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G. Prisoner dilemma 39 hold in many quantum system. More over, some physi- cist believed that phase transition is due to the competi- H. Cournot duopoly 39 tion between two competing orders in a physical system, but people can always propose some anomalous examples I. The Atiyah-Singer index theorem 39 which can not be explained by two competing orders. The occurrence mechanism of phase transitions is still J. Topological quantity 40 not clear in countless systems. Similar critical phenom- ena arose in a broad physical and social systems. Many K. Linear response and Green function 41 unrelated models covering physics, chemistry, biology present similar scaling laws. Anyone who saw this can L. Green function 42 not help asking why. Is there a ’theory of everything’ that can give us an universal explanation? Such a theory References 42 sounds like superstring theory. What I am trying to do in this paper is not to establish the superstring theory of phase transition, but to present 1. INTRODUCTION an universal theoretical explanation to phase transition occurred in different systems based on renormalization When physicists encounter millions of interacting group theory and game theory, and beyond the two, such molecules or atoms, they can not control the trajec- as topology, quantum field theory. tory or momentum of each individual particles, so they The first step is to break the envelop of physics and ex- study the macroscopic states at different temperature or tend the concept of phase transition from physical system other physical parameter. In most cases, there would be to complex system including chemistry system, biologi- some significant change of the macroscopic states at cer- cal system, social system, economic system, et al. Phase tain value of parameters, they call it a phase transition. transition is a sudden jumping from one stable state to Phase transition are common phenomena in all branches another. A two player game always has two stable states, of physics. People’s interest in phase transition can trace the winning state and losing state. Thus we can define back to thousands of years ago. A recent example is the phase transition as a war game. If we check the war game superfluid-Mott-insulator phase transition occurred in a carefully, one would see it has all the the same phenom- gas of ultracold atoms in optical lattice[1]. ena as phase transition occurred in physical system. War Statistic mechanics were developed to provide a theo- is a conflict between millions of soldiers who are armed retical description of phase transition. The occurrence of and well organized. The butterfly effect is a basic char- phase transition is related to the singularity of statisti- acter of war going on. The final destiny of the fighter is cal functions in the thermodynamic limit(see Ref. [2] for determined by some minor accidental event. When the review). However statistical mechanics is not powerful two groups with equivalent force are fighting against each enough to predict all different kind of phase transition. other but keeping at a draw state, if any one of them is For classical Hamiltonian systems, the hypothesis con- reinforced a little, he will win the war in a few seconds. necting phase transitions to the change of configuration This is a phase transition. space topology was proposed recent years(see Ref. [3] and So we can take phase transition as a war game, the references therein for review). Topological order also rise strategies of the players extended the strategy base man- in quantum systems (see Ref. [4] for review), such as ifold. When we are studying the state evolution of the the fractional quantum Hall system. Different fractional game corresponding to different strategy, the topology of quantum Hall effect states all have the same symmetry, the strategy base manifold comes in. The phase tran- that is beyond the Landaus symmetry breaking theory. sition is a war game between new phase and old phase, The topological order in quantum system is related to each of them is governed by a kind of dominant inter- degenerate ground states, quasiparticle statistics, edge action(it may has many minor affiliated companions). It states, or momentum topology[5], et al. These topolog- will be shown in the main tex of this paper, the surviving ical order theory told us many interesting phenomena strategy of the two phases carry opposite winding num- in some special models and special systems. However ber, the sum of these winding numbers is a topological it is still far from providing us an universal explana- number on the strategy base manifold. After the wind- tion to phase transition occurred in all different branches ing number are annihilated by pairs, the last one winding of physics, much less in other complex system beyond number around the last surviving strategy is decided by physics. the topological number, this also decided who will be the Generally people believed that it is the thermal fluctu- winner between the old phase and new phase. It is in ation that drives the transition from one phase to another this sense, we call it a topological phase transition. In in classical phase transition. As temperature is lowered, fact, phase transition is always related to the topology of the thermal fluctuations are suppressed. The quantum the strategy base manifold when the strategy base mani- fluctuation began to play a vital role in quantum phase fold is compact. In some cases, the base manifold is open transition. Unfortunately this kind of argument does not and noncompact , there is no need to consider the effect 3 of topological constrain. In fact, the compact strategy such as volume, pressure, temperature, density, crystal manifold like a finite region confined by fencing, it put structure, index of refraction, chemical potential, and so some constrains on the way we choose strategy. There is forth. One of the main task of physicist is to find the rela- singular point we can never eliminated smoothly in strat- tions among these macroscopic variables via presumably egy space, that is the fundamental origin of topological well known microscopic interactions between particles. phase transition. A stable phase has self-restoring ability when it is dis- The paper is organized as follows: turbed from equilibrium states. During a relative long In section 2, the most general conception of phase tran- life time, the chemical makeup does not decompose, and sition in complex system is defined. the physical properties keep a good manner. If we know In section 3, we established the game theory of renor- the position and momentum of every particles in a dy- malization group transformation, and find the general namic system at a given time, the evolution of an integral solution of renormalization group transformation equa- system is exactly predictable. Unfortunately, there is few tions. The fundament classification of phase transition integrable many body system. The position and momen- through symmetry losing is presented. tum of particles can not be exactly measured at the same In section 4, topological current theory of phase tran- time due to the Heisenberg uncertainty principal. So we sition is established, this theory spontaneously produced define the state vector of a stable phase by the complete an universal equation of phase coexistence. A conjecture set of observables. on the universal scaling law is proposed base on a topo- Let ~x be n independent states of a complex system. logical hypothesis in fractal strategy space of game the- For a physical system consists of N interacting particles, ory. Further more, we established the evolution equation the components of an arbitrary vector ~xi are consists of phases and the quantum phase coexistence equation. of 3N position coordinates q1, q1, ..., q3N ,3N momentum In section 5, we developed the quantum statistics of coordinates p1,p1, ..., p3N , and a vector of other physical many player game, and proposed a conjecture to find parameters ~γ, such as temperature γ1 = T , pressure γ2 = the fixed point of a many player game using quantum P , particle density γ3 = N, volume γ4 = V , chemical density matrix. More over, a new quantity to measure potential γ = µ, , conductivity γ = σ, susceptibility 5 ··· j the entanglement of quantum states in a game is found. γk = χ, and so on. Here the state vector ~x = (~q, ~p) is The single direction of renormalization group follows the a much more general conception than that of statistical second thermodynamic law, the renormalization flows to physics. It indicates the information inside the black box increase the entropy as well as the quantum entangle- of any complex system. ment. More over, the coexistence state of multi-player The response of the complex system are induced by game is discussed. external input vector ~γ. For physicist, γi represents In section 6, we developed the game theory of phase those familiar external applied magnetic field, electric transition in classical many body system as well as quan- field, pressure, neutrino current, electric current, detect- tum many body system. A new holographic topolog- ing laser beams, heaters, and so forth. For chemist or ical quantity to characterize momentum space is pro- biologist, the input vector represents something like en- posed. We studied the quantum many body theory of zymes, chemical accelerator. war game and gave a new pairing mechanism base on Not all the components of the state vector are observ- prisoner dilemma. able under perturbation. The output vector ~y represents The last section is devoted to a brief summary and those observables that people can definitely detected in outlook. laboratory. The output is strongly depend on what peo- ple want to study. For example, there are circulatory systems, nervous systems and digestive systems within 2. PHASE TRANSITION AND WAR GAME human body. If we are studying the human population, there is no need to take into account of these subsystems; 2.1. Phases of complex system one only counts the people, the output vector covers the number of people, distribution of people, etc. If the sub- A complex system consists of many different elements ject is about flu’s spread, it may be best to discuss the that are connected or related, it appears like a black box immune subsystem. to us. One can obtain the information within the complex For a general dynamic system system by its responses to external perturbation. These responses and perturbations are macroscopic variables. d~x in = F (~x,~γ), O~ = F (~x,~γ), (2.1) Different stable phases of complex system are character- dt in ut out ized and distinguished by these observables. For most condensed matter physicists, a liter of wa- the output vector is not always differentiable for all de- ter is a complex system, since it is hardly possible to grees of differentiation on the whole range of the param- 27 find exact analytical solution for the motion of 10 H2O eter space M(~x,~γ). There exist critical points (~x∗,~γ∗) molecules. We can characterize its different phases by ob- at which the Ck output functions blow up. The whole serving its chemical composition and physical properties, input space is divided into separated blocks by these crit- 4
state vector space. As the input vector changes within a domain, the system is in a stable state, it represents a kind of physical order. When the input vector jumps from one domain to another, the system jumps from one stable phase to another. The domain wall is the exact phase boundary at which the old phase becomes unsta- ble and decays, but the new phase comes into being and finally leads to new stable order. Phase transition occurs FIG. 1: The whole phase diagram is split into discrete do- everywhere in nature. Every phase transition indicates mains. The stable phase exist in the inner region. The do- a revolution induced by the interaction between the sys- mains wall is the phase coexistence region. In each domain, tems and their environment. the fittest states of the old there is a dominant interaction which is the king ruling over domain is replaced by the fittest states of the new do- the other weaker interaction. There is a war going on at main. the phase coexistence boundary. When we tune the physi- The lifeless nature world is much more intellectual cal parameters, we are helping a certain interaction to fight than most physicists thought. What physicist measures against the others, this certain phase would grow stronger and stronger, it will finally unified the whole phase diagram. in experiment is always the observable in equilibrium. Quantum theory tell us the energy levels of molecules and atoms are discrete and quantized, we can calculate the transitions probability between those levels involv- ical points, ing the absorption or emission of photons. But we don’t 0 know how and why that occurs. In principal, the non- M (~r)= (r∗, r∗) (r∗, r∗) (r∗ , r∗ ) , (2.2) { 1 2 ∪ 2 3 ···∪ n n } equilibrium process of many particle system follows the here we denote (r := (~x,~γ)). The stable phase are defined same rules that govern the living systems, such as ants in these discrete blocks. The output functions present society, honeybees group, traffic system, etc. very good behavior within the blocks but diverge at the If we focus on the behavior of particles before they very boundary. These boundaries are where the phase reach equilibrium, one would doubt that electrons are transition occurs. possibly intelligent particles. One simple example is two This general mathematical definition of phases for resistors connected in parallel in electric circuit. As all complex system applies for many different fields. The knew, the current is proportional to the inverse of resis- most familiar Ehrenfest’s definition[6] of phase transition tivity in subways. In the beginning, the electron moves in thermodynamics is a good exemplar. The output vec- together in the main path. When they reach the bifur- tor is only a function—free energy. The input vectors are cation point, they split into two subways. Less current thermodynamic quantities, temperature T and pressures in the strong resistive way, and larger current in the less P . For the zeroth order stable phase the free energy of resistive one. Like the cars in traffic, if all the electrons the two phases F (T, P ) is C∞ in the whole input space choose the less resistive way, they block each other until (T, P ) [ , + ]. For the first order phase transi- ∈ −∞ ∞ it becomes more resistive than the other subway. Then tion, the free energy F (T, P ) is continuous in the region some electrons withdraw and transferred to the other T [T ,T ], P [P , P ], but the first order derivative is ∈ i f ∈ i f way. There is no traffic jam in electric current network, not continuous, because the cars are everywhere under the control of a ∂F ∂F ∂F ∂F global electric field. A = B , A = B , ∂T 6 ∂T ∂P 6 ∂P The non-equilibrium dynamic process shows up in the TA (Ti,Tc),TB (Tc,Tf ), vicinity of critical point, at which a small quantitative ∈ ∈ change of a parameter would results in a qualitative P (P , P ), P (P , P ). (2.3) A ∈ i c B ∈ c f change of the global behavior of a complex system. Bi- The first order stable phase blocks are divided into ological systems that adapt to their environment thrive, smaller blocks by the second order phase transition. those that fail to evolve fade away. The environment are The phase transition of complex system may be defined external input of the biological system. A stable phase from the divergence of the Ck output functions. They of a biological system only exist in a finite region of en- could be any observable functions. For mathematician, a vironment parameters, so does a lifeless physical system. k stable phase is marked by a C function in the discrete The stable phase of a collection of H2O molecules is wa- blocks on the input vector space. The critical point is ter between 0◦C 100◦C, below 0◦C(273 K) is ice crystal, ∼ the phase boundary between the separated blocks. and it becomes gas above 100◦C. Like any living creatures in nature, a lifeless physical system of interacting H2O molecules evolutes following 2.2. Phase transition the rule— ”survival of the fittest”. They took differ- ent collective structures according to different pressure, The stable output states of a dynamic complex sys- temperature, impurity, radiation, gravitation field, elec- tem are confined in different domains in the whole input tric field, magnetic field, density, etc. The stablest phase 5 at certain range of the parameter space survives as the soldiers who are fighting with each other. Since they do fittest structure, the other phases in this domain fade not hate each other personally, they behave as indistin- away. Out of this special domain, the molecules have to guishable identical particles, and fight as a whole. Renor- reorganize their collective motion pattern in order to fit malization theory simplifies the war between millions of the new environment. When the new phase is born, the soldiers to a war between thousands of companies, to a old phase dies. war between hundreds of regiments, finally to the war The phase transition most scientists observed in between two army. It is the war between two generals, physics, chemistry, biology, complex network, economics, it is also the war between hundred of colonels as well as is a revolution. Phase transition is a war between the captains. force··· of old phase and the force of new phase. The funda- The mean field theory view the war as a fight between mental particles are millions of armed soldiers. They are two full generals dressed up by millions of soldiers. This is divided into different large-scale armed groups who are correct in most cases, but it is not always accurate, for the fighting with one another. There is always a critical re- general’s strategy is carried out by hundreds of colonels gion in which the fighters bet their bottom dollar on one and captains instead of the elementary soldiers. At the war, the winners take everything, the losers get nothing. critical point, the correlation between the members of This critical point is the phase boundary. Sometimes, an army extends to its maximal value. As the butterfly the turning point is not so definite that it broadens into effect says, the battle may be lost due to a nail which fail a finite region, this indicates a crossover transition. to fix the shoes of the horse. For want of the horse, a rider is lost. The lost of one rider may directly leads to the loss of a battle. 3. RENORMALIZATION GROUP THEORY The renormalization group starts from the most fun- AND GAME THEORY damental particles of the army: soldiers. The first order renormalization is to reduce the interaction between mil- 3.1. Game theory and Renormalization group lions of soldiers to hundred of captains which are dressed up by soldiers. The second order renormalization pro- cedure is to identify effectively the captains belonging Wilson’s renormalization group theory provides a fun- to one regiment with one colonel. This particle-blocking damental non-perturbative approach to quantum critical process may continue, and finally end up with full gen- phenomena[7]. One basic character of quantum many erals. body system is that the microscopic particles are not This explanation of renormalization group theory from distinguishable. Army is the best social system for sim- war between armies is not merely a parable. There is a ulating collective behaviors of quantum many particles rigorous mathematical correspondence between the game in physics. Usually the soldiers are identical particles in theory of war and renormalization group theory. Let’s the eyes of a general, but are distinguishable particles for take the two-dimensional Ising model on triangle lattice a sergeant. The hierarchy structure of army plays the as one example. The soldiers are spins σ , the battle field same role as quantum numbers in physics. The statis- i is the triangular lattice, the Hamiltonian of this game tics of particles is scale dependent. One example is two reads column of dipolar Bosonic atoms obeying anyonic statis- tics due to the long range interaction. There are other N N biology system which similar to human society, such as H = γ σ σ + γ σ , (σ = 1). (3.1) 1 i j 2 i i ± ant group, honeybee group, we mainly take the army as
1 βH(γi(t)) This decision rule matrix of the two players is just the F = lnZ, Z = Tre− , (3.15) renormalization group transformation matrix −β ˆ ˆ 0 fγ1 using the rescaled Hamiltonian H′( σi′ ,γ1′ ,γ2′ ,N ′) and U = ˆ (3.7) { } fγ2 0 the H′( σi ,γ1,γ2,N). Since the Hamiltonian is of the same form{ } as that before the scale transformation, so which can be deduced from the decimation procedure of does the free energy. Comparing the coefficient function renormalization group transformation. The players may of the spin-spin coupling term of the effective Hamil- play many steps to reach an agrement. Each time we tonian, we get the transformation function UL. In the group the spins in a sum by Kadanoff blocks, the origi- vicinity of the non-trivial fixed point ~γ∗, nal degree of freedom is decimated into the fewer degree of freedom. Every block is now a new giant elementary ~γ∗ = UˆL~γ∗, (3.16) particle whose spin is determined by the majority rule. The Kadanoff-blocking process is actually a game pro- we perform Taylor expansion around ~γ∗ = (K1∗,γ2∗), and cess. After the first step of Kadanoff-blocking, the two make a truncation to the first order(the simplest case). player accomplished the first round of game, and their The renormalization group transformation is identical possible strategy space is reduced to a smaller one due with coordination transformation, to the information they get from the first round game. ′ ′ ( ∂γ1 ) ( ∂γ1 ) The rescaled Hamiltonian is of the same form as the γ1′ γ1∗ ∂γ1 ∂γ2 γ1 γ1∗ − = ∂γ′ ∂γ′ − . (3.17) original one, γ2′ γ2∗ ( 2 ) ( 2 ) γ2 γ2∗ − ∂γ1 ∂γ2 ! − ′ N /2 N ′ ∗ T H′ = γ1′ σi′σj′ + γ2′ σi′. (3.8) We denote δγ′ = (γ1′ γ1∗,γ2′ γ2∗) and δγ = (γ1 T − − −
a b a b fA(fB(γ0 )), γ2 = fB(γ1 )= fB(fA(γ0)), i.e., of the product or service he could offer. So a complete difference equations of this game is a ˆ ˆ a γ¯2 fAfB 0 γ¯0 b = b . (3.20) a a b b a b γ¯2 0 fˆ fˆ γ¯0 γn = fA(γn 1,γn 1), γn = fB(γn 1,γn 1). (3.26) B A − − − − 2 In fact, the players of a game mainly concerns about This equation has a more clear expression ~γ2 = Uˆ ~γ0 after we introduced the game operator Uˆ and strategy profit difference between last round and next round, the most efficient way is to take the profit of last round as a vector ~γ, a datum mark. So we can always find a term like γn 1 on − ˆ a the right hand of Eq. (3.26). Thus the game difference ˆ 0 fA γ¯n U = , ~γn = . (3.21) equation is transformed into differential equations, fˆ 0 γ¯b B n a a a γn γn 1 a b Suppose the players paly alternatively, Alice is in the γ′n = lim − − = FA(γn 1,γn 1), h 0 h − − even periods and Bob in the odd periods, when Bob → b b b b γ γ produces γ2n 1 in the period 2n 1, Alice produces n n 1 a b − − γ′n = lim − − = FB(γn 1,γn 1), (3.27) a b h 0 h − − γ2n = fA(γ2n 1) in the period 2n, Alice then changes → − b a her production rate and produces γ2n+1 = fB(γ2n), and h is the step size of the tuning parameter. This equation a b so on. The sequences of γ2n and γ2n 1 satisfies the re- may be viewed as renormalization group transformation cursion relation, − equation. The two functions FA/B form the game op- erator which map one state in the strategy space into 2zn+1 + zn = u, (3.22) another. Its trajectory depicts the renormalization flow. The strategy space of two player game is two dimensional This sequences converge to u . u is the fixed point of 3 3 parameter space. The corresponding game operator is a ˆ n+1 U , (n ), general 2 2 matrix. This two dimensional matrix is an → ∞ × n+1 element of renormalization group. For some special case, lim ~γn = lim Uˆ ~γ0 = γ∗. (3.23) n n if the game operator is an element of spin(3), we can →∞ →∞ expand it in terms of traceless Pauli matrix. Now we see this game operator is the same as bargain The game operator of a N-player game is a traceless game in the sense of the renormalization group transfor- N N matrix, mation. × fˆ fˆ fˆ 11 12 ··· 1N fˆ fˆ fˆ 3.2. Solutions of Renormalization group ˆ 21 22 2N U = . . ···. . . (3.28) transformation equation in game theory . . .. . ˆ ˆ ˆ fN1 fN2 fNN The game in reality always experience a dynamic pro- ··· cess. For example, the bargain on price between the seller Following the same procedure as two player game, we can and buyer still exist in primary market. The advent of su- get an equivalent N dimensional difference equations. We permarket drive this kind of observable negotiation pro- summarize the output field and input field into one state cess to the backstage market investigation. The operator vector γ = (O~ ut, ~rin), a general system is governed by of the supermarket adjust its next day’s merchandize dis- the differential equations, tribution according to last day’s sell. A good customer also compares today’s price with previous price to buy ∂sγi = Lˆ(γ, ∂γ)γi, i =1, 2, ..., n, (3.29) what he needs at a lower price. This is a mutual inter- acting process which may be expressed by a nonlinear where s is the tuning parameter, it could be any input game operator, field. In fact, the output and input are not absolutely dis- tinguished, that depends on what we have known, what a ˆ a γ¯n 0 fA γ¯n 1 we still do not know. We usually take those we can ma- b = b− , (3.24) nipulate as input, and those we will detect as output. γ¯n fˆB 0 γ¯n 1 − When physicist study phase transition of a system, they where γa is the seller, γb is the buyer. This game has an usually tune some external parameter, such as temper- equivalent representation by a pair of difference equation, ature, magnetic field, et al. They investigate how the system response according to different physical parame- a b b a γn = fA(γn 1), γn = fB(γn 1). (3.25) ters so that we may control it for practical purpose. For − − example, superconductor physicist study how the con- In fact, the buyer not only consider the seller’s price, ductivity behaves when people raise the temperature as but also consider money amount he could pay. On the high as room temperature, they do not care much about other hand, the seller must reflet the amount and quality the time. If the parameter s is taken as time, it is just 9 the conventional dynamic system, a physicist’s approach to be, it lies in the hand of this point. Any minor devia- to its solution is the algebra dynamic arithmetic[10]. tion from this point would decide who is winner, who is The game operator Lˆ(γ, ∂γ) is the infinitesimal gener- loser. ator of the translational transformation with respect to In the region of stable phase, the state vector γ of the parameter s. For a given initial strategy γ0, the evolution game is Cp continuous function, i.e., the derivative of the dpγ of the system along s follows state vector dsp is continuous up to the pth order. This n differentiability of the pth order derivative beaks at some ˆ ∞ s γ(s)= eL(γ,∂γ )sγ0 = Lˆn(γ, ∂ )γ0. (3.30) singular points, at which we would observe a pth order n! γ n=0 phase transition. X This equation enveloped the whole process of renormal- We usually take the pth order non-differentiability of ization group transformations. Take the the bargain output vector to measure phase transition. These output ~ game as an example, n = 0 is the first round of bargain, vectors Out(~u) are the subset of the state vector, usually the buyer and seller put their initial cards on the table. they are the cost function of the game. The control vec- n = 1 is the second round of bargain, after they have set- tors ~u are game players. For a quantum many body | i ~ tled the main business down, they began the bargaining system, the output vectors Out(~u) could be any statis- on minor business now, the amplitude of this round is 1. tical observables or any external response, such as free The amplitude of the third round of bargain is 1/2!. As energy O1 = F , ground state energy O2 = Eg, thermal the renormalization group transformation goes on, the potential O3 = Ω, susceptibility O4 = χ, specific heat amplitude of the nth round of bargain decays following O5 = CH , correlation length O6 = ξ, compressibility 1/n!. When they made an agrement on all the problems O7 = κT , . The players of the game is control vector ··· from the dominant ones to the ignorable ones, peace ar- ~u , its component includes all input variables, for ex- | i rived. ample, temperature u1 = T , pressure u2 = P , effective The ending point of this game is fixed point which is external magnetic u3 = γ2 = µB/kBT , spin-spin inter- given by the exact evolution state vector γ(s). The un- action u4 = γ1 = J/kBT , electric field u5 = E, chemical stable fixed point corresponds to the Nash equilibrium potential u5 = µ, . ··· point, any one of the players takes one more aggressive The Nash equilibrium of this N-player game is to find step to increase his own profit would result in the break- the eigenvector of the operator Uˆ n+1 in strategy space. down of the deal. The Nash equilibrium point is where This is tantamount to solve the equation the phase transition occurs. The stable fixed point indi- cates stable phase. (Uˆ n+1 I) ψ =0. (3.31) − | i This equation indicates an infinitesimal transformation 3.3. Symmetry losing as a classification of phase around the identity. A finite transformation is con- transition structed by the repeated application of this infinitesimal transformation. For a realistic game, it is always impos- ˆ n+1 We gave a very general definition of phase transition sible to get an exact matrix of limn U up to the →∞ in section (2.2), phase transition is a game between old infinite order. The output function encounter divergence phase and new phase. Whenever the winner becomes on the input base manifold. These singular points are loser or vices versa, phase transition occurs. Transition where the transition from loser to winner occurs. Out of is always accompanied by the transfer of award from loser these singular regions, the output function has very good to winner. The award is quantized, so we observe sudden behavior. change of output across the phase boundary. Renormalization group is a semigroup, because its el- Each time we make a renormalization group transfor- ements have no inverse. The ordinary group is a subset mation, the players accomplished one round of game. If of the Renormalization group. So they only provide us someone lose, someone must win. The temporary phase some basic understanding to phase transition in certain boundary invades from the winner into the loser. The special cases. amplitude of boundary’s change in the nth round of game In critical phenomena, the correlation length between is proportional to 1/n!. If n = 1, that is the 1th order particles goes to infinity at the phase transition point[2], phase transition, the amplitude of sudden change is most there is an obvious conformal transformation symmetry. significant, the phase boundary is roughly fixed except This is a kind of symmetry when we take the physical some unsettled regions. Then we need the second round particles as players of a game. But a practical physical of combat to negotiate the main part of the unsettled re- system always contain billions of particles, this is not a gion. As this kind of negotiation goes on to higher order, good choice. the unsettled boundary becomes smaller and smaller, its A more practical way to study quantum many body amplitude decays following 1/n!. When all the unsettled system as a game is taking the different interactions as boundary are fixed, we reached a fixed point. If this fixed players. The interaction parameters form the strategy point is stable, we are in a stable phase. If this is an un- vector of the game. As we know, the strategy vector stable fixed point, we are at a critical state, to be or not at fixed point is invariant under the operation of game 10 operator. The holonomy group at the Nash equilibrium example, for n = 0, U (0) = I, it yields F A = F B. For point is a special subset of the game operator space. We n = 1, U = I + Lθˆ , then (I + Lθˆ )F A = (I + Lθˆ )F B , we may check the discontinuity of the output field under derived holonomy group to check the phase transition point. The A B strategy space is expanded by the strategies of all play- F = F , ⇒ ers, the holonomy group transformation is actually the ∂F B ∂F A ∂F B ∂F A =0, =0. (3.35) transformation on the strategy manifold. ∂T − ∂T ∂P − ∂P Lie group is more familiar to most physicists, it is also a subset of renormalization group. When the game has Therefore the essence of Ehrenfest’s definition for dif- a continuum of players, the strategy space construct a ferent order of phase transition actually depend on how manifold. The payoff functions is a cross section of the many order of the symmetry of free energy is pre- (p) fibre bundle established on this base manifold. We can served during the phase transition. We denote U = p 1 ˆ p define the order of phase transition from the loss of Lie 0 p! (Lθ) , Eq, (3.34) may be reduced to group symmetry. P ( a ization group transformation point out where the Nash we set up the vielbein eµ on the surface, the Riemannian equilibrium point is. If the vector field is continuous ev- curvature tensor can be expressed in terms of the gauge a b ab erywhere on the whole base manifold, one observe noth- field tensor as Rµνλσ = eλeσFµν . We introduce the ing. If only there appears a discontinuity, then we make SO(2) spin connection for− the tangent vector Eq. (4.2), sure that there is something happening. The fundamen- ~ tal vector field to detect the discontinuity is φ = φ1T1 + φ2T2. (4.6) p a δ [U(θ)Out(~γ)] Then we can define an Gauss mapping ~n with n = φ~ = . (4.1) a a a a a δθp θ=0 φ / φ , n n = 1, where φ = √φ φ (a = 1, 2). Considering|| || the symmetry of|| the|| unit vector field ~n, we ab We first take two component of this vector field to study introduce the SO(2) spin connection ωµ , the covariant the pth order phase transition. They are the pth order a a ab b derivative of ~n is defined as Dµn = ∂µn ωµ n . Notic- tangent vector field on Out(~γ), ing SO(2) is homeomorphic to U(1). There− is a one to p 1 a p 1+a one correspondence between SO(2) spin connection and φ = ∂ − − ∂ − O (~γ), (a = b) 1 γ1 γ2 ut 6 U(1) connection. Both of them have only one indepen- p 1 b p 1+b φ2 = ∂γ1− − ∂γ2− Out(~γ), (4.2) dent component. As shown by Duan et al[12], one con- sider parallel transportation from D φ = ∂ φ iA φ =0 i i − i where p is arbitrary number, γi are input vectors. Here and its conjugate equation Diφ† = ∂iφ† +iAiφ† = 0, it is a b we first take the simplest case that has only two input easy to obtain the U(1) gauge potential Ai = ǫabn ∂in . field as an example to present the basic relation between It was proved that the Gaussian curvature G can be topology and phase transition. expressed as In the neighborhood of the critical point, the manifold µν is approximately holomorphic to flat Euclidean space, the 1 ǫ Ω= Fµν , Fµν = (∂µAν ∂ν Aµ), (4.7) translation operator ∂γ is good enough to describe the −2 √g − transportation of output field. But topology concerns here Fµν is U(1) gauge field tensor. Substitute the U(1) about the global geometry of the manifold, we have to a b walk out of the vicinity of the saddle point, then we need gauge potential Ai = ǫabn ∂in into the Gaussian cur- to introduce the covariant derivative, vature, one may express the Gaussian curvature into a topological current D = ∂ + iA ,D = ∂ + iA . (4.3) γ1 γ1 γ1 γ2 γ2 γ2 2 a b ij ∂n ∂n Ω= ǫ ǫab , (4.8) Aγ1 and Aγ2 is the gauge potential which connects the ∂γi ∂γj i,j,a,b=1 vector field between different domains. The covariant X derivative of the tangent vector field on Out(~γ) is This topological current appeared in a lot of condensed matter systems. In differential geometry, the integral of D φ = ∂ φ + iA φ . (4.4) γi i γi i i i the gauge field 2-form is the first Chern number C1 = M Ω, it is the Euler characteristic number on a compact The commutator of the two covariant derivative produce Riemannian manifold. the Gaussian curvature on the two dimensional Rieman- R From the unit vector field ~n, one sees that the zero nian manifold, [D ,D ]= iΩ with where Ω = ∂ = i j − ij ij ij points of φ~ are the singular points of the unit vec- ∂iAj ∂j Ai. The Gaussian curvature may decompose − 0 tor field ~n which describes a 1-sphere in the φ~ vec- into the product of two principal curvatures, κ1(γ ) and 0 tor space. In light of Duan’s φ mapping topological κ2(γ ). When κ1 > 0 and κ2 > 0, it is a elliptic surface, a − a current theory[12], using ∂ φ = ∂iφ + φa∂ 1 and when κ1 > 0 and κ2 = 0, it is parabolic, for κ1 > 0 and i φ φ i φ k k k k k k κ < 0, it is hyperbolic. If the Gaussian curvature is the Green function relation in φ space : ∂a∂a ln φ = 2 − || || zero, it means the manifold is flat. 2 ~ ∂ 2πδ (φ), (∂a = ∂φa ), one can prove that As it is well known, Euler characteristic number on the compact two-dimensional surface is defined by Gaussian 2 ~ φ 2 ~ 1 2 1 2 Ω= δ (φ)D( )= δ (φ) φ , φ , (4.9) curvature G, i.e., χ(M)= 2π M G√gd x. In the differ- γ { } ential geometry, the Euler characteristic is just a special R 1 2 jk a b case of the Gauss-Bonnet-Chern theorem which defines where D(φ/q) = 2 i,j,a,b=1 ǫ ǫab∂j φ ∂kφ is the Ja- a topological invariant on the 2n dimensional surface. In cobian vector. In the extra-two dimensional space, this Jacobian vector is justP the Poisson bracket of φ1 and φ2, terms of the Riemannian curvature tensor, the Gaussian 1 2 1 2 curvature Ω is written as φ1, φ2 = ( ∂φ ∂φ ∂φ ∂φ ). As we defined above, { } i ∂γi ∂γj − ∂γi ∂γj ~ 1 ǫµν ǫλσ the vector fieldP φ is the tangent vector field on the out- Ω= Rµνλσ . (4.5) put manifold. This tangent vector field may be viewed −4 √g √g as a projection of a source vector field, Ψ~ which satisfy As all know, the Riemannian curvature tensor correspond Ψcos~ θ = φ,~ θ is angle between the vector Ψ and the tan- to the gauge field tensor in gauge field theory[11]. When gent plane at point p, Obviously when φ = 0, the source 13 vector field points vertically up. If we draw the configu- here βk is the Hopf index and the Brouwer degree ration of the vector field around point p, one would see ηk =signD(φ/q)zk = 1, i.e., signdetMδF (zk) = 1. a Skyrmion configuration. There is a sharp peak around .Applying Morse theory,± we can obtain the topological± the point at which φ = 0. These sharp peaks bears a charge of the transition points from Eq. (4.8). topological origin. Following Duan’s φ mapping method, it is easy to − Eq. (4.9) provides us an important conclusion imme- prove[12] that βkηk = Wk is the winding number around diately: Ω = 0, iff φ~ =0; Ω = 0, iff φ~ = 0. In other the kth critical point. The winding number measure how words, the solutions of6 the equations6 many times the vector flow surround the isolated surviv- p 1 a p 1+a ing strategy. When the output manifold extend a com- φ = ∂ − − ∂ − O (~γ)=0, (a = b) 1 γ1 γ2 ut 6 pact oriented Riemannian manifold in strategy space, the p 1 b p 1+b total winding number φ2 = ∂γ1− − ∂γ2− Out(~γ)=0, (4.10) decides whether the phase transition exist or not. If l 2 Eq. (4.10) has solutions, it means that there are some Ch = Ωd q = Wk (4.12) points on the domain wall at which the tangent vector Z Xk=1 field φ~ is continuous. According to our general definition about phase transition, a phase transition is a revolu- is just the Euler number. Euler number is a topolog- tion, it happens when the system can not survival with- ical number, it strongly relies on the topology of the out changing itself to fit the new environment. If there is base manifold. Especially when the output manifold is a any strategy that the system could take to survive, it will compact orientable Riemannian manifold, such as sphere, not take any risk to face revolution. Therefore as long torus, or a disc with boundary, and so on, the total topo- logical charge of the transition points is the Euler num- as there exist solutions for φ~ = 0, we will not observe ber. The Euler number of a 2-sphere is Ch = 2. Accord- any sudden change, this state is marked by a non-zero ing to Eq. (4.12), we see if there are two peaks of the Euler number. On the contrary, if Eq. (4.10) has no vector field distributed on output manifold, each point is solution over all strategy space, this means the tangent assigned with a winding number W = 1 to make sure the vector field across the domain wall encounter a barrier, Euler characteristic number of the 2-sphere. If there is it has to jump over the barrier with finite hight to access only one peak, the winding number must be 2. another domain. In the game theory, this indicates the system can not find any strategy to help itself move from As we know, each peak represents a surviving strategy, more strategies provide more surviving opportunities for one domain to another domain, a reform or revolution is required. This indicates a phase transition, this state of the old phase. But there is a topological constrain from the base manifold which requires that the sum of winding system is marked by a zero Euler number. Now we see Eq. (4.9) actually describes topological number around each strategy must be equal to the Euler number. If we require the winding number must be posi- configuration of vector flow around the surviving strat- egy. Each surviving strategy present a peak on the tan- tive, we see if there are four strategies, each of them must carries half winding number W = 1/2, (i =1, 2, 3, 4). gent vector plane. More peaks means longer life time for i the old phase, if peaks become less and less, that means The optimal distribution of the four strategies should make them separated as far as possible. For if they are the old phase is dying. The system becomes more and at a crowd, it would be dangerous, their enemy does not more unstable, and began to collapse. When there is no peak left, the system has to start a revolution up, the need to spend much energy to block them all, then the old phase dies. The four critical points of the same sign system is totally unstable now, chaos effect come into action. repel each other to reach the minimal of the system’s total energy, so they will be separated as far as possi- The number of solutions of equation φ = 0 counts the number of surviving strategy for the old phase. ble. When the equilibrium is reached, the most likely distribution is the four critical points are situated at the The implicit function theory shows, under the regular condition[13] D(φ/γ) = 0, we can solve the equations vertices of a tetrahedron. As for the two strategies case, one sits at the North Pole, the other sits at the South φ = 0 and derive n isolated6 solutions, which is denoted as ~z = (γ1,γ2), (k =1, 2,...n) At the critical point z , Pole. If there is only one strategy with W = 2, it is k k k k unstable and is apt to split into two or four. If there the Jacobian D( φ ) can be expressed by Hessian matrix γ is a strategy with W = +3, the strategy with negative φ MδO (~γ) of δOut(~γ), i.e., D( ) zk = detMδO (~γ)(zk). winding number would appear, but these state are very ut γ | ut According to the δ function theory[14], one can expand unstable. − 2 ~ 2 ~ l δ (~γ ~zk) The output manifold may jump from a sphere to a δ(φ) at these solutions, δ (φ) = βk φ− . Then k=1 D( ) z | γ | k torus, and to a torus with many holes, the Euler num- the topological current becomes P ber would jumps from Ch = 2 to Ch = 0, then to l Ch = 2(1 h) with h as the number of holes of the detMδOut(~γ)(zk) 2 2 1 1 − Ω= βk δ(γ γk)δ(γ γk), torus. The topological change of base manifold would detM (zk) − − k=1 | δOut(~γ) | either kill the old phase or save its life, so topology plays X (4.11) a very important role in phase transition. 14 history of a system. The renormalization group trans- formation theory told us a phase transition point is the Nash equilibrium solution of a game. As shown in the bargain game, the profit of seller is the loss of the buyer and vice versa. When the seller takes proper strategies to maximize his profit, the buyer is trying to minimize it by taking strategies from a different space. So the seller is approaching to the maximal point of the payoff func- tion, in the meantime the buyer is looking for its minimal point. A Nash equilibrium appears at their intersection. The Nash equilibrium point is a saddle point, the output field reaches its maximal point in γ1 direction, but get a FIG. 2: (a) The distribution of four surviving strategy with minimal value in γ2 direction. The derivative of the out- topological charge +1/2 on a 2-sphere. (b) Two surviving put field corresponds to γ1 and γ2 must be of opposite strategies with topological charge W = +1. sign. This leads to the coexistence equation for different phases. In previous sections, when solving the equation of tan- What we presented in the discussions above is the sim- gent vector field φ = 0 to find the surviving strategies, we applied a regular condition D(φ/γ) = 0, which comes plest case, there is only one output field with two input 6 vectors. For the most general case, the output vector has from the implicit function theorem[13]. When the regular m components with m input vectors. There exist a m condition is violated, i.e., D(φ/q) = 0, a definite solution dimensional tangent vector field for every component of of equation φ = 0 is not available. Then the branch pro- the output vector, cess of the solutions function occurs. A mathematical demonstration of the branch process could be found in i p 1 ai1 p 1 ai2 p 1 aim Ref.