigated utilizing microwave irradiation as heating source to achieve short reaction times

Chiral phase transitions and quantum critical points of the D3/D7(D5) system with mutually perpendicular E and B fields at finite temperature and density

We study chiral symmetry restoration with increasing temperature and density in gauge theories subject to mutually perpendicular electric and magnetic fields using holography. We determine the chiral symmetry breaking phase structure of the D3/D7 and D3/D5 systems in the temperature-density-electric field directions. A magnetic field may break the chiral symmetry and an additional electric field induces Ohm and Hall currents as well as restoring the chiral symmetry. At zero temperature the D3/D5 system displays a line of holographic BKT phase transitions in the density-electric field plane, while the D3/D7 system shows a mean-field phase transition. At intermediate temperatures, the transitions in the density-electric field plane are of first order at low density, transforming to second order at critical points as density rises. At high temperature the transition is only ever first order.Comment: 15 pages, 7 figures, v2: Added a referenc

Criticality, Scaling and Chiral Symmetry Breaking in External Magnetic Field

We consider a D7-brane probe of $AdS_{5}\times S^5$ in the presence of pure gauge $B$-field. The dual gauge theory is flavored Yang-Mills theory in external magnetic field. We explore the dependence of the fermionic condensate on the bare quark mass $m_{q}$ and study the discrete self-similar behavior of the theory near the origin of the parametric space. We calculate the critical exponents of the bare quark mass and the fermionic condensate. A study of the meson spectrum supports the expectation based on thermodynamic considerations that at zero bare quark mass the stable phase of the theory is a chiral symmetry breaking one. Our study reveals the self-similar structure of the spectrum near the critical phase of the theory, characterized by zero fermionic condensate and we calculate the corresponding critical exponent of the meson spectrum.Comment: 29 pages, 9 figures. Accepted in JHEP. Updated to mach the published version. One figure added, some definitions improve

Four-Dimensional Superconformal Theories with Interacting Boundaries or Defects

We study four-dimensional superconformal field theories coupled to three-dimensional superconformal boundary or defect degrees of freedom. Starting with bulk N=2, d=4 theories, we construct abelian models preserving N=2, d=3 supersymmetry and the conformal symmetries under which the boundary/defect is invariant. We write the action, including the bulk terms, in N=2, d=3 superspace. Moreover we derive Callan-Symanzik equations for these models using their superconformal transformation properties and show that the beta functions vanish to all orders in perturbation theory, such that the models remain superconformal upon quantization. Furthermore we study a model with N=4 SU(N) Yang-Mills theory in the bulk coupled to a N=4, d=3 hypermultiplet on a defect. This model was constructed by DeWolfe, Freedman and Ooguri, and conjectured to be conformal based on its relation to an AdS configuration studied by Karch and Randall. We write this model in N=2, d=3 superspace, which has the distinct advantage that non-renormalization theorems become transparent. Using N=4, d=3 supersymmetry, we argue that the model is conformal.Comment: 30 pages, 4 figures, AMSLaTeX, revised comments on Chern-Simons term, references adde

Strongly bound mesons at finite temperature and in magnetic fields from AdS/CFT

We study mesons in N=4 super Yang-Mills theory with fundamental flavors added at large 't Hooft coupling using the gauge/gravity correspondence. High-spin mesons are well described by using semiclassical string configurations. We determine the meson spectrum at finite temperature and in a background magnetic field.Comment: 15 pages, 11 figures; v2: references adde

Holographic Flavor Transport in Arbitrary Constant Background Fields

We use gauge-gravity duality to compute a new transport coefficient associated with a number Nf of massive N=2 supersymmetric hypermultiplet fields propagating through an N=4 SU(Nc) super-Yang-Mills theory plasma in the limits of large Nc and large 't Hooft coupling, with Nf << Nc. We introduce a baryon number density as well as arbitrary constant electric and magnetic fields, generalizing previous calculations by including a magnetic field with a component parallel to the electric field. We can thus compute all components of the conductivity tensor associated with transport of baryon number charge, including a component never before calculated in gauge-gravity duality. We also compute the contribution that the flavor degrees of freedom make to the stress-energy tensor, which exhibits divergences associated with the rates of energy and momentum loss of the flavor degrees of freedom. We discuss two currents that are free from these divergences, one of which becomes anomalous when the magnetic field has a component parallel to the electric field and hence may be related to recent study of charge transport in the presence of anomalies.Comment: 27 page

Dual expansions of N=4 super Yang-Mills theory via IIB superstring theory

We examine the dual correspondence between holographic IIB superstring theory and N=4 super Yang-Mills theory at finite values of the coupling constants. In particular we analyze a field theory strong-coupling expansion which is the S-dual of the planar expansion. This expansion arises naturally as the AdS/CFT dual of the IIB superstring scattering amplitudes given a genus truncation property due to modular invariance. The space-time structure of the contributions to the field theory four-point correlation functions obtained from the IIB scattering elements is investigated in the example of the product of four conserved stress tensors, and is expressed as an infinite sum of field theory triangle integrals. The OPE structure of these contributions to the stress tensor four-point function is analyzed and shown not to give rise to any poles. Quantization of the string in the background of a five-form field strength is performed through a covariantized background field approach, and relations to the N=4 topological string are found.Comment: LaTeX, 40 pages, 5 figure

Spectral Flow on the Higgs Branch and AdS/CFT Duality

We use AdS/CFT duality to study the large N_c limit of the meson spectrum on the Higgs branch of a strongly coupled, N=2 supersymmetric SU(N_c) gauge theory with N_f =2 fundamental hypermultiplets. In the dual supergravity description, the Higgs branch is described by SU(2) instanton configurations on D7-branes in an AdS background. We compute the spectral flow parameterized by the size of a single instanton. In the large N_c limit, there is a sense in which the flow from zero to infinite instanton size, or Higgs VEV, can be viewed as a closed loop. We show that this flow leads to a non-trivial rearrangement of the spectrum.Comment: v2; 16 pages, 3 figures, LaTeX + JHEP class, 3 refs added, accepted for publication by JHE

Thermalization from gauge/gravity duality: Evolution of singularities in unequal time correlators

We consider a gauge/gravity dual model of thermalization which consists of a collapsing thin matter shell in asymptotically Anti-de Sitter space. A central aspect of our model is to consider a shell moving at finite velocity as determined by its equation of motion, rather than a quasi-static approximation as considered previously in the literature. By applying a divergence matching method, we obtain the evolution of singularities in the retarded unequal time correlator $G^R(t,t')$, which probes different stages of the thermalization. We find that the number of singularities decreases from a finite number to zero as the gauge theory thermalizes. This may be interpreted as a sign of decoherence. Moreover, in a second part of the paper, we show explicitly that the thermal correlator is characterized by the existence of singularities in the complex time plane. By studying a quasi-static state, we show the singularities at real times originate from contributions of normal modes. We also investigate the possibility of obtaining complex singularities from contributions of quasi-normal modes.Comment: 35 pages, 4 figure