text:
MANYUK VOLODYMYR V.
Study and Preservation of Geosites
As.Prof., Ph.D., Volodymyr V. Manyuk, Director of the Research Institute of Geology, Docent of the Department of Geology and Hydrogeology, Oles Honchar Dnipro National University, Ukraine.
ResearchGate: Account
Google Scholar: Account
Chief Editor: Journal of Geology, Geography and Geoecology
Executive Editor: Journal of Geology, Geography and Geoecology
Deputy Chief Editor: Geology and Ore Content of Ukraine (Geology and Ore-bearing of Ukraine)
Member of Editorial Board: Sustainable Geoscience and Geotourism (SGG, SciPress Ltd, Switzerland)
Contacts: +38-067-947-45-04; +38 095 824 61 77;
vgeoman@gmail.com
Web: http://manuk-geo.ucoz.ua
1. Publications in peer-reviewed scientific journals
Manyuk, V., 2004. Geological heritage of a southern part of Ukraine. Natural and Cultural landscapes. – Proceedings of a conference, Dublin castel, 2002, 3-98. Edited by Matthew Parkes.
Manyuk, V., 2004. Geoscientific significance and tourist values of Zmeynyi (Snake) island. Proceeding of the Conference "Geological heritage concept, conservation and protection policy in Central Europe.-Polish Geological Institute Special 2004. Papers, 13, Cracow-Warzawa, 155-161.
Manyuk, V., 2005. Preservation of the geological heritage of Ukraine: new steps towards the creation of a database of geosites.// Uppsala, Sweden, ProGEO NEWS, 2, 1-8.
Manyuk V., 2006. Stratigraphic control of distribution of useful minerals in the entrails of the earth of Dnipropetrovsk oblast. Bulletin of Dnipropetrovsk University. Series: geology, geography. Dnipropetrovsk, 10(1), 10-15.
Manyuk, V., 2007. The problem of creation of Network National Geoparks in Ukraine. Bulletin of Dnipropetrovsk University. Series: geology, geography. Dnipropetrovsk, issue 9, 11, 63-67.
Manyuk, V., Golturenko, I., 2009. Geological heritage of the south-western Ukraine (Odessa, Mykolaiv and Kherson region). Dnipropetrovsk, Bulletin of Dnipropetrovsk University. Series: geology, geography. V.17, issue.11, 3/2, 24-28.
Manyuk, V., 2015. Geological Heritage and Geoscience education in Ukraine. Dnipropetrovsk, Bulletin of Dnipropetrovsk University. Series: geology, geography, V.23 (1), issue. 17, 75 - 82.
Manyuk, V., Samoilova, K., Samoilych, D. 2015. Geodiversity: new category in the development of the Nature reserve fund of Ukraine. Dnipropetrovsk, Bulletin of Dnipropetrovsk University. Series: geology, geography, V.23 (1), ussue 17, 104 - 107.
Manyuk, V., 2015. Objects of Geological Heritage in Kirovograd Region as a Basis for Development of Geotourism. Materials of the International Science. Conf., October 27-30. Kyiv: NNPM NAS of Ukraine, 146 - 149.
Manyuk, Vol., Manyuk, Vad., 2016. Scientific results of monitoring objects of geological heritage in Dnipropetrovsk oblast. Bulletin of Dnipropetrovsk University. Series: geology, geography. Dnipropetrovsk, V.24 (1), 18. 65 – 70.
Manyuk, V., 2016. Geological heritage as an integral part of the Nature Reserve Lands of Kharkiv oblast. Bulletin of Dnipropetrovsk University. Series: geology, geography. Dnipropetrovsk, V.24 (1), 18. 71-81.
Manyuk, V., 2015. Objects of Geological Heritage in the Kirovohrad Region as Basis for Geotourism Development. Bulletin of the National Museum of Natural History. V. 13, Kyiv: 10-16.
Manyuk, V., Overview of the geological heritage of Kharkiv oblast. Materials of the II International Scientific and Practical Conference (May 5-7, 2016, Lviv). - Lviv: Scientific and production center "Maps and Atlases", 2016. - P. 43-46.
Manyuk, V.V., 2016. Study and preservation of geosites: training course for geology students. Geoheritage, Springer Berlin Heidelberg, doi 12371, 10.1007/s 12371 – 015 – 0147 – y. –Vol.8, 181 – 187.
Manyuk V.V., Manyuk Vad.V., 2016. Geological and geomorphological studies for projecting the "Orilskyi" national geopark. Bulletin of Dnipropetrovsk University. Series: geology, geography. Dnipropetrovsk, V.24 (2), 19, 70 – 91.
Manyuk, V.V., 2017. Geological journeys of Valerian Domger to the basins of the rivers Omelnyk, Samotkan and Domotkan.Bulletin of Dnipropetrovsk University. Series: geology, geography. - Dnipropetrovsk, V.25 (2), 21. - P. 89-98.
Yaholnic, O.V., Manyuk, V.V., 2017. Legal aspects of creating geoparks in Ukraine. Bulletin of Dnipropetrovsk University. Series: geology, geography. - Dnipropetrovsk, V.25 (2), 21, 136-145.
These materials can be accessed at http://manuk-geo.ucoz.ua/
2. Research monographs, chapters in collective volumes, manuals, guidebooks, teaching aids, tutorials, textbooks and any translations thereof
Geosites of Ukraine. Geological landmarks of Ukraine. The collective of authors, in three volumes, ukr. and english languages, Kyiv: DIA, 2006.-1, 1-320.
Manyuk, Vad.V., Manyuk, Vol.V., 2008. Natural heritage of Dnipropetrovsk region. Dnipropetrovsk: Version, PP "Oberemok", 1- 20 (in Ukrainian).
Manyuk, V.V., Spylchak V.O., Sukach V.V., Necryach A.I., 2007. State geological map of Ukraine scale 1: 200000, sheet M - 36 - XXXVI (Dnipropetrovsk). Central Ukrainian series. Explanatory note. State Geological Service, KP "Yuzhnukrgeologiya", UkrDGRI. 1-116. (in Ukrainian).
Manyuk, V.V., Shapar A.G, Skrypnyk O., Smetana S.M. Catalog of unique natural objects of Dnipropetrovsk region. Dnipropetrovsk: IPAPE NASU, 2008.-56s. (Ukr. And Eng.).
Geosites of Ukraine. Geological landmarks of Ukraine. The collective of authors, in three volumes, ukr. and english languages, Kyiv: 2009.-V. 3, 1-200.
Manyuk, Vad.V., Manyuk, Vol.V., 2010. The Nature Reserve Lands of Dnipropetrovsk (on 1.12.2010). [educational manual] Dnipropetrovsk, 2010, 116p. (in Ukrainian).
Gritsenko V., Derevskaya E., Konstantinenko L., Manyuk, V., 2011. Nature monuments are a vivid evidence of the evolution of the Earth. Geological excursion. - K.: Logos, 32 p.
Manyuk, Vad.V., Manyuk, Vol.V., 2011. Geological monuments (geosites) of Dnepropetrovsk region. Teaching tutorial.- Dnipropetrovsk: 60p. (in Ukrainian).
Manyuk, Vad.V., Manyuk, Vol.V., 2011. The Nature monuments of Dnipropetrovsk region. Teaching tutorial.- Dnipropetrovsk: 2011, 120 p. (in Ukrainian).
Geosites of Ukraine. Geological landmarks of Ukraine. The collective of authors, in three volumes, ukr. and English languages, Kiev :, 2011.-Vol. IV, 280 p. (Ukrainian, English).
Manyuk, V.V., VV Shevchuk, V.V., 2014. Crimean geological survey practice: teaching. Manual. Kyiv, PC "Kyiv University", 151p.
3. Main publications in Ukrainian and Russian (articles)
Manyuk, V.V., 1994. Upper sarmatian bioherm in area joint of the Ukrainian shield and Prechernomorian of a depression. Stratigraphic and paleontological of research in Ukraine, Kiev: 66-73 (in Ukrainian).
Konenkova I.D., Bogdanovitch Е.M., Koralova V.V. Manyuk, V.V., 1995. Paleontological characteristic of upper Eocene deposits of Middle Pridneprovian. Depos. in DNТB Ukraine, № 1950. Uk. 95, 18 p. (in Ukrainian).
Manyuk, V.V., 1996. Paleogeographical reconstruction of conditions sedimentogenesis in the Palaeogene and Neogene basins of the Middle Pridneprovian. Abstract. Dissert. Candidate of geological science. Kharcov,16 p. (in Ukrainian).
Konenkova, I.D., Bogdanovitch, Е.M. Manyuk, V.V., 1996. The new data on distribution upper Eocene of deposits within the limits of the Middle Pridneprovian.Reports NAS of Ukrainem, 6, 100-104 pp. (in Ukrainian).
Manyuk, V.V., 1997. To stratigraphy Palaeogene and Neogene of deposits of a Southern slope of the Ukrainian shield. Urgent problems of geology, geography and protection of an environment. Dnipropetrovsk: DSU, 24-34 pp. (in Ukrainian).
Manyuk, V.V., 1998. To a problem of the reconstruction of physiographic conditions of sedimentogenesis in Paleogene basins of the Middle Pridneprovian. Bulletin of the Dnipropetrovsk University: Geology, geography. 2, Dnipropetrovsk, DSU, 57-62pp. (in Ukrainian).
Manyuk, V.V., 1999. The features of Eocene stratigraphy of Nicopolean basin of the manganese ore. Proceedings of a conference NMAU. Dnipropetrovsk, 6, Vol.2, 118-123pp. (in Ukrainian).
Manyuk, V.V., 1999. Source of raw materials of Dnipropetrovsk. Proceedings КТU, series a geology and mineralogy. Krivoy Rog, 25-34pp. (in Ukrainian).
Manyuk, V.V., 1999. Geoconservation of a geological heritage of Dnipropetrovskian region. Journ. Sviata Sprava, 3, Dnipropetrovsk, 19-25pp. (in Ukrainian).
The geological heritage of Dnipropetrovsk region. 1999. Urgent problems of geology, geography and ecology. Dnipropetrovsk, Navch. kniga, v.3, 81-90.
Manyuk, V.V., 1999. The problem of preservation of geological monuments of nature of Dnipropetrovsk region for local lore and tourism needs. Tourist-regional studies. -Sb. scientific. Works, vol.2 .- Federation of Trade Unions of Ukraine, Institute of Tourism. Кyiv, PE Karmalita, 354-367 (in Ukrainian).
Barg I.M., Manyuk, V.V., 2001. About a new find of the rests of a giant deer in Quaternary sediments near Dnipropetrovsk. Collected articles of Palaeontology., Lvov, 33, 91-95. (in Ukrainian).
Manyuk, V.V., 2001. Tourist aspects of use of geological monuments of a nature. Steady development of tourism at coast of the Black sea., Odessa, 50-59 (in Ukrainian).
Manyuk, V.V., 2002. Geological monuments of Dnipropetrovsk region in natural environment and life of the man. Almanac of local lore: Sicheslavcshina, 4 - Treasures Pridneprovian of steppe. Dnipropetrovsk, 5-31pp. (in Ukrainian).
Manyuk, V.V., Manyuk T.V., 2003. The past and future of an island Snake. Proceedings of NMA, 17, v.1, Dnipropetrovsk, 539-547pp. (in Ukrainian).
Manyuk, V.V., 2004. The scientific basis and practical experience in research stratigraphic and paleontological of the natural monuments of southern and east regions of Ukraine. Problems of Phanerozoic stratigraphy of Ukraine. Kiev, IGS, 253-259pp. (in Ukrainian).
Manyuk, V.V., 2005. The Some problems questions of biostratigraphical subdivision of Miocene Northern Prichernomorian and Southern slope of the Ukrainian shield. Biostratigraphical criterions of a subdivision and correlation of deposits Phanerozoic of Ukraine.-Proceedings of IGS NASU, Kiev, 317-321(in Ukrainian).
Manyuk, V.V., 2005. About necessity of a detailed subdivision of Quaternary sediments in process geological survey. // Materials scientific - practical conference. " A Modern condition and tasks of development of regional geological researches. "Rivne, September 8-15, Kiev, 78-183 (in Ukrainian).
Manyuk, V.V., 2005. Abiotic the factors of the control Late sarmatian education of bioherms in Northern Prichernomorian. Theses of the report LI of session Paleontological society: Paleontological chronicle of regional and global events.- St.-Petersburg, 82-84 (in Ukrainian).
Manyuk, V.V., 2005. Oncoid bodies of biogerms in the upper Sarmatian of the Middle Prydniprov'ya. Problems of paleontology and biostratigraphy by the Proterozoic and Phanerozoic Ukraine. Kyiv, 264-269 (in Ukrainian).
Bogdanovich, E.M., Konenkova, I.D., Ivanova, T.A., Bondar, O.V., Manyuk, V.V., 2006. On the stratigraphical position and paleontological features of the Eocene deposits of the Dnepropetrovsk region (the Visirka quarry). Problems of paleontology and biostratigraphy by the Proterozoic and Phanerozoic Ukraine. Kyiv, 188-192 (in Ukrainian).
Manyuk, V.V., 2006. Geological research in the Zaporizhzhya and Donetsk regions for the purpose of inspection, inventory and further preservation of geological monuments of nature. Coll.science.works to the 75th anniversary of professor O.P. Fisunenko . Lugansk: Alma-Mater, 82-93 (in Ukrainian).
Manyuk, V.V., 2005. Geological journey to life long. To the 175th anniversary of the birth of MP Bardot de Marne. Paleontological collection, Lviv, 37, 125-130 (in Ukrainian).
Manyuk, V.V., 2006. Historical aspects of Ukraine's attach to the European Association for the Conservation of Geological Heritage (ProGEO). Mineral Resources of Ukraine, 4, 8-11 (in Ukrainian).
Manyuk, V.V., 2007. Historic role of paleontological studies by Valerian Domger. Paleontological studies in Ukraine: history, current state and perspectives. Kyiv: IGN NANU, 470-474 (in Ukrainian).
Manyuk, V.V., 2008. Historical background for the creation of the national geological park "Eagle Nest". Protection and management of non-living objects in protected territories: Mater. international scientific-practical conference. Grimailov-Ternopil, Dzhura, 75-181(in Ukrainian).
Manyuk, V.V., 2008. Unique object of the geological heritage "Kakhovsky cliffs." Vestnik DNU, Series: geology, geography, issue 10, volume 16, 3/2, 5-14 (in Ukrainian).
Manyuk, V.V., 2009. Natural beauty and geological value of the Kakhovsky Reservoir Coast. Ecology and Local History Journal "Holy activity - XXI", 1-2, 33-39 (in Ukrainian).
Manyuk, V.V., 2009. 33rd International Geological Congress: afterword. 2009. Materials of the international conference "The Miners' Forum 2009", V. 4, 277p., Dnipropetrovsk: National Mining University,111-116 (in Ukrainian).
Manyuk, Vol.V., Manyuk, Vad.V., 2009. On the need to preserve technogenic objects of the geological heritage in the Middle Dnieper. Visnik DNU, seriya: geology, geography. V.17, issue11, 3/2, 28-32(I n Ukrainian).
Manyuk, V.V., Barg I.M., 2009. Stratigraphy of Paleogene deposits of the southern slope of the Ukrainian Shield (Nikopol-Marganets region). Visnik DNU, series: geology, geography. V.17, issue 11, 3/2, 3-12 (in Ukrainian).
Manyuk, V.V., 2009. The Nature Paleontological monuments of Dnipropetrovsk and Zaporizja oblast: actual station and conservation problem. Paleontology, 41, 109 – 118 (in Ukrainian).
Manyuk, V.V., 2010. The geological past of the unique archaeological monument of the nature of Ukraine "Stone Tomb" Vestnik Lvov. Nation. University, series of geol., 24, 223-228 (in Ukrainian).
Manyuk, V.V., 2010. Paleontological monuments of the nature of Dnipropetrovsk and Zaporizhzhya regions: the present state and the problem of conservation. Paleontological collection, 42, Part 2, 108 - 117 (in Ukrainian).
Manyuk, V.V., 2010. The page of history N.P. Barbot de Marny travels. Sicheslavschina: Local history almanac. Publication 5, Dnipropetrovsk: DОUNL, 55 – 62 (in Ukrainian).
Pazynych V., Manyuk, V., Stetsuk, 2011. Newest data about Kodake burial soil age and Dneprovian bouldering loam (to problem of Stratigraphy Upper Pleistocene).Аrcheology and old history of Ukraine. 6, Аrcheology: from sources to reconstructions. – К.: IA NАNU, 86 – 90 (in Ukrainian).
Manyuk, V.V., Manyuk Vad. V., 2012. Natural monuments of Steppe Podneprovye: Past, Present and Prospects. History and Culture Pridneprovya: none known and little-known pages. Dnipropetrovsk, Nat. Mining University, 107 – 114 (in Ukrainian).
Manyuk, V.V., 2012. "Stone Grave" in Ukraine: geological and archaeological heritage of world significance. Bulletin of the National Museum of Natural History, Kyiv, 10, 69-74 (in Ukrainian).
Manyuk, V.V., 2013. Peculiarities of Quaternary deposits in typical sections of the Middle Dnieper. Bulletin Dnipropetrovsk National University. Series: geology, geography. Dnipropetrovsk, V.21, Edition15, 3/2, 8 – 14 (in Ukrainian).
Manyuk, V.V., 2014. Geological heritage Poltava region. International Conference: Science and Society. Donetsk, SIC Knowledge, part 3, 57 – 62 (in Ukrainian).
Manyuk, V.V., 2014. Geological component in the formation of the nature reserve fund of Ukraine. / / Bulletin of the DNU, series: geology, geography. - Dnipropetrovsk, V.22, issue16, 3/2, 29 – 33 (in Ukrainian).
Manyuk, V.V., 2014. Quaternary deposits in the Rybalsky quarry of Dnepropetrovsk city. Bulletin of the DNU, series: geology, geography. - Dnipropetrovsk, V.22,16, 3/2, 107 - 111(in Ukrainian).
Manyuk, V.V., 2015. Mount Pivicha is a unique object of geological heritage of Ukraine. Coll. Center of Sciences. publications: "Science in an era of imbalances". K .: Center for Scientific Publications, 21 – 25 (in Ukrainian).
Manyuk, V.V., 2015. Geological monuments of the Crimea. Geology and ore-bearing of Ukraine. - Dnipropetrovsk, Vol. 1, 1, 51 – 54 (in Ukrainian).
Manyuk, V.V., Manyuk Vad.V., 2017. Geological past of rock massif "Stone graves" Science. Works of the All-Ukrainian Scientific-Practice. Confer.: Natural, historical and cultural heritage of the area of the reserve "Stone graves", May 25-27, 2017, works of the "Stone graves" department of the USSR Academy of Sciences of Ukraine, no. 4, series "Conservation Biology in Ukraine", Zaporozhye: Wild Field, 89 - 98.
Manyuk, V.V., 2017. On reassigning the location of the southern border of the Dnieper (Riss) Glaciation. Bulletin of Dnipropetrovsk University. Series: geology, geography. - Dnipropetrovsk, V.25 (2), 21, 99-110.
4. Abstracts of oral presentations
Osaulenko V.T., Manyuk, V.V., 1996. About the Ukrainian special geological terminology. Problems of the Ukrainian scientific and technical terminology. The thesis of reports 4-th International scientific conference. Lviv, 244-245pp. (in Ukrainian).
Osaulenko V., Manyuk, V.V., Kovalchuk M., 1998. Аbout necessity of creation of the encyclopaedic dictionary on geology. In Bulletin: The problems of Ukrainian terminology. Proceeding of the fifth International conference. Lvov, Polytechnic, 336, 116-119pp. (in Ukrainian).
Manyuk, V.V., 1999. James, facing and ornamental stones of Dnipropetrovsk region. Bulletin of the First international conference: Precious and decorative stones., Kiev, 59-60pp. (in Ukrainian).
Manyuk, V.V., 2001. Quaternary geology of Dnipropetrovsk region. 2001. Volume of Abstracts: The Ukraine Quaternary Explored. Kyiv, 54p.
Manyuk, V.V., 2002. Geological Heritage of the Dnipropetrovsk region. Natural and Cultural landscapes: the geological foundation. Dublin, Ireland, 25-26.
Manyuk, V.V., 2003. Features of preservation of geological monuments of a nature within the framework of the international program "Geosites". The Problem nature-study, steady development and technogenous of safety regions. Proceedings of second International conference: Dnipropetrovsk, 161-163pp. (in Ukrainian).
Manyuk, V.V., 2003. Geological and Historical a Heritage of Zmeynyi (Snake). Geological Heritage concept, conservation and protection policy in Central Europe.-International Conference, Krakow, Poland, 36-37pp.
Manyuk, V.V., 2004. New strategy for conservation of geosites in Ukraine. Geology-marker of cultural and geological heritage geosites under threat.- Proceeding 32 th , International geological congress Florence.
Manyuk, V.V., 2004. New strategy for conservation of geosites in Ukraine. Geology-marker of cultural and geological heritage geosites under threat. Proceeding 32 th , International geological congress Florence.
Manyuk, V.V., 2005. Peculiarity of Geoconservation in Ukraine. IV International Symposium ProGeo on the Conservation of the Geological Heritage. – Braga, Portugal, University of Minho, 26.
Manyuk, V.V., 2005. The problem of creating and improving a network of objects of geological heritage: European experience. Problems of nature use, sustainable development and technogenic safety of regions. - Materials of the Third International. Practical Conference, Dnipropetrovsk, 110-112. (in Ukrainian).
Manyuk, V.V., 2005. Inventory and cataloguing of a geological heritage of Southern and East Ukraine.ProGEO WG-1 subregional meeting and field trip.-Tirana, Albania, 20-21.
Manyuk, V.V., 2006. Inventory and cataloguing of a geological heritage of Southern and East Ukraine. ProGEO Symposium "Safeguarding our Geological Heritage", Kyiv – Kamianets-Podil'sky, 28-30.
Manyuk, V.V., 2006. Potential objects for creation of a Network National Geoparks in Ukraine. ProGEO Symposium "Safeguarding our Geological Heritage", Kyiv – Kamianets-Podil'sky, 30-32.
Manyuk, V.V., 2007. Geological monuments of the Crimea as the determining component of the natural environment. Problems of nature use, sustainable development and technogenic safety of the regions: materials of the Fourth International Scientific and Practical Conference, Dnipropetrovsk, 134-136 (in Ukrainian).
Manyuk, V.V., 2008. Historical background for the creation of the national geological park "Eagle Nest" . Protection and management of non-living objects in protected territories: Mater. international scientific-practical conference. Grimailov-Ternopil, Dzhura, 175-181(in Ukrainian).
Manyuk, V.V., 2008. The problem of creation of a Network National Geoparks in Ukraine. Proceeding 33 th International Geological Congress, Norway, Oslo, 6-14 August.
Manyuk, V.V., 2009. Use of geological monuments of a nature at realization educational practice. Geology at school both high school: geology and civilization: materials of a conference. Volume ІІ Under edited by Е. M. Nesterov. Saint Petersburg, RSРU by A. I. Hertsen, 410-412 (in Russian).
Manyuk, V.V., 2009. Tourist and recreational activity as important components of monitoring of geological nature monuments. Materials of the All-Russia Scientific and Practical Conference devoted to the 50th anniversary of the discovery of A.V. Rumin Palaeolithic painting in the Shulgan-Tash cave (Kapova). The natural and cultural heritage of the Southern Urals as an innovative resource. - Ufa: DesignPolygraph Center, 82-87 (in Russian).
Manyuk, V.V., 2010. The questions of Geological heritage monitoring of Crimea. MCTN, VII Scientist readings for memory professor M.V. Muratov. - Моscow, 83-86 (in Russian).
Manyuk, V.V., 2011. Kam'jana Mogyla (Stone Tomb) in Ukraine: geological and archeological heritage of global significance. Geological monuments are a vivid testimony to the evolution of the Earth. – Collect. materials of the 2nd scientific-practical conference. – Kyiv, Logos, 6-8.
Manyuk, V.V., 2012. Unique objects of geological heritage (geosites) and geological practice. Materials of the IV International Conference. Crimea, Trudolyubovka. - Simferopol, DIAIPI, 192 – 194 (in Russian)..
Manyuk, V.V., 2014. Geological heritage Poltava region. International Conference: Science and Society. - Donetsk, SIC Knowledge, - part 3, 57 - 62.
Manyuk, V.V., Yaroschuk, 2014. The current state of the movement for the preservation of geological heritage in Ukraine. Proceedings of international science. confer. "The role higher education institutions in the development of geology", Kyiv, 24-25 (in Russian).
Manyuk, V.V., 2014. Experience in studying and using geotourism ojects of the Middle Dnieper. Geotourism: practice and experience. Materials of the international scientific conference. Ed. LZ Skakun, IM Bubniak - Lviv, Scientific and production firm "Maps and Atlases", 99-102.
Manyuk, V.V., 2016. Overview of the geological heritage of Kharkiv oblast. Materials of the II International Scientific and Practical Conference (May 5-7, 2016, Lviv). - Lviv: Scientific and production center "Maps and Atlases", 43-46.
6. Professional Career (Positions held)
2017 – present – Director of the Scientific Research Institute of Geology Oles Honchar Dnipro National University
2017 – present - Deputy Chief Editor of the scientific journals "Geology and Ore Content of Ukraine" https://gocu.com.ua
2015 – present - Executive Editor of the scientific journals "Bulletin of the Dnipropetrovsk University: Geology, geography." https://geology-dnu.dp.ua
2001 – present – member of European Association for the Conservation of the Geological Heritage
1996 – present - Oles Honchar Dnipro National University, Department of geology-geography, Assistant Professor in the Chair of Geology and Hydrogeology, Ukraine. Teaches the following subjects: "Monitoring geological heritage sites", "Regional Geology", "Geotectonic", "Geomorphology", "Economic Geology", " Geology of Minerals Deposits", "Geological prospecting (Drilling and Mining)", "Physical Geology", "Structural Geology", "Prospecting and Exploration of Minerals Deposits"
1993-1996 - Oles Honchar Dnipro National University, Department of geology-geography, Senior Lecturer in above-mentioned department, teaching the above disciplines, Ukraine
1984-1993 – Novomoskovsk geological expedition, senior technician-geologist, geologic engineer, Ukraine
1976-1984 – Dnipropetrovsk geophysical expedition, senior technician-geologist, senior geologist, Ukraine
1972–1976 – Uchkoschkonska geological expedition, technician-geologist, senior technician-geologist, Kyrgyzstan
1996 - successfully defended dissertation Paleogeographic reconstructions of sedimentogenesis conditions in the Paleogene and Neogene basins of the Middle Dnieper
1978 - 1984 - Student specializing in Geology (geological survey, prospecting and exploration of mineral deposits), Maxim Gorky Kharkiv State University.
1969–1972 – Student specializing in Geology, Kryvyi Rih mining college, Ukraine
Volodymyr V. Manyuk, Date and Place of Birth – 26.01.1952, Bagan, Novosibirsk region, Russia.
Citizenship Ukraine.
Tel.: +38-050-824-61-77; +38 067 947-45-04
E-mail: vgeoman@gmail.com
Volodymyr V. Manyuk, Oles Honchar Dnipro National University, Gagarin Ave., 72, Dnipropetrovsk, 49010, Ukraine.


text:
package rx.internal.operators;

import static org.junit.Assert.assertEquals;

import java.util.concurrent.TimeUnit;
import java.util.concurrent.atomic.AtomicInteger;

import org.junit.Test;

import rx.Observable;
import rx.functions.Action1;
import rx.functions.Func1;

public class OperatorParallelTest {

    @Test
    public void testParallel() {
        int NUM = 1000;
        final AtomicInteger count = new AtomicInteger();
        final AtomicInteger innerCount = new AtomicInteger();
        Observable.range(1, NUM).parallel(
                new Func1<Observable<Integer>, Observable<Integer[]>>() {

                    @Override
                    public Observable<Integer[]> call(Observable<Integer> o) {
                        return o.map(new Func1<Integer, Integer[]>() {

                            @Override
                            public Integer[] call(Integer t) {
                                try {
                                    // randomize to try and force non-determinism
                                    // if we see these tests fail randomly then we have a problem with merging it all back together
                                    Thread.sleep((int) (Math.random() * 10));
                                } catch (InterruptedException e) {
                                    System.out.println("*********** error!!!!!!!");
                                    e.printStackTrace();
                                    // TODO why is this exception not being thrown?
                                    throw new RuntimeException(e);
                                }
                                //                                System.out.println("V: " + t  + " Thread: " + Thread.currentThread());
                                innerCount.incrementAndGet();
                                return new Integer[] { t, t * 99 };
                            }

                        });
                    }
                })
                .toBlocking().forEach(new Action1<Integer[]>() {

                    @Override
                    public void call(Integer[] v) {
                        count.incrementAndGet();
                        //                System.out.println("V: " + v[0] + " R: " + v[1] + " Thread: " + Thread.currentThread());
                    }

                });
        System.out.println("parallel test completed ----------");

        // just making sure we finish and get the number we expect
        assertEquals("innerCount", NUM, innerCount.get());
        assertEquals("finalCount", NUM, count.get());
    }

    @Test
    public void testParallelWithNestedAsyncWork() {
        int NUM = 20;
        final AtomicInteger count = new AtomicInteger();
        Observable.range(1, NUM).parallel(
                new Func1<Observable<Integer>, Observable<String>>() {

                    @Override
                    public Observable<String> call(Observable<Integer> o) {
                        return o.flatMap(new Func1<Integer, Observable<String>>() {

                            @Override
                            public Observable<String> call(Integer t) {
                                return Observable.from(String.valueOf(t)).delay(100, TimeUnit.MILLISECONDS);
                            }

                        });
                    }
                }).toBlocking().forEach(new Action1<String>() {

            @Override
            public void call(String v) {
                count.incrementAndGet();
            }

        });

        // just making sure we finish and get the number we expect
        assertEquals(NUM, count.get());
    }
}

redpajama_set_name: RedPajamaGithub


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\begin{document}

\title{
Perturbative non-Fermi liquids from dimensional regularization
}

\author{Denis Dalidovich$^{1}$ and Sung-Sik Lee$^{1,2}$\\
{\normalsize{$^1$Perimeter Institute for Theoretical Physics,}}\\
{\normalsize{31 Caroline St. N., Waterloo ON N2L 2Y5, Canada}}
\vspace{0.2cm}\\
{\normalsize{$^2$Department of Physics $\&$ Astronomy, McMaster University,}}\\
{\normalsize{1280 Main St. W., Hamilton ON L8S 4M1, Canada}}
}

\date{\today}

\begin{abstract}

We devise a dimensional regularization scheme for
quantum field theories with Fermi surface
to study scaling behaviour of non-Fermi liquid states
in a controlled approximation.
Starting from a Fermi surface in two space dimensions,
the co-dimension of Fermi surface is extended to
a general value while the dimension of Fermi surface is fixed.
When Fermi surface is coupled with a critical boson centred at zero momentum,
the interaction becomes marginal at a
critical space dimension $d_c=5/2$.
A deviation from the critical dimension
is used as a small parameter for a systematic expansion.
We apply this method to the theory where two patches of Fermi surface
is coupled with a critical boson,
and show that the Ising-nematic critical point is described
by a stable non-Fermi liquid state
slightly below the critical dimension.
Critical exponents are computed up to the two-loop order.

\end{abstract}

\maketitle

\section{Introduction}

It is of central importance in condensed matter physics
to understand universal properties of phases
using low energy effective theories.
In critical states of quantum matter,  effective theories
take the form of quantum field theories
which describe low energy degrees of freedom and their interactions.
Although the most generic critical state of electrons in solids is metal,
quantum field theories of metals are less well understood
compared to relativistic field theories
due to low symmetry and
extensive gapless modes
that need to be kept in low energy theories.

In Fermi liquid metals\cite{LANDAU}, quasiparticles provide a single-particle basis
in which the low energy field theories can be diagonalized\cite{POLCHINSKI_FL,SHANKAR}.
In non-Fermi liquid states, there exist no such single-particle basis,
and the low energy physics is described by genuine interacting quantum
field theories.
Non-Fermi liquid states can arise
when Fermi surface is coupled with a gapless boson
in many different physical contexts.
A boson can be made gapless either
by fine tuning of microscopic parameters
entailing non-Fermi liquid state at a quantum critical point,
or as a result of dynamical tuning,
which gives rise to non-Fermi liquid phases
within an extended region in the parameter space.
The examples for the former case include
heavy fermion compounds near magnetic quantum critical points\cite{LOHNEYSEN,COLEMAN},
quantum critical point for Mott transitions\cite{Senthil_Mott,Podolsky},
and the nematic quantum critical point \cite{ogankivfr,metzner,delanna,kee,lawler,rech,wolfle,maslov,quintanilla,yamase1,yamase2,halboth,jakub,zacharias,kim,huh}.
The $\nu=1/2$ quantum Hall state\cite{HALPERIN}
and Bose metals
which support fractionalized fermionic excitations
along with an emergent gauge field\cite{MOTRUNICH,LEE_U1,PALEE,MotrunichFisher}
are among the examples for the latter case.

From earlier works\cite{holstein,reizer,lee89,leenag},
it was pointed out that the low energy properties of Fermi surface
can be qualitatively modified by the coupling with gapless boson.
In three space dimensions,
logarithmic corrections arise
due to the Yugawa coupling\cite{holstein,reizer,Kachru}.
In two space dimensions,
theories of non-Fermi liquids
flow to strongly interacting
fixed points at low energies.
For chiral non-Fermi liquid states\cite{SSLee},
where only one patch of Fermi surface is coupled with a critical boson,
exact dynamical information can be extracted
thanks to the chiral nature of the theory\cite{shouvik}.
It is much harder to understand
non-chiral theories which
include two-patches of Fermi surface
with opposite Fermi velocities.
One strategy to make a progress in non-chiral theories is
to deform the original theory
into a perturbatively solvable regime in a continuous way.

There can be different and complimentary ways
to obtain perturbative non-Fermi liquid states.
One attempt is to introduce a large number of flavors\cite{ALTSHULER,polchinski,ybkim}.
This is arguably the most natural extension
in the sense
that extra flavors do not introduce
qualitatively new element to the theory
except that the flavor symmetry group is enlarged.
However,  it turns out that even the infinite flavor limit is not
described by a mean-field theory
due to a large residual quantum fluctuations of Fermi surface\cite{SSLee,metlsach1}.
One way to achieve a controlled expansion
is to deform the dynamics of the theory,
for example the dispersion of a critical boson,
to suppress quantum fluctuations at low energies\cite{nayak,mross}.
One can also try to access
two-dimensional non-Fermi liquid states
by increasing the number of one-dimensional chains,
where bosonization provides a controlled analytical tool\cite{Jiang}.
Finally, one can modify the dimension of spacetime continuously
to gain a controlled access to non-Fermi liquid states.
In doing so, one can extend either
the dimension of Fermi surface\cite{Chakravarty}
 or the co-dimension\cite{senshank}.
In this paper, we devise a dimensional regularization scheme
where the co-dimension of Fermi surface is extended to
obtain a perturbative non-Fermi liquid state
which describes two patches of Fermi surface
coupled with a critical boson.
This scheme has an advantage that Fermi surface remains one-dimensional
and has only one tangent vector.
Because fermions in one region of the momentum space
near the Fermi surface
are primarily coupled with the boson whose momentum
is tangential to the Fermi surface,
fermions in different momentum patches
(except for the ones in the exact opposite direction)
are decoupled from each other in the low energy limit.
Because of this, one can focus on
local patches in the momentum space,
which allows one to develop a systematic
field theoretic renormalization group scheme.
If the dimension of Fermi surface is extended,
each patch has more than one tangent vectors.
Since one can not ignore couplings between different patches,
the whole Fermi surface has to be included in the low energy theory.
Recently, a non-Fermi liquid state was studied
through a Wilsonian renormalization group scheme
in $d=3-\epsilon$ space dimensions
with co-dimension of Fermi surface fixed to be one\cite{Fit}.

The paper is organized as follows.
In Sec. II, we introduce a $(2+1)$-dimensional theory
which describes two patches of Fermi surface coupled
with a critical boson.
Depending on the way the boson is coupled with the patches,
the theory describes either the Ising-nematic critical point
or the quantum electrodynamics with a finite density.
In this paper, we will focus on the Ising-nematic theory.
We then generalize the theory to a $(d+1)$-dimensional theory
with general space dimension $d$.
For this, we first combine particle in one patch
and hole in the opposite patch to construct
a spinor with two components.
In this representation, fermionic excitations near the Fermi surface
in the original $(2+1)$-dimensional theory
can be formally viewed as $(1+1)$-dimensional
Dirac fermions with a continuous flavor that
corresponds to the momentum along the Fermi surface.
Then the Dirac fermion
is extended to general dimensions,
where the energy of fermion disperses linearly
away from the gapless point in $(d-1)$ directions.
Physically, this describes a one-dimensional Fermi line
embedded in $d$-dimensional momentum space.
In $d=3$ with $SU(2)$ flavor (spin) group,
this theory describes a $p$-wave spin-triplet
superconducting state which supports a line node.
The Yugawa coupling between fermion and boson
becomes marginal at the critical dimension, $d_c=5/2$,
and non-Fermi liquid states arise in $d<5/2$.
Using $\epsilon = 5/2-d$ as  an expansion parameter,
one can access the non-Fermi liquid state perturbatively.
The following section is devoted to the RG analysis of the theory
based on the dimensional regularization scheme.
In Sec. III. A, the minimal local action is constructed.
In Sec. III. B, the symmetry of the regularized theory is discussed.
Because the extension to higher dimensions involves
turning on flavor non-singlet superconducting order parameter,
the regularized theory breaks the charge conservation and some of the flavor symmetry.
In Sec. III. C, the renormalization group equation is derived using the
minimal subtraction scheme.
In Sec. III. D, we demonstrate that the expansion is controlled
in the small $\epsilon$ limit with fixed $N$
where $N$ is the number of fermion flavour.
In Sec. III. E, we summarize the computation of the counter terms up to the two-loop level.
Some three-loop results are also included.
Based on the results, the dynamical critical exponents and the anomalous  dimensions are computed in Sec. III. F.
While the interaction modifies
the dynamics of  fermion in a non-trivial way,
the boson does not receive a non-trivial quantum correction
up to the three-loop diagrams that we checked.
In Sec. IV, the scaling forms of the thermodynamic quantities
and $2k_F$ scattering processes are obtained.
We finish with a summary and some outlook in Sec. V.
Details on the computation of Feynman diagrams
can be found in the appendices.

\section{Model}

\begin{figure}
\begin{center}
        \includegraphics[height=5cm,width=6cm]{FS_two_patch.eps}
\end{center}
\caption{The right-moving and left-moving modes near the Fermi surface
can be combined into a two-component Dirac fermion.}
\label{fig:FS_two_patch}
\end{figure}

We consider a theory where two patches of Fermi surface
are coupled with one critical boson in $(2+1)$-dimensions,
\bqa
S & = &  \sum_{s=\pm} \sum_{j=1}^N \int \frac{d^3k}{(2\pi)^{3}}
\psi_{s,j}^\dagger (k)
\Bigl[
 i k_0   +  s  k_1 +  k_2^2  \Bigr] \psi_{s,j}(k) \nonumber \\
 &+& \frac{1}{2} \int  \frac{d^3k}{(2\pi)^{3}}
 \left[ k_0^2 + k_1^2 +  k_2^2 \right] \phi(-k) \phi(k) \nonumber \\
 &+&  \frac{e}{\sqrt{N}} \sum_{s=\pm} \sum_{j=1}^N
\int \frac{d^3k d^3q}{(2\pi)^{6}}  ~
\lambda_s ~ \phi(q) ~  \psi^\dagger_{s,j}(k+q) \psi_{s,j}(k).
\label{act0}
\end{eqnarray}
Here $\psi_{+,j}$ ($\psi_{-,j}$)
is the right (left) moving fermion with flavor $j=1,2,..,N$
whose Fermi velocity along the $k_1$ direction is positive (negative).
The momenta are rescaled in such a way
that the absolute value of Fermi velocity
and curvature of the Fermi surface are equal to one.
$\phi$ is a real critical boson,
and $e$ is the fermion-boson coupling.
Although the velocity of boson is in general different from that of fermion,
the dynamics of boson is dominated by
particle-hole excitations of fermion at low energies.
As a result, the bare velocity of boson,
which is also set to be one in the action,
does not matter for the low energy effective theory.
$\lambda_s$ controls the way the fermions are coupled with the boson.
The case with $\lambda_+ = \lambda_- = 1$ describes
the Ising-nematic critical point.
The coupling with $\lambda_+ = -\lambda_- = 1$ describes
the quantum electrodynamics at a finite density,
where $\phi$ corresponds
to the transverse component of the U(1) gauge field.
The action in Eq. (\ref{act0})
admits a self-contained
renormalization group analysis\cite{metlsach1}.

In the following, we will focus on the Ising-nematic critical point.
Quantum phase transitions to nematic states with
broken point group symmetry\cite{pomeran} have been observed
in cuprate superconductors\cite{ando,hinkov,kohsaka,taill},
ruthenades\cite{borzi}, and pnictides\cite{fang1,xu1,chuang1,chu1}.
The Ising-nematic order parameter is
represented by a real scalar boson
which undergoes
strong quantum fluctuations
at the quantum critical point.

Because the energy of fermion disperses only in one direction near the Fermi surface,
the $(2+1)$-dimensional fermion can be viewed as $(1+1)$-dimensional Dirac fermions,
where the momentum along the Fermi surface is interpreted as a continuous flavor.
To make this more precise, the right and left moving fermions are combined into
one spinor (see Fig. \ref{fig:FS_two_patch}),
\bqa
\Psi_j(k) = \left(
\begin{array}{c}
\psi_{+,j}(k) \\
\psi_{-,j}^\dagger(-k)
\end{array}
\right).
\end{eqnarray}
In this representation, the action in Eq. (\ref{act0}) becomes
\bqa
S & = &  \sum_{j} \int \frac{d^3k}{(2\pi)^{3}}
\bar \Psi_j(k)
\Bigl[
 i k_0 \gamma_0  + i (   k_1 +  k_2^2 )  \gamma_1 \Bigr]
\Psi_{j}(k) \nonumber \\
 &+& \frac{1}{2} \int  \frac{d^3k}{(2\pi)^{3}}
 \left[ k_0^2 + k_1^2 +  k_2^2 \right] \phi(-k) \phi(k) \nonumber \\
 &+&     \frac{e}{\sqrt{N}} \sum_{j}  \int \frac{d^3k dq}{(2\pi)^{6}}
\phi(q) \bar \Psi_{j}(k+q) W \Psi_{j}(k),
\label{act1}
\end{eqnarray}
where $\gamma_0 = \sigma_y$, $\gamma_1 = \sigma_x$
are the gamma matrices for the two component spinor,
and $\bar \Psi \equiv \Psi^\dagger \gamma_0$.
$W = i \gamma_1$ ($W = \gamma_0$)
for the Ising-nematic system (quantum electrodynamics).
For the rest of the paper, we will focus on the Ising-nematic case.
The fermionic kinetic term is indeed identical
to that of the $(1+1)$-dimensional Dirac fermion
where the location of Dirac point
depends on $k_2$.

\begin{figure}
\begin{center}
        \includegraphics[height=6cm,width=14cm]{FS_general.eps}
\end{center}
\caption{
(a) The one-dimensional Fermi surface embedded in the $d$-dimensional momentum space.
(b) For each $k_d$, there is a Fermi point at
$(k_1, k_2, ..., k_{d-2}, k_{d-1}) = (0,0,.., 0, - \sqrt{d-1} k_d^2)$
which is denoted as a (red) dot.
Around the Fermi point, the energy disperses linearly
like a two-component Dirac fermion in
$(d-1)$-dimensional space.
}
\label{fig:FS_general}
\end{figure}

Now we promote the theory to
general dimensions.
The action that describes Fermi surface
with a general co-dimension is written as
\begin{eqnarray}\label{act4}
S & = &  \sum_{j} \int \frac{d^{d+1}k}{(2\pi)^{d+1}} \bar \Psi_j(k)
\Bigl[
i \vec \Gamma \cdot \vec K + i \gamma_{d-1} \delta_k \Bigr] \Psi_{j}(k) \nonumber\\
&+&
\frac{1}{2} \int  \frac{d^{d+1}k}{(2\pi)^{d+1}}
 \left[ |\vec K|^2 + k_{d-1}^2 +  k_d^2 \right] \phi(-k) \phi(k) \nonumber \\
 &+&     \frac{ie}{\sqrt{N}}\sqrt{d-1} \sum_{j}
\int \frac{d^{d+1}k d^{d+1}q}{(2\pi)^{2d+2}}
\phi(q) \bar \Psi_{j}(k+q) \gamma_{d-1} \Psi_{j}(k).
\end{eqnarray}
Here $\vec K \equiv (k_0, k_1,\ldots, k_{d-2})$
represents frequency and $(d-2)$ components of
the full $(d+1)$-dimensional energy-momentum vector,
$(k_0, k_1, ..., k_d)$.
$k_1, .., k_{d-2}$ are the newly added directions
which are transverse to the Fermi surface.
$\delta_k =  k_{d-1}+ \sqrt{d-1} k_d^2$
is the energy dispersion of the fermion within
the original two-dimensional momentum space.
The gamma matrices associated with $\vec K$
are written as
$\vec \Gamma \equiv (\gamma_0, \gamma_1,\ldots, \gamma_{d-2})$.
Since the actual space dimension of
interest lies between $2$ and $3$,
the number of spinor components is fixed to be two.
We will use the representation
where $\gamma_0 = \sigma_y$ and $\gamma_{d-1} = \sigma_x$ are
fixed in general dimensions.

The spinor has the energy dispersion with two bands,
\bqa
E_k = \pm \sqrt{ \sum_{i=1}^{(d-2)} k_i^2 + \delta_k^2  }.
\end{eqnarray}
The energy vanishes if
\bqa\label{fersur2}
k_i &= &0,\qquad \mbox{for}\,\,\, i=1,\ldots,d-2\, \nonumber \\
{ k}_{d-1} & =& - \sqrt{d-1} k_d^2.
\end{eqnarray}
Therefore the Fermi surface is a one-dimensional manifold
embedded in the $d$-dimensional momentum space
as is shown in Fig. \ref{fig:FS_general}.

\begin{figure}
\begin{center}
        \includegraphics[height=6cm,width=8cm]{FS_3d.eps}
\end{center}
\caption{
Fermi lines in three dimensional momentum space can be
obtained by turning on
$p$-wave superconducting order parameter that
gaps out the cylindrical Fermi surface except
for the line node denoted by the thick (red) line.
}
\label{fig:FS_3d}
\end{figure}

Before we delve into the RG analysis in an abstract dimension $d$,
we consider a concrete physical realization
of the theory in $d=3$ with $N=2$,
where two flavors represent spin $1/2$ degree of freedom.
In the basis
where $\gamma_0 = \sigma_y$, $\gamma_1 = \sigma_z$, $\gamma_2 = \sigma_x$,
the quadratic action for the fermions becomes
\bqa
S & = &
\int \frac{d^4k}{(2\pi)^{3}}
\Bigg\{
\sum_{s=\pm}
\sum_{j=\uparrow, \downarrow}
\psi_{s,j}^\dagger (k)
\left(  i k_0   +  s  k_2 + \sqrt{2} k_3^2  \right) \psi_{s,j}(k)  \nonumber \\
&& - k_1
\left(
\psi_{+,\uparrow}^\dagger(k)
\psi_{-,\uparrow}^\dagger(-k)
+\psi_{+,\downarrow}^\dagger(k)
\psi_{-,\downarrow}^\dagger(-k)
+ h.c.
\right)
\Bigg\}.
\end{eqnarray}
The last term represents a pairing
which can be written as
\bqa
 i \Bigl( \psi_{+,\uparrow}^\dagger(k), \psi_{+,\downarrow}^\dagger(k) \Bigr)
\Bigl( \vec d(k) \cdot \boldsymbol \sigma  \Bigr)
\sigma_y
\left(
\begin{array}{c}
\psi_{-,\uparrow}^\dagger(-k) \\
\psi_{-,\downarrow}^\dagger(-k)
\end{array}
\right) + h.c.
\end{eqnarray}
with $\vec d(k) = i  k_1 \hat y$.
It is noted that $\vec d(k)$ transforms as a vector
under $SU(2)$ spin rotations.
Therefore this describes a $p$-wave spin triplet superconducting state.
Without the pairing term,
one has the cylindrical Fermi surface
with co-dimension one at $\pm k_2 + \sqrt{2} k_3^2 = 0$.
The pairing gaps out the Fermi surface
except for the line node at $k_1=0$.
This is illustrated in Fig. \ref{fig:FS_3d}.
The triplet pairing breaks the $U(1)$ symmetry associated with the fermion number conservation to $Z_2$
and the $SU(2)$ spin rotational symmetry to $U(1)$.
The theories in general dimensions
continuously interpolate the Fermi surface in two dimensions
to the triplet superconducting state in three dimensions.
In terms of symmetry, the theory in $d=2$ is a
special point with an enhanced symmetry.

\section{Renormalization Group}

\subsection{Minimal Action}

To start with a renormalization group analysis,
we first focus on the quadratic action in Eq.~(\ref{act4}).
The leading terms in the quadratic action
are invariant under the scale transformation,
\begin{eqnarray}
\vec K & = & \frac{\vec K'}{b}, \label{s1} \\
k_{d-1} & = & \frac{k_{d-1}'}{b}, \label{s2} \\
k_d & = & \frac{k_d'}{\sqrt{b}}, \label{s3} \\
\Psi (k) & = & b^{\frac{d}{2}+\frac{3}{4}} \Psi' (k'), \label{s4} \\
\phi (k) & = & b^{\frac{d}{2}+\frac{3}{4}} \phi'(k'). \label{s5}
\end{eqnarray}
Under the scaling with $b>1$,
the fermion-boson coupling scales as
\begin{equation}
e' = b^{\frac12 (\frac52 -d)} e.
\end{equation}
The coupling $e$ is irrelevant for $d>d_{{\rm cr}}=5/2$,
and is relevant for $d<5/2$.
This allows  one to access an interacting non-Fermi liquid state
perturbatively in $d = 5/2 -\epsilon$
using $\epsilon$ as a small parameter.

We note that
$|\vec K|^2 \phi^* (k)\phi(k)$
and
$k_{d-1}^2 \phi^* (k)\phi(k)$
are irrelevant in the low energy limit.
Only $k_d^2 \phi^* (k)\phi(k)$
is kept in the quadratic action of the boson.
The frequency dependent self energy of the boson will
be dynamically generated by particle-hole fluctuations.
Therefore, the minimal local action is given by
\begin{eqnarray}\label{act5}
S & = &  \sum_{j} \int \frac{d^{d+1}k}{(2\pi)^{d+1}} \bar \Psi_j(k)
\Bigl[
i \vec \Gamma \cdot \vec K
+ i \gamma_{d-1} \delta_k \Bigr] \Psi_{j}(k) \nonumber\\
&+&
\frac{1}{2} \int  \frac{d^{d+1}k}{(2\pi)^{d+1}} ~~
   k_d^2  \phi(-k) \phi(k) \nonumber \\
 &+&     \frac{ie \mu^{\epsilon/2}}{\sqrt{N}}\sqrt{d-1} \sum_{j}
\int \frac{d^{d+1}k d^{d+1}q}{(2\pi)^{2d+2}}
\phi(q) \bar \Psi_{j}(k+q) \gamma_{d-1} \Psi_{j}(k),
\end{eqnarray}
where a mass scale $\mu$ is introduced
to make the coupling constant dimensionless.
Short-ranged four fermion interactions
and the $\phi^4$ term for the boson are not included
in the minimal action because they are irrelevant near $d=5/2$.
There is no further BCS instability
that gaps out the line nodes near $d=5/2$
because the density of state vanishes at Fermi energy.