[12]. This bifurcation may be understand from game φ = ∂γ1− − ∂γ2− − ...∂γ−m − Out(~γ), theory. Under the regular condition D(φ/γ) = 0, if the solutions of φ = 0 exist, that means the old6 phase still where i ai = 0. We can define a gauss map, i.e., an unit vector field ~n with na = φa/ φ , nana =1, where has strategies to survive, if there is no solutions, the old φ = P√φaφa (a =1, 2, ..., m). Following|| || the topological phase can not find any strategy to make a living, it has ||field|| theory[12], we can find a topological current, to die. When the regular condition fails, D(φ/γ) = 0, the old phase and new phase is at an equilibrium war, m a b c if the old phase win, it find ways to survive, if it loses, ij...k ∂n ∂n ∂n Ω= ǫ ǫab...c ... , (4.13) no surviving strategy exist, the old phase dies. There- ∂γi ∂γj ∂γk i,j,...,kX;a,b,...,c=1 fore, it is at the very battlefield of D(φ/γ) = 0, the two phases have equivalent power, nobody wins, nobody lose, where (a, b, ..., c = 1, 2, ..., m), (i, j, ..., k = 1, 2, ..., m), but they are fighting against each other. Several roads ǫab...c is antisymmetric tensor. On even dimensional branched out of this battlefield, to be or not to be, the manifold, this topological current is exactly equivalent old phase has to make a choice when passing this critical to the Riemanian curvature tensor which directly leads region. to the Gaussian curvature in two dimensions. Applying In two dimensional input space, the coexistence Laplacian Green function relation, it can be proved that curve equation D(φ/γ) = 0 is just the familiar ~ φ φ i Ω= δ(φ)D( γ ), where the Jacobian D( γ ) is defined as Poisson bracket for the tangent vector field φ = p 1 i1 p 1 i2 p 1 im ∂γ1− − ∂γ2− − ...∂γm− − Out(~γ), m a b c φ ij...k ∂φ ∂φ ∂φ D( )= ǫ ǫab...c ... . 1 2 γ ∂γi ∂γj ∂γk φ , φ =0. (4.14) i,j,...,k;a,b,...,c=1 { } X It is an unification of the special coexistence equations of In m =2n dimensional manifold, it was proved that the different order phase transition. As all know, in quantum topological charge of this current is the Chern number mechanics, if two operator Aˆ and Bˆ commutate with each 2 l Ch = Ωd q = k=1 Wk, which is the sum of the wind- other, i.e., [A,ˆ Bˆ] = 0, they share the same eigenfunction. ing number around the surviving strategies for multi- 1 2 R P Eq. (4.14) means that the two classical field φ and φ player game. are commutable at the phase transition point. When the game has three players, the output field is a function of three parameters, each of them holds the 4.2. The universal equation of coexistence curve in life of an old phase. The three phases intersects with phase diagram one another at the coexistence points which sit at the solutions of A phase transition is a war, is a game, is a revolution. No matter where it takes place, it becomes landmark in φi, φj , φk =0, (4.15) { } 15 where φi, φj , φk is the generalized Poisson bracket. Its quantum{ correspondence} id the Jacobi identity [A, [B, C]] + [C, [A, B]] + [B, [C, A]] = 0. (4.16) For a n-player game, we need to introduce a n- dimensional renormalization group transformation on the output manifold. The transformation operator expand the tangent vector space around the identity on the man- ifold. We denote a vector operator as L~ , a group element iθ~n L~ is given by U = e · . The basic tangent vector field for phase transition is FIG. 3: The saddle surface of the free energy around the critical point. p ˆ ~ δ [U(θ)δ 0 O 0 ] φ = (φ 1 , φ 2 , ..., φ )= h | | i . γ γ γn δθp |θ=0 (4.17) Recalling the Ehrenfest equations The most general definition of generalized Poisson bracket[15] for n component vector field is dP αB αA dP CB CA = − , = p − p , (4.23) dT κB κA dT T V (αB αA) ∂(φ1, φ2, ..., φn) − − φ1, φ2, ..., φn = . (4.18) { } ∂(γ1,γ2, ..., γn) it is easy to verify that the equation above is in consistent with the bifurcation condition Eq. (4.22). So the bifur- The coexistence surface equation for n-player game is the cation equation D(φ/q) = 0 is an equivalent expression Jacobian field for n-component output field, it is equiva- of the coexistence curve equation. The solution of this lent to the n-dimensional generalized Poisson bracket, equation is a two dimensional coexistence surface, the n ∂φ1 ∂φ2 ∂φc Ehrenfest equations actually indicates the normal vector φ1, φ2, ..., φn = ǫij...k ... =0. of phase A and B is of equivalent value with opposite { } ∂γi ∂γj ∂γn i,j,...,kX direction, i.e., ~nB = ~nA , that means they reach bal- (4.19) ance on this surface.| | −| | This coexistence equation of m vector field φi, (i = For the first order phase transition, we chose the vector { 0 1, 2, ..., m) may be decomposed as a group equation of order parameter as φ = ∂ δF , here ′0′ means no deriva- } i j two field equations φ , φ =0,i,j =1, 2, ..., m where tive of the free energy. The generalized Jacobian vector {{ } } i and j must runs over all component of the vector field. of the first order phase transition with φ = ∂0δF is given In order to verify the universal coexistence curve by equation, we take the two-phase coexistence equation φ1, φ2 = 0 as an example, and apply it to thermo- φ ∂F B ∂F A ∂F B ∂F A { } D( ) = ( ) + ( )=0, (4.24) dynamic physics. The output field is the difference of q ∂T − ∂T ∂P − ∂P A B free energy Oˆut(γ) = δF = F F , the input vector are temperature γ = T and pressure− γ = P . It will in mind of the relation ∂F = S and ∂F = V , and 1 2 ∂T − ∂P be shown, the universal coexist equation φ1, φ2 = 0 φ considering D( q ) = 0, we have unified all the coexistence equations in classical{ } phase transitions. dP (SB SA) = − . (4.25) We first verify the second order phase transition. The dT (V B V A) order parameter of the second order phase transition is − 1 2 φ = ∂T δF and φ = ∂P δF , substituting them into the This is the famous Clapeyron equation. The critical point Jacobian vector is a saddle point. So it is the maximal point of free energy difference in γ direction, and it is minimal point in γ ∂φ1 ∂φ2 ∂φ1 ∂φ2 1 2 φ1, φ2 = D(φ/q)= =0, (4.20) direction, their first order derivative must obey { } ∂T ∂P − ∂P ∂T dδF dδF and using the relations < 0. (4.26) dγ1 dγ2 A B Cp Cp A B ∂T ∂T δF = − , ∂P ∂P δF = V (κT κT ), The first order phase transition requires they must share T − the same absolute value at the critical point, then ∂ ∂ δF = V (αB αA), (4.21) P T − dδF dδF we arrive + =0. (4.27) dγ1 dγ2 V D(φ/q)= (CB CA)(κB κA) (V αB V αA)2. T p − p T − T − − In fact, the free energy difference between the two sides (4.22) of the coexistence acts as the phase potential, its first 16 derivative is force, the force of the two phases must be of the same value but pointing in the opposite direction at the critical point. The bifurcation equation D(φ/q) = 0 can be natu- rally generalized to a higher-order transition, it also leads to the coexistence curve of the higher order transition. We consider a system whose free energy is a function of temperature T and magnetic field B, then the Clausius- Clapeyron equation becomes dB/dT = ∆S/∆M. If the entropy and the magnetization are continuous− across the phase boundary, the transition is of higher order. For the pth order phase transition, the vector field is chosen 1 p 1 2 as the (p 1)th derivative of δF , φ = ∂T− δF, φ = p 1 − 1 2 ∂B− δF . Substituting (φ , φ ) into Eq. (4.20), we arrive p p p 1 p 1 ∂ δF ∂ δF ∂∂ − δF ∂∂ − δF FIG. 4: In analogy with Newtonian mechanics, the difference D(φ/q)= p p p 1 p 1 =0(4.28). ∂T ∂B − ∂B∂T − ∂T∂B − of output between two players is equivalent to gravitational 2 potential V , the first order phase transition means the two ∂ F CB Considering the heat capacity ∂T 2 = T and the sus- phases have the same potential. For p = 2, the order parame- 2 − A p−1 B p−1 ∂ F ter field, φ = ∂γA Out, φ = ∂γB Out, represents the force ceptibility ∂B2 = χ, the bifurcation condition D(φ/q)= 0 is rewritten as of phase A and B. When they reach balance, the two phase coexist on the hypersurface. This is the first order phase dB p ∆∂p 2C/∂T p 2 transition. For p = 3, φA and φB represents the accelera- = ( 1)p − − . (4.29) dT − T ∆∂p 2χ/∂Bp 2 tion, although the force is not equal, but the acceleration are c − − the same, this is the second order phase transition. On the This equation is in perfect agreement with the equations two sides of the coexistence surface, the tangent vector are in Ref. [16]. continuous, but the normal unit vector has a sudden jump. In mind of our holographic definition of phase transi- tion Eq. (3.36), we may also derive the holographic coex- istence equation using the fundamental order parameter Now we see, the universal coexistence equation not field, only reproduced all the coexistence equations of classical δp[R(θ)(F A F B)] δp[R(θ)δF ] phase transition in physics, but also gave more equations φ~ = − = , that have never been appeared before. This indicates δθp δθp that the game theory of renormalization group transfor- n 1 R(θ)δF = (Lθˆ )nδF, mation has very broad applications. n! 0 For the magnetic field and temperature depended free X ∂ ∂ energy F (B,T ), the scaling laws were derived in Ref. Lˆ = γ2 γ1 . (4.30) [16], the exponents is defined as ∂γ1 − ∂γ2 p 2 p 2 To study the pth order phase transition, one need to ex- ∂ − C µ ∂ − χ κ p 2 = a− , p 2 = a− . (4.34) pand the group element R(θ) to the p the order, and split ∂T − ∂B − it into the real part and imaginary part, i.e., The equivalent expression in terms of the vector field φ~ δp[R(θ)δF ] φ~ = = φ1 + iφ2. is δθp θ=0 1 µ 2 κ (4.31) ∂ φ = a− , ∂ φ = a− . (4.35) T P Then the coexistence curve equation is This scaling law is only a special case, there were many other scaling laws in various physical system, the same ∂φ1 ∂φ2 ∂φ1 ∂φ2 φ1, φ2 = =0. (4.32) value of critical exponents falls into the same universality { } ∂γ1 ∂γ2 − ∂γ2 ∂γ1 class. In the next section, we shall discuss the scaling laws Under this definition, it is easy to see that Kunmar’s re- based on the most general definition of phase transition. sult (4.29) is only special case of a series of coexistence equations for the pth order of phase transition, the com- plete coexistence curve equations are given by 4.3. Universal scaling laws and coexistence equation around Nash equilibrium point i 2p i j p j k p k ∂ δF ∂ − δF ∂ ∂ − δF ∂ ∂ − δF i 2p i j p j k p k =0, ∂T ∂B − − ∂B ∂T − ∂T ∂B − The phase transition of a physical system occurs at (i, j, k =1, 2, ..., p). (4.33) the critical point where the correlation length between 17 particles becomes infinite[2]. It is assumed that the elliptic surface, when κ1 > 0 and κ2 = 0, it is parabolic, free energy is a generalized homogeneous function, i.e., for κ1 > 0 and κ2 < 0, it is hyperbolic. Usually the F (λaγ1, λbγ2) = λF (γ1,γ2). The quantity defined by local manifold on a two dimensional output manifold is the free energy obey power laws around the critical point. hyperbolic at a phase transition point. In fact, if we carry From the two scaling exponent a and b, one may derive out the derivative of Eq. (4.36) to the seconde order, it those critical exponents which obey some equalities. The spontaneously leads to, systems with the same scaling law fall into an universality 2 ˆ 2 ˆ class. ∂γ1 Out = κ1, ∂γ2 Out = κ2, (4.37) When it comes to the general phase transition defined as a war game in this paper, all the critical phenomenons they are actually intrinsic geometric constant of the in physical reappeared. During the war, the correlation neighbor manifold around the critical point. between the all the members of the participants becomes Recall the game theory of renormalization group trans- infinity, people may not know each other, but every tiny formation in the first section, one would see that the work they do may cause great effect to the final results. game operator is a nonlinear operator, it defines an iter- If we focus on the individual person in battle field, one ative map for the game process. The dimension of the in- would see two opposite soldiers are fighting. Then we finite small neighboring manifold around the Nash equi- go to a larger scale, we see two companies are fighting. librium point can be exactly calculated from the game We can continue to magnify the scale, the participants operator. For most nonlinear game operators, the di- who are fighting range from hundreds of people to mil- mension of the manifold around the Nash equilibrium is lions of people, range from a small village to the whole fractal instead of integer. There are only a few very sim- world. No matter from which scale we see it, it is the ple cases that one can find an integer dimension. But the same war. At the critical point, the participants of the game operator in that cases is too trivial to give us any in- war have equal strength, if any one of them make a tiny teresting phenomena. According to the experiments and mistake(the mistake may come from an unimportant sol- numerical calculation of physics, we can make a general dier), the whole army will lose the war, so the correlation hypothesis that the neighboring output manifold around length between soldiers goes to infinity. In this sense, the the critical point of phase transition, namely around the output field of the war should be a generalized homoge- Nash equilibrium point of a non-cooperative game, has neous function at the critical point, it obeys the relation fractal dimension. i Oˆut(λ γi)= λOˆut(γi). In the vicinity of the phase transition point,an approx- According to the topological current of phase transi- imation of the scale invariant output manifold is a com- plex function in fractal space, tion, we established the tangent vector field of Oˆut. The tangent vector field is the projection of physical field con- d ˆ di j dk dm figuration on the strategy manifold. The physical field di- Out(γ)= κiγi + κm[f(γj + γk ...)] + ..., (4.38) vergent at some singular points where the tangent vector field vanished. It was proved that the nontrivial Rieman- where d1, d2, ..., dn are fractal dimensions wit respect to nian curvature just around these surviving strategy. The different input parameters. Recall that most of the phys- integral of the Riemannian curvature is a topological in- ical observables in statistical mechanics are defined by variant, the critical exponent should bear a topological the second order derivative of free energy, we can define origin. the observables of a complex system by the second order According to the universal definition of phase transi- derivative of the output field. The free energy is only tion, the phase transition point is a Nash equilibrium a special component of the output manifold in physics. solution. A special two dimensional output manifold is When we study the most general complex system, as long a saddle surface in the vicinity of the critical point. The as people can measure it, we can take any order of deriva- output manifold is maximum for one parameter, but min- tive of the output field as observables. These observables imum for another parameter. are just the components of the tangent vector field of the Suppose the topological dimension of the manifold output field. A simple example of the tangent vector field around the critical point is integer, i.e., D = 1, 2, ..., n, is we may introduce a local coordinates to approximately 2 ˆ ∂ Out(γ) di 1 dj 1 express the output manifold in the vicinity of the critical φij = = diγi − + dj γj − + . (4.39) point as ∂γi∂γj ··· κ κ According to the topological phase transition theory, the Oˆ = 1 γ2 + 2 γ2, (4.36) ut 2 1 2 2 tangent vector field satisfy the phase coexistence equa- tion φ1, φ2, ..., φn = 0 at the critical point, we can de- { } where γ1 is the infinite small variable, based on which the rive a constrain on these vector field. We substitute the 1 1 local coordination is r1 = (δr1 + Γ γiγj )√gii + . We explicit expansion of the observable quantities into the 2 ij ··· may abandon the quadratic term of γi. Eq. (4.36) is the coexistence equation, it would lead us to a constrain on local approximation of the output manifold. κ1 and κ2 the fractal exponent index. These constrain relations are are principal curvature. When κ1 > 0 and κ2 > 0, it is a just the scaling laws. Therefore scaling laws come from 18 the coexistence equation of the crossing defined physical commutable in the vicinity of critical point. This is in quantities. consistent with our picture of war game at the phase The scaling relations found in various physical system transition point. Further more, one may choose different are probably the simplest relations on the fractal space tangent vector field for the coexistence equation, then one extended by two parameters. One can reach all kinds of may obtain all different scaling relations in the vicinity different scaling relations[2] in statistical mechanics by of critical point. taking the output field as free energy Oˆut(γi) = F (γi), and taking the input γi as physical parameters, such as temperature T , pressure P , magnetic field B, and 4.4. The symmetry of Landau phase transition so on. In the vicinity of a second order phase tran- theory and symmetry of game theory sition, the divergent physical quantity are defined by the second order derivative of the free energy. Such The Landau theory of continuous phase transition the- as the susceptibility χ = ∂2F/∂H2, H is magnetic ory provides a basic description to the phase transition − field. Each divergent quantity is characterized by a crit- characterized by spontaneous symmetry breaking. Take ical exponent, this critical exponent comes from fractal its application to the structure phase transitions as one space. The two component coexistence φ1, φ2 = 0 pro- example, it derives several important features, namely, { } duced the scaling relations, such as the Fisher relation the change in the crystal’s space group, the dimension νd =2 α, Widom relationγ ˆ = β(δ 1), Rushbrooke re- and symmetry properties of the transition’s order param- − − lation α+2β+ˆγ = 2, and so on. We take the Rushbrooke eter, and the form of the free energy expansion. It has relation as example to verify the coexistence equation. the same range of validity as the mean-field approxima- The Gibbs free energy is G = U TS, its differentia- tion in microscopic theories. A central assumption of the − tion is dG = SdT + V dP MdH. Experimental and Landau theory is that the free energy can be expanded − − numerical calculation found that three thermodynamic as a Taylor series with respect to the order parameter η: quantity obey the following scaling laws in the vicinity 2 3 4 of critical point, F (P,T,η)= F0 +A(P,T )η +B(P,T )η +C(P,T )η +... (4.44) ∂G M = ( ) T β, in which the phases are marked by the order parameter η. − ∂H ∼ | | The symmetry in Laudau theory talks about the invari- 2 ∂ G α ance of this free energy when we do some transformation CP = T ( )P,H T − , − ∂T 2 ∼ | | on the order parameter field η′ Uη. This symmetry → ∂G γ is the same conception as that in quantum field theory, χ = ( ) T − . (4.40) 4 1 2 1 2 2 λ 4 − ∂H ∼ | | such as the ψ model, = 2 (∂µψ) 2 m ψ 4 ψ . If the equations of motionL derived from− this Lagrange− is in- The fundamental vector field can be taken as variant under some transformation ψ ψ′, we call such → ∂G ∂G a transformation a symmetry transformation. φ1 = ( ), φ2 = ( ). (4.41) But in this paper, the symmetry we are talking about ∂H ∂T in the game theory of phase transition is a different con- Substituting the two vector field into the coexistence ception. equation φ1, φ2 =0, one may derive The object of our research is a very general system { } O~ ut = Fout(~x,~γ), ~γ is the input vector. In game theory, ∂2G ∂2G ∂2G ∂2G =0. (4.42) the input vector represent the input strategies of different ∂T 2 ∂H2 − ∂T∂H ∂H∂T players. In physics, the input vector are physical oper- ~ Now we substitute the thermodynamic quantities (4.40) ational quantity. Out is output vector, it encompasses into the coexistence equation, it yields all the information the observer can received by send- ing different inputs. In physics, the outputs are physical α γ 2 2β 2 observables. The state ~x represent the inner states of T − − = β T − . (4.43) | | | | system, it plays a similar role as the order parameter When T 0, we may ignore the coefficient β2 at the field η in the free energy expansion equation (4.44). In right hand→ side of Eq. (4.43), then we obtained the Rush- our game theory of topological phase transition theory, brooke relation α+2β+ˆγ = 2. Other scaling relation can all the state vector have been integrated out, the funda- be verified following similar procedure. These relations mental starting point is the output field O~ ut = Fout(~γ), were firstly found by computational simulation and ex- the symmetry we mentioned in this theory is about the periments. Therefore the scaling law of universal phase transformation invariant property of O~ ut under the trans- transition in a general complex system has solid numeri- formation ~γ∗ U~γ. For example, in the free energy cal and experimental foundation. Here it must be pointed equation (4.44),→F (P,T ) is output, P and T are the two out that the commutable relation ∂ ∂ = ∂ ∂ have players, we study the symmetry of F (P,T ) when T T ′ T H H T → been used in the calculation. This suggests that the par- and P P ′. The order parameter η is not an operation tial differential corresponding to different variables are quantity,→ it describes the inner state of the system. 