Isaac Newton once asked himself that any object which is placed in our hands, if the same object is located in another part in space, would that object show the same properties assuming that space and matter are homogeneous and isotropic. Great philosophers and scientists like Galileo Galilei, René Descartes, and others may not have used the term symmetry directly, but it would be certainly not wrong to assume that they thought of something like that. The word symmetry is relevant to our lives. Those who are dealing with different branches of physics, they understand the special importance of the term symmetry. In a sense, this is slightly exaggerated to state that the study of physics means the study of symmetry. At least that's how Philip Anderson, a Nobel laureate physicist, described symmetry in his famous article *More Is Different*, published in Science, 1972. While discussing various aspects of low temperature, it is clear that with lowering of temperature for many systems properties of matter changes. The phase transitions occur in different systems depending on the environment and parameter ranges. So far, well-known examples of these phase transitions are water to ice, normal conductor to superconductor (a state with zero resistivity), normal liquid to superfluid (a state with zero friction), or different magnetic phase transitions (paramagnetic to ferromagnetic, antiferromagnetic etc.). The topic phase transition is in general of great interest for basic research. In particular for many electron systems, with lowering temperature or varying other parameters, such as external magnetic field, pressure (hydrostatic, uniaxial pressure) lead to emergent phases. These are not well understood things, and many bright minds from different parts of the world are interested to understand the origin of these emergent phenomena.

To grasp the concept of symmetry, the basic understanding of phase transition is crucial. Let us begin with one real life example. If we change the water temperature, say, from 30°C to 32°C, do we observe any noticeable change in the state of the water! No certainly not. We will not recognize anything. Water molecules will move at a slightly higher speed, which we may not perceive with our naked eye. It will not change the state of the water; the water will remain the same. There will be no change in its color, no change in its shape. The container it was kept in will remain the same. If the container is changed, the water will take the shape of the new container. The state of the water will not change. Even if we change the location keeping the temperature same the state of the water will not change. Similarly, if we apply small external pressure to the water, the state of the water will not change. The water remains water. But what if we change the temperature of the same water from 1°C to -1°C, we will notice a dramatic change of the state (shape and size) of the water. The water then turned into ice. This is called one kinds of phase transition. Matter goes from one state to another. This transition or changes of the states can be observed visually or in the laboratory. But the question remains why does this happen!

If we want to understand the origin of the phase transition of matter, say water to ice, normal conductor to superconductors, normal fluid to superfluid and different kinds of magnetic phase transitions, the word *symmetry* comes up again and again. Basically, different rules of symmetry are used to explain the properties of phase transitions. Not that physicists, chemists, material scientists, and mathematicians are only interested to know the origin of different emergent phenomena in many body systems. Philosophers are also particularly attracted to this aspect of research. Even in a simple solid which contains many molecules, atoms, they are strongly entangled giving rise to emergent properties with changing different parameters.

Symmetry arguments are carefully used to explain these phenomena. At least it would be rational to state that the mathematical interpretation of symmetries helps us to understand these exotic phases and phase transitions in condensed matter systems. In summary one can state that when matter goes to a new state, the symmetry of the system also changes. When the symmetry is broken, reduced or changed, the properties of matter also change dramatically. The state of the matter changes, accordingly.

When discussing symmetry one cannot simply ignore the name of Hermann Weyl. He was a German mathematician, physicist and philosopher. Herman Weyl made outstanding contributions to various subjects such as space, time, physics, philosophy, logic, symmetry, and the history of mathematics. One cannot simply forget his contributions in all these fields. However, he understood symmetry somewhat differently. His book *The Symmetry* clearly proves his grasp on the subject. In his words beauty and symmetry are bound together. Symmetry is a vast topic; this has an undeniable significance in art and nature. However, the root of symmetry is mathematics. So to understand symmetry properly, one must take the help of mathematics. The symmetry has special importance not only in physics, chemistry, and mathematics but also in our life. Symmetry arguments can potentially be used to understand our life, the origin of life. Thus we need to understand the concepts of symmetry. Let us first try to know what the meaning of the word symmetry!

To simply put, if an object looks the same when viewed in different ways, from different perspectives, and different operations on this, then we can say that the object is symmetrical. In more technical words it can be said that after changing an object in a different way, such as rotation, inversion, translation (change of place), change of time (time reversal), if the properties of the object remains unchanged, and then it can be said that the object has preserved symmetry. The word symmetry comes from the Greek word *symmetros*, symmetros, its first part syn means the same, and the second part metros mean measure, meaning well-proportioned or harmonious. Symmetry is a subject of comparison. This means that it is only possible to tell whether an object is symmetrical when it is compared with respect to something else, essentially an asymmetric object. For example, if we have a round soccer ball in our hand, if the soccer ball is the same color and if there is no difference in the surface of the soccer ball, if we keep spinning the soccer ball differently, no matter how we spin it, the ball will look the same to us before and after the spin or after all these operations. We cannot distinguish between its initial and final states. We would say that football has maintained its symmetry after all such operations and/or transformations. A question remains, however, with respect to what the object keeps this symmetry. To simply put, symmetry can only be measured when compared to an asymmetric object, an asymmetric animate or inanimate object, this also applies to space and time.

Let us try to understand symmetry with a few more simple examples. If we imagine a circle with a plane passing through its center, no matter how many planes we draw with respect to that center, the circle will be symmetrical about that plane. If we look at an equilateral and a right triangle differently, we will find them similar. But if we take a quadrilateral, we cannot distinguish anything between its initial and final states, if the quadrilateral is rotated 180 degrees about its center. Similarly for a square, that rotation would be 90 degrees. But it can be said that the square will retain its symmetry after 90 degree rotation. The subject can be thought of in terms of many other such geometrical objects. A ring is symmetrical about the circle. The object can be made a little more complicated. A solid cylinder and a hollow cylinder can roll in the same way on a table. As an outsider without having the prior knowledge about the constructions of these cylinders, it is difficult to identify the nature of these cylinders. However, their properties will not be the same in water. For example, a hollow cylinder can float in a liquid, but a solid cylinder will sink in a liquid.

Now let us view the symmetry from a bit different perspective. We have already mentioned symmetry and beauty are greatly entangled. In our common sense, a system looks beautiful or attractive when the system possesses some kind of symmetry. In majority of the cases the lack of symmetry can lead to partial loss of attractions. The system can be inanimate or animate. If one looks at the yellow sunflower in bloom, the flower petals maintain a strange but appealing symmetry. In nature many flowers are not only beautiful in their colors and scent, but they are also mostly symmetrical. The flower's stamen has a wonderful structure. It also has the symmetry. Beeswax has hexagonal symmetry. Rotational symmetry is present in ornaments. Human body structure is symmetric. Butterflies are highly symmetric. In fact humans and other species are evidently more attracted to symmetrical patterns. Of course we compare these with asymmetric patterns. The question is why! Why do we like symmetrical patterns! Why do we like symmetrical objects! How our brain and our phycology is inclined so. Is there a part of our brain highly responsive with the symmetrical patterns! The reason is certainly scientific.

One reason that comes to our mind is how the evolutionary aspect is related to the symmetry and living objects. From the beginning of our life there has been a continuous evolutionary process not visible at the macroscopic level, but certainly at the microscopic level. As an important part of the biological processes this is rather conventional wisdom to accept that to attract or influence each other is rather spontaneous. It is natural to accept that some invisible biological signals are sent to each other to influence each other. There symmetry comes very naturally as a part of evolution. Light signal, which is made out of photons and nothing but the frequency, comes from an object and falls into the eyes of a living object. Then the light signal goes to the brain. Some special section of our brain is being activated through this specific frequency. This is how we react to animate or inanimate objects. There is an invisible message being transferred from one object to another. In this way symmetry comes out of each other's necessity, which brings about certain configurations. The whole point can be rephrased in this way that preserving symmetry means obeying some natural laws.

*In the figure, the infinite numbers of planes pass through the center of the circle. The system is symmetric with respect to each plane. *

*Several common symmetries are compared in Fig. (a) A square symmetry resulting from its 90 degree rotation. (b) The rectangle is symmetrical after 180 degree rotation. (c) The square is symmetric with respect to the dotted line that is to say reflection symmetry. (d) This figure is symmetric due to 90 degree rotation. But it has no symmetry in any reflection. (e) The circle is symmetrical no matter how it is rotated. (f) Equal symmetry of ring with respect to circle. (g) This image has no symmetry. (Image idea of Juan Maldacena)*

Universal laws of physics are symmetric in space and time. In simple picture universal laws remain the same when changing locations. They are independent on time. That is to say if the laws are valid today, then these will be valid in the future, and these were valid in the past. For example, the hydrogen molecule obeys the same laws of physics on Earth and billions of light-years away. The fundamental laws of physics have no preferred directions or no preferred orientations. Fundamental laws retain their symmetry even if they are turned upside down. All those basic principles of science, whether measured in a room in Kolkata, measured in the Himalayas or in the farthest quasar, the principles remain the same. These are the basic principles of nature. It is not the constitution of any country, or any religion, or Chankya's economic theory, which will have to be modified with respect to time, place, environment and socio-political system. If Chanakya's economic theory that had been practiced during Chandragupta's rule in the Maurya Empire is to be used in the modern era then many additions are necessary. Basic principles of science are only basic principles of science. A mango falls to the ground or a rock falls to the ground when it is thrown from the sky, or a hand bounces when it hits a solid wall, a man recoils when a gun is fired, the principle that the earth revolves around the sun, etc all follows the same laws of gravitation provided by the Isaac Newton. The planets revolve around the Sun, the principle of the ebb and flow of the tides, the principle that when a moving horse stops suddenly, the rider falls forward, they also follow the laws of Gravitation. Those laws remain unchanged across space and time. To simply put they are symmetric. Isaac Newton proposed these fundamental laws of nature in the seventeenth century. Still this is valid and it will continue to do so in the future as well. So does Einstein's theory of relativity, that the speed of light is constant, or nothing can be faster than the speed of light or that space and time are twisted together because of heavy mass. These are all basic principles and they are all symmetrical.

At the outset Einstein first attempted to develop the theory of relativity by placing symmetry on various reference frames. He wanted to find a method in which all the basic principles of nature would be the same with respect to all observers, whether the observer stays in a fixed position, moves with a constant velocity or the observer moves with acceleration.

Einstein, however, first took the path shown by James Clerk Maxwell when it came to the question of symmetry. Einstein was born in 1879. James Clerk Maxwell died in the same year. Maxwell constructed a set of mathematical equations. Whenever a stationary charge gains momentum, it produces a current and the current creates the magnetic field. Maxwell combined all the electrical and magnetic properties known at the time into a set of equations. These are famous Maxwell’s equations.

In the following we will discuss some very important symmetry of physics and how one can classify them. The geometrical objects have spatial symmetry. The laws of nature are essentially a physical symmetry. Discrete symmetry, continuous symