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高考英语气候变化的全球影响与应对单选题30题1.Climate change refers to long-term shifts in temperature and weather patterns. What is one major cause of climate change?A.Natural disastersB.Human activitiesC.Astronomical eventsD.Animal behaviors答案：B。
本题主要考查对气候变化原因的理解。
选项A 自然灾害不是气候变化的主要原因；选项C 天文事件对气候变化影响较小；选项D 动物行为对气候变化影响微乎其微。
而人类活动，如燃烧化石燃料、砍伐森林等，是导致气候变化的主要原因之一。
2.Which of the following is a sign of climate change?A.Frequent rainstormsB.Clear skiesC.Calm windsD.Warm winters only答案：A。
气候变化的表现包括极端天气增多，频繁的暴雨是其中之一。
选项B 晴朗的天空不是气候变化的标志；选项C 平静的风也不是；选项D 仅仅是暖冬不能全面代表气候变化，气候变化有多种表现。
3.What does the term “greenhouse gases” refer to?A.Gases that cool the planetB.Gases that are only produced by natureC.Gases that trap heat in the atmosphereD.Gases that are harmless to the environment答案：C。
“greenhouse gases”指的是温室气体，能在大气中捕获热量。
选项A 温室气体是使地球升温而非降温；选项B 温室气体也有人为产生的；选项D 温室气体对环境并非无害。
临海的变化六级英语作文## The Shifting Sands of Coastal Landscapes The rhythmic ebb and flow of the tide against the shore has sculpted coastlines for millennia, creating diverse and dynamic ecosystems. Yet, these landscapes, once considered immutable features of our planet, are undergoing a period of unprecedented change. Human activity and the escalating climate crisis are acting as powerful forces, reshaping coastlines at an alarming rate. This essay delves into the multifaceted nature of these changes, exploring their causes, consequences, and potential solutions. One of the most visible impacts on coastal areas is the **(1) rising tide** (meaning: increasing amount) of sea levels. As global temperatures climb, glaciers and ice sheets melt, adding vast quantities of water to the oceans. This, coupled with the thermal expansion of seawater, is causing sea levels to rise at an accelerating pace. The consequences are dire for low-lying coastal communities, which face increased risks of flooding, erosion, and saltwater intrusion into freshwater sources. Island nations, in particular, are at the **(2) mercy** (meaning: under the control) of this rising tide, with their very existence threatened by the encroaching sea. Human activities further exacerbate the pressures on coastal environments. Unplanned and unsustainable development along coastlines destroys natural habitats like mangroves and sand dunes, which act as buffers against storms and erosion. The construction of seawalls and other hard engineering structures may offer temporary protection, but often disrupts natural sediment transport processes, leading to unintended consequences elsewhere. Additionally, pollution from industrial and agricultural sources degrades water quality, harming marine life and disrupting delicate ecological balances. The economic ramifications of these changes are far-reaching. Coastal communities reliant on tourism and fisheries face declining incomes as beaches erode and fish stocks dwindle. The costs of adapting to rising sea levels and mitigating the impacts of coastal erosion are substantial, placing a significant burden on governments and local communities. Moreover, the loss of coastal ecosystems, which provide valuable services such as carbon sequestration and storm protection, has wider implications for global climate regulation and human well-being. Despite the challenges, there is a growing recognition of the need for sustainable coastalmanagement practices. Implementing nature-based solutions, such as restoring wetlands and planting mangroves, can help to protect coastlines while enhancing biodiversity. Embracing renewable energy sources and reducing greenhouse gas emissions are crucial to mitigating climate change and slowing the rate of sea-level rise. Additionally, investing in early warning systems and disaster preparedness measures can help communities adapt to the changing coastal environment and build resilience. Ultimately, protecting our coastlines requires a **(3) paradigm shift** (meaning: a fundamental change in approach) in how we interact with these dynamic environments. We must move away from short-term, exploitative practices and embrace a more holistic and sustainable approach that recognizes the interconnectedness of human activities and the natural world. By working together, we can ensure that future generations can continue to enjoy the beauty and bounty of our coastlines.。
光照影响大型海藻英文英文回答：Light is a primary factor influencing the distribution and productivity of large seaweeds. These large algae, including kelps, laminaria, and fucus, are essential components of coastal ecosystems, providing food and habitat for a wide range of marine organisms. The availability and quality of light can significantly impact their growth, reproduction, and overall health.Light Intensity and Quality:Large seaweeds require sufficient light intensity for photosynthesis, the process by which they convert sunlight into energy. The optimal light intensity for different species varies, but most prefer well-lit environments. However, excessive light intensity can lead to photoinhibition, where the photosynthetic apparatus is damaged by high levels of radiation.Light quality, primarily determined by the wavelength, also influences seaweed growth. Certain wavelengths, such as blue and red light, are more effectively absorbed by pigments involved in photosynthesis than others. The spectral composition of light thus affects the photosynthetic efficiency and growth rate of seaweeds.Water Column Properties:The underwater light environment is influenced by several water column properties that affect light availability and penetration. These include:Water depth: As light passes through water, it is absorbed and scattered, resulting in a decrease in intensity with increasing depth.Turbidity: Suspended particles in the water, such as sediment and phytoplankton, can scatter and absorb light, further reducing its availability for seaweeds.Wave motion: Waves can cause the water column to move, resulting in variable light conditions for seaweeds.Seaweed Adaptations:Large seaweeds have evolved various adaptations to optimize their light utilization:Leaf morphology: Seaweeds often have broad, flat leaves that maximize light absorption.Pigmentation: They possess a range of photosynthetic pigments that absorb different wavelengths of light.Physiology: Seaweeds can adjust their photosynthetic rates and pigment composition based on light availability.Light and Algal Distribution:The distribution of large seaweeds is strongly influenced by light availability. They are typically found in shallow waters where light penetration is sufficient forphotosynthesis. In deeper waters or regions with high turbidity, seaweed abundance and diversity decrease.Climate Change Impacts:Climate change is projected to alter light availability and quality in coastal waters due to factors such as sea level rise, increased wave energy, and changes in water clarity. These changes could potentially affect the growth, reproduction, and distribution of large seaweeds.中文回答：光照对大型海藻的影响。
海洋对英国历史的影响英文版The ocean has a far-reaching impact on British history. It can be said that without the ocean, there would be no current British history. First of all, during the ice age, the ice cover covered the whole British Isles. As a result, the native animals and plants in Britain almost disappeared. Except for the red grouse, other animal and plant species are no different from those in the European continent. In ancient times, due to the influence of the North Atlantic warm current. The climate of Britain belongs to temperate marine climate. It is mild throughout the year, with no severe cold in winter and no intense heat in summer.In the eyes of people living in the "bitter and cold land" at that time, Britain was a treasure land to the letter. In addition, the south of Britain is a fertile plain, suitable for living. To the north are high mountains. This "Pearl" has attracted one nation after another to colonize, and the continuous collision and integration between various nations are developing. The modern British nation gradually formed. The nature of British marine civilization has brought her the spirit of daring to take risks and being proactive. The nature of an island country also prompted Britain to develop a prosperous maritime transport and a strong navy. Although Britain lost the hundred year war, it lost all its territory in France. But this also made her no longer care about mainland affairs, but look at the sea. Through the maritime triangle trade, Britain's industry and Commerce continued to develop, defeated the Spanish Armada, and established colonies in India, North America and other places, reflecting the strength of Britain in those days.Finally, the Empire was born.。
受海平面上升的影响英语作文The Impact of Sea Level Rise.Sea level rise is one of the most significant consequences of climate change, and its impact is felt across the globe. As the planet warms, the polar ice caps melt, adding vast amounts of water to the oceans. This rise in sea level has the potential to reshape coastlines, inundate low-lying areas, and disrupt ecosystems in profound ways. In this article, we will explore the various impacts of sea level rise and consider the measures that can be taken to mitigate its effects.One of the most immediate impacts of sea level rise is the inundation of coastal cities and towns. As the sealevel rises, it claims more and more land, eventually rendering some areas uninhabitable. This is particularly problematic in densely populated areas, where the loss of land can lead to displacement and social upheaval. For example, in Bangladesh, a country that is already highlyvulnerable to flooding, sea level rise could submergeentire islands and coastal communities, displacing millions of people.In addition to displacing people, sea level rise also poses a significant threat to infrastructure. Ports, roads, and bridges are all vulnerable to inundation, and their destruction can have far-reaching economic consequences. For instance, the Port of Miami is one of the busiest ports in the United States, handling billions of dollars worth of goods each year. However, sea level rise threatens to inundate the port, potentially disrupting global trade and causing significant economic losses.Ecosystems are also profoundly affected by sea level rise. Coastal wetlands, mangroves, and coral reefs act as natural buffers against storm surges and flooding. However, as the sea level rises, these ecosystems are inundated, reducing their ability to perform these functions. This not only has immediate consequences for coastal communities but also has longer-term impacts on global climate patterns and biodiversity.The impact of sea level rise is not evenly distributed across the globe. Developing countries and island nations are particularly vulnerable, as they often lack the resources to adapt to these changes. This can lead to a vicious cycle of poverty and environmental degradation, as communities are unable to invest in resilience measures due to limited financial resources.In response to these challenges, it is crucial to take action to mitigate the effects of sea level rise. One effective measure is to reduce greenhouse gas emissions, which are the primary driver of climate change. By transitioning to clean energy sources and implementing energy-efficient policies, we can slow the rate of sealevel rise and reduce its impact.In addition, investment in resilience infrastructure is crucial. This includes building sea walls, floodgates, and other barriers to protect coastal communities from inundation. It also involves investing in sustainable drainage systems and green infrastructure that can absorband retain water, reducing the risk of flooding.However, it is important to note that resilience infrastructure alone is not a silver bullet. These measures can be expensive and difficult to implement, particularlyin developing countries. Therefore, it is also necessary to focus on adaptive measures that help communities adapt to the new realities of sea level rise. This could include relocating vulnerable communities to safer areas, developing flood-resilient housing, and improving early warning systems for flooding and storm surges.In conclusion, sea level rise is a significant and growing threat that requires urgent action. It poses a profound challenge to coastal communities, infrastructure, and ecosystems around the world. By reducing greenhouse gas emissions, investing in resilience infrastructure, and promoting adaptive measures, we can mitigate the impact of sea level rise and protect the future of coastal communities and the planet at large.。
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Succession, Climax, and EcosystemsIn the late nineteenth century, ecology began to grow into an independent science from its roots in natural history and plant geography. The emphasis of this new "community ecology" was on the composition and structure of communities consisting of different species. In the early twentieth century, the American ecologist Frederic Clements pointed out that a succession of plant communities would develop after a disturbance such as a volcanic eruption, heavy flood, or forest fire. An abandoned field, for instance, will be invaded successively by herbaceous plants (plants with little or no woody tissue, shrubs, and trees, eventually becoming a forest. Light-loving species are always among the first invaders, while shade-tolerant species appear later in the succession.Clements and other early ecologists saw almost lawlike regularity in the order of succession, but that has not been substantiated. A general trend can be recognized, but the details are usually unpredictable. Succession is influenced by many factors: the nature of the soil, exposure to sun and wind, regularity of precipitation, chance colonizations, and many other random processes.The final stage of a succession, called the climax by Clements and early ecologists, is likewise not predictable or of uniform composition. There is usually a good deal of turnover in species composition, even in a mature community. The nature of the climax is influenced by the same factors that influenced succession. Nevertheless, mature natural environments are usually in equilibrium. They change relatively little through time unless the environment itself changes.For Clements, the climax was a "superorganism," an organic entity. Even some authors who accepted the climax concept rejected Clements' characterization of it as a superorganism, and it is indeed a misleading metaphor. An ant colony may be legitimately called a superorganism because its communication system is so highly organized that the colony always works as a whole and appropriately according to the circumstances. But there is no evidence for such an interacting communicative network in a climax plant formation. Many authors prefer the term "association" to the term "community" in order to stress the looseness of the interaction.上述内容就是小编为大家准备的托福阅读模考软件TPO19文本+解析的部分内容,但是仅仅只是部分内容,大家可以点击下载按钮进行免费的下载,希望本资料对大家的备考有帮助。
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We already know that the underwater world is anything but silent. It is full of natural sounds produced by animals and by the motion of the atmosphere and water. But today's researchers are increasingly concerned about the man-made noises under the water. They are confusing-and even killing-fish,whales and other marine animals.Although hearing is important to all animals, the special qualities of the undersea world emphasize the use of sound. Underwater sound allows marine animals to gather information and communicate at great distances and from all directions. Loud underwater noises can cause damage to their hearing. And nowadays it's becoming too much for marine wildlife. Evidence shows that whales will swim hundreds of miles, rapidly change their depth, and even beach themselves to get away from the sounds of sonar.You might think lower noises, typically caused by shipping or construction, do not pose a threat to marine animals. In fact, while they may not kill the animals directly, they can disturb their ability to locate food, find mates and avoid predators(掠食动物).Scientists looked at the effects of noise from small motorboats on coral-reef fish at the Great Barrier Reef in Australia. In the lab, the fish had been trained to recognize the smell of three common predators as a threat. Some fish were trained in the presence of boat noise, while the others had training with the natural sounds of the ocean. It turned out that the first group showed no fright reactions when exposed to the predator. However, the other group was scared. And under the influence of underwater noise on this group，only 20 percent of fish survived, compared to almost 70 percent of unexposed fish.Humans rely on the ocean for its natural resources and biodiversity. The ocean also plays an important role in regulating(调节)temperatures. As Cousteau said，"For most of history, man has had to fight nature to survive; in this century he is beginning to realize that, in order to survive, he must protect it.整个课堂设计以三个主题展开：分析文本体裁；解构概要写作步骤；总结概要写作原则。
城市化对地理环境的影响英语作文Urbanization, the process of population concentration in cities, has profound impacts on the geographical environment. As cities expand and develop, they alter landscapes, ecosystems, and natural resources. This essay explores the various ways in which urbanization influences the geographical environment.One of the most evident impacts of urbanization is the transformation of land use patterns. As cities grow, they consume vast amounts of land for infrastructure, residential areas, and industrial zones. This leads to the conversion of natural habitats such as forests, wetlands, and agricultural land into built-up areas. Consequently, biodiversity is often reduced, and ecosystems are fragmented, affecting the overall balance of the environment.Moreover, urbanization affects water resources and hydrological systems. The increased demand for water in urban areas puts pressure on freshwater sources. Rivers are often dammed or diverted to meet the needs of growing populations, leading to changesin water flow patterns and altering aquatic ecosystems. Additionally, urbanization exacerbates issues such as water pollution and flooding due to inadequate wastewater treatment and impermeable surfaces that prevent natural infiltration.Air quality is another critical aspect influenced by urbanization. The concentration of industrial activities, transportation emissions, and energy consumption in cities contribute to air pollution. Pollutants such as particulate matter, nitrogen oxides, and volatile organic compounds have adverse effects on human health and the environment. Urban heat islands, where cities experience higher temperatures than surrounding rural areas, further exacerbate air quality issues and impact local climates.Furthermore, urbanization alters the natural landscape, leading to changes in soil composition and erosion patterns. The extensive paving and construction in cities reduce the amount of permeable surfaces, disrupting the water and nutrient cycles in soils. This can result in soil degradation, loss of fertility, and increased susceptibility to erosion during heavy rainfall events. Soil sealing also limits opportunities for vegetation growth, further impacting biodiversity and ecosystem services.Additionally, urbanization influences the energy dynamics of regions. Cities are major consumers of energy, primarily sourced from fossil fuels and electricity grids. The construction and operation of urban infrastructure, transportation networks, and buildings require significant energy inputs, contributing to greenhouse gas emissions and climate change. Moreover, the spatial concentration of energy demand in cities necessitates long-distance transportation of energy resources, leading to energy losses and environmental impacts associated with extraction and transport.In conclusion, urbanization profoundly shapes the geographical environment through changes in land use, water resources, air quality, soil dynamics, and energy consumption. While cities drive economic development and provide opportunities for social advancement, they also pose significant challenges for sustainable environmental management. Addressing the environmental impacts of urbanization requires holistic planning, innovative technologies, and proactive policies to ensure the long-term well-being of both urban residents and the surrounding ecosystems.。
模拟试题一Part I.单选(30%)1.There’s something wrong with the thermometer. Don’t worry . I’ll make it works 。
2.Now people can learn huge amounts of information from the computer。
3.The expert you saw at the liaison meeting is a friend of mine 。
4.You have worked more carefully this week, for there are few mistakes in your calculation 。
5.Although the three workers got very tired in repairing the boiler, none of them would giveit up。
6.Now computers can work out problems much faster than human beings。
7.There is too much cream on the cake。
8.Don’t worry. It is not difficult for us to treat the fault。
9.Ms, Chang doesn’t look well today . What is wrong with her?10.I’d like to lend you my electrical engineering handbook, but you’d better not lend it toothers。
11.I think English is quite different from Chinese。
12.— Excuse me, is this the way to Shanghai power Company?—No, I’m afraid it isn’t 。
香料对航海的影响英语作文The Influence of Spices on Navigation。
Spices have played a significant role in the history of navigation. Their influence on the exploration and navigation of the world has been profound and far-reaching. In this essay, we will explore the impact of spices on navigation, from the ancient world to the present day.In ancient times, spices were highly valued for their aromatic and preservative properties. They were used to flavor food, preserve meat, and mask the taste of spoiled food. However, their scarcity and high demand made them extremely valuable commodities. Spices such as cinnamon, pepper, cloves, and nutmeg were highly sought after in Europe, and their trade routes became the focus of early exploration and navigation.The search for spices led to the Age of Exploration, as European powers sought to establish direct trade routes tothe spice-producing regions of Asia. The Portuguese were the first to venture out into the unknown seas, followed by the Spanish, Dutch, and English. These voyages were driven by the desire to find new sources of spices and establish control over the lucrative spice trade.The quest for spices also led to the discovery of new lands and the mapping of the world. Explorers such as Christopher Columbus, Vasco da Gama, and Ferdinand Magellan set out on their voyages in search of spices, but ended up discovering new continents and oceans. The spice trade routes became the highways of exploration, connecting the East and the West and opening up new trade opportunities.The influence of spices on navigation can also be seen in the development of navigation techniques and tools. Navigating the vast oceans required precise knowledge of the stars, the use of compasses, and the development of accurate maps. The need to find new routes to the spice-producing regions spurred advancements in navigation, leading to the development of more accurate maps and navigation instruments.The impact of spices on navigation can also be seen in the cultural exchange and globalization that resulted from the spice trade. The trade routes brought together people from different continents, leading to the exchange of goods, ideas, and technologies. The spice trade also had aprofound impact on the economies of the countries involved, leading to the rise of powerful trading empires and the spread of wealth and prosperity.In conclusion, the influence of spices on navigationhas been profound and far-reaching. From the ancient worldto the present day, spices have shaped the course of history, leading to the exploration of new lands, the development of navigation techniques, and the globalization of trade. The search for spices has driven the exploration and navigation of the world, leaving a lasting impact onthe history of mankind.。
a r X i v :0802.4102v 1 [c o n d -m a t .m e s -h a l l ] 27 F eb 2008Influence of Landau-level mixing on Wigner crystallization in grapheneC.-H.Zhang 1and Yogesh N.Joglekar 11Department of Physics,Indiana University-Purdue University Indianapolis,Indianapolis,Indiana 46202,USA(Dated:February 27,2008)Graphene,with its massless linearly-dispersing carriers,in the quantum Hall regime provides an instructive comparison with conventional two-dimensional (2D)systems in which carriers have a nonzero band mass and quadratic dispersion.We investigate the influence of Landau level mixing in graphene on Wigner crystal states in the n th Landau level obtained using single Landau level approximation.We show that the Landau level mixing does not qualitatively change the phase diagram as a function of partial filling factor νin the n th level.We find that the inter-Landau level mixing,quantified by relative occupations of the two Landau levels,ρn +1/ρn ,oscillates around 2%and,in general,remains small (<4%)irrespective of the Landau level index n .Our results show that the single Landau level approximation is applicable in high Landau levels,even though the energy gap between the adjacent Landau levels vanishes.PACS numbers:73.20.Qt,73.43.-fI.INTRODUCTIONWigner crystallization,where the density profile of carriers in a system develops a periodic spatial modulation spontaneously,is a classic example of interplay between (classical)repulsive potential energy and the (quantum)kinetic energy associated with localization of carriers as the density of carriers is varied.1,2,3Although predicted in 19341this phenomenon has defied direct experimental observation in bulk systems and conventional 2D systems.In quantum Hall systems,where the kinetic energy of carriers is quantized and quenched,Wigner crystallization is induced by a competition between the electrostatic and exchange interactions as the partial filling factor νin a given Landau level is varied.(In the quantum Hall regime,Wigner crystallization depends only on the filling factor and can occur at any carrier density.4)Wigner crystallization in the lowest Landau level has been inferred via transport measurements,5and the anisotropic transport observed 6in high Landau levels can be interpreted 7in terms of anisotropic Wigner crystal ground states.A direct observation of the Wigner crystal,via local carrier density,however,has not yet been possible.Graphene,with its massless carriers on the surface,is a unique and ideal candidate for this purpose.8,9Recent studies,using Hartree-Fock mean-field theory in the single-Landau-level approximation (SLLA)10or exact diagonalization in the single-Landau-level subspace 11have predicted that Wigner crystal states will appear as ground states over a range of partial filling factor νin a given Landau level.In this paper,we examine the validity of the single-Landau-level approximation.Let us first recall the relevant results for a conventional 2D system in perpendicular magnetic field B with partial filling factor ν≤1in the Landau level n .Thus,the actual filling factor for spinless carriers (with no other degeneracies)is n +ν.For this system,the difference between energies of the adjacent Landau levels is ∆E n =E n +1−E n = ωc where ωc =eB/mc is the cyclotron frequency,m ∼0.5m e −0.1m e is the band mass of the carriers,and m e is thebare electron mass.We remind the Reader that ∆E n = 2/ml 2B is (approximately)the quantum kinetic energy of a particle with mass m in a box with size l B =B ,as B →∞the amplitudefor inter-Landau level transitions vanishes and the SLLA becomes a good approximation.12A corresponding analysis for graphene shows the stark difference between the two systems.The gap between the adjacent Landau level energies in graphene is ∆E n =E n +1−E n = ω[ 2n ]where ω=v G /l B is the cyclotron frequency,v G ∼c/300is the speed of massless carriers in graphene,and c is the speed of light.It follows that the ratiog n =V cǫ v G12(n +1)−√2n ,as n →∞.Therefore,inter-Landau level transitionsbecome increasingly important as the Landau level index n increases,irrespective of the magnetic field;even in the lowest Landau level,the ratio g ∼e 2/ǫ v G =αG ∼1is not small (αG is the fine structure constant for graphene).This analysis suggests that the SLLA is not reliable in graphene for any B and that it gets worse with increasing n since the energy gap ∆E n →0.In the following we show that,contrary to the expectations from a simple analysis presented above,the effect of Landau level mixing in graphene remains small and SLLA remains applicable.The outline of the paper is as follows.In Sec.II,we briefly describe the Hartree-Fock approximation with Landau-level mixing and outline our approach.The details presented in this section are essentially identical to those in our earlier work.10In Sec.III,we present the results obtained without and with Landau-level mixing.Wefind that the Landau-level mixing does not qualitatively change the phase diagram of the system.We quantify the mixing using off-diagonal self-energy and relative occupation of Landau levels n and n+1.We compare the results for Landau level mixing as a function of n in graphene with those for conventional2D systems.We summarize our conclusions in Sec.IV.II.MICROSCOPIC HAMILTONIAN AND HARTREE-FOCK APPROXIMATIONLet us consider graphene in a strong perpendicular magneticfield B in the quantum Hall regime.The single-particle states of the non-interacting system are given by|n,k,σ where(n,k)denote the Landau level and intra-Landau level indices,andσ=±correspond to the two inequivalent valleys,K and K′=−K,in the Brillouin zone.The details presented in this section follow closely Ref.[10].The Hamiltonian for the system,including the Coulomb interaction isˆH=Nφ nσ(E n−µ)ˆρσ,σn,n(0)+1Nφ k,k′e−i√(1−δnn′,0)[F n,n′(q)+F n−1,n′−1(q)].(5)2We recall that F n,n′(q)is a linear combination of the form factors for a conventional2D system,10F n≥n′(q)= n! (iq x−q y)2 (n−n′)L(n−n′)n q2whereH n 1n 3,n 2n 4(q )=1(2π)2V (k )e −il 2B k ×q ·ˆz F n 1,n 4(k )F n 3,n 2(−k ),(11)and ρσ1,σ2n 1,n 2(q )= ˆρσ1,σ2n 1,n 2(q ) are the density matrix elements which should be determined self-consistently from Eq.(7).Thedensitymatrix is obtained from the equal-time limit (τ→0−)of the single-particle Green’s functionG σ1,σ2n 1,n 2(k 1,k 2;τ)=− T c n 1k 1σ1(τ)c †n 2k 2σ2(0) .(12)The equation of motion for the Green’s function in Fourier space is given by 10δσ1,σ2δn 1,n 2δq ,0=[iωn −(E n 1−µ)]G σ1,σ2n 1,n 2(q ,iωn )−σ3n 3q ′Σσ1n 1,σ3n 3(q ,q ′)G σ3,σ2n 3,n 2(q ′,iωn )(13)and the Hartree-Fock self-energy matrix is (p =q −q ′)Σσ1n 1,σ3n 3(q ,q ′)= m 1m 3H n 1m 1,n 3m 3(−p )ρm 3,m 1(p )−X n 1m 1,n 3m 3(−p )ρσ1,σ1m 3,m 1(p ) δσ1,σ3−X n 1m 1,n 3m 3(−p )ρ¯σ1,σ1m 3,m 1(p )δσ3,¯σ1 e iiωn −ωk +µ(18)which,in turn,leads to the self-consistent density matrixρσ1,σ2n 1,n 2(q )=kV σ2,n 2(q ,k )V ∗σ1,n 1(0,k )f (ωk −µ),(19)where f (x )=θ(−x )denotes the Fermi function at zero temperature.The chemical potential µis determined by theconstraint that the total occupation in the two Landau levels is equal to the partial filling factor,σ[ρσ,σn,n (0)+ρσ,σn +1,n +1(0)]=ν.(20)Using the self-consistent density matrix (19),we calculate the Hartree-Fock mean-field energy E HF for various triallattice configurations to obtain the ground state crystal structure.III.RESULTSWe consider mean-field Wigner crystal lattices with two primitive lattice vectors a1=(a,b/2),a2=(0,b)and define the lattice anisotropy asγ=b/a.Note that the triangular lattice(γ=2/√2πN e/νγand b=aγ.The reciprocal lattice vectors are Q mn=m b1+n b2where b1=(2π/a)(1,0)and b2=(2π/a)(−1/2,1/γ) are the reciprocal lattice basis vectors.We determine the optimal lattice structure by choosing theγ(0<γ≤2/√3=1.15is a constant.At higher values ofν,the anisotropy increases leading to a quasi-striped structure for the ground state.We see from Fig.2that the region of stability of the triangular lattice increases when inter-Landau level transitions are taken into account.Results in Figs.1and2suggest that the effect of Landau-level mixing is not dominant in higher Landau levels, even though the energy gap between adjacent Landau levels becomes smaller.To understand this unexpected result, we recall that the inter-Landau level transitions from n→n+1are determined by the off-diagonal self-energy matrix elements and the gap between adjacent Landau levels,Σσn,σn+1/∆E n.It follows from Eqs.(14,10,11)that for large nΣσn,σn+1∼ρσ,σn,n+1√n+1.Wefind that this asymptotic behavior is reproduced by our results.We quantify the Landau-level mixing by the ratio of relative occupations of the two levels in question,ρn+1/ρn whereρm= σρσ,σm,m(0). Left panel in Fig.3shows the ratioΣn,n+1/∆E n as a function of Landau level index n for graphene(solid red)and the0 0.2 0.4 0.6 0.8 1 1.2Without MixingWith Mixing 0 0.2 0.4 0.6 0.8 1 1.2 0.1 0.20.3 0.4 0.5ν(a) n=2(b) n=3γWithout Mixing With MixingFIG.2:(Color Online)Ground state lattice anisotropy γ(ν)in graphene for n =2(top)and n =3(bottom)Landau levels.The solid (red)line shows the result without mixing and the dashed (green)line indicates the result with Landau level mixing.γ=2/√n +1dependence at large n .We recall that this ratio for a conventional 2D systemdepends on the magnetic field B or the magnetic length l B .Our results are for g =l B /a B =0.67or l B ∼35˚A .(This g =V c /∆E n for a conventional 2D system is equal to the g =αG in graphene with ǫ=3.3as the dielectric constant.)The right panel in Fig.3shows the corresponding relative occupations for graphene (solid red)and the conventional 2D system (dotted green).The fact that this ratio,in the presence of inter-Landau level mixing,is small (ρn +1/ρn ≤4%)provides complementary support for the validity of SLLA in graphene.0.01 0.02 0.03 0.04 0 2 46 8 10 12 14 16Σ+,n ;+,n +1(0,Q 01)/∆E nLandau Level Index nν = 0.5, g=0.67graphene usual 2DEG0.01 0.02 0.03 0.040 2 46 8 10 12 14 16ρn +1/ρnLandau Level Index nν = 0.5, g=0.67graphene usual 2DFIG.3:(Color Online)Left:Σ+,n ;+,n +1/∆E n as a function of n for partial filling factor ν=0.5in graphene (solid red)and a conventional 2D system (dotted green).In graphene,the ratio is small for all n and shows that SLLA is applicable even in high Landau levels when ∆E n →0.In conventional 2D systems,the ratio decays monotonically as 1/√not qualitatively change the phase diagram of the system,although it shifts upwards the critical values offillingfactorνat which transitions from one lattice structure to another occur.We quantify the Landau-level mixing in terms of off-diagonal self-energy and relative occupation numbers,and show that it remains small as a function of theLandau level index n.Thus we conclude that SLLA provides a reliable description of Wigner crystal ground states in graphene.We emphasize that our results for graphene are independent of the magneticfield B.For conventional2D systems, the Landau-level mixing depends on the magneticfield and can be important at weakfields B≤B c when themagnetic length becomes larger than the Bohr radius of the massive carriers,l B≥a B for B≤B c.The absence of a corresponding criticalfield B c in graphene is due to the massless nature of the carriers.Our conclusions do notdepend,qualitatively,on the range of the interaction V(q)because the large-q scattering is strongly suppressed by the form factors F(q)that decay exponentially with q.In this paper,we have ignored transitions to next-higher Landau levels12n→n+k,because the amplitude for them vanishes rapidly:Σn,n+k/∆E nk∼n for large n≫k.Therefore,it is sufficient to consider the Landau-level mixing only between adjacent levels.Since carriers in graphene are on the surface,in contrast to those in the conventional2D system,it is an ideal candidate for direct observation of the local carrier density structure.9Our results provide further support for the existence of triangular Wigner lattice as the ground state at smallνand anisotropic ground states in high Landau levels forν→1/2.10,11A direct observation of carrier density in graphene in the quantum Hall regime will verify(or falsify)our conclusions.1E.Wigner,Phys.Rev.46,1002(1934).2A.L.Fetter and J.D.Walecka,Quantum Theory of Many-Particle System(Dover Publications,Inc.,NewYork,2003).3B.Tanatar and D.M.Ceperley,Phys.Rev.B39,5005(1989).4H.A.Fertig in Perspectives in Quantum Hall Effects Eds.S.Das Sarma and A.Pinczuk(Wiley and Sons,New York,1997). 5R.L.Willett,H.L.Stormer,D.C.Tusi,L.N.Pfeiffer,K.W.West,and K.W.Baldwin,Phys.Rev.B38,7881(1988).6M.P.Lilly,K.B.Cooper,J.P.Eisenstein,L.N.Pfeiffer,and K.W.West,Phys.Rev.Lett.82,394(1999).7H.A.Fertig,Phys.Rev.Lett.82,3693(1999);A.H.MacDonald and M.P.Fisher,Phys.Rev.B61,5724(2000).8K.S.Novoselov,A.K.Geim,S.V.Morozov,D.Jiang,M.I.Katsnelson,I.V.Grigorieva,S.V.Dubonos and A.A.Firsov, Nature438,197(2005);Y.Zhang,Y.-W.Tan,H.L.Stormer and P.Kim,Nature438,201(2005).9J.Martin,N.Akerman,G.Ulbricht,T.Lohmann,J.H.Smet,K.von Klitzing,and A.Yacobi,Nature Physics,4,144(2008). 10C.-H.Zhang and Y.N.Joglekar,Phys.Rev.B75,245414(2007).11Hao Wang,D.N.Sheng,L.Sheng,and F.D.M.Haldane,cond-mat/0708.0382.12A.H.MacDonald,Phys.Rev.B30,4392(1984).13A.A.Koulakov,M.M.Fogler,and B.I.Shklovskii,Phys.Rev.Lett.76,499(1996).14M.O.Goerbig,P.Lederer,and C.M.Smith,Phys.Rev.B69,115327(2004).15J.J.Sakurai,Modern Quantum Mechanics(Addison-Wesley,1995).。