19 The Landau theory of phase transition can be sum- The coexistence curve equation φ1, φ2 = 0 is an univer- marized as a differential game(see Appendix D,E). The sal equation, it holds for this two{ player} game. Consider order parameter is the state vector of the game. In the the two special vector field Eq. (4.47), we arrived a so- frame work of our topological current theory of phase phisticate coexistence equation, transition, the tangent vector field φi consists of the com- 2 2 2 2 2 2 2 plete set of phase dynamics system. The vector field 4η ∂T C[∂P A(T, P )+2∂P Cη ] = [∂P a(P )+2∂P ∂T Cη ] . i q 1 p 1 δF (P,T,η) If we have obtained the explicit relation of A(T, P ) and φ = ∂ − ∂ − ( ) (4.45) P T δη C(T, P ), then we can depict the phase diagram in T P plane for different state vector η. − describes how the two player T and P behave across the state vector space η. The free energy is the output func- tion. F (P,T,η) should be written as an expansion in the 4.5. Symmetry and evolution of phases in game even powers in the spirit of Gintzburg-Landau formal- theory ism, for F (P,T,η) must be gauge invariant with respect to the order parameter η. Then the series of free energy In game theory, the output vector are payoff functions, in even powers truncated at the fourth order is the inputs are strategies. A symmetry transformation in 2 4 strategy space does not always lead to different output, F (P,T,η)= F0 + A(P,T )η + C(P,T )η , (4.46) the Nash equilibrium solution is invariant fixed point un- A special phase vector field may be chosen as der continuous transformation. As shown in Ising model, 1 1 2 the renormalization group transformation keep mapping φ =2∂T [A(P,T )+2C(P,T )η ]η, a pair of strategy to another pair, no matter where it 2 1 2 φ =2∂P [A(P,T )+2C(P,T )η ]η. (4.47) starts, it always flows to the fixed point. The phase of a system evolutes in the direction of If φ1 and φ1 do not commute with each other, both the renormalization group transformation flow. The direc- two players have surviving strategy which are the solu- tion of renormalization group transformation flow is de- tions of φ1 = 0, φ2 = 0, these solution sit at some iso- termined by the Second Law of Thermodynamics, which lated points states all physical systems in thermal equilibrium can be Tk = Tk(η), Pk = Pk(η), k =1, 2, ..., l, (4.48) characterized by a quantity called entropy, this entropy cannot decrease in any process in which the system re- Each of these isolated solutions has a winding number mains adiabatically isolated. We can ignore the thermo- Wk. These surviving strategy pair (T ,P ) are changing dynamics, and grab the central point that an isolated sys- according to the state vector η. The surviving strategy tem only evolutes in the direction of increasing entropy. of new phase and old phase carry opposite winding num- This statement of the Second Law of Thermodynamics ber, they are annihilating and generating at the phase may also hold in game theory, but notice that the entropy coexistence region. The sum of these winding number is here in our theory is not the conventional conception de- a topological quantity which is determined by the topol- fined in statistical physics, since we are not manipulating ogy of the state vector manifold. . the physical particles, it is the entropy of players ~γ in- Usually A(T, P ) has the form of a(P )(T Tc) near stead. In physics, the players are those physical observ- − the critical temperature Tc, and C(P,T) is supposed to ables. An isolated system evolves spontaneously toward be weakly dependent on the temperature, i.e., ∂T C 1. a maximal symmetry. ≪ Considering Eq. (4.47), the surviving strategy for the old The entropy of the multi-player game measures indis- phase and new phase can be derived from tinguishability of players. Let’s look at an ancient battle- field with two armies ready to fight, before the beginning η[a(P )+2∂ C(P,T )η2]=0, T of the combat, the two troops are organized in ordered η[∂ a(P )(T T )+2∂ C(P,T )η2]=0. (4.49) P − c P states separated by vacant glacis, every part of them has One sees that η = 0 corresponds to a stable phase solu- special functioning. A bystander can tell which is which. tion of the equation above. For the case η = 0, there are Once the combat begins, they rush at each other and two solutions: 6 fused into one. In this case, the bystander is unable to distinguish them, it is completely in chaos. When the war a(P ) 1/2 ∂ a(P )(T T ) 1/2 finished, the loser surrendered, winner gathered, the two η = − , η = P c − , ± 2∂ C ± 2∂ C troops translated into another ordered states. The sur- T P (4.50) viving troops are distinguishable again, but those dead if we chose C = exp(T + P ) and a = exp(P ), it turns can never be back to ordered states again. into a familiar form High symmetry leads to high entropy. If we do some transformation to a system in chaos, it won’t make any 1/2 1/2 a(P ) a(P )(Tc T ) difference. As in the war, the two armies become a mix- η = − , η = − . 1 ± 2C 2 ± 2C ture, it is an unstable equilibrium state with the highest (4.51) symmetry. The higher symmetry a system owns, the 20 more unstable it would be. The state before and after space. As shown in section (3.2), a multi-player game can the war are both of less symmetry states. Like the be- be equivalently expressed as a group of differential equa- ginning of universe, all different conflicting interactions tions. The evolution of output vector O~ ut is governed by confined in a singular point, reach an unstable equilib- the differential equations, rium state. It has the highest space time symmetry. A minor imbalance results in the Big Bang. The explosion ∂sO~ ut = Lˆ(γ, ∂γ)O~ ut, i =1, 2, ..., n. (4.54) breaks the singularity apart into all different symmetry zone. Each symmetry group brings about a stable phase. If we take the tuning parameter s as temperature T , then The stable phases at different stage of history may be we understand the evolution of phase with respect to found following the group chain of the original symmetry temperature. In physical system, we can obtain the func- group. A high symmetry group usually has a series of tion of linear response by perturbation theory. The re- subgroup. A physical system liken to stay in the most sponse function O~ ut may have sophisticate relation with fundamental symmetry state. People love peace, but hate input parameters, O~ ut(γ). Some input parameters are war. They like to live at the most stable point of the function of temperature γi = γi(γj ). For example, dur- system. A war cost too much energy. The subgroup ing the isothermal process, the product of pressure and chain of a given group is always finite, volume is a constant P V = constant, so P and V are U U1 U2 U3... Up. (4.52) related by this equation. We take the γi(γj ) as funda- ⊃ ⊃ ⊃ ⊃ mental variables, γj is the tuning parameter. Then the For example, SO(4, 2) constitutes a dynamical group for generalized momentum is defined as the Hydrogen atom[15]. It has many four subgroups[17], ∂φ(γ, ∂γ γ,γj ) SO(4, 2) SO(4, 1) SO(4) SO(3) SO(2). P = j , (4.55) ⊃ ⊃ ⊃ ⊃ γi i (4.53) ∂(∂γj γ ) This group chain may be followed by point group. For instance, the U(1) symmetry group may be decreased where φ(γ, ∂γj γ,γj) is a reference function like La- to Tn which means a physical system is invariant when grangian function. We can further find a vector field 2π φH (γ, ∂γ γ,γj ) which is similar to the Hamiltonian func- it rotates n around one axel. So we can roughly find j the stable phase according to the group chain. Generally tion in Lagrangian mechanics. For a given field φH , the ˆ speaking, Lie group is highes rank of group. The discrete game operator L(γ, ∂γ ) bear an expression in terms of group are imbedded in Lie groups. The most stable phase Poisson bracket, focus on the lowest symmetry. ∂φH ∂ ∂φH ∂ A game system evolutes following the first principal, Lˆ(γ, ∂γ)= . (4.56) the players take different strategies to ground state. The ∂Pγi ∂γ − ∂γ ∂Pγi N players represents N interaction, or N input vectors. In the beginning, each of them occupies a small domain This formula of game operator leads to another equiva- in phase diagram and is the ruler in his own domain. lent representation of the differential equation , There is temporary balance between the different play- ∂ O = φ , O . (4.57) ers. The war is going on all the time between neighboring γj ut { H ut} domains. Whenever a player loses, he dies out, his ene- mies absorbed his domain. When a player fails, an old This equation actually describes the game process be- phase died, and a new phase is born, this indicates a tween φH and Out, each of them navigates one stable phase. If they are in the same complete set, φ , O = phase transition. The balance on the boarder between { H ut} different domains represent the phase coexistence region, 0, then the two phases coexist. In fact, the Hamiltonian on which Nash equilibrium is reached. But this state is function does not play any special role in our topological an unstable equilibrium state, there is constantly minor current theory of phase transition, it is just one of the conflicting to destroy this balance. Then the war goes tangent vector field in Lie algebra space. on, the winners gain advantage, and becomes stronger and stronger, the war will not stop until he unified the domains as one. 4.6. Phase coexistence boundary and unstable The phase diagram of a physical system is depicted vacuum state in the frame of some static physical parameter. We can view the evolution of phase as an continuous tuning of As we defined in the beginning, phase transition is a the physical parameter. For example, if we continuously transition from one stable state to another stable state, raise temperature, we will see single curve in the phase there is a critical point at which the old stable state col- diagram bifurcates at certain temperature, this is the lapsed and the new state arise from the shambles and evolution of physical phase, although it is not the usual grow to a stable state. A oversimplified model is take two evolution in the common sense of time. states as a vector of two component (x, y), the evolution This phase evolution with respect to temperature can operator is a diagonal 2 by 2 matrix Lˆ = diag( 1, 1) be viewed as a generalized dynamic process in parameter at the coexistence region. The evolution of the− two 21 p ˆ 1 2 n δ [U(θ)Out] phase follows the equation ∂γ~r = L~r, which we estab- (φ , φ , ..., φ )= δθp θ=0. For this equation is an- lished in last section. Then in the vicinity of the coexis- tisymmetric, exchanging any| two of the players would tence region, we would see the old phase decays follow- add a ( 1) to the output. As shown in the topologi- aγ − ing x e− , in the meantime the new phase blows up cal current of phase transition, the sum of the winding y e+∝aγ , where a> 0. numbers around the surviving strategies is a topological ∝The mechanical simulation of the phase coexisting number—-Chern number. The sign of each winding num- state is a ball on the maximal tip of the parabola f(z)= ber is determined by the sign of φ1, φ2, ..., φn . Every φi z2. At Nash equilibrium point, a player take the best field is a player, if exchange any{ two of them,} the winner strategy− to minimize his own damage, and obtain his becomes loser, the loser turns into winner. Therefore if maximal profit in the meantime. In fact, it is the profit φi has a positive winding number, his opponent must has difference between them that they fight for. When the a negative one. The strategy for φi to survive is the anti- two conflicting force reach a balance, the game arrived strategy of his opponents. During phase transition, the at a Nash equilibrium. old phase is the opponent of the new phase. The wind- j j The Nash equilibrium state is an unstable maximal ing number of the new phase’s strategy γ is WNew > 0, point in the mechanical potential of the output field. Let the winding number around the surviving strategy γi of ˆi i i Out(~γ) be the output function of the player γ , the ef- the old phase is WOld < 0. The topological constrain fective potential for two players in a game is ∆Oij = suggests ˆi ˆj Out(~γ) Out(~γ). Nash equilibrium sits at the minimal − i point of δOij . Notice that the player γ ’s profit is the N = W i + W j . (4.61) | | j ChernNumber Old New damage of player γ , so ∆Oij = ∆Oij . We may es- i j tablish a general equation of motion− for player γi’s profit X X function, We may view each strategy of the new phase as one parti- 2 i i cle with topological charge WNew, and its corresponding ∂ i Oˆ = ∂ i V (Oˆ ), (4.58) γ ut γ ut anti-particle is the anti-strategy of the old phase which carries a negative winding number. The soldiers of the V (Oˆi ) is a general potential which is self-consistently ut new phase are the strategies, they carry positive wind- decided by other players, usually the player γi sits at ing number, rush at the coexist curve to fight against the some of its minimal points, while the other players occu- anti-strategy of the old phase. The particles and antipar- pied its maximal point. The typical profit function Oˆi ut ticles annihilate at the phase boundary, so the coexisting of player γi is a kink solution, phase boundary behaves as vacuum. This war is going i ∗ ∗ i on under the constrain of Eq. (4.61). eγ γ eγ γ ˆi i − − µ O = tanh(γ γ∗)= i ∗ − ∗ i . (4.59) The generator of translation group along γ is i∂ µ . ut ± − eγ γ + eγ γ γ − − The evolution of the output on the strategy space follows where γ∗ is the optimal strategy. Eq. (4.59) is the fa- the classical Hamiltonian-Jacobi equation, miliar kink solution of quantum tunnelling problems[18]. Any minor deviation from the Nash equilibrium solution i∂γµ Oˆut(θ) = Oˆut(θ) ,L , (4.62) would results in drastic increase or decrease of the profit h i {h i } function. Therefore the Nash equilibrium solutions of a The second quantization of this equation is just the game play the same role as the vacuum solution to quan- Heisenberg equation i∂tOˆut = [Oˆut,H], which has a more tum tunnelling. A collection degenerate vacua means general covariant form there are a series of Nash equilibrium solutions with the same optimal value. ˆ ˆ ˆ The game of phase transition has two different equi- i∂γµ Out = [Out, Pµ], (4.63) librium states. One is all different interactions reach an µ agreement to maintain peace, this is the optimal strat- here γ = (t,~γ), Pµ = (H, P~). In four dimensional egy of cooperative game. It is the trivial vacuum. The Minkoveski space time, the Hamiltonian is the generator other is that no peace agreement is derived, a war breaks of translation group with respect to time t. While the out. At the critical point, different phases coexisted but momentum operator Pˆµ is the generator of space trans- against each other. This is the Nash equilibrium state. lation group. Integrating the covariant Heisenberg equa- It is an unstable vacuum state. tion (4.63), we arrive The coexistence equation of n-different phases we ob- tained in previous sections, iPγˆ µ iPγˆ µ Oˆ(γ)= e Oˆ(0)e− . (4.64) φ1, φ2, ..., φn =0, (4.60) { } If the output is expressed by quantum operators, its evo- describes a Nash equilibrium state, or in other words, lution in strategy space is governed by renormalization the coexisting unstable vacuum state, here φ~ = group transformation 22 2 1 iθ~n L~ iθ~n L~ (iθ) U(θ) Oˆ U − (θ)= e · Oˆ e− · = Oˆ + iθ [~n L,~ Oˆ ]+ [~n L,~ [~n L,~ Oˆ ]] + . (4.65) h i h i h i · h i 2! · · h i ··· For the most simple Lie group SO(2) whose Lie algebra This operator is defined from by vectors of Lie algebra, has only one generator Lˆz, this transformation equation we call it phase coexistence operator. According to the takes a very simple form, group representation theory[20], we can always find the representation of coexistence operator, 1 Uδ Oˆ U − n h i 2 Dˆ ψ = D ψ . (4.71) (iθ) | ni | ni = δ Oˆ + iθ [Lˆz, δ Oˆ ]+ [Lˆz, [Lˆz, δ Oˆ ]] + . h i h i 2! h i ··· If n = 2, Dˆ n is just the commutator of two operators In quantum mechanics, if a group transformations U which represent two stable phases. The phase boundary commutes with Hamiltonian Hˆ ,[U,H]= HUˆ UHˆ = 0, between phase A and phase B is given by the zero modes − i.e., H = H = UHUˆ , then Hˆ is invariant under trans- of the commutator of the two phase operators. When ′ † ˆ ˆ formation of group U, they share the same eigenfunction. φA and φB are not commutable, the eigenfunctions of In a general game theory, the Hamiltonian has nothing the phase coexist operator may be divided into positive special. The generators of Lie algebra plays the role of modes and negative modes, Hamiltonian operator. The quantization of the coexis- Dˆ n ψ = signW ψ . (4.72) tence equation reads | ni | ni ˆ1 ˆ2 ˆn in which W is the winding number around the surviving [φ , φ , ..., φ ]=0. (4.66) strategy. sign(W )=+1 > 0 corresponds to stable phase A and sign(W ) = 1 < 0 corresponds phase B, while Each operator φˆi represents a quantum operator, if pt (W ) = 0 indicates the− phase boundary. One of the most they commute with each other, they share the same pt familiar example is to take the phase operator as the eigenspace. Eq. (4.66) is the quantum coexistence equa- three component of angular momentum operator, φˆ = tion. A ˆ ˆ 2 If the φˆi are vectors expanded in the tangent space Lx and φB = Ly, then D = Lz. Lz ψm = m ψm , m = 0, . m = 0 is the coexistence eigenvalue,| i | mi= 1 of Lie group around the identity, they are vectors of Lie { ±} ± algebra. For Mathematician, this coexistence equation represents the two stable phases. corresponds to an invariant Cartan space of Lie alge- In topological quantum field theory, the topological in- bra. The coexistence phase space is the eigenspace of formation of the zero mode is described by the Atiyah- Cartan subalgebra. The Cartan subalgebra is the maxi- Singer index theory. The strategy space of the game of mal Abelian subalgebra. An arbitrary Lie algebra vector phase transition can be split into positive eigenspace and commuting with this subalgebra is still a Lie vector in negative eigenspace. The surviving strategies of the new phase carry positive winding number, we call them pos- the same space. The elements Cj in this commutative + subalgebra must satisfies itive modes φ . The strategies of the old phase is the enemy of new phase, they carry negative winding num- [Ci, Cj ]=0, (i, j =0, 1, 2, ..., l). (4.67) ber, so we call them negative modes φ−. As shown by Eq. (4.9), our topological current theory of phase transi- l is the dimension of the Cartan subalgebra space. It is tion proved that the Euler characteristic number on two also the number of coexistence phases. For the coexist- dimensional strategy space is, ing point of three phases, we define the three operator commutator as 2 ~ 1 2 2 ~ 1 2 Ch = δ (φ+) φ+, φ+ δ (φ ) φ , φ , { }− − { − −} Z Z [Ci, Cj , Ck] i j = W+ W . (4.73) − − = [Ci, Cj ]Ck + [Cj , Ck]Ci + [Ck, Ci]Cj i j X X =0, (i, j, k =0, 1, 2, ..., l). (4.68) For the 2n-player game, the topological Chern number It has a straight forward generalization for the points at on this 2n dimensional strategy manifold is determined which n phases intersects, by Atiyah-Singer theorem, ˆ n ˆ n ˆ n [C , C , C , ..., C ]=0, (i, j, k, ..., f =0, 1, 2, ...).(4.69) ChBi = IndexD = dimD+ dimD , (4.74) i j k f − − So we have a well defined quantum operator where Dˆ n is th quantized phase coexistence operator. The positive modes behaves as particles, and the negative Dˆ n = [φˆ1, φˆ2, ..., φˆn], (4.70) modes are antiparticle, they coexist in vacuum. They are 23 born by pairs from vacuum, and annihilate by pairs to ment between different phases reaches a maximal point. vacuum. Every local maximal point of the entanglement indicates In the language of Atiyah-Singer index theorem(see ap- a transition, or a war. In order to give an exact pre- pendix section I), the topological index is the difference diction on where or when the war arise, we need to find between the number of positive eigen-modes and nega- some detectable quantity to measure the entanglement. tive eigen-modes. The positive modes is the surviving So that we can quantitatively determine the position of strategy for the new phase, the negative modes is the the phase transition point. surviving strategy for the old phase. Thus the topologi- The most familiar quantity for physicist is the von Neu- cal index counts how many extra strategies the new phase mann entropy. Quantum statistics suggests that the von have after his soldiers rushed at front and annihilated at Neumann entropy of a pure ensemble is zero. Entropy is a the coexistence phase boundary. We first assume every quantity to measure how disorder a system is. More dis- surviving strategy has a topological W = 1 for con- order means higher entropy. The von Neumann entropy venience, the Euler number Ch = +2| on| a sphere, this measure the disorder of the mixed states. The von Neu- topological constrain says that when all the soldier of the mann entropy is a relatively small quantity if the system old phase died, there are at least two positive soldiers of is in a stable phase. When the phase transition occurs, the new phase left. The victory of the new phase has been it would reach a climax point. determined by topology of the strategy manifold. If it is In fact, any sensitive output function corresponding to on a torus, the Euler number is zero, Ch = 0. The new a group of inputs can be used as measure of entangle- phase is not that lucky now, his total number of soldiers ment. If the inputs have strong relation between them, is identical with that of the old phase. When the war one minor change would definitely change the others, and break out, the strategy-anti-strategy pair annihilate on this would leads to the response of many outputs. One the phase coexistence boundary, the final result is a draw. can see this from a war, it is at the war that the ignor- When the base manifold of strategy space is a torus with able individuals began unite to work as a team, they are n-holes , the Euler number is Ch = 2(1 n),n> 1, the strongly correlated and entangled. final victory goes to the old phase, he has− at least 2(1 n) In our topological current of phase transition, a quan- surviving soldiers when the new phase is extinguished.− tity to measure entanglement may be defined as 1 Eent = exp [ D(φ/γ)/(4h)] (5.1) 5. PHASE TRANSITION AND 2√πh − ENTANGLEMENT IN GAME THEORY where D(φ/γ)= φ1, φ2, ..., φn is the coexistence equa- tion, and h is step{ size of renormalization} group trans- 5.1. Phase transition and entanglement formation. As shown in the previous section, the phase evolution equation in parameter space is ∂γj Out = The participants of a war entangled with each other φH , Out = LOˆ ut. So Eq. (5.1) is some kind of prop- before the war breaks out, otherwise there will not be a {agator of} phases similar to the propagator in physics war at all. In fact, a game is played by many interacting iHt U(t) = e− . We rewrite the coexistence equation players. During the game, a player choose strategy ac- D(φ/γ) as Det Φ, where Φ is the matrix extended by cording to other players’ strategy, so he could never be a j ∂γi φ . Recall the definition of Pfaffian for matrix M, free particle unless he is not in the game. [Pf M]2 =Det M, we can decompose the coexistence The entanglement we are talking about here is a much equation as D(φ/γ)=[Pf Φ]2. The entanglement equa- more general conception, it includes the relation, inter- tion (5.1) reads, action or connection between the elements of a system. The quantum entanglement in physics is one special case. 1 (Pf Φ)2 When the players are at peace, the entanglement be- Eent = exp [ ]. (5.2) 2√πh − 4h tween them is kept at a stable level. As the imbalance be- tween different players increase, the war is coming, their A typical physical example of Pfaffian is the fermionic entanglement grows stronger and stronger. The whole parity. In the game theory here, a Pfaffian is a sum over system becomes more and more unstable. When the all partition of the players into pairs, exchanging any two critical point arrived, a negligible event trigged the war, of them contributes a minus sign. For a bargain game which spread the whole system through the strong en- between buyer and seller, the step size h is amplitude of tanglement. During the war, all the players summon up the unfixed money they are fighting for. The smaller h their internal strength to collect information from all the they are negotiating on, the more information they are other players, and make the best strategy to win the war, exchanging so that they could persuade each other to the entanglement reach a climax. The external response accept his offer. This means the entanglement between during the war also reached the strongest level. When them increases when h approaches to zero. So the entan- the war is over, everything is in order, their entangle- glement increases when the renormalization group trans- ment gradually decay to another stable level. formation goes on. The entropy increases in the mean Therefore, phase transition occurs when the entangle- time. The maximal entanglement appears at the phase 24 coexistence state, standing on how the entanglement grows stronger and stronger in a game process. 1 (Pf Φ)2 The most fundamental principal for a physical system Eent = lim exp [ ]= δ(Pf Φ). (5.3) h 0 √ − 4h is the Principle of Least Action. Newton’s mechanics is → 2 πh unified in Hamilton’s principle of least action as well as Here Pf Φ = 0 is an equivalent expression of the coexis- in Gauss’s principle of least constraint. Maxwell’s equa- tence equation. The matrix Φ in Eent is constructed by tions can be derived as conditions of least action for elec- output vector field. The output vector φ~ could be any tromagnetic field propagation. When light goes through physical parameters. For the most simple case of two optical systems, it always take the path of least time, phases, φ~ has two component φA and φB, the Pfaffian is and takes short cuts in glass and water where light trav- just the Poisson bracket. els slower. The most stable state of many body system is Besides the statistical observable or external response, the ground state, which possesses the least energy. The the output vector can also be chosen as the conventional ground state is the final game results of the particles in order parameter which is a quantity used to characterize many body system. Every particle wants to stay at the the structural and inner order changes of physical system most stable point, it is the medal all the other particles at the phase transition point. In the superconductor- struggle to get, this drives the particles in the game. insulator transition, upon tuning some parameter in the The Principle of Least Action of a cooperative game Hamiltonian, a dramatic change in the behavior of the is to maximize the profit of the whole group. The Prin- electrons occurs, the order parameter of this quantum ciple of Least Action of a non-cooperative game is that phase transition φ = ∆eiθ is the energy gap function of all players only take strategies to maximize his payoff cooper pair theory. For the ultracold Bosonic atom gas function or to minimize his loss function. Of course his confined in an optical lattice, the order parameter is the personal strategy must take into account of other play- mean field value of the operator of Bosons φ = b = b† . ers’ strategy, because if the whole group breaks down, he There were some numerical calculations in theh Hubbardi h i will get nothing. model, it shows the entanglement follows different scal- The players may be classified by their statistics in anal- ing with the size on the two sides of the critical point ogy with the statistics of particles in physical system. As denoting an incoherent quantum phase transition[21]. all know, there are two types of elementary particles: Entanglement state is the most important resource for Fermions and Bosons. There are anyons whose statistics quantum information technology. The entanglement be- stands between fermions and bosons, we first put that tween the different stable phases is a good candidate for aside for convenience. The combination of fermions may quantum computer. The most entangled states exist at form boson. Particles obeying Bose statistics shows a sta- the critical point of phase transition. The critical point is tistical attractive interaction. While the fermions demon- where the war breaks out, one of the fundamental char- strated a statistical repulsive interaction which comes acter of war is chaos, thus any minor change of parameter from the Pauli exclusion principle. would leads to totally different results. Quantum com- The predetermined physical environment is the game putation using entangled states is extracting information rule of the players. Players choose their states accord- from chaos, and control its output in a exact way. In ing to its interaction with external field and other play- fact, no matter it is classical system or quantum system, ers. We can alternatively specify each strategy profile entanglement is the basic source of information manip- (s1,s2, ..., sN ) by the occupation number (n1,n2, ..., nN ), ulation, the best entanglement states exist in a chaos which means there are ni players take strategy si. Since state. The biggest problem for present quantum comput- all men are born free and equal, there would be no payoff ing schemes is decoherence of entanglement, the quantum difference if they choose the same strategy. entanglement decays rapidly as time goes on. The players in a non-cooperative game behaves as However, the entanglement we present in this section fermions in physical system. If we exchange any two of is independent of time, it only relies on the physical pa- the players, the difference of profit between them would rameter. It would be much easier to tune the physical change a sign. They do not share the same occupation. parameters that to fight against time. So the entangle- So the expected number of players in noncooperative ment in the vicinity of phase transition is very promising game is fermi-dirac distribution, candidate for quantum computation. gi ni = , (5.4) β(u0 ui) e − +1 5.2. Quantum states and Nash equilibrium solution where ui is the benefit the player could get by strategy of games state si. The players in cooperative game are altruistic players, they concern for the welfare of others to maxi- We have shown that phase transition occurs at mize the benefit of the whole group. They do not care the Nash-equilibrium point of a non-cooperative game. too much about their own profit. so exchanging two of While the most entangled states just arise from this them does not make any difference, they like to share Nash-equilibrium point. We need to get a deeper under- with other players. In this case, the distribution obey 25 Bose-Einstein statistics, altruistic players; (3) Alice is selfish player and Bob is altruistic players; (3) Alice is altruistic player and Bob is gi ni = . (5.5) selfish players. Selfish player take strategy to maximize eβ(u0 ui) 1 − − his own benefit, and altruistic players tries to maximize However, it is naive to say a player in reality is altru- the other’s benefit. istic player who only concerns about others or the ego- If we know which kind of players Alice and Bob are, istical player who only concerns about himself. Every we can find the Nash equilibrium fixed point, the two real player is in a mixed state of the altruistic and the players does not need to commute with each other. For selfish. We consider a rational player, each time he sets case (1), Alice knows Bob would certainly choose confess a step further by choosing a particular strategy, he has that can maximize his benefit, the only Nash equilibrium to confront a wining result or a losing results. When he solution is that she also confess, so they reach the Nash loses, his benefit is transferred to other players, we call equilibrium point (a, a). For case (2), the Nash equilib- him altruistic, on the contrary, we say he is selfish. We rium is neither of them confess. In case (3), Alice does expressed the altruistic state as 1 and the selfish state not confess, Bob confess. In case (4), Alice confess, Bob as 0 , | i does not confess. The above is the case when the two | i players are in pure states, Alice(bob) is either altruistic altruistic = 1 , selfish = 0 . (5.6) 1 or selfish 0 . We denotes the four mixed self-states | i | i | i | i as| i | i 1 and 0 form an orthogonal basis for the self-state |spacei of| player,i i.e., 1 0 = 0. Because when he faces 1 A 1 B, 1 A 0 B, 0 A 1 B, 0 A 0 B. (5.9) h | i | i | i | i | i | i | i | i | i a particular pure strategy, there is only two possibilities: The Nash equilibrium solution of the four states are sum- take it or give it up. An arbitrary mixed self-state vector marized in table (5.10) is the linear combination of the two basis, Alice Bob equilibrium point m = a 1 + b 0 , (5.7) | i | i | i 0 0 (a, a) |0i |1i (0, c) (5.10) where a and b are complex number. m is a unit vector, | i | i | i 1 0 (c, 0) m m = 1, it is equivalent to a2 + b2 = 1. m means | i | i h | i 2 | i 2 1 1 (b, b) the player has the possibility of a to be altruistic and b | i | i to be selfish. This table is the density matrix of the prisoner dilemma. A game consists of N players is physically equivalent The statistical weight of 1 A 1 B is the Pareto optimal to a many body system with N particles. Each player point of collective payoff.| Thei | i statistical weight of self- represents a particle, the self state is the quantum states state 0 0 is Nash equilibrium. For the other two | iA| iB of the particle. A statistic distribution of the N particle states 1 A 0 B and 0 A 1 B, the two equilibrium points states can be present in an ensemble which is a collection appear| ini the| i off diagonal| i | i elements of the payoff matrix. of identically prepared physical system. For a n player The above is the simplest case of prisoner’s dilemma, game, the global strategy state space is the product space there are only two available strategies: confess or non- of the n players strategy space, confess. Now we consider a more complex case: the avail- able strategies of Alice is eA1,eA2, ..., eAq and Bob’s S = S(1) S(2) ... S(n). (5.8) { } ⊗ ⊗ ⊗ strategies are eB1,eB2, ..., eBp . The payoff matrix is now a q p bi-matrix.{ } The payoff function is a set value map from this Hilbert × space to a number in Euclidean space. The payoff func- Bob Bob ··· tion plays the role of a negative Hamiltonian in may body eB1 eBp physical system. ··· Alice eA1 (A11, B11) (A1p, B1p) (5.11) We first take the prisoner’s dilemma to illustrate the ··· ...... basic phenomena of quantum states in non-cooperative . . . . . game theory. Alice eAq (Aq1, Bq1) (Aqp, Bqp) ··· In conventional text books about prisoner’s dilemma, the two players are rational players(see Appendix G). The prisoner dilemma told us, the fixed point solution When they are prevented from cooperation, both of them of a game can be determined by self-state of players, would confess to minimize his own loss, both of them namely, they are altruistic or selfish. A player could spend b years in jail. If Alice and Bob communicate and be in a mixed state of altruistic and selfish. For each pure strategy ek , we introduce a fractional number ρ cooperate with each other, they would not confess so that | i k they only serve a corresponds to physical observables, such as Hamil- tonian, angular momentum, the given mixed strategy is just the eigenfunction with the fixed point solu- tion as the eigenvalue. We can define the density matrix using the strategy vector s , ρ = T r( s s ). The von Neuman entropy S(ρ(γ))| =i T r(ρ(γ) lg| ρih(γ|)) measures the entanglement between the− strategy vectors. Game theory of Hohenberg-Kohn theorem A many electron system can be mapped into a n 1 player game, each player carries 2 spin. The ground state energy of an electronic± system is completely determined by the minimization of the total energy as a functional of the density function. The exter- nal potential together with the number of electrons completely determines the Hamiltonian, these two quantities determine all properties of the ground state. Hohenberg-Kohn theorem states: the external potential FIG. 5: This is the payoff table which is rotated by 45 degrees V is determined, within a trivial additive constant, by in clock wise direction. Each black dot represents a pair of the electron density. This theorem insures the that there payoff value, the elliptic black point in the middle represent the Nash-Equilibrium(N.E.) points. The game begins at an can not exist more than one external potential for any arbitrary pair of strategies, there is an imbalance between given density. Alice’s payoff and Bob’s payoff in the begging. Then the losing The electrons are players, the many body wave func- player takes a better strategy to eliminate the imbalance in tion are the strategy space. The external potential comes the next step, the winning player also choose his best response from external constrain. For example, in the prisoner to keep his predominance. This game process is actually a dilemma, the police could put the two prisoners together renormalization group transformation, it finally converges at or separate them. If the two prisoner are put together, the Nash-Equilibrium point. they would conspire to remain silent so that both of spend less time in jail. If the police put them in separated rooms, they can not communicate with each other. Since From 0 to 1 record how a player grows from a self- they do not trust each other, they would confess. So we ish player| i to| ai altruistic player step by step. Once we see external potential from police decided the behavior know the self-state of a player, we know the probability of the player, or in another point of view, the external he would play a certain strategy. As shown in Prisoner potential transform the altruistic into the selfish by sepa- dilemma, the two player is either altruistic or selfish, the rating them, and transform the selfish into the altruistic probability of confess as well as not-confess is either 0 by putting them together. Therefore there is only one or 1. A vector of this altruistic degree could be the self- self-characteristic vectors for a given external potential. characteristic vector of a player, the equilibrium point of This is Hohenberg-Kohn theorem in game theory. a game is utterly relies on this self-characteristic vector. There exists a one-to-one correspondence between the self-characteristic vector of players and the fixed point of 5.3. Renormalization group transformation and a game. quantum entanglement This self-characteristic vector of one player uniquely determined the probability he choose certain strategy. In fact, the von Neuman entropy measures the entan- Thus when we introduce the mixed strategy for two play- glement between the self-characteristics vectors. Every ers Alice and Bob, self-characteristics states vector is a mixed states of al- q truistic and selfish. If the player is in a complex mixed s = ρ ek , ρ =1, (5.13) states of altruistic and selfish, it is hard to predict where | Ai Ak| Ai Ak he ends. The Hohenberg-Kohn theorem told us the ex- k=1 k Xp X ternal potential uniquely determined the state vector of s = ρ ek , ρ =1. (5.14) players, as shown in prisoner dilemma. This suggests us | Bi Bk| Bi Bk a possible way to operate entanglement state using ex- k=1 k X X ternal potential. the self-characteristic vector of Alice and Bob have The entanglement states grows stronger and stronger been uniquely defined by ρAk and ρBk . So every following the step of renormalization group transforma- mixed state has an unique{ fixed} point{ in} the payoff tion. For example, in the bargain game, the seller first matrix. In physics, payoff function or loss function gave a price to see if the buyer take it. The buyer thought 27 it too expensive, he feedback his price to the seller. Then 5.4. Topological phase in strategy space for the buyer and seller both know each other at the first multi-player game step. As this bargain goes on, they know each other bet- ter and better. In other words, the entanglement between We study the topological quantity in the strategy space them grows stronger and stronger. When this entangle- of n-player game in this section. The total payoff function ment reach a maximal point, a Nash equilibrium state is of the n players could be view ed as Hamiltonian matrix. arrived. At this point, if the seller raise one penney, the buyer won’t buy it, on the other hand, if the buyer lower H = uˆ1, uˆ2, ..., uˆn . (5.15) the price one penney, the seller will not sell his product. { } When the game passed over the Nash equilibrium, the A strategy vector of the many players’s strategy space entanglement decreased dramatically. reads The entanglement in non-cooperative game is much (1) (2) (n) s = si ,sj ...sk , (5.16) stronger than the entanglement in cooperative game. In { } last section, we know if the self-character of players is (n) sk is the kth strategy of the nth player.u ˆn maps this determined, we can uniquely find an equilibrium solution. strategy vector into the payoff value of the nth player, We call the altruistic player an angle player, while the i.e.,u ˆns = un. When the Hamiltonian matrix of the selfish a devil player. The angel player who manifests payoff function operates on this vector, it produces the goodness, purity, and selflessness, behaves like bosons. eigenvalues While the devil player are fermions. H s = u ,u , ..., u . (5.17) The devil player do not trust each other, they most { 1 2 n} likely to betray in the game. So two devil players would The players fight against each other to reduce the differ- try their best to bound them together and form a pair, ence between their payoffs they increase communication and cooperation to prevent his companion from betray. The entanglement between ∆1 = u1 u2, ∆2 = u2 u3, , ∆n = un u1. (5.18) them is very strong, this entanglement would reach a − − ·· − maximal when the trial is around the corner. In a quantum system, up could be interpreted as the pth But angle players always trust their companions. Ev- energy level. The Nash equilibrium solution is governed erything they do is to increase the welfare of other play- by the coexistence equation ers. The angel player does not need to communicate too D(∆/s)= ∆ , ∆ , ..., ∆ =0. (5.19) much. They behave much like non-interacting, indistin- { 1 2 n} guishable particles. So angle player is boson. An un- The war between energy levels breaks out in the degen- limited number of bosons may occupy the same state at erated eigenspace. The players take their strategy to de- the same time. At low temperatures, bosons can behave crease the energy gap between them. Those points at very differently than fermions; all the particles will tend which the energy gap vanishes are the core center of vor- to congregate together at the same lowest-energy state tex, they are the energy level crossing point. The ith to warm each other, this is Bose-Einstein condensate. player choose strategies according to other players’ strat- For a given many particle system, the entanglement egy, thus his strategy vector is a map from the other play- can be measured by the particle-particle correlation ers’s eigen-strategy to his own space ψi(γ). In analogy length. the longer correlation length there exist be- with conventional Berry phase of quantum mechanics, we tween particles, the stronger entanglement they have. proposed a general topological quantity(See Appendix J). we can read the entanglement out following Kadanoff The topological phase of many players reads block transformation. For example, in the two dimen- Ω = iǫ ∂ ψ (γ) ∂ ψ (γ) ... ∂ ψ (γ) ∂ ψ (γ) . sional Ising model, the spins are players of game. They i tj...klh γt i | γj i i h γk i | γl i i are mixed type of players between angle and devil. Their Ω is actually the Riemannian curvature tensor in the self-characteristic state vector ψ can be expanded by i | i strategy vector bundle. This quantity is equivalent to a four entangled eigenstates of the Bell operators: ψ± = 1 1 topological current of the payoff functions ( ), φ± = ( ). √2 | ↓i| ↑i±| ↑i| ↓i √2 | ↓i| ↓i±| ↑i| ↑i The density matrix, describing all the physical vari- p Ωi = δ(∆i ∆j )d∆1 d∆2 d∆n 1 d∆n. ables accessible to entanglement state ψ , is given by − ∧ ∧ · · · − ∧ j=i ρ = T r( ψ ψ ). Then we can calculate| i von Neuman Y6 | ih | entropy S0(ρ(γ)) = T r(ρ(γ) lg ρ(γ)) for the first or- From our general generalization of conventional Berry − der Kadanoff block transformation. Then we construct phase, the most general topological quantity is the Chern the entanglement states between two blocks and obtained character. In this strategy vector bundle, the Chern char- the second order von Neuman entropy S1. We can con- acter can be derived from the exponential map, tinue this renormalization group transformation, and fi- nally derive an exact von Neuman entropy. Ch(x)= Exp(d ψ(γ) d ψ(γ) ). (5.20) h | ∧ | i 28 These topological quantities are good candidate to mea- However the physical system we encountered always sure the entanglement between different players in strat- have a very larger number of particles, usually N 1023. egy space. One can see from Eq. (5.20), topological It is impractical to control the position and momentum∼ charges focused on the equilibrium solutions where all of particles. Therefore we take a different way to model players get the same payoff, there is no difference among the many body system. We focus on a few interaction pa- any two players. But remember these equilibrium point rameters and integrate the configuration space out. For are isolated point, they are the center of vortex. A tiny example, we choose the Hamiltonian, deviation from these point would break the equilibrium. N 1 H = p2 + γ1V (q , ..., q )+ γ2V (q , ..., q )+ .... 