For $d > 5/2$, the interaction is irrelevant
and the low energy physics is governed by the scaling
in Eqs. (\ref{s1})-(\ref{s5})
with the dynamical critical exponent $z=1$.
For $d < 5/2$, the scaling will be modified such that
the interaction plays the dominant role.
In order to see this, one can choose an alternative scaling,
\begin{eqnarray}
\vec K & = & \frac{\vec K'}{b^z}, \label{s21} \\
k_{d-1} & = & \frac{k_{d-1}'}{b}, \label{s22} \\
k_d & = & \frac{k_d'}{\sqrt{b}}, \label{s23} \\
\Psi (k) & = & b^{\frac{(d-1)z}{2}+\frac{5}{4}} \Psi' (k'), \label{s24} \\
\phi (k) & = & b^{\frac{(d-1)z}{2}+\frac{5}{4}} \phi'(k'). \label{s25}
\end{eqnarray}
The condition that the interaction is kept marginal
at the expense of making
the first quadratic term in Eq. (\ref{act5}) irrelevant
uniquely fixes the dynamical critical exponent to be
$z = \frac{3}{3- 2\epsilon}$ at the tree level.
As will be shown later,
this is indeed what we obtain from a full-fledged computation
modulo a correction coming from
the anomalous dimension of the boson field.

\subsection{Symmetry}
\label{sec:symm}

In this section, we discuss about the symmetry
of the action in Eq. (\ref{act5}).
The $(2+1)$-dimensional theory in Eq. (\ref{act1})
has two classical $U(N)$ symmetries given by
\bqa
U_A(N) :
\Psi_i & \rightarrow & [ e^{i \theta_\alpha T^\alpha } ]_{ij} \Psi_j, \label{u11} \\
U_B(N) :
\Psi_i & \rightarrow & [ e^{i \sigma_z \varphi_\alpha T^\alpha } ]_{ij} \Psi_j, \label{u12}
\end{eqnarray}
where $T^\alpha$ with $\alpha = 1, 2, .., N^2$ represent
$N \times N$ Hermitian matrices.
$U_A(N)$ and $U_B(N)$ respectively
correspond to the global and axial symmetry groups
of the underlying $(1+1)$-dimensional theory
when $k_2$ is interpreted as an internal flavor.
The generators of the two $U(N)$ groups are given by
\bqa
j_{A0}^\alpha & = &
\psi_{+,i}^\dagger T_{ij}^\alpha \psi_{+,j}
- \psi_{-,i}^\dagger (T^{\alpha})^{T}_{ij} \psi_{-,j}, \label{jA0} \\
j_{B0}^\alpha & = &
\psi_{+,i}^\dagger T^\alpha_{ij} \psi_{+,j}
+ \psi_{-,i}^\dagger (T^{\alpha})^{T}_{ij} \psi_{-,j}, \label{jB0}
\end{eqnarray}
where $(T^{\alpha})^{T}$ denotes the transpose of $T^\alpha$.
In general dimensions,
only the $U_A(N)$ symmetry is kept.
The axial $U_B(N)$ symmetry is absent in $d>2$
because fermions with opposite chiralities are mixed.
The inability to keep both symmetries is related to
the chiral anomaly in $(1+1)$ dimensions.

The theory in general dimensions retain
the Ward identity and the sliding symmetry
of the $(2+1)$-dimensional theory\cite{metlsach1}.
In the Ising-nematic case,
the boson couples to the $(d-1)$-th component of the
$U_A(1)$ current.
This implies the Ward identity
\bqa
\Gamma(k,0) & = &  \frac{ \partial G^{-1}(k) }{\partial k_{d-1} },
\label{WI}
\end{eqnarray}
where $\Gamma(k,q)$ is the fermion-boson vertex function,
and $G(k)$ is the fermion propagator.
The theory also has the sliding symmetry
along the Fermi surface given by
\bqa
\Psi( \vec K, k_{d-1}, k_d ) & \rightarrow &
\Psi( \vec K, k_{d-1} - \sqrt{d-1}( 2 \theta k_d + \theta^2), k_d + \theta ), \nonumber \\
\phi( \vec Q, q_{d-1}, q_d ) & \rightarrow &
\phi( \vec Q, q_{d-1} - 2 \sqrt{d-1} \theta q_d, q_d ).
\end{eqnarray}
As a result, the fermion propagator
depends on $k_{d-1}$ and $k_d$ only through $\delta_k$,
and the boson propagator is independent of $q_{d-1}$,
\bqa
G( \vec K, k_{d-1}, k_d ) & = & G( \vec K, \delta_k ), \nonumber \\
D( \vec Q, q_{d-1}, q_d ) & = & D( \vec Q, q_d ).
\end{eqnarray}
Finally, the action respects the
$(d-1)$-dimensional rotational symmetry
in the space of $\vec K$
and the time-reversal symmetry.

\subsection{Renormalization Group Equation}

\begin{figure}
\begin{center}
        \includegraphics[height=4cm,width=8cm]{bos1.eps}
\end{center}
\caption{The one-loop boson self-energy.}
\label{fig:RPA}
\end{figure}

We use the field theoretical renormalization group approach
to study the scaling behaviour of the theory
in $d<5/2$, using $\epsilon=\frac52 -d$ as a perturbative parameter.
At each order in the loop expansion,
we add counter terms to cancel
divergent terms in $1/\epsilon$
using the minimal subtraction scheme.
The bare propagator for fermions is given by
\begin{equation}\label{propf}
G_0 (k) =\frac{1}{i} \, \frac{\vec \Gamma \cdot \vec K +
\gamma_{d-1} \delta_k}
{\vec K^2  + \delta_k^2}.
\end{equation}
Since the bare kinetic term of boson depends only on $k_d$,
one has to include the lowest order quantum correction
to ensure IR and UV finiteness.
Therefore, we use the dressed propagator for boson
which includes the one-loop self-energy
as is shown in Fig.~\ref{fig:RPA},
\begin{equation}\label{babos}
D_1(k) = \frac{1}{k_d^2 - \Pi_1 (k)} =
\frac{1}{ k_d^2  + \beta_d e^2 \mu^{\epsilon} \frac{|\vec K|^{d-1}}{|k_d|} },
\end{equation}
where
\begin{equation}\label{betad}
\beta_d = \frac{\sqrt{d-1} \Gamma^2 (d/2)}
{2^d \pi^{(d-1)/2}\, |\cos(\pi d/2)|\, \Gamma(\frac{d-1}{2}) \Gamma (d)}.
\end{equation}
We use the sign convention where the self energy subtract the bare action
in the dressed propagator as $D(k)=\frac{1}{D_1^{-1} (k) -\Pi (k)}$,
$G(k)=\frac{1}{G_0^{-1} (k) -\Sigma (k)}$,
where $\Pi(k)$ and $\Sigma(k)$ are the self energies of
boson and fermion respectively.
The one-loop boson self-energy $\Pi_1 (k)$ is finite
for $2\le d < 3$.
At $d=5/2$, it has the same scaling dimension as $k_d^2$ as expected.
For computation of $\Pi_1(k)$, see Appendix~\ref{app:oneloopbos}.

We note that the inclusion of the one-loop boson self energy
in the zero-th order quantum effective action
is nothing but a rearrangement in the perturbative expansion
of a local theory.
This is because the non-local self energy
is dynamically generated
from the local action.
The fact that  the one-loop boson self energy has to be included
from the beginning has some consequences.
First, the `loop-expansion' we are going to use is defined
modulo the inclusion of the one-loop self energy of boson.
For examples, the diagrams in Fig. \ref{fig:fermvert} are regarded
as one-loop diagrams although the boson propagators in the diagrams
already include the RPA sum of boson self energy.
Second, the dynamics of boson has a intrinsic crossover scale
at $k_d \sim e^{2/3} |\vec K|^{(d-1)/3}$ which goes to zero
in the weak coupling limit.
Because of this, the actual parameter that controls the loop expansion
is not $e$ as will be discussed in Sec. III D in more detail.

The counter terms take the same form
as the original local action,
\begin{eqnarray}\label{act6}
S_{CT} & = &  \sum_{j} \int \frac{d^{d+1}k}{(2\pi)^{d+1}} \bar \Psi_j(k)
\Bigl[
i A_1 \vec \Gamma \cdot \vec K
+ i A_2 \gamma_{d-1} \delta_k
 \Bigr] \Psi_{j}(k)
\nonumber\\
&+&
\frac{A_3}{2} \int  \frac{d^{d+1}k}{(2\pi)^{d+1}}~~
k_d^2 \phi(-k) \phi(k) \nonumber \\
 &+&    A_4 \frac{ie \mu^{\epsilon/2} }{\sqrt{N}}\sqrt{d-1} \sum_{j}
\int \frac{d^{d+1}k d^{d+1}q}{(2\pi)^{2d+2}}
\phi(q) \bar \Psi_{j}(k+q)
\gamma_{d-1} \Psi_{j}(k),
\end{eqnarray}
where
\bq
A_n = \sum_{k=1}^\infty \frac{Z_{n,k}(e)}{\epsilon^k}.
\end{equation}
In the mass independent minimal subtraction scheme,
the coefficients $Z_{n,k}(e)$ depend only on the coupling.
$A_n$ and $Z_{n,k}$ can be further expanded in the number of loops.
We use $A_n^{(L)}$ and $Z_{n,k}^{(L)}$ to denote $L$-loop contributions
modulo the one-loop self energy of boson which
is already included in Eq. (\ref{babos}).
Note that the $(d-1)$-dimensional rotational invariance
in the space perpendicular to the Fermi surface
guarantees that  $\Gamma_i K_i$
are renormalized in the same way
for $0\leq i \leq (d-2)$.
Similarly, the sliding symmetry along the Fermi surface
guarantees that the form of $\delta_k$ is preserved.
However, $A_1$ and $A_2$ are in general different
due to a lack of the full rotational symmetry in the $(d+1)$-dimensional spacetime.
This will leads to a non-trivial dynamical critical exponent
as will be shown later.
The Ward identity in Eq. (\ref{WI}) forces $A_4 = A_2$.

Adding the counter terms to the original action,
we obtain the renormalized action
which gives the finite quantum effective action,
\begin{eqnarray}\label{act7}
S_{ren} & = &  \sum_{j} \int \frac{d^{d+1}k_B}{(2\pi)^{d+1}}
\bar \Psi_{Bj}(k_B)
\Bigl[
i  \vec \Gamma \cdot \vec K_B +
i \gamma_{d-1} \delta_{k_B}  \Bigr] \Psi_{Bj}(k_B)
\nonumber\\
&+&
\frac{1}{2} \int  \frac{d^{d+1}k_B}{(2\pi)^{d+1}} ~
k_{Bd}^2 \phi_B(-k_B) \phi_B(k_B) \nonumber \\
 &+&     \frac{ie_B }{\sqrt{N}}\sqrt{d-1} \sum_{j}
\int \frac{d^{d+1}k_B d^{d+1}q_B}{(2\pi)^{2d+2}}
\phi_B(q_B) \bar \Psi_{Bj}(k_B+q_B)
\gamma_{d-1} \Psi_{Bj}(k_B),
\end{eqnarray}
where
\bqa
\vec K & = & \frac{Z_2}{Z_1} \vec K_B, \nonumber \\
k_{d-1} & = & k_{B, d-1}, \nonumber \\
k_d & = & k_{B,d}, \nonumber \\
\Psi(k) & = &  Z_\Psi^{-1/2} \Psi_B(k_B), \nonumber \\
\phi(k) & = & Z_\phi^{-1/2} \phi_B(k_B), \nonumber \\
e_B & = & Z_3^{-1/2} \left( \frac{Z_2}{Z_1} \right)^{(d-1)/2} \mu^{\epsilon/2} e
\end{eqnarray}
with $Z_n  =  1 + A_n$,
$Z_\Psi =  Z_2 \left( \frac{Z_2}{Z_1} \right)^{(d-1)}$ and
$Z_\phi = Z_3 \left( \frac{Z_2}{Z_1} \right)^{(d-1)}$.
In Eq. (\ref{act7}), there is a freedom to change
the renormalizations of the fields
and the renormalization of momentum
without affecting the action.
Here we fix the freedom by requiring
that  $\delta_{k_B} = \delta_k$.
This amounts to measuring scaling dimensions
of all other quantities
relative to that of $\delta_k$.