2 i 1 1 N 2 1 N i=1 6. GAME THEORY OF MANY BODY PHYSICS X The interaction potential relies on n players above con- 6.1. Cooperative game and classical many body figuration space, system 1 2 n V (~q)= γ V1 + γ V2 + ... + γ Vn. (6.3) The particles of a physical is the players of a game. They act following the least principal, the aim of the where γ , γ , ..., are physical parameters, and { 1 2 } players is to decrease the total energy. Some of them Vi = Vi(q1, ..., qN ) is the potential functions. We may united to form a subgroup due to local potential. integrate out all the uncontrollable position and mo- Two subgroups may fuse into one when this fusion can mentum variables in the Helmoltz free energy on make both of them more stable. It will be shown the configuration space, then the effective output function many player cooperative game(see Appendix F) is just a for a finite player game is derived, F (γ1,γ2, ...) = 1 physics system. (2β)− Log(π/β) f(β,γ1,γ2, ...)/β. with − 1 − N We consider a 3-person cooperative game 1, 2, 3 . f(β,γ1,γ2, ...)= Log d q exp[ βV (~q, γ1,γ2, ...)]. N − We choose the payoff function of this game{ as} The free energy function is merely one special compo- the conventional Hamiltonian, the first step of the nent of output vector fieldR corresponding to Hamiltonian cooperative game is to find out all subgroups function. We can choose any output function Out on , 1, 2, 3, (1, 2), (1, 3), (3, 2), (1, 2, 3) . The energy func- the strategy manifold, and choose a arbitrary component {∅tion maps each subgroup to a real number,} of the compete set whose members are in involution as Hamiltonian function φH , then we introduce the renor- ε =0, ε1, ε2, ε3, ε12, ε23, ε13, ε123. (6.1) malization group transformation upon the output vector ∅ field. The basic tangent vector field φ~ to characterize the This energy mapping may have various formulas. The topology of the extended strategy space is given by the total energy of the three particle is E(N) = ε1 + ε2 + variational principal ε3 + ε12 + ε13 + ε23 + ε123. The super-additivity requires ε + ε ε . e(23) = ε ε ε is the difference of p N N p 2 3 23 23 2 3 δ [UOˆ ut(γ1,γ2, ...)] d qd p exp[ βφH ]δ [UOˆ ut] payoffs called≤ excess. In fact,− this− is a nuclear reaction in = − . δθp N dN qdN p exp[ βφ ] physics. In game theory, people are looking for optimal R − H self-content energy vector of the n particles ε∗ and the A special subset of renormalizationR group transformation excess value pair ǫ . In the eyes of physicist, this is group ˆ is Lie group Uˆ = eiθL. Now we can apply the topologi- expansion algorithm.∗ cal phase transition theory to study the phase transition For a standard Hamiltonian system, on the strategy space extended by the interaction pa- N rameters γ = (γ1,γ2, ...). The conventional conjugate 1 H = p2 + V (q , ..., q ), (6.2) momentum and position variables have been integrated 2 i 1 N i=1 out, they are out of the set of players for the phase tran- X sition game. Of course, people can bring them in the we take N particles as players, their position qi and mo- game, in that case, the number of players is too large to mentum P i as their strategy, the energy is the payoff. manipulate, it does not tell us any practical information. The target of this game is to find the stable ground states. Generally a general dynamic system can be viewed as To investigate the phase transition of this game in a non-cooperative game between players γ1, γ2, ... γn. the frame work of our topological phase transition the- If we view the classical system as a coopera- ory, one only need to replace the output function Oˆut(γ) tive game, we can divided the effective interac- with the Hamiltonian H, or any other independent in- tion parameters γ ,γ , ..., γ into several groups { 1 2 n} tegrals in the complete set (φ1 = H, φ2, ..., φn) which (γ1,γ2), (...γi), ..., γn , and consider the interaction be- are in involution. The strategy vectors is taken as tween{ these groups.} Then we analyze their coalition, γ = (q1, ..., qN ,p1, ..., pN ). Then all the conclusion on their excess value pair, their payoff, etc. We could get topological phase transition we obtained in the previous the most effective information about the system through section holds here. the phase coexistence equation. 29 6.2. Topological phase transition of quantum many 25 body system 20 15 J 10 The topological phase transition theory in this paper W<0 W>0 aims at the most general systems, no matter they are 5 0 classical physical systems or quantum physical systems, 0 2.5 5 7.5 10 12.5 15 or biological system, or social system. For a quantum U many body system, any output that responses corre- sponding to certain input can be used to detect a phase FIG. 6: The phase diagram of the Bose-Hubbard Hamiltonian transition. The output Oˆut could be any physical observ- for g=1. Inside the curve is the Mott-insulating phase, outside the curve is the superfluid phase. able, such as the correction to ground state energy δEg, the correction to thermal potential δΩ, the single-particle current operator Jˆ = d3xj, the number density oper- ator < ρˆ(x) >, the spin density operator < σˆ(x) >, the From our definition of the first order phase transition, the free energy F , susceptibilityR χ, specific heat C , corre- H boundary between the superfluid and the Mott insulator lation length ξ, compressibility κ , , and so on. The T phases should be decided by the equation δE = 0. We input are any physical parameters, such··· as magnetic field, g have plotted the phase diagram(FIG. 6). This phase di- electric field, radiation, temperature, pressure, on-site re- agram is in agrement with the familiar phase diagram of pulsive interaction, chemical potential, density, potential, superfluid-Mott-insulator phase transition[23]. It is also scattering length, neutral current, electron wave, probing the same as the phase boundary obtained from the min- laser beams, enzymes, ..., and so forth. Our topological imizing the free energy[24], where they take the hopping phase transition theory can be successfully applied to ex- term as perturbation. plain the quantum tunnelling of the magnetization vector in ferromagnetic nano-particles[22]. In this phase diagram, each phase is assigned with a We take a specific model to demonstrate the appli- winding number. The winding number βkηk = Wk is cation in the following. A gas of ultracold atoms in the generalization of the Morse index in Morse theory. Its absolute value βk measures the strength of the phase. an optical lattice has provided us a very good exper- 1 2 ηk =sign φ , φ = 1, represents different phases. In imental observation of superfluid-Mott-insulator phase { } ± 1 2 transition[1]. The system is described by the Bose- the dark area circulated by the curve, ηk =sign φ , φ = +1, it represents the Mott-Insulator phase, while{ outside} Hubbard model 1 2 the curve, ηk =sign φ , φ = 1, it represents the su- U perfluid phase. On the{ curve,} φ−1, φ2 = 0, that is where H = J a†aj + a†a†aiai µa†ai, (6.4) − i 2 i i − i a quantum phase transition takes{ place.} i,j i i hXi X X Most of the recent phase diagram of the quantum phase where the sum in the first term of the right-hand side transition are focused on the first order. As for the pth- is restricted to nearest neighbors and ai† and ai are the order quantum phase transition, it is characterized by creation and annihilation operators of an atom at site i a discontinuity in the pth derivative of the variation of respectively. J is the hopping parameter. U corresponds ground state energy. The order parameter field of the to the on site repulsion between atoms, and µ is the chem- pth order Quantum phase transition can be chosen as ical potential. This Hamiltonian admits two conflicting the (p-1)th derivative of δEg with respect to U and J, 1 p 1 2 p 1 forces, J and U, player J drive the particles hoping from i.e., φ = ∂J− δEg, φ = ∂U− δEg. The equation of one site to another, but U repulse any particles jumps coexistence curve is given by φ1, φ2 = D( φ ) = 0, to his sits, this prevent the particles from moving site { } γ by site. The quantum phase transition is governed by ∂pδE ∂pδE ∂∂p 1δE ∂∂p 1δE the two players, when the hoping player is dominated, 1 2 g g − g − g φ , φ = p p p 1 p 1 =0(6.6), the ultralcold atoms can easily hop in the optical lattice, { } ∂J ∂U − ∂U∂J − ∂J∂U − this is the superfluid phase.When the on-site repulsive here we have taken γ1 = U, γ2 = J. From this equation, force is dominated, the particles is strongly repulsed by one can arrive the phase diagram of the pth order quan- its neighbors, so it would be difficult for him to move to tum phase transition from the equation above, and find other house, then the ultralcold atoms have to stay at some new quantum phases. home, this is the Mott-insulator phase. The above discussion is based on the correction to Using mean-field approaches, the ground state energy ground state energy, now let’s choose the output field of Bose-Hubbard model can be generally expressed as as thermal potential, Oˆ = ∆Ω. One can do some per- the functional of ε , J and U, i.e., E = E (ε ,J,U). ut i g g i turbation calculation up to the second order, and derive Following perturbation theory up to the second order[23], the Green function, the variation of ground state energy δEg is g g +1 1 2 nα nα+1 G− (p,iω )= ǫ ǫ (α + 1) − (6.7) δEg = [ + + 1], (6.5) n − {iω αU + µ} U(g 1) J J Ug α n − − − X − 30 The two parameters are µ.U . The correction of the This topological action measures the non-trivial topology thermodynamic potential{ is } of Green function which maps the momentum space to an external response. The Green function correspond- 1 1 ∆Ω=Ω Ω0 = ln[ βG− (p,iωn)]. (6.8) ing to the exactly solvable terms of hamiltonian can − β − p,n X be exactly derived, the non-exactly solvable part is al- The first order phase transition is given by ∆Ω = 0. ways calculated using perturbation theory. According 1 Put Eq. (6.7) into ∆Ω = 0, we have G− = T. At to Dyson equation, the difference between the inverse 1 − zero temperature, T = 0, it leads to G− = 0. This of exact Green function and the exactly solvable Green is exactly the condition for the superfluid-mott-insulator function gave us the correction to energy, phase transition in Ref. [25]. 1 1 δE = G(p, E)− G (P, E)− = Σ(p, E). (6.11) Now we see different physical observables are only dif- − 0 ferent outputs, they lead to the same phase boundary. usually people call it self-energy Σ(p, E). In this case, the nontrivial topological current of the N-point Green function focus on 6.3. Elementary excitations and momentum space Bp = T r[d Σ d Σ d Σ d Σ 2p], (6.12) One of the best merits of quantum many body theory h | ∧ | i···∧ h | ∧ | i on lattice is we do not need to consider canonical po- again we have the holographic topological action sition coordination and its conjugate canonical momen- i tum. The wave vector has only three spatial components. Ch(M)Σ = T r(exp d Σ(p, E) d Σ(p, E) ). (6.13) π h | ∧ | i Although we can not manipulate them, they are very good candidates for the player of a game. The self-energy here is the output vector field. The mo- The inverse of the wave vector is wave length which mentum vector and frequency are the strategies. The is quantized due to boundaries in a finite solid state lat- surviving strategies are where the stable elementary exci- tice. Like the string fixed at both ends, the wave vector tations arise. One may apply a Lie group transformation of the standing wave modes only take a few discrete val- to self-energy to check if there the discontinuity exist. ues. The standing wave patterns of the electron wave Usually we take SO(4) whose element can be written as and atomic wave in lattice is viewed as elementary exci- θLˆ n 1 ˆ n ˆ U(θ) = e = 0 n! (Lθ) , where L is the generator of tations, or quasi-particles. If we think the electrons as SO(4), it is operator constructed from the momentum soldiers, the elementary excitation is the collective danc- vector P and energyP E. If there is discontinuity, we in- ing of millions of soldiers. When the soldiers are confined ~ δU(θ)Σ(p,E) troduce the vector field φ = δθ . Then we can in a square battlefield, they self-reorganized into certain use the phase coexistence equation φi, φj , ..., φk =0to dancing modes to avoid crashing each other. In fact, the decide the phase boundary. { } independent spatial components of wave vector are their We always assume the periodic boundary condition in commanders, this is a war game between wave vectors. solid state physics, this boundary condition created a These oscillation modes, or elementary excitations, are torus lattice manifold. This naturally leads to a topo- the surviving strategies of the commanders in this bat- logical constrain in momentum space. If we take the tle. components of momentum as the player of a game, their In quantum many body system, the elementary exci- strategy space is the momentum space. If the momentum tations sit at the singular points of the green function. space is noncompact, there is no topological constrain on The external response can be derived from green func- strategies. If the momentum space is a compact mani- tion by linear response theory(see Appendix K). These fold like sphere or torus, there is a nontrivial topological responses are the output vector field of a quantum many constrain on strategy manifold. body game. For the most general case, we set the To avoid the non-figurative reasoning, we consider the Green function on a m dimensional momentum space, superconducting pairing Hamiltonian [26], p (p = E,p ,p , ..., p ). In analogy the topologi- ≡ 0 1 2 m cal current of Riemannian curvature tensor on m dimen- σ He = (tij ciσ† cjσ +∆ij ci† cj† +∆ij∗ cj ci ) µ ciσ† ciσ. sional manifold, we introduce the topological current of ↑ ↓ ↓ ↑ − ij iσ X X N-point Green function G in momentum space, (6.14) B = ǫ ∂ G ∂ G ... ∂ G ∂ G , (6.9) If the system is translationally invariant, then tσ = tσ(i G tj...kl pt pj pk pl ij − h | i h | i j) and ∆ij = ∆(i j). We further require that hopping where we denote ∂pl G = ∂pl G(p) and ∂pk G = − | i h | rate for the up-spin and down spin is the same, t↑ = t↓ , ∂ G†(p). The inner product between Bra and Ket is ij ji pl h| |i then the up-spin and down-spin have the same kinetic an integral, i.e., ∂p G ∂p G = dp∂p G† (p)∂p Gi(p). ik rj σ t j t N j energy ξ↑(k)= ξ↓(k)= ξ(k)= e− · t (j) µ. The Use this topologicalh current,| i we can construct a more j − R momentum representation of Hamiltonian He is simpli- general topological action fied as P i Ch(M)= T r(exp d G(p) d G(p) ). (6.10) H (k)= R~(k) ~σ, (6.15) π h | ∧ | i e · 31 where ~σ = (σx, σy, σz) are the Pauli matrices, and R~ = quasi-particle or collective oscillation modes do not cost (Rx, Ry, Rz) = [Re∆(k), Im∆(k), ξ(k)] with ∆(k) = energy any more. Those solutions are given by R~ = 0. ik rj − j e− · ∆(j) as the gap function. Using the Bo- These periodically distributed extremal points are the basic channels through which the electron like to travel. goliubovP transformation, αk = ukck vkc† k, αk† = − − Each of these channels may be view as the center of cy- u∗c† v∗c k and considering [αk,He]= Ekαk for all k, k k − k − clone, they carry a winding number. The sum of the the effective Hamiltonian becomes He = k Ekαk† αk + winding numbers of all these channels is under a topo- const with Ek 0. The effective energy of quasi- logical constrain which comes form the topology of mo- ≥ 2 2 P particle is Ek = ξk + ∆k . The Anderson’s pseu- mentum manifold. dospin vector[27] | | p The topology of momentum manifold strongly depend on boundary condition. It seems strange for a brick of (Re∆k, Im∆k, ξk) ~n = − . (6.16) material with a rough surface to have some periodic or E k other boundary condition for the wave function of elec- The pseudospin vector ~n describes a mapping from k- trons. In fact, like the droplet of water in vacuum without space to a sphere S2 in the pseudospin space[28]. This gravitational field, their surface are spontaneously com- map defined a topological number to classify the topology pact sphere. Compact boundary is one way for a system of the momentum space. to fit the environment under certain environment. In As observed in the calculated results for the berry other worlds, compact boundary condition is the result phase curvature in SrRuO3. It has a very sharp peak of evolution following the Principal of Least Action(The near the Γ-point and ridges along the diagonals[29]. The principal we talked bout here is the generalized principal origin for this sharp structure is the degeneracy and band of least action in the first several sections of this paper). crossing. The transverse conductivity σxy is given by[30] Therefore he electrons are organized into certain com- pact boundary condition by themselves in order to save 1 σxy = Ωxy(k). (6.17) energy or to form a more stable state. The topology of eβ(εk µ) +1 k − their momentum space is modified according to exter- X nal inputs, such as chemical potential, magnetic field, et We applied the topological current theory[12] to the mo- al. Anyway, if the Hamiltonian relies on many physi- mentum space of this pairing Hamiltonian, and proved cal parameter γi,i = 1, 2, ... , i.e., H = H(γ1,γ2, ...), ~ that the curvature only exist at the solutions of R = 0, the coexisting boundary{ at which} they reach a balance is i.e., given by Ω = W δ(kµ kµ) (6.18) D(φ/γˆ ) = [φˆ , φˆ , ..., φˆ ]=0. (6.19) xy l − l 1 2 n k X If one of the operator is taken as Hamiltonian itself, φˆ = ˆ While Wl is just the winding number of R~ around the H, this is nothing but the complete set of dynamic system l-th solution R~(kl) = 0. in parameter space. If we take the whole two dimensional system as a war, the soldiers are the two momentum components 6.4. Quantum many body theory of game ~k = (pl,p2). Each pi has a strategy space which is his available value across the whole momentum space. pi fights against each other to steer the navigation of The player of war consists of a large number of soldiers quasi-particle. If the whole system is homogeneous and which act like identical particles. The soldiers split into external potential is not anisotropic, pl and p2 are ex- several subgroups, inside every subgroup there is a com- changeable. In real space, the electrons in a homogeneous mon agreement constituted and obeyed by all the mem- system has no bias direction, they circular around each bers of the association. Thus we may take mean field other. Two dimensional system is rather special, the elec- approach inside the subgroup whose member move in a trons have to dance around each other to avoid crashing. mean-field potential formed by all the other members. When the external magnetic field is present, the elec- Generally speaking, the subgroup consists of angel trons change their pattern of motion in real space, they players is stronger than that of devil players. Angle play- are adjusting their wavelength and oscillation modes in ers are bosonic players. The collective strategy is the the mean time. The electrons are spontaneous organized symmetric combination of the personal strategy, into collective oscillating modes in which any two of them m! form pairs. S (1, 2, ..., n) = P ( s(1) s(2) .... s(n) ). | Bos i n! | i| i | i The interference of the electron wave forms periodic rQ distribution in the two dimensional momentum plane. The devil payer are fermionic players. Their collective Generally speaking, p1 and p2 are much like coworkers, strategy is a totally anti-symmetric strategy, which can they balance their inner conflicting profits to fit the envi- be written as Slater determinant, ronment. The best surviving strategy focused on the ex- 1 S = ε εα1α2...αn ( s(i) s(j) .... s(n) ). tremal point of the external potential, at these points, the | F eri n! ij...k | α1 i| α2 i | αn i X 32 The above discussion is only suitable for some oversim- plified system. When it comes to a general multi player game, we do not distinguish which one is angel player and which one is devil player. They are all rational players. In every round of the game, the player choose one strategy from his strategy space, so do other players. When they check their payoffs, what someone win is what someone lose. We may call the individual loser an angel player, and call the individual winner a devil player. In fact, the players share the same strategy space. A FIG. 7: Two spins in magnetic field is a good demonstration of prisoner dilemma in physics. wining player takes good strategies, a losing player takes those bad ones. It does not make any sense to talk about the good or the bad of the strategy itself, the same strat- egy may bring different results in different situations for realize a prisoner dilemma. A direct physical observation a certain layer. Every combinatorial series of strategies is that the particles with spin is lean to form pairs, and corresponds to a payoff distribution to individual play- the magnetic field would strengthen the entanglement ers. If we take particles as player, a physics system always between two particles with opposite spins. contain millions of particles, it is impossible to calculate A physical system to demonstrate the Prisoner’s every particle’s payoff. That is why we need statistics dilemma is two spins in magnetic field. Two players are physics through which we summarize the information of the two particles: Alice and Bob, they both carry mag- payoffs into a few statistical observables, but we lost the netic dipole momentum µ. The strategies: confession and information on individual payoffs. We choose a more accusation corresponds to spin up and spin down, the ex- practical way to grab the key information of many parti- ternal magnetic field is external law. Their loss function cle system. The player of game is taken as the different is their energy in this system. They fight to minimize interactions that govern the microscopic behavior of par- their loss function. The payoff matrix is ticle. A vacuum state 0 is defined by a state that no player Bob Bob | i confirm his strategy. When the ith player has chosen ↓ ↑ (i) Alice ( µB, µB) (µB ∆, µB + ∆) his strategy sα1 , and place his card on the table, it | i Alice ↓ (µB−+ ∆, µB− ∆) −(µB, µB) means that a strategy sα is generated from vacuum, i.e., ↑ − (i) sα =s ˆ† 0 . One player’s strategy is not isolated, it | i iα| i ∆ is the energy shift due to the interaction between the is always entangled with other players’s strategy, since two particles. This is the classical picture of two-particle whenever he makes the decision he must reflects other game. There was experimental realization of prisoner player’s choice, the collective strategy is highly entangled dilemma on nuclear magnetic resonance quantum com- states. The anti-strategy of the ith player comes from the puter [31]. combination strategy he received from the rest players, A general n-player prisoner dilemma may be summa- we denote it as rized into a n-spin Hamiltonian. A player has two strate- gies: spin-up and spin-down. The payoff function is the Si = 0 Fˆ(ˆs , sˆ , ..., sˆ , ...sˆ ), (k = i). h | h | α1 α2 αk αn 6 Hamiltonian, When the ith player encounter the jth player, his i ij i j H = hiσ + J σ σ . (6.20) i (i) z z z payoff is uij = T rk=j [ S sα ]. We can further define i ij 6 h | i X X the density matrix ρ = j pj Sj Sj , and introduce the familiar Von Neumann entropy| ih | to measure the If we take the local coupling J ij = J and consider only entanglement. P the nearest neighbor interaction, this Hamiltonian be- comes traditional Ising model. The payoff matrix of this Pairing mechanism in the war game n-player prisoner dilemma game for Ising model is diag- onal block. A group of soldiers in a war also encounter As shown in the prisoner dilemma, two altruistic prisoner dilemma. Two soldiers helping each other is players can get their best payoff without communication stronger than the two that they do not help each other. and cooperate, they trust each other and both choose We consider a war on two dimensional lattice. The sol- the strategy for the welfare of other people. But two diers are particles, they are divided into two equal groups, selfish player can not get the best payoffs unless they A and B. The particles of A is the anti-particle of B and communicate and cooperate, in order to prevent the vice versa. In the beginning, A and B are separated two other player’s betray, they would like to bound into parts on the lattice. When the war break out, they rush a pair so that any betray would damage himself, this to each other and began the combat. A particle only increases the entanglement between them. in physics, fight against his nearest antiparticles and help his near- two spin in magnetic field is a perfect physical system to est friend neighbors. If the soldiers are identical particles 33 at the same level, the interaction between particles does tlefield. We may take Kadanoff block procedure to sum- not go too far, it is confined in a local region. If we marize all the many player interaction to the commander. assume all the soldiers only fight against enemy soldiers The many body interactions, such as for his own survival, they do help friend neighbors, the n † † dynamic process only includes V sˆBi ... sˆBj ... sˆAk ... sˆAl , εij sˆ† sˆB , εji sˆ† sˆA , Ai A, Bj B¯ , (6.21) are renormalized into partition function. The war is go- { Ai j Bj i ∈ ∈ } ing on between the renormalized commanders, SB and where s represents the soldiers. This is a system com- SA, where posed of free particle and antiparticles. In fact, some soldiers of G may help each other, they grouped into (ˆs† ... sˆ† ) Sˆ , (ˆs† ... sˆ† ) Sˆ . (6.23) Bi Bj B Ai Aj A pairs to fight. So does the soldiers of G¯. Then we have → → to consider the pair interactions. The war now contains Again we may consider the pairing interactions among three solders interaction and four soldiers interaction, these commanders, 3 ε sˆ† sˆ , ε sˆ† sˆ , V Sˆ† Sˆ† Sˆ , ... , ε Sˆ† Sˆ† Sˆ , ij Ai Bj ji Bj Ai S Ai Ak Bj ji Bj Bk Ai 3 4 4 V sˆ† sˆ† sˆ , ... , ε sˆ† sˆ† sˆ , ˆ† ˆ† ˆ ˆ ˆ† ˆ† ˆ ˆ Ai Ak Bj ji Bj Bk Ai VS SAi SAj SBk SBl , V SBi SBj SAk SAl . 4 4 † † † † (6.24) V sˆAi sˆAj sˆBk sˆBl , V sˆBi sˆBj sˆAk sˆAl . (6.22) Repeating this renormalization group transformation, we transform the sophisticate many body problems into If we recall the BCS pairing mechanism in superconduc- something we can handle. This is the fundamental spirit tor theory, the war between electrons does not have three- of various numerical renormalization group method de- particle interaction. Even two electrons separated far veloped in quantum many body system. from each other, they can act as one pair. They do not The phase boundary appears when the fighting groups even know each other, since identical particle are indistin- reach a balance at certain frontiers regions. Friend and guishable. This interesting phenomena in the quantum enemy are maximally entangled in the coexistence state. world may found some naive analogy in war. During the One can not tell who is who. In order to study the sym- war game, we can assume a pair of soldier are confined metry transformation at the coexistence boundary, we in the lattice, they kept running from one place to an- introduce the position displacement operator and its con- ther, but helping each other all the time. They shot any jugate momentum operator, enemy that try to kill his partner. This is a long range interaction. ˆ ˆ ˆ ˆ Sj† + Sj i(Sj† Sj ) There are two types of pairs, angle pair and devil pair. Xj = , Pj = − . (6.25) The angle pair is more powerful than the devil pair as √2 √2 a whole. They love each other and trust each other, so Then the angular momentum operator is given by L = they can separated from each other in a longer distance. ij (X P X P ). The generator of the symmetry trans- But devil player have to stay closer to prevent his partner i j j i − ˆ from betraying him for his own survival, their correlation formation may be denoted as L = i ij θLij , through which we derived the Lie group element length is shorter. P There are hierarchical structure in army. The soldiers † † iθLˆ i P θ(Sˆ Sˆ SˆiSˆj ) are organized into different groups in which a large num- U = e = e− ij i j − . (6.26) ber of them fight as a whole. These hierarchical struc- ˆ ˆ ˆ ture are well kept in the region far from the coexistence The quantum output vector Out(Si, Sj, ...) is expressed boundary, but when combat begins in one battlefield, by the operator of renormalized soldiers. Then we can all these hierarchical structure becomes meaningless, the detect the phase transition by the transformation general or the captain at high plays no different role as ˆ 1 iθ Lˆ ˆ iθ Lˆ an ordinary soldier at the lowest level, they are just the U(p)OutU(−p) = e · Oute− · (6.27) same individual fighters with arm. It is the hierarchi- cal structure or inner structure of a system that differs This transformation equation determined the evolution it from others. When the phase transition occurs, all of the quantum output in the vicinity of phase coexis- these hierarchical structure are broken, men are born to tence region. Inside one phase, or inside one army, this equal, men are also died to equal. This is the origin of transformation is continuous up to infinite order. When universality class. never the symmetry lost, an enemy popped out and ini- For a further consideration of war game, we have to tiated the revolution. take into count of different hierarchical structures. The The uncertainty principal shows up across the whole commanders are the critical players, they are dressed up phase diagram. In the phase diagram, the war game is in by a group soldiers, and fight as a whole around the bat- different squeeze state for different stage of combat. The 34 transformation operator (6.26) is a special squeeze oper- 7. SUMMARY ator. A general squeeze state[37] generated from vacuum is (1)game theory of general phase transition and † † ∗ i P (θSˆ Sˆ θ SˆiSˆj ) ψ = e− ij i j − 0 . (6.28) Renormaization group transformation | i | i When the two players began rushing toward each other Phase transition is a much more universal phe- but still do not touch each other, their fluctuation of nomena than the conventional phase transitions in spacial position is much higher than that of their force. physics. Similar sudden changes arise from evolution When they meet in the phase coexistence boundary, the of biomolecule, self-organization in nanoscale systems, fluctuation of spatial position is around the boundary, it cosmic evolution, and so forth. The general phase is much lower than the conflicting force. At the Nash transition breaking the envelop of physics can be viewed equilibrium state, the phase boundary is almost sta- as a non-cooperative game of many players. The players tionary, but both of the two players summon up their try different strategies to win. Whenever a winning maximal force to against each other, they reach a dan- player fails and becomes a loser, a phase transition gerous equilibrium state on the boundary. This is an would occur. extremal squeezed state, the fluctuation of position is When the players represent different interactions be- highly squeezed, but the fluctuation of conflicting force tween the elements of a complex system, a winning player is very strong. is a dominant interaction which governs one stable phase So we may take two Hermitian operators γ and P , γ γ of the system, the other phases represented by loser’s in- denotes the position in phase diagram, Pγ represents the teraction are suppressed. When we tune the interaction corresponding force. The commutation relation derive parameters, we are changing the strategies of the players, from the second quantization is [γ, Pγ ] = iC. We can the winning interaction may becomes weaker and weaker. calculate the uncertainty of an operator A by (∆A)2 = 2 2 There is a critical point at which the losing players grows ψ A ψ ( ψ A ψ ) . According to Heisenberg uncer- stronger enough to balance the previous winning player, htainty| | principal,i− h | ∆| γi∆P 1 C . The uncertainty of the γ ≥ 2 | | then we arrived the Nash equilibrium point. At this point position operator is , no one wins, and no one losses, it is the coexistence 1 point of the new phase and old phase. The Nash equilib- ∆γ C (6.29) ≪ 2| | rium point is an unstable saddle point, any tiny deviation at the coexistence state. While the uncertainty of corre- would decided the fate of the coexisting new phase and 1 old phase, there is only one winner left to dominate the sponding momentum is much stronger ∆Pγ 2 C . The angel pairing and devil pairing are≫ two| stable| phase structure of the system. squeezed states. The angle pairing has less uncertainty Usually the Nash equilibrium of a war game is not de- about their trustworthiness, so they have larger uncer- rived in one round of combat. The players have to play tainty on their spatial connection. But the devil pair many rounds of negotiation to find the optimal strategy. has larger uncertainty on his partner’s trustworthiness, The renormalization group transformation theory is ac- so they strengthen their spatial connection, this reduced tually one theoretical description of this kind of game their spatial uncertainty. No matter which kind of pair- process in physics. In physics, the players are the inter- ing it is, the total information in the two pairing states action parameter or physical parameter. For example, should be conserved. If we can make sure a system doe we take the Ising model as a game, the two players are not lose information, we can maintain the pairing to a the spin coupling interaction and external magnetic field. stable level. Superconductor is such a system, below the The stable phase is determined by the dominant interac- critical temperature, no chaos invades into it, no infor- tion. mation escapes out of it. There is a constant electric At the first round of renormalization group transfor- flow. mation, the two player accomplished the first round of According to quantum field theory, symmetry means combat, they made an agreement on the most part of conserved quantity, a loss of symmetry indicates a loss boundary between their domains. But the unsettled part of conserved quantity. What the old phase lost becomes is occupied by the winner. So the loser initiated the sec- the generator of the new phase. The squeeze transforma- ond round of renormalization group transformation to tion on output field is a way to detect the loss of infor- get some of his losing back. As this transformation goes mation. Imbalance is the original engine of development. on, the unsettled boundary between the players becomes For physicist, the physical measurement always affect the less and less. Finally, they reach the Nash equilibrium output state itself. We can not detect a state without point, that is the fixed point of the renormalization group changing it, this minor change is neglectable when it is transformation. This point is where the phase transition done in a stable phase. But at critical region, the mi- occurs. nor effect of detection might induce significant effect. In Usually to obtain the exact critical point, one has to other words, when a war is going on, once the spy step perform the renormalization group transformation to into the frontier, he is force to fight for his own survival, infinite order. However, it is too far to reach in reality, it would be too hard for him to sent out any information. we always cut it off at certain order. It inevitably gives 35 birth to a loser and a winner to certain order. Although By substituting these fractal polynomials into the uni- the players at that order maybe are fighting for only versal coexistence equation, one can derive the scaling one thousandth of a penny, but the loser is loser, winner relations. As we shown, the coexistence operator has a is winner, the title is fixed. The order of cutoff is the degenerated subspace, the coexistence equation bear a order of phase transition. we can take renormalization topological origin, so these scaling relations are universal. group transformation on output function to observe different order of phase transition. When we apply (3)Many body physics of games this definition to thermodynamics physics, it naturally unified Ehrenfest definition. Game theory is actually one different way to see many body system. For example, a collection of millions (2)Universal coexistence equation and of ultralcold atoms trapped in optical lattice can be topological conjecture of scaling laws in described by Bose-Hubbard model. There are two key fractal space parameter, the hopping parameter J and the on site repulsive interaction U. We can model this system The game is played on the strategy base manifold, as a game between U and J. Player J command the so a phase transition is related to the topology of the particles hoping from one site to another, drive them strategy base manifold. If the strategy space is noncom- into superfluid phase. While U direct the soldiers to pact, we do not need to consider topology at all, in that stick to the lattice site. If J wins, the ultracold atoms cases, the phase transition governed by game dynamics. becomes superfluid. If U wins, the ultracold atoms are We may call it a non-topological phase transition. When confined in the lattice, they form Mott-insulator phase. the strategy manifold is compact, there would be a The transition point is the Nash equilibrium point of topological constrain on the player’s strategies, then the this game. The two phases coexist at the critical point. topological effect would appear in phase transition, we Another way to model the millions of ultracold atoms is call it a topological phase transition. to take every atom as a player, and their states at lattice The topology is intimately related to the fixed points sites are strategy space, this reproduced the standard of renormalization group transformation flow on strat- quantum many body theory. egy manifold. We introduced the general flow vector A practical approach to a given many body system is field, and showed that the sum of the winding number first to find out the main different interactions that are around the surviving strategies is a topological number. players of the game. Secondly, we choose some output These surviving strategies with opposite charges annihi- quantity to measure the states. Physicist usually take late at the universal coexistence curve, whose equation is the external response function, these output field are the φ1, φ2, ..., φ2 = 0, φi are the component of the flow vec- payoff function of the game. The tangent vector field es- tor{ field of output} field. In the thermodynamics physics, tablished on the hypersurface of these response function the output field may be chosen as free energy, the players is the fundamental vector field to study the topological are temperature and pressure, one can verify that this co- effect of the strategy space. Applying the universal co- existence equation unified all the coexistence equations existence equation, we can find out the basic structure at different order classical phase transition. The phases of different phases. This scheme holds both for classical separated by the coexistence curve are assigned with dif- systems and quantum systems. ferent winding number of opposite sign. Game theory only provide us a mathematical frame The strategy space is a fractal dimensional space. It work to understanding the general property of a system. is easy to see this point from a war game. We focus The topological phase transition theory developed in this on the battlefield, it is a war between two armies. If paper is a general mathematical results, it does not de- we look closer, it is the war between corps. We may pend on any specific model in physics, biology or social continue the magnify the battle field, one would see, it is system. When it comes to some specific physical system, a war at all different scales, it is the combat between two we may apply the conventional theory for that special individual soldiers at any local region. A war game may system to derive the specific relation between input pa- be summarized into the war between two commanders rameters and output field, or one may directly conduct by Kadanoff-block procedure. There is an intrinsic some experiments to find some approximated relation. fractal structure around the critical point. So we make As long as one derived the specific relation between out- the hypothesis that a local neighborhood on the strategy put and input, just substitute it into the universal coex- base manifold is homeomorphic to a fractal dimensional istence equation, one can get basic phase diagram. space. Then the output field may be expanded by the In fact, quantum many body theory may provide us polynomial of power functions with fractal dimension new understanding to many player games. We developed index. In analogy with the observables defined in the density matrix theory of many player game, it was the statistical mechanics, we introduce the tangent shown there is a one-to-one correspondence between fixed vector field of the output field as observables. These point of many player game and the self-character vector tangent vector field may be approximated by fractal of the player. Every quantum phase may be represented polynomials in the vicinity of Nash equilibrium solution. by a state function, these state functions are the players 36 of multi-player game. A player’s strategy must take into We rewrite Ehrenfest’s definition into more compact for- account of the other players’s strategy. So game theory mulism. The pth-order quantum phase transition is char- is born to be an entangled theory. This may help us to acterized by a discontinuity in the pth derivative of dif- understand the entanglement among coexistence phases. ference of free energy δF , We find a new quantity to measure the entanglement p 1 m p m 1 of different phases. The maximal point of entanglement ∂ 1− δF =0, ∂ 1 ∂ 2− − δF =0, (m =1, 2, ..., p 1), γ γ γ − is the Nash equilibrium point at which different phases p 1 ∂γ2− δF =0. coexist. The maximal entanglement between phases is p m p m controlled by interaction parameter instead of time, so ∂γ1 δF =0, ∂γ1 ∂γ2− δF =0, (m =1, 2, ..., p), p 6 6 the quantum entanglement between different quantum ∂γ2 δF =0. (A5) phases are good candidates for quantum computation. 