The finite renormalized Green's function
is defined by
\bqa
&& \Bigl<
\bar
\Psi( k_1 )  .. \bar \Psi(k_m) ~
\Psi( k_{m+1} )  .. \Psi(k_{2m}) ~
\phi( k_{2m+1} ) .. \phi(k_{2m+n})
\Bigr> \nonumber \\
&& = G^{(m,m,n)}( \{ k_i \}; e, \mu ) ~~
\delta^{d+1} \left( \sum_{i=1}^{m} k_i  - \sum_{j=m+1}^{2m+n} k_j \right),
\end{eqnarray}
where the flavor and spacetime indices of fermions
are suppressed.
It is related to the bare Green's function defined by
\bqa
&& \Bigl<
\bar \Psi_B( k_{B1} )  .. \bar \Psi_B(k_{Bm}) ~
\Psi_B( k_{B m+1} )  .. \Psi_B(k_{B 2m}) ~
\phi_B( k_{B 2m+1} ) .. \phi_B(k_{B 2m+n})
\Bigr> \nonumber \\
&& = G_B^{(m,m,n)}( \{ k_{B i} \}; e_B ) ~~
\delta^{d+1} \left( \sum_{i=1}^{m} k_{B i}  - \sum_{j=m+1}^{2m+n} k_{B j} \right),
\end{eqnarray}
through the multiplicative renormalization,
\bqa
G^{(m,m,n)}(\{ k_i \}; e, \mu )
=
Z_\Psi^{-m}
Z_\phi^{-\frac{n}{2}}
\left( \frac{Z_2}{Z_1} \right)^{d-1}
G_B^{(m,m,n)}( \{ k_{Bi} \}; e_B ).
\end{eqnarray}
Using the facts that the bare Green's function
is independent of $\mu$
and that $G^{(m,m,n)}$ has the engineering
scaling dimension $-(2m+n)\frac{4-\epsilon}{2}+(3-\epsilon)$,
one obtains the renormalization group equation,
\bqa
&&
\Bigg\{
\sum_{i=1}^{2m+n} \left(
z(e)\, \vec K_i \cdot \nabla_{K_i}
+ k_{i, d-1} \frac{\partial}{\partial k_{i,d-1}}
+ \frac{ k_{i,d} }{2} \frac{\partial}{\partial k_{i,d}}
\right)
 - \beta(e) \frac{\partial}{\partial e}
 - 2m \left[ -\frac{4-\epsilon}{2} +  \eta_\psi \right] \nonumber \\
&& - n \left[ -\frac{4-\epsilon}{2} +  \eta_\phi \right]
- \left[ z(e) \left( \frac{3}{2} -\epsilon \right) + \frac{3}{2} \right]
\Bigg\}
G^{(m,m,n)}(\{ k_i \}; e, \mu )
=0.
\label{RGeq}
\end{eqnarray}
From the equation,
it is clear that the dimension of $\vec K$ is renormalized to $z$,
and the total dimension of spacetime becomes
$-\left[ z \left( \frac{3}{2} -\epsilon \right) + \frac{3}{2} \right]$
accordingly.
Here  $z$ is the dynamical critical exponent,
$\beta$ is the beta function,
and $\eta_\psi$ ($\eta_\phi$) is
the anomalous dimensions for fermion (boson)
which are given by
\bqa
z(e) & = & 1 - \frac{ \partial \ln (Z_2/Z_1) }{\partial \ln \mu}, \nonumber \\
\beta(e) & = &  \frac{\partial e}{\partial \ln \mu}, \nonumber \\
\eta_\psi(e) & = & \frac{1}{2} \frac{ \partial \ln Z_\Psi}{\partial \ln \mu}, \nonumber \\
\eta_\phi(e) & =& \frac{1}{2} \frac{ \partial \ln Z_\phi}{\partial \ln \mu}.
\label{beta1}
\end{eqnarray}
We use the convention that the beta function describes
the flow of the coupling with increasing energy scale.
These four equations can be rewritten as
\bqa
\beta ( Z_1 Z_2^{'} - Z_2 Z_1^{'} ) + Z_1 Z_2 (z-1) & = & 0, \nonumber \\
e \left[ -\frac{\epsilon}{2} z + \frac{3}{4} (z-1) \right] Z_3
- \left[ Z_3 - \frac{e}{2} Z_3^{'} \right] \beta & = & 0, \nonumber \\
Z_2 \eta_\psi - \frac{\beta}{2} Z_2^{'} + \left( \frac{3}{4} - \frac{\epsilon}{2} \right) (z-1) Z_2 & = & 0, \nonumber \\
Z_3 \eta_\phi - \frac{\beta}{2} Z_3^{'} + \left( \frac{3}{4} - \frac{\epsilon}{2} \right) (z-1) Z_3 & = & 0,
\label{beta2}
\end{eqnarray}
where primes represent derivatives with respect to $e$.
One can readily see that
the regular part of Eqs. (\ref{beta1})
in the $\epsilon \rightarrow 0$ limit
requires the solutions of the form,
\bqa
z & = & z^{(0)}, \nonumber \\
\beta & =  & \beta^{(1)} \epsilon + \beta^{(0)}, \nonumber \\
\eta_\psi & = & \eta_{\Psi}^{(1)} \epsilon + \eta_{\Psi}^{(0)}, \nonumber \\
\eta_\phi & = & \eta_{\phi}^{(1)} \epsilon + \eta_{\phi}^{(0)}.
\end{eqnarray}
Using this form,
one can solve Eqs. (\ref{beta2}) at each order in $\epsilon$.
$z, \beta, \eta_\psi, \eta_\phi$
are determined from the simple poles of the counter terms as
\bqa
z & = & \frac{2}{ 2 + e ( Z_{1,1}^{'} - Z_{2,1}^{'} ) }, \label{zcal} \\
\beta & = &
\left(
-\frac{\epsilon }{2} z    + \frac{3}{4} ( z - 1 )
\right) e
- \frac{z e^2}{4} Z_{3,1}^{'}, \label{bcal} \\
\eta_\psi & = & -\frac{( z - 1) (3-2\epsilon)}{4} - \frac{z e}{4}  Z_{2,1}^{'}, \label{psical} \\
\eta_\phi & = & -\frac{( z - 1) (3-2\epsilon)}{4} -
\frac{z e}{4}  Z_{3,1}^{'}. \label{phical}
\end{eqnarray}
In Eq. (\ref{zcal}), we see that the dynamical critical exponent
is renormalized by quantum effects.
In other words, the first $(d-1)$ components
of the energy-momentum vector
acquires an anomalous dimension $(z-1)$.
The anomalous dimension of spacetime
affects the scaling dimension of the coupling
and the anomalous dimensions of the fields
in Eqs. (\ref{bcal}), (\ref{psical}) and (\ref{phical}).
Once $Z_{n,m}^{'}$ are computed,
one can obtain the beta functions
and the critical exponents.

The theory has the Gaussian fixed point at which
$e=0, z=1, \eta_\psi= \eta_\phi = 0$.
As a small coupling is turned on,
the theory flows to an interacting fixed point with $e^* \neq 0$ at low energies.
The condition that the beta function vanishes at the interacting fixed point
determines the dynamical critical exponent to be
\bqa
z^* = \frac{3}{3 - 2 \epsilon - e^* Z_{3,1}^{'}}.
\end{eqnarray}
It is remarkable that the dynamical critical exponent at the fixed point
is independent of $Z_1$ and $Z_2$.
If $Z_{3,1}=0$, $z^*$ is exactly given by
$z^* = \frac{3}{3 - 2 \epsilon }$
which monotonically increases from $1$ to $3/2$
as $d$ changes from $5/2$ to $2$.
At the fixed point,
the scaling dictates the form of the two-point functions as
\bqa
D(k) & = & \frac{1}{ ( k_d^2 )^{1 - (z-1)(3/2-\epsilon) - 2 \eta_\phi} }
~~ f \left( \frac{| \vec K |^{1/z} }{k_d^2} \right),  \label{Dk0} \\
G(k) & = & \frac{1}{ | \delta_k |^{1 - (z-1)(3/2-\epsilon) - 2 \eta_\psi} }
~~ g \left( \frac{ | \vec K |^{1/z} }{\delta_k} \right), \label{Gk0}
\end{eqnarray}
where $f(x)$ and $g(x)$ are universal cross-over functions.
The flavor and the spinor indices are suppressed in $g(x)$.
If the anomalous dimensions are large enough,
the singularity in the Green's functions can in principle turn
into an algebraic gap\cite{UNP}.
However, the anomalous dimensions are small near the
upper critical dimension.

\subsection{Expansion parameter}

We take the small $\epsilon$ limit with fixed $N$.
In this section, we show that the loop expansion is controlled in this limit.
Although the bare fermion-boson vertex includes $e$,
it is not the actual expansion parameter.
This is due to the fact that the boson propagator
includes the self energy which vanishes
in the $e \rightarrow 0 $ limit.
To examine this issue more closely,
let us consider a boson propagator
which carries an internal momentum $k$
within a diagram.
The integration over $k$ is of the form,
\bqa
\int dk ~
F( k, \{ q_i \} ) \frac{1}{ k_d^2  + \beta_d e^2 \mu^\epsilon
\frac{|\vec K|^{d-1}}{|k_d|} },
\end{eqnarray}
where $\{ q_i \}$ is a set of other internal and external momenta,
and $F( k, \{ q_i \} )$ represents the contribution from other propagators.
When $k_d$ can be arbitrarily small in magnitude,
the integration is in general IR divergent
in the $e \rightarrow 0$ limit.
The IR divergence is cut-off at a scale
$k_d \sim e^{2/3} |\vec K|^{(d-1)/3}$,
and the result of the integration becomes order of $e^{-2/3}$.
Therefore, each boson propagator contributes
an IR enhancement factor of $e^{-2/3}$
provided that the internal momentum
that runs through each boson propagator
is allowed to vanish {\it independently}.

If there is a kinematic constraint that keeps $k_d$
from becoming arbitrarily small in magnitude,
$k_d$ integration is convergent in the $e \rightarrow 0$ limit.
Then there is no IR enhancement factor.
However, one still has to worry about UV divergence
in the $e \rightarrow 0$ limit.
In particular, the integration over $\vec K$ can be UV divergent
without the self energy in the boson propagator.
In the presence of the self-energy,
quantum corrections to marginal operators
can have at most log divergences by power counting.
In the $e \rightarrow 0$ limit,
they can have power-law UV divergences
because the boson propagator no longer depends on $\vec K$.
The degree of UV divergence for marginal operators
is {\it at most}  $I_b$,
where $I_b$ is the number of internal boson propagators.
This is because only the boson propagator depends on $e$,
and each boson propagator carries
the scaling dimension $-1$.
In the presence of the boson self energy,
the power-law divergence is cut-off at a scale
$|\vec K| \sim e^{-2/(d-1)} k_d^{3/(d-1)}$.
In $d=5/2$, the UV divergence in the $e = 0$ limit
can introduce an enhancement factor of $e^{-4/3 I_b}$.
However, we emphasize that this is an upper bound for the
enhancement factor.
Typical diagrams have weaker UV divergence
in the $e \rightarrow 0$ limit due to kinematic constraints,
which results in a smaller enhancement factor.

In the presence of the IR and UV enhancement factors,
a $L$-loop diagram goes as
\bqa
e^{2L - Y I_b} = e^{ ( 2 - Y ) L - ( E_f/2 - 1) Y  }.
\label{e}
\end{eqnarray}
Here $2/3 \leq Y \leq 4/3$ is the average enhancement factor per boson propagator,
which is specific to each diagram.
The identity $I_b = L - 1 + E_f/2$ is used,
where $E_f$ is the number of external fermion lines.
From explicit calculations,
we will see that all diagrams up to two-loop level
have $Y=2/3$.
At the three-loop order, we will see an example where $Y=1$.
At present, we don't have any example with $Y>1$.
Up To the three-loop diagrams that we have checked,
all $L$-loop diagrams are suppressed by $e^{4/3 L}$,
compared to the bare action and the one-loop self energy of boson.
This suggests that
the actual average enhancement factor
may be strictly smaller than $4/3$.
Although we don't know the precise expansion parameter,
all $L$-loop diagrams are suppressed at least by
the factor of $e^{ 2/3 L}$,
and the loop expansion is controlled.

\subsection{Computation of counter terms}

In this section, we summarize the results of the counter terms
computed up to two loops.
Some three-loop diagrams are also computed.

\subsubsection{One-loop level}

\begin{figure}
\begin{center}
\includegraphics[height=4cm,width=8cm]{ferm1.eps}
\end{center}
\vspace{0.7cm}
\begin{center}
\includegraphics[height=4cm,width=8cm]{vert1.eps}
\end{center}
\caption{(a) The one-loop fermion self-energy.
(b) The one-loop vertex correction. }
\label{fig:fermvert}
\end{figure}

The one-loop self energy of boson has been
already taken into account in the dressed propagator, $D_1(k)$.
The one-loop fermion self energy shown in
Fig.~\ref{fig:fermvert} (a) is given by
\begin{equation}\label{fermiloop}
\Sigma_1 (q) = \frac{(ie)^2 \mu^{\epsilon}}{N} (d-1)
\int \frac{dk}{(2\pi)^{d+1}}
\gamma_{d-1} G_0 (q-k)\gamma_{d-1} D_1 (k),
\end{equation}
where
\begin{equation}\label{propbos}
D_1 (k) =\frac{1}{k_d^2 +\beta_{d} e^2 \mu^{\epsilon}
\displaystyle\frac{|\vec K|^{d-1}}{|k_d|} }.
\end{equation}
As is computed in Appendix~\ref{app:oneloopferm},
the resulting self energy is given by
\begin{equation}
\Sigma_1(q) =
\left(  - \frac{e^{4/3}}{N} \frac{u_1}{\epsilon} + \mbox{finite terms}  \right)
(i \vec \Gamma \cdot \vec Q )
\end{equation}
with
\bqa
u_1 = \frac{1}{ 2^{5/2} \pi^{3/4} \sqrt{3} \beta_{5/2}^{1/3} \Gamma(3/4) }
& \approx  & 0.08758634.
\end{eqnarray}
It is noted that
not only the UV divergent part but also the finite part in
$\Sigma_1(k)$ is proportional to $\vec \Gamma \cdot \vec Q$.
This fact simplifies the calculation at higher loops as will be discussed in the next section.
To cancel the UV divergence,
we only need the counter term of the form,
\begin{equation}\label{actct}
S_{CT}^{(1loop)}  =  \sum_{j} \int \frac{dk}{(2\pi)^{d+1}} \bar \Psi_j(k)\,
i A_1^{(1)}
(\vec \Gamma \cdot \vec K )
\, \Psi_{j}(k)
\end{equation}
with
\begin{equation}
A_1^{(1)}  = - \frac{e^{4/3}}{N} \frac{u_1}{\epsilon}.
\end{equation}

The absence of $\delta_k$ dependence in the fermion self energy
combined with the Ward identity in Eq. (\ref{WI})
implies that there is no vertex correction
at the one-loop.
This is explicitly checked in Appendix~\ref{app:oneloopvert}.

\subsubsection{Two-loop level}

\begin{figure}
\begin{center}
\includegraphics[width=15cm]{bos2.eps}
\end{center}
\caption{The diagrams for two-loop boson self energy.}
\label{fig:bos2}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=15cm]{ferm2.eps}
\end{center}
\caption{The diagrams for two-loop fermion self energy.}
\label{fig:ferm2}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=15cm]{vert2.eps}
\end{center}
\caption{The diagrams for two-loop vertex corrections.}
\label{fig:vert2}
\end{figure}

The two-loop diagrams are listed in
Figs.~\ref{fig:bos2} ,\ref{fig:ferm2} and~ \ref{fig:vert2}.
The black circles in
Figs. \ref{fig:bos2} (d)-(e), \ref{fig:ferm2}(c) and
\ref{fig:vert2}(i)-(j) denote the one-loop
counter term for the fermion self energy,
\begin{equation}\label{2legv}
i A_1^{(1)} \bar \Psi (\vec \Gamma \cdot \vec K ) \Psi.
\end{equation}
To examine which diagrams can give non-zero contributions,
we first note that the fermion self-energy of the form
\begin{equation}\label{eliashferm}
\Sigma (k) =-i \xi (K) [\vec \Gamma \cdot \vec K]
\end{equation}
with $K=|\vec K|$
solves the Eliashberg equations for the bosonic and
fermionic self-energies.
If one uses
the dressed fermionic propagators
\begin{equation}\label{fullferp}
G(k) = [G_0^{-1} (k) -\Sigma (k)]^{-1}
\end{equation}
in lieu of the bare one,
one obtains the same self energies, $\Sigma_1 (q)$ and $\Pi_1 (q)$
which are obtained by using $G_0$.
This can be understood from the fact
that the dependence on $\xi (K)$ drops out
in Eqs.~(\ref{fermiloop}) and (\ref{bosloop})
once $k_{d-1}$ and $k_d$ are integrated out.
We also note that
the full one-loop fermion self-energy in Eq.~(\ref{sigma3})
has the form of Eq.~(\ref{eliashferm}).
As a result, the diagrams
in Figs. \ref{fig:bos2}(b), (c) and \ref{fig:ferm2}(b)
vanish
because they can be obtained by expanding the dressed propagators
in powers of $\Sigma_1 (q)$ in the corresponding expressions for
the one-loop diagrams.
Since the one-loop counter term is also of the same form,
the diagrams in Figs. \ref{fig:bos2}(d)-(e) and
\ref{fig:ferm2}(c) vanish as well.
This feature can be checked by explicit computation.
We thus conclude that the only two-loop diagrams
that need to be computed
for the self-energies are those in
Figs.~\ref{fig:bos2}(a) and \ref{fig:ferm2}(a).
The vertex correction can be obtained from the Ward identity.

The diagram in Fig. \ref{fig:bos2}(a) is computed in Appendix~\ref{app:twoloopbos}.
Although it is UV finite,
it renormalizes $\beta_d$ in the boson propagator
by a finite amount, $\beta_d^{6a}  \sim O(e^{4/3}/N)$.
Once this correction is fed back to the one-loop fermion self energy in Eq. (\ref{fermiloop}),
we obtain a correction to the UV-divergent fermion self energy,
\bqa
\Sigma^{6a}_2(k) &=& -\frac{\beta_d^{6a} }{3 \beta_d} \Sigma_1(k) \nonumber \\
&=& \left(  - \frac{e^{8/3}}{N^2} \frac{u_2^{\rq{}}}{\epsilon} + \mbox{finite terms}  \right)
(i \vec \Gamma \cdot \vec K ),
\end{eqnarray}
 where
 \bqa
u_2^{\rq{}}  \approx 0.0016449.
\end{eqnarray}

The two-loop fermion self-energy
in Fig. \ref{fig:ferm2}(a)
is given by
\begin{equation}\label{twoloopf}
\Sigma^{7a}_2 (q) = \frac{(ie)^4 \mu^{2\epsilon}}{N^2} (d-1)^2
\int \frac{dp dl}{(2\pi)^{2d+2}} D_1 (p) D_1 (l)
\gamma_{d-1} G_0 (p+q)\gamma_{d-1} G_0 (p+l+q)\gamma_{d-1} G_0 (l+q)\gamma_{d-1}.
\end{equation}
The computation described in Appendix~\ref{app:twoloopferm}
results in
\begin{equation}\label{twoloopf2}
\Sigma_2^{7a} (q) =
-  \frac{ e^{8/3} }{N^2} u_2
 ( i \vec \Gamma \cdot \vec K )
-  \frac{ e^{8/3} }{N^2} v_2  ( i \gamma_{d-1} \delta_k )
+ \mbox{ finite terms},
\end{equation}
where
\bqa
u_2 &  \approx & - 0.0194218 \nonumber \\
v_2 &  \approx & 0.000867775.
\end{eqnarray}

From the Ward identity, one has to include the vertex correction
at the two-loop level.
The counter terms that are necessary
to cancel the UV divergence at the two-loop level is given by
\bqa\label{actct2}
S_{CT}^{(2loop)}  &=&  \sum_{j} \int \frac{dk}{(2\pi)^{d+1}} \bar \Psi_j(k)\,
[i A_1^{(2)}( \vec \Gamma \cdot \vec K )+ i A_2^{(2)}
\gamma_{d-1} \delta_k ]\, \Psi_{j}(k) +\nonumber \\
 &+&    A_2^{(2)} \frac{ie \mu^{\epsilon/2} }{\sqrt{N}}\sqrt{d-1} \sum_{j}
\int \frac{dk dq}{(2\pi)^{2d+2}}
\phi(q) \bar \Psi_{j}(k+q) \gamma_{d-1} \Psi_{j}(k),
\end{eqnarray}
where
\bqa\label{Z1and2}
A_{1}^{(2)} & = &  -  \frac{ e^{8/3} }{N^2} ( u_2 + u_2^{\rq{}} ), \nonumber \\
A_{2}^{(2)} & = &  -  \frac{ e^{8/3} }{N^2} v_2.
\end{eqnarray}

\subsubsection{Three-loop Aslamazov-Larkin-type
contribution to boson self-energy}

\begin{figure}
\begin{center}
\includegraphics[width=11cm]{ALbos.eps}
\end{center}
\caption{Aslamazov-Larkin-type contributions to boson self-energy.
Diagrams (a) and (b) correspond to the particle-particle and
particle-hole channels respectively.}
\label{fig:ALbos}
\end{figure}

The number of diagrams increases dramatically at higher loops.
This makes it hard to go beyond the two-loop level systematically.
It is of interest, however, to consider some three-loop diagrams that
can potentially contribute to  anomalous dimension of boson through
a non-trivial correction to $Z_3$, given that $Z_3=1$ up to the two-loop order.
For this, we consider the  Aslamazov-Larkin-type diagrams shown in Fig.~\ref{fig:ALbos}
which is the lowest known diagrams that renormalize
the boson kinetic term.
Metlitski and Sachdev evaluated these diagrams in
Ref.~\cite{metlsach1},
and showed that they introduce a finite
renormalization to the kinetic energy of bosons,
which violates the genus expansion in the two-patch theory.
A finite quantum correction to the kinetic energy is also found by Mross et. al
in Ref. \cite{mross}.

To extract the term that renormalizes the $q_d^2$ term in the boson action,
we compute the diagrams at $\vec Q =0$.
Details of computation are presented in Appendix~\ref{app:ALbos}.
The final result can be written as
\bqa\label{ALres}
\Pi_{AL} (q_d) & = & q_d^2 \cdot \frac{e^6}{N}
\left( \frac{\mu}{q_d^2} \right)^{3\epsilon}
(d-1)^{3d/2} \int \frac{d\vec P d\vec K d\vec L}{(2\pi)^{3(d-1)}}
\frac{{\cal J}_d (L)}
{\{(P+ |\vec P +\vec L| +K +|\vec K +\vec L|)^2 +1\} } \nonumber \\
&\cdot&
\frac{\left( [\vec K \cdot (\vec K +\vec L)] -K\, |\vec K +\vec L| \right)\,
\left( [\vec P \cdot (\vec P +\vec L)] -P\, |\vec P +\vec L|
\right)}
{2 P K\,|\vec P +\vec L|\,|\vec K +\vec L|
\left[ P+ |\vec P +\vec L| +K +|\vec K +\vec L| \right]},
\end{eqnarray}
where
\bqa
{\cal J}_d (L)  &=& \int_0^1 \frac{dl_d}{2\pi}
\frac{2^{3d-6} \, [l_d (1-l_d)]^{2(d-1)}}{ l_d^{4-d} +\beta_d e^2
\left(\mu/q_d^2\right)^{\epsilon}\, [2\sqrt{d-1} (1-l_d) L]^{d-1} }\nonumber \\
&\cdot& \frac{1}{(1-l_d)^{4-d} +\beta_d e^2
\left( \mu/q_d^2\right)^{\epsilon}\, [2\sqrt{d-1} l_d L]^{d-1}}.
\end{eqnarray}
Here $\vec P, \vec K, \vec L$ have been rescaled to be dimensionless
in the unit of $q_d^2$.
We see that
$\Pi_{AL} (q_d) \propto q_d^2$
in the  $\epsilon\rightarrow 0$  limit.
One can also check that the coefficient of the $k_d^2$ term
is finite at $d=5/2$.
To see this,
we introduce a $9/2$-dimensional vector $\vec X =(\vec L, \vec P, \vec K)$.
Since ${\cal J}_{5/2}(L)$ decays as $1/L^3$
in the $\vec L \rightarrow \infty$  limit,
Eq.~(\ref{ALres}) behaves as $\int dX /X^{5/2}$ at large momenta,
which is UV convergent.
Infrared convergence is explicit as well.
To estimate the dependence on $e$,
we  note that ${\cal J}_{5/2} (L)$ has a non-trivial dependence on $e$,
and behaves differently depending on whether $L$ is large or small compared to
$L_* =1/e^{4/3} \gg 1$ (in the unit of $q_d^2$) :
\bqa\label{jint1}
{\cal J}_{5/2} (L) \approx \left\{ \begin{array}{cl}
C_1 ,& \quad L \ll L_* \\
\displaystyle\frac{C_2}{e^4 L^3}, & \quad L \gg L_* , \end{array} \right.
\end{eqnarray}
where $C_1$ and $C_2$ are constants
which are independent of $e$.
It is not an easy to perform the integration over $\vec P$, $\vec K$ explicitly.
However, based on the scaling arguments,
one can write that
\begin{equation}\label{alint1}
\Pi_{AL} (q_d) =\frac{e^6}{N} \cdot q_d^2 \, \int dL ~ L^{1/2} {\cal F} (L)
{\cal J}_{5/2} (L),
\end{equation}
where ${\cal F} (L)$ can be considered approximately constant when $L >> 1$.
Breaking the integration into the regions $0<L<L_*$ and $L_*<L<\infty$,
and taking into account the asymptotics given by Eq.~(\ref{jint1}),
we obtain
\begin{equation}
\Pi_{AL} (q_d) = C_3 \frac{e^4}{N} \cdot q_d^2.
\end{equation}
$C_3$ is a numerical constant independent of $e$.
Thus the Aslamazov-Larkin diagrams contribute
 a finite renormalization to the boson kinetic term.
Therefore, we still have $Z_3=1$.
It is an outstanding question
whether $Z_3$ ever receives a non-trivial quantum correction,
and, if so, at which order the first quantum correction appears.

\subsection{Critical exponents}

Collecting all the results,
the counter terms up to the two-loop level are given by
\bqa
Z_{1,1} & = &  - \frac{ e^{4/3} }{N} u_1 -  \frac{ e^{8/3} }{N^2} ( u_2 + u_2^{'} ), \nonumber \\
Z_{2,1} & = &  -  \frac{ e^{8/3} }{N^2} v_2, \nonumber \\
Z_{3,1} & = & 0, \nonumber \\
\end{eqnarray}
where
\bqa\label{uvnum}
u_1 & \approx  & 0.08758634, \nonumber \\
u_2 & \approx &  - 0. 0194218, \nonumber \\
u_2^{'} & \approx &   0.0016449, \nonumber \\
v_2 & \approx  & 0.000867775.
\end{eqnarray}
The beta function becomes
\bqa
\beta & = & -\frac{\epsilon}{2} e
+ 0.02920 \left( \frac{3}{2} - \epsilon \right) \frac{e^{7/3}}{N}
- 0.01073 \left( \frac{3}{2} - \epsilon \right)  \frac{e^{11/3}}{N^2}
\end{eqnarray}
which has a stable interacting fixed point at
\bqa
\frac{ e^{* 4/3} }{N} & = & 11.417 \epsilon + 55.498 \epsilon^2.
\end{eqnarray}
Therefore we conclude that the theory flows to a
stable non-Fermi liquid state in the low energy limit.
To the two-loop order, the dynamical critical exponent and
the anomalous dimensions at the critical point are given by
\bqa
z & = & \frac{3}{3 - 2 \epsilon},  \label{zcal2} \\
\eta_\psi & = & -\frac{\epsilon}{2} + 0.07541 \epsilon^2, \label{psical2} \\
\eta_\phi & = & -\frac{\epsilon}{2}, \label{phical2}
\end{eqnarray}
and the propagators are given by
\bqa
D(k) & = & \frac{1}{  k_d^2 }
~~ f \left( \frac{| \vec K |^{1/z} }{k_d^2} \right), \label{Dk} \\
G(k) & = & \frac{1}{ | \delta_k |^{1-0.1508 \epsilon^2 } }
~~ g \left( \frac{ | \vec K |^{1/z} }{\delta_k} \right). \label{Gk}
\end{eqnarray}
It is noted that the contribution of the dynamical critical exponent
to the anomalous dimensions of the fields,
that is the first term in Eqs. (\ref{psical}) and (\ref{phical}),
drops out in the two-point functions because of the cancellation
with the dynamical critical exponent in the delta function
which enforces the energy-momentum conservation
in the Green's functions.
However, the contribution of the dynamical critical exponent
shows up in higher point functions.

The upper bound on the enhancement factor discussed in Sec. III D
suggests that there can be, in principle,
quantum corrections of the order of
$e^{2} \sim \epsilon^{3/2}$ at the three-loop order,
which is larger than the corrections at the two-loop order.
However, this does not mean that the expansion is uncontrolled.
If $L$-loop corrections are indeed suppressed only by
$e^{2/3L} \sim \epsilon^{L/2}$,
one has to compute up to $2n$-loop level in order
to compute critical exponents to the order of $\epsilon^n$.

\section{Physical Properties}

\subsection{Thermodynamic quantities }

Observables that are local in momentum space,
such as the self energy of a fermion near the Fermi surface
and scattering amplitudes with small momentum exchange,
are insensitive to other modes which are separated in the momentum space.
This is due to the emergent locality in the momentum space\cite{polchinski,LEE2008},
which makes the patch description valid in non-Fermi liquid states.
Therefore  temperature dependences of the quantities that are local in momentum space
are solely dictated by their scaling dimensions.

The scaling of thermodynamic quantities are different
from those observables that are local in momentum space.
This is because all low energy modes near the Fermi surface contribute
to the thermodynamic responses.
In order to examine the scaling behavior of thermodynamic quantities,
we consider the free energy density at finite temperature.
The scaling dimension of the free energy density
is set by the dimension of  spacetime, $(d-1)z + 1 + 1/2$.
If the free energy was insensitive to all UV cut-off scales,
one would have the form of $f(T) \sim T^{ (d-1) + \frac{3}{2z}}$.
However, this is not the case in theories with Fermi surface
because the free energy is a global quantity
which depends on all low energy modes around the Fermi surface.
Since low energy effective theory is local in momentum space,
the singular part of the free energy linearly depends on the size of the Fermi surface\cite{LEE2008}, which then leads to a violation of hyperscaling.
In our local patch description,
the size of the Fermi surface is set by the largest
momentum $\Lambda$ along the $k_d$ direction.
Because $\Lambda$ has scaling dimension $1/2$,
the free energy density should scale
as $f(T) = \Lambda T^{ (d-1) + \frac{1}{z}}$.

Let us also consider an external field $h^\alpha$
which sources the flavor quantum number given by
\bqa
\rho^\alpha & = &
\psi_{+,i}^\dagger T^\alpha_{ij} \psi_{+,j}
+
\psi_{-,i}^\dagger T^\alpha_{ij} \psi_{-,j}.
\label{rho}
\end{eqnarray}
Note that $\rho^\alpha$ is the physical flavor quantum number
under which $\psi_{+,i}$ and $\psi_{-,j}$
transform in the same manner.
Although all components of $\rho^\alpha$ are conserved in $d=2$,
only parts of them are conserved in $d>2$
due to the absence of the axial flavor symmetry.
In the example of $d=3$ with $N=2$,
this can be easily understood from the fact that
the spin triplet pairing leaves
only $\sigma^y$ as a conserved flavor
among $T^\alpha = \{ I, \sigma_x, \sigma_y, \sigma_z \}$.
For general $N$, only the flavor density with anti-symmetric $T^\alpha$
is related to the conserved charge density,
\bqa
\rho_a^\alpha = j_{A0}^\alpha, \mbox{~~ for $(T^\alpha)^T = -T^\alpha$}.
\end{eqnarray}
The symmetric flavor density $\rho_s^\alpha$ with $(T^\alpha)^T = T$  is not a conserved density.
Although $\rho_a^\alpha$ is not a density of a conserved current,
it is related to the $(d-1)$-th component of the  conserved current
\bqa
\rho_s^\alpha = j_{A, d-1}^\alpha,
\end{eqnarray}
where $j_{A,d-1} = i \bar \Psi_i T^\alpha_{ij} \gamma_{d-1} \Psi$.
Because $\rho^\alpha_a$ and $\rho^\alpha_s$ are parts of different components of the conserved current, the fields that couples to them have different scaling dimensions,
\bqa
[ h_a^\alpha] =  z, ~~ [h_s^\alpha] = 1.
\end{eqnarray}

From the above considerations, we write the scaling form of the free energy density as
\bqa
f(T,h_s,h_a) = \Lambda T^{ (d-1) + \frac{1}{z} }
\tilde f \left( h_s T^{-1/z}, h_a T^{-1} \right).
\end{eqnarray}
This leads to the scaling behavior of the specific heat
and the flavor susceptibility,
\bqa
c & \sim & T^{(d-2)+\frac{1}{z}}, \label{C} \\
\chi_{ss} & \sim & T^{(d-1)-\frac{1}{z}}, \label{chi++} \\
\chi_{aa} & \sim & T^{(d-3)+\frac{1}{z}}.\label{chi--}
\end{eqnarray}
Note  that the flavor susceptibility is anisotropic
because of the absence of the full flavor symmetry.
Nonetheless, their scaling dimensions
are completely set by the dynamical critical exponent
because they are parts of the conserved current.
In $d=2$, Eqs. (\ref{C}) and (\ref{chi--}) are consistent
with the results obtained for
the specific heat and the susceptibility of conserved spin
in Ref. \cite{Senthil2008}.
For $d>2$, the low temperature response functions are suppressed by
a higher powers of temperature because of the suppression
of density of state with a larger co-dimension of Fermi surface.

\subsection{$2k_F$ scattering}

In order to examine how the back-scattering is affected by interaction
in the non-Fermi liquid state, we add an operator which carries
momentum $2 k_F$,
\bqa\label{scat2k1}
S_{2k_F} & = & -2 \mu r \sum_j \int \frac{dk}{(2\pi)^{d+1}} (
(\psi_{+,j}^\dagger (k) \psi_{-,j} (k) + \psi_{-,j}^\dagger (k) \psi_{+,j} (k) ),
\end{eqnarray}
where $r$ is the source.
In the spinor representation,
Eq.~(\ref{scat2k1}) can be written as
\bqa
S_{2k_F} & = & i \mu r \int \frac{dk}{(2\pi)^{d+1}} (
\Psi^T (k) \gamma_0 \Psi (-k) + \bar{\Psi} (k) \gamma_0 \bar{\Psi}^T (-k) ).
\end{eqnarray}
To cancel UV divergences,
we need to add a counter term of the same form,
\bqa
S_{2k_F}^{CT} & = & i \mu r (Z_r - 1) \int \frac{dk}{(2\pi)^{d+1}}
( \Psi^T (k) \gamma_0 \Psi (-k) + \bar{\Psi} (k) \gamma_0 \bar{\Psi}^T (-k) ),
\end{eqnarray}
which renormalizes the insertion into
\bqa
S_{2k_F}^{ren} & = & i r_B \int \frac{dk_B}{(2\pi)^{d+1}} (
\Psi_B^T (k) \gamma_0 \Psi_B (-k) + \bar{\Psi}_B (k) \gamma_0 \bar{\Psi}_B^T (-k) ),
\end{eqnarray}
where $r_B = \mu Z_r Z_\psi^{-1}  \left( \frac{Z_2}{Z_1} \right)^{(d-1)} r$
with $Z_r = 1 + Z_{r,1}/\epsilon + ...$.
The beta function of $r$ is given by
\bqa
\beta_r & = & - ( 1 - \gamma_r ) r,
\end{eqnarray}
where $\gamma_r  =  \frac{e}{2}z ( Z_{r,1}^{'} - Z_{2,1}^{'} )$ is
the anomalous dimension of the operator. We can easily calculate $Z_{r,1}$ at
the one-loop level. The diagrams that renormalize $r$ are shown in
Fig.~\ref{fig:scatt2k}.
\begin{figure}
\begin{center}
        \includegraphics[height=4cm,width=14cm]{scatt2k.eps}
\end{center}
\caption{The one-loop diagrams that renormalize the $2k_F$ scattering
amplitude $r$.
}\label{fig:scatt2k}
\end{figure}
Calculations are done in Appendix~\ref{app:oneloop2kscatt},
where it is shown that
\begin{equation}
Z_r = 1+ \frac{e^{4/3}}{N} \frac{u_r}{\epsilon},
\end{equation}
with the positive value of $u_r$ given by Eq.~(\ref{ur1}).
From this we obtain the anomalous dimension,
\bqa
\gamma_r = 2 \epsilon + 11.2059 \epsilon^2.
\end{eqnarray}
As expected, the quantum correction suppresses the $2k_F$ scattering at low energies.

\section{Conclusion}

In summary, we develop a dimensional regularization scheme
where Fermi surface of dimension one
is embedded in general dimensions
by combining low energy fermionic excitations on opposite patches
of Fermi surface into a Dirac fermion.
When Fermi surface is coupled with a critical boson whose momentum
is centered at zero,
the Yugawa coupling becomes marginal at a critical space dimension $d_c=5/2$.
Using $\epsilon = 5/2 - d$ as a perturbative parameter,
we show that the Ising-nematic phase transition
is described by a stable non-Fermi liquid fixed point
near the critical dimension.
Critical exponents and
temperature dependences of thermodynamic quantities
are computed to the two-loop order.

The dimensional regulairzation scheme is complimentary
to other expansion schemes\cite{nayak,mross}.
The pro of the dimensional regularization scheme
is that the locality is maintained in the regularization.
The con is that some symmetry is broken by regularization.
In the expansion scheme based on dynamical modification,
one has to give up some locality in the action,
but the original symmetry can be easily kept.
Despite the difference in the approach,
both schemes provide similar conclusions
regarding the existence of stable non-Fermi liquid fixed points
in the perturbative limit and the absence of
anomalous dimension of boson up to the three-loop order.
In the dimensional regularization scheme,
there is a room for the boson to acquire
a non-trivial anomalous dimension
through a renormalization of the kinetic  term
because all operators in the local action
can in principle receive quantum corrections
unless protected by a symmetry.
It is an open question at which order
the anomalous dimension first shows up.

The dimensional regularization scheme
may be applied to different systems.
However, the direct application of this scheme
to quantum electrodynamics at finite density
is subtle because of the fact that the superconducting order
introduced by the dimensional regularization scheme gaps out
the gauge field.
It will be interesting to find an alternative scheme where
the global $U(1)$ symmetry is preserved by regularization.
For quantum critical points
associated with spin/charge density wave\cite{metlsach2},
the critical dimension turns out to be $d_c=3$
in the dimensional regularization scheme.
In this case, one does not need to break
the global $U(1)$ or the spin rotational symmetry
because one can linearize the dispersion
of fermions near the hot spots\cite{shouvik2}.

\section{Acknowledgment}

We would like to thank
Grigory Bednik,
Matthew Fisher,
Patrick Lee,
Sri Raghu,
Subir Sachdev
and
T. Senthil
for useful comments and discussions.
The research of SSL was supported in part by
the Natural Sciences and Engineering Research Council of Canada,
the Early Research Award from the Ontario Ministry of Research and Innovation
and the Templeton Foundation.
Research at the Perimeter Institute is supported
in part by the Government of Canada
through Industry Canada,
and by the Province of Ontario through the
Ministry of Research and Information.

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