6 Further more, if we take two particles as the two play- If the pth derivative of δF becomes continuous, the phase ers of prisoner dilemma, it would be found that there are transition jumps to the p + 1th order. two types of entangled pairs: the angel pair and devil pair, the angel pair love each other, their entanglement is a little bit weaker. While the devil players do not trust APPENDIX B: DIFFERENTIAL GEOMETRY OF each other, they inclined to bounded together to prevent FREE ENERGY the other from betray, it is a strong entangled pair. This may help us to understand pairing mechanism in super- We demonstrate the basic conception of differential ge- conductor. ometry on the free energy manifold expanded by two thermal variables temperature T and pressure P . In differential geometry, the open set δF (T, P ) in a 8. ACKNOWLEDGMENT manifold may be mapped to an open set in three di- mensional Euclidean space through the homeomorphic The author expresses his gratitude to professor Yue Yu mapping fδF : δF (T, P ) XδF , X = X1(γ1,γ2)i + → 0 for his support. This work was supported by the National X2(γ1,γ2)j + X3(γ1,γ2)k, At each point γ in δF there Natural Science Foundation of China. exist a set of tangent vectors. The two basis of the tan- gent vector space on this manifold are APPENDIX A: EHRENFEST’S DEFINITION ∂ ∂ ∂ ∂ e1 = = , e2 = = , (B1) ABOUT ORDER OF PHASE TRANSITION ∂γ1 ∂T ∂γ2 ∂T here we have defined γ = T (temperature) and γ = The zeroth order phase transition defines that the free 1 1 P (pressure). An arbitrary tangent vectors may be ex- energy of the two phases F (T, P ) are not equal, pressed as F (T, P )=F (T, P ). (A1) A 6 B ∂X ∂2X Xγi = , Xγiγj = (B2) For the first order phase transition, the free energy of the ∂γi ∂γi∂γj two phases FA and FB is continuous, but the first order derivative is not continuous, All the tangent vectors to the surface δF at x denoted form a tangent vector space denoted by TxδF , it is par- dFA dFB dFA dFB allel to the tangent plane to δF at x. FA(T, P )= FB(T, P ), = , = . dT 6 dT dP 6 dP The metric tensor is given in a bilinear form by the (A2) inner product of two basis vectors, gij = A B A B A B E = Xγ1 Xγ1 , F ′ = Xγ1 Xγ2 , G = Xγ2 Xγ2 , Cp = Cp , α = α , κ = κ , (A3) · · · 6 6 6 g11 = E, g12 = g21 = F ′, g22 = G. (B3) in which the specific heat Cp, α, κ are defined as: The second fundamental form is dS ∂2F Cp = T ( )P = T 2 , L = X n = X n dT − dT γ1γ1 · − γ1 · γ1 2 1 ∂V 1 ∂ F M = Xγ1γ2 n = Xγ1 nγ2 α = ( )P = , · − · V ∂T V ∂T∂P N = X 2 2 n = X 2 n 2 . (B4) γ γ · − γ · γ 1 ∂V 1 ∂2F κ = ( )P = . (A4) n −V ∂P −V ∂P 2 with as the normal vector of the surface, The higher order of phase transition is defined from the Xγ1 Xγ1 n = × . (B5) discontinuity of higher order derivative of free energy. X X | γ1 × γ1 | 37 Considering the second order of phase transition, the first strategy means, the players come up with one of her fea- fundamental form and the second fundamental form in- sible actions with a pre-assigned probability. Each mixed cludes more information about the variation of the free strategy corresponds to a point P of the mixed strategy energy than Ehrenfest’s definition. It is more reasonable simplex. to choose the Gaussian curvature, m 2 m LN M Mp = P = (p1,...,pm) R : pq 0, pq =1 , Ω= − , (B6) { ∈ ≥ } EG F 2 q=1 − ′ X (C1) It is positive for spheres, negative for one sheet hyper- whose corners are the pure strategies. For the case of two bolic surface and zero for planes. The principal curva- players, a strategy profile is a pair of probability vector 0 0 0 tures, κ1(γ ) and κ2(γ ), of X at X(γ ) are defined as (p, q) with p Mp and q Mq. The expected payoffs the maximum and the minimum normal curvatures at of player 1 and∈ 2 are expressed∈ as X(γ0), respectively. The directions of the tangents of the two curves that are the result of the intersection of m 0 0 u (p, q)= p Aq = p A q , (C2) the surface X(γ ) and the planes containing n(γ ). Then 1 · i ij j ij X m Ω(γ1,γ2)= Ω(γ1,γ2) p = κ1(γ1,γ2)κ2(γ1,γ2) p. | | u (p, q)= p Bq = p B q . (C3) p p 2 · i ij j X X ij (B7) X The cross section of the curvature X at γ0 may be ex- panded as parabola by ignoring the higher order, This payoffs function relies on the entangled strategies of the two players. κ1 2 κ2 2 The essence of a game is the payoff function defined X = r1 + r2. (B8) 2 2 over different strategy profile of the players. In the lan- 1 1 guage of quantum statistics, the strategy space of the ith where ri = (dγ1 + 2 Γij dγidγj )√gii + . When κ1 > 0 ··· player SA is the Hilbert space of the i particle. and κ2 > 0, it is a elliptic surface, when κ1 > 0 and κ2 = 0, it is parabolic, for κ1 > 0 and κ2 < 0, it is hyperbolic. The covariant derivative may be defined as APPENDIX D: DIFFERENTIAL GAME j a j k Dγi φ = ∂γi φ +Γikφ , (B9) The differential game is applied to model dynamic conflicts[38], such as the labor-employers relations in eco- here we have introduced the Christoffel-Levi-Civita con- nomic processes, the bull and fighter in bull-fighting. The nection Γk , which is defined as ij labor and employer are called players in differential game. We take the two player game as an example. There are k 1 kl ∂gil ∂gjl ∂gij Γij = g ( + + ). (B10) two players, one is called Alice, the other player is Bob. 2 ∂γj ∂γi ∂γl Alice has a strategy space SA(s1,s2,...,sN ), and Bob’s strategy space is SB. We introduce the state vector ~x APPENDIX C: DEFINITION OF GAME THEORY which characterize the conflict to convert a real life con- flict to a differential game model. For example, Alice and Bob are concerned with the motion of a point in a plane, A game is an abstract formulation of an interactive the state vector denotes the location of the point x(t) decision situation with possibly conflicting interests, it at the time t. All possible state vector is a subset of a involves of a number of agents or players, who are al- n-dimensional Euclidean space. lowed a certain set of moves or actions. The payoff func- The influence of the decisions of the evolution of the tion specifies how the players will be rewarded after they state is described by the equations of motion, have performed their actions. Let n = 1,...,N denote the players; The ith player’s strategy, Si, is her proce- x˙ = f(x, sA, sB). (D1) dure for deciding which action to play, depending on her information. The strategy space of the ith player When the two players realize their strategy (sA, sB) at = e ,e ,...,e is the set of strategies available to Si { i1 i2 im} time t, the outcome of the differential game is functional her. A strategy profile (s1,s2,...,sN ) is an assignment on the state space, U[x, sA, sB]. If the outcome satisfies of one strategy to each player. ui(s1,...,sN ) is player for a pair of strategy (sA∗ , sB∗ ) satisfies, i’s payoff function (utility function), i.e., the measure of her satisfaction if the strategy profile (s1,...,sN ) gets realized. U[x, sA∗ , sB∗ ] U[x, sA∗ , sB∗ ] U[x, sA, sB∗ ], (D2) The strategy assigned to each player could be pure ≤ ≤ state or mixed state. A mixed strategy is a probabil- then this strategy pair is the optimal play for the two ity distribution over pure strategies. Playing a mixed players, U[x, sA∗ , sB∗ ] is called the optimal outcome at 38 ∂V (T,ψ∗) x. The optimal outcome is a function in the Euclidean The conjugate momentum of ψ is p(T ) = ∂ψ . For space, it is called the value function, a stochastic differential game, the Hamilton function is J(x)= U[x, sA∗ , sB∗ ]. (D3) (t,ψ,γ)= V (t, ψ)f(t,ψ,γ)+ g(t,ψ,γ), (E6) Hs ∇ψ There is a theorem states that if the value function of a differential game exists it is unique[38]. The saddle the Hamilton-Jacobi-Bellman equation reads relation implies that a solution of a differential game is a Nash equilibrium. Neither player Alice nor player Bob 2 can improve their guaranteed results. ∂V (t, ψ) 1 ij ∂ V + σ + min s(t,ψ,γ)=0. (E7) ∂t 2 ∂ψi∂ψj H APPENDIX E: THE CONTINUOUS-TIME INFINITE DYNAMIC GAME For a given two person game, suppose that the strat- egy pair (γ1∗,γ2∗) provides a saddle-point solution, and ψ∗(t) denoting the corresponding state trajectory, then We present the basic conception of the continuous-time the Hamilton function satisfies, infinite dynamic game in this section. Suppose for a given many-body system, there are n player which are denoted (t, ψ∗,γ∗,γ ) (t, ψ∗,γ∗,γ∗) (t, ψ∗,γ ,γ∗), as N = 1, 2, ..., n , ψ is the state function of this game, H 1 2 ≤ H 1 2 ≤ H 1 2 the state{ space is} an entangled space of many Hilbert (E8) (γ∗,γ∗) is the solution of the Issac equation min max = space. The evolution equation of this game is governed 1 2 H by the differential equation, ∂V /∂t, V (ψ) is the value function. It represents the coexistence− hyper-surface extended in ψ space. dψ i = f(t,ψ,γ (t),γ (t), ..., γ (t)). (E1) dt 1 2 n γ1,γ2, ..., γn is the strategy profile of the n players. γi are real or complex numbers. A cost function of the game is APPENDIX F: BRIEF INTRODUCTION TO COOPERATIVE GAME a map from the strategy space Γ = Γ1 Γ1 ... Γn to a real number, Ei : Γ R(i N) for⊗ each⊗ fixed⊗ initial 0 0 → 0 ∈ strategy profile γ1 ,γ2 , ..., γn. The minimum cost-to-go Coalition is a crucial ingredient in a n-person cooper- from any initial state and any initial time is described by ative game. When a subset g of the n player forms a the so-called value function which is defined by coalition and all its members act together, we assign a T real number E(g) to each possible coalition, E(g) mea- V (t, ψ) = min[ g(s, ψ(t),γ(t))ds + q(T, ψ(T ))], (E2) sures the payoff when this coalition acts together. If we γ(t) Zt can find an imputation vector ε = (ε1,ε2, ..., εn) with real components, which satisfies εi E( i ), i N and satisfying the boundary condition V (T, ψ) = q(T, ψ), g ≥ { } ∀ ∈ ε1 + ε2 + ... + εn = E(N), it is the Pareto optimality. is a map from g : (t, ψ(t),γ(t)) R. The application of → An imputation x represents a realizable way that the n the principal of optimality leads to the Hamilton-Jacobi- player can distribute the total payoff E(N). Bellman equation, Usually, two coalitions S and T are inclined to united, ∂V (t, ψ) ∂V (t, ψ) if their union brings them better payoffs, this is the super- = min[ f(t,ψ,γ)+ g(t,ψ,γ)]. (E3) − ∂t γ ∂ψ additivity of a game, E(S T ) E(S)+ E(T ). Suppose a coalition S has m players,∪ usually≥ there is a difference A theorem states that, if a continuously differentiable between the payoff of the coalition and the sum of payoff function V (t, ψ) can be found that satisfies the Hamilton- of each individual in this coalition, the excess is defined Jacobi-Bellman equation, with corresponding boundary by, condition V (T, ψ) = q(T, ψ), then it generates the op- timal strategy through the static minimization problem e(S, ~ε )= E(S) ε , (F1) defined by the right hand side of Eq. (E3). We introduce a − i i S the Hamilton function X∈ ∂V (t, ψ) A n-person cooperative game has 2n subsets, i.e., it has (t,ψ,γ)= f(t,ψ,γ)+ g(t,ψ,γ), (E4) n n n 2 2 2 2 H ∂ψ 2 coalitions. Let D2n 2 : R − R − be a mapping which arranges elements− of a 2n →2-dimensional vector the minimizing γ will be denoted by γ∗, then in order of decreasing magnitude.− For a certain set ~ε of ∂V (t, ψ) payoff vector, the nucleolus over ~ε is the solution mini- (t,ψ,γ∗)+ =0, (E5) mizing the vector of the excesses, H ∂t 39 Nc(N, E, ~ε)= εa ~ε D2n 2(e(S1,εa), ..., e(S2n 2,εa)) D2n 2(e(S1,εb), ..., e(S2n 2,εb)) , b n. (F2) { ∈ | − − ≤ − − } ∀ ∈ In a game (N, E), the ǫ-core Cǫ(N, E) is the set of all A consistent pair of strategies following the constrain pre-imputations ε, satisfying that all the excess function ¯ are not greater than ǫ, i.e., fA(sB) := s¯A SA uA(¯sA,sB)= inf uA(sA,sB) . { ∈ | } f¯B(sA) := s¯B SB uB(¯sB,sA)= inf uB(sA,sB) (G2). C (N, E)= ~ε ε(N, E) e(S, ~ε ) ǫ, S N . (F3) { ∈ | } ǫ { a ∈ | a ≤ ∀ ⊂ } is called a non-equilibrium (Nash equilibrium). In the where the set of pre-imputations ε(N, E) is ε(N, E) = prisoner’s dilemma, the Nash equilibrium is that both Al- n ~εa R εai = E(N),εai 0 . ice and Bob confesses, they both spend b years in prison. { ∈ | i ≥ } The most important quantity we want to find is Besides the Nash equilibrium, there is still a better P the ground state payoff vectors ~εg, i.e., the least strategies for them. If Alice and Bob communicate and core Cǫg (N, E), which satisfies e(S, ~εg) ǫg = cooperate with each other, they would not confess so that ≤ min max e(S, ~ε). In fact, this is equivalent to the self- they only serve a minimize ǫg (F4) A market comprised of two sellers and many compet- subject to E(S) ε ǫ , S N, (F5) itive buyers is know as a duopoly. The buyers can not − i ≤ g ∀ ⊂ i influence the price or quantities offered, it is assumed that X the collective behavior of the buyers is fixed and known. ε1 + ε2 + ... + εn = E(N), (F6) The competitive and cooperative behavior between the ε 0,i =1, 2, ..., n. (F7) i ≥ sellers determines the price. There is a lemma says the unique payoff vector ε mini- Two players are each manufacturers of the same single ∗ commodity, the loss functions are cost function which mizing ε can always be determined by at most n steps in algorithm computation[38]. depends on the production of the two players. We denote the quantities of this commodity produced by the two players by x R+ and y R+. The price p(x, y) is an affine function∈ of the total∈ production x + y, APPENDIX G: PRISONER DILEMMA p(x + y) := α β(x + y), (H1) The Prisoner’s dilemma is the most famous paradigm − to study game theory. It occurs between two prisoners: and the cost function uA and uB of each manufacturer are Alice and Bob, who are accomplices to a crime which affine functions of the production fAx = ax + b, fB(y)= leads to their imprisonment. Each has to choose between ay + b. Alice’s net cost is equal to f (x, y) := f p(x + A A − the strategies of confession or accusation. If neither con- y)x, fA(x, y) := fB p(x+y)y. Eliminating the constant fesses, moderate sentences (a years in prison) are handed terms which does not− modify the game, the loss function out. If Alice confesses and Bob accuses him, Bob is free reduced to (0 years in prison) and Alice is sentenced to c>a years in prison. If both confess, they will each have to serve b uA(x, y) := x(x + y u), (H2) − years in prison, where a KerD = ξ Γ(E) Dξ =0 . (I2) B2 = d ψ d ψ d ψ d ψ { ∈ | } h | ∧ | i∧ h | ∧ | i 4 = ǫtjkl ∂Rt ψi ∂Rj ψi ∂Rk ψi ∂Rl ψi d R. On a real oriented compact smooth n =2l-manifold M, h | ih | i (J5) the Atiyah-Singer index theorem states that the Euler’s characteristic Ch(M) is the sum of the Betti number, It is natural to arrive at the pth order generalization, p p Ch(M)= Index(Λ, d)= ( 1) dimRH (T (M), R). B = d ψ d ψ d ψ d ψ . (J6) − dR p p h | ∧ | i···∧ h | ∧ | i X (I3) 2p It is just the Gauss-Bonnet-Chern theorem. on the four In mind of our new| definition{z of the general} Berry phase, dimensional manifold, there are more interesting case. the p-form expression of Eq. (J6) is Such as the Dirac operator, there are k zero modes of fermions coupled to the k-instanton gauge field in the B1 = δ(E E )dδE dδE (J7) fundamental representation. Atiyah-Singer index theo- i j − i i ∧ i j=i rem which states that the the index of the Dirac operator Y6 is minus the first Chern class, where ν+ is the number of positive chirality zero-modes and ν is the number of p − B = δ(E E )dδE dδE dδE dδE (J8) negative chirality zero-modes. i.e., i j − i i ∧ i ∧ · · · i ∧ i j=i Y6 1 IndexD = Ch = ν+ ν = Ω. (I4) According to Eq. (J2), the complete Chern character can − − 2π ZM be expressed into beautiful form, i Ch(M) = T r[exp( δ(E E )dδE dδE )] APPENDIX J: TOPOLOGICAL QUANTITY π j − i i ∧ i i 1 i2 = k + B + (B )+ . (J9) Chern character 2π 1 2! (2π)2 2 ··· For a complex vector bundle on the base space-time man- ifold M, whose structure group is the general k dimen- When the correction to the ith energy level δEi is ex- sional complex linear group GL(k,c), there exists the panded up to the pth order, in mind of Eq. (??), it is Chern character, which is an invariant polynomial of the easy to verify that group GL(k,c), dδE dδE dδE dδE i ∧ ∧···∧ ∧ Ch(M) = T r(exp Ω) m π = dδE(m) dδE(m) dδE(m) dδE(m). (J10) i 1 i2 | ∧ {z ∧···∧} ∧ = k + T rΩ+ T r(Ω Ω)+ (J1). 2π 2! (2π)2 ∧ ··· This equation suggests that if we want to find out the pth order of phase transition, the highest order of per- While this invariant polynomial may be expressed into a turbation to the ith energy level must be extended to the more familiar generalized Berry phase, pth order. Berry Phase 1 ˆ ChB (M) = B0 + B1 + B2 + . (J2) For the time dependent Hamiltonian operator Hi(R(t)), 2! ··· the eigenvalues and eigenvectors at time t can be ex- pressed as a function of t, Ei(R(t)) and ψi(R(t)). My Generalization of Berry Phase In this section, we generalize the conventional Berry Hˆ (R(t)) ψ (R(t)) = E (R(t)) ψ (R(t)) . (J11) i | i i i | i i 41 According to Berry’s definition[32], there was a induced perturbation and the observable. If Oˆ is a local observ- gage potential able, He(t) represents an external source which couples linearly to the observable, He = Ofˆ (x, t). The coeffi- ~ ~ A(R(t)) = i ψi(R(t)) ~ ψi(R(t)) . (J12) cient of proportionality between the change in the expec- h |∇R| i tation value 0 O(x, t) 0 and theR force f(x, t) defines a The first Chern number of Berry phase can be rewritten generalized susceptibilityh | | i by these coordinates in R space as 0 1 i iωτ ˆ ˆ Ch = A~(R(t)), χ = ~ dτe 0 [O(x′,t′), O(x, t)] 0 . (K3) 1 2iπ ∇× − h | | i ZR Z−∞ = iǫijk ∂Rj ψ(R(t)) ∂Rk ψ(R(t)) . (J13) h | i G(x′, x)o = [Oˆ(x′,t′), Oˆ(x, t)] is the retarded Green func- The Chern number is actually the topological quantiza- tion. Under Fourier transforms, this Green function can tion of the magnetic field be mapped into its momentum space, 1 ~ ′ B = A = iǫtjk ∂Rj ψi ∂Rk ψi ik(x x) iωt ∇× h | i G(k,ω)= d(x x′) dte − e G(x′, x)o (K4) ψ ∂ H ψ ψ ∂ H ψ − i Rt p p Rj i Z Z = iǫtjk h | | ih | 2 | i (Ep Ei) p=i X6 − In fact, the correction to the ground state energy can ψi ∂Rt H ψp ψp ∂Rj H ψi also be expressed by Green function in momentum space. = Imǫtjk h | | ih | 2 | i. − (Ep Ei) Usually, it is easy to get δEg from the familiar formula, p=i X6 − (J14) 0 He n n He 0 δEg = 0 He 0 + h | | ih | | i + . (K5) h | | i E0 En ··· There has been some evidences pointing out that the n=0 X6 − Berry curvature is singular at the points where the energy bands touches[33]. We can write the Hamiltonian with a variable coupling constant λ as Hˆ (λ) = Hˆ0 + λHˆe,so that Hˆ (1) = Hˆ and Hˆ (0) = Hˆ0. Then the correction to the ground state APPENDIX K: LINEAR RESPONSE AND energy is, GREEN FUNCTION 1 1 V dλ 3 ∞ iω Let H0 be the full Hamiltonian describing the system δEg = i 4 d k dωe η ±2 (2π ) 0 λ in isolation. One way to test its properties is to couple the Z Z Z−∞ ~2k2 system to a weak external perturbation and to determine (~ω )T rGλ(k,ω). (K6) how the ground state and the excited states are affected − 2m by the perturbation. The Hamiltonian H for the system weakly coupled to an external perturbation, which we Therefore, the topological quantum phase transition is will represent by a Hamiltonian He, is H = H0 + He. intimately related to the topology of momentum space. Let Oˆ(x, t) be a local observable, such as the local den- In fact, the correction to the ground state energy is not sity, the charge current, or the local magnetization. The the only physical quantity that can be used to study the expectation value of the observable in the exact ground quantum phase transition, other physical observables also present a sudden change at the phase transition point. state 0 , under the action of the weak perturbation He, is modified| i as The thermal dynamic potential is another good candi- date, t 1 0 Oˆ 0 = 0 Oˆ 0 + i~− dt′ 0 [H , Oˆ] 0 + (K1). h | | ie h | | i h | e | i ··· 1 dλ Zt0 δΩ=Ω Ω = − 0 ± λ Z0 If we only retain the linear term in He, the first-order ~2 2 3 1 ∂ k λ change in a matrix element arising from the external per- d x lim lim [ ~ + + µ]T rG (xτ, x′τ(K7)′). 2 − ∂τ 2m turbation is expressed in terms of Heisenberg operator of Z the interacting, δ 0 Oˆ 0 0 Oˆ 0 e 0 Oˆ 0 , i.e. h | | i ≡ h | | i − h | | i There also exist many other physical observables, such t as the single-particle current operator, Jˆ = d3xj, 1 δ 0 Oˆ 0 = i~− dt′ 0 [He, Oˆ] 0 . (K2) h | | i h | | i jαβ = αβ ψα† Jαβψβ, whose expression in terms of Zt0 R Green function is APPENDIX L: GREEN FUNCTION The terms in the series for 0 S(+ , ) 0 are called vacuum polarization terms.h There| ∞ was−∞ theorem| i states We first show that the Green function follows a sim- that the vacuum polarization diagrams exactly cancel the disconnected diagrams in the expansions for G(p,t t′). ilar Schr¨odinger equation. For a system described by − a Hamiltonian H which can not be solved exactly, the A great simplification is that we sum over only usual approach is to set, H = H0 + V, where H0 is the linked (connected) distinct graphs. It turn out that unperturbed part which may be solved exactly. The term this corresponds exactly to cancelling the factor of V represents all the interactions. The wave function is 0 S(+ , ) 0 in the denominator. From the discus- governed by the interaction, sionsh | above,∞ −∞ one| i can that the unperturbed part of the Hamiltonian H plays a trivial role in the quantum phase ˆ 0 i∂tψ(t)= V (t)ψ(t). (L1) transition, it is the interaction part He that decide the orientation of the evolution of a physical system. If we The wave functions of the Schr¨odinger equation, introduce the self-energy function Σ(p, E) to absorb all ˆ i∂tψ = Hψ, are time dependent, ψ(t) = U(t)ψ(0) = the interaction, the exact Green function can obtained iH0t iHt e e− ψ(0).The operator U(t) obeys a differential from the Dyson’s equation, equation which can written in the interaction represen- tation, i∂ U(t) = Vˆ (t)U(t). The operator U(t) in the t 1 1 interaction representation has a expansion, U(t)=1+ G(p, E)− = G0− (P, E)[1 G0(p, E)Σ(p, E)]. (L7) ( i)n t t t − ∞ − ˆ ˆ ˆ n=1 n! 0 dt1 0 dt2 dtn 0 T [V (t1)V (t2) V (tn)], ··· ··· 2 T is the time-ordering operator. U(t) may be abbre- we expand G in the power series G(p, E)= G0 + G0 Σ+ P R R R t 3 2 viated by writing it as U(t) = T exp[ i dt1Vˆ (t1)].. G Σ + . The self-energy is a summation of an infi- − 0 0 ··· The S matrix changes the wave function ψ(t′) into ψ(t) nite number of distinct diagrams in the series. However it R may be defined from U(t), ψˆ(t) = S(t,t′)ψˆ(t′). The is impossible to get Σ(p, E) exactly, one must be contend time-ordered operator also obeys S(t,t′)= U(t)U(t′). It with an approximation results. Usually, the higher order is easy to verify S(t,t′) obeys of phase transition means the higher order of self-energy terms must be included. The Green function of an inter- 1 i∂tS(t,t′)= Vˆ (t)S(t,t′). (L2) acting electron system is Gr(p,ω) = [ω Ek Σ]− . The − − 1 corresponding spectra function is A(p,ω)= π ImGr, This equation is intimately related to the computation − of ground state (or vacuum) expectation values of time ordered products of field operators in the Heisenberg rep- 1/πImΣr A(p,ω) = lim − 2 2 = δ(ω Eq′ ). ImΣr 0 (ω E ) + ImΣ − resentation → − q′ r (L8) N G (x1, x2, ..., xN )= 0 T [φˆ(x1)φˆ(x2)...φˆ(xN )] 0 , (L3) where E′ = E ReΣ (p,ω). The spectra function h | | i q q − r represents a resonance peak with width 2Γq. One can 2 In particular the 2-point Green function, G (x1, x2) = associate each peak with a quasi-particle. The life- 0 T [φˆ(x1)φˆ(x2)] 0 , is known as the Feynman Propaga- time of quasiparticle is infinite for a vanished ImΣr, h | | i 1 tor. For an interacting case, the Green function reads[39] τq = limImΣr [2ImΣr]− . So the quasiparticles on the fermi surface is a kind→ of ∞ topological excitation, the ˆ ˆ 12 0 T [φ(x1)φ(x2)S(+ , )] 0 external perturbation does not shorten its life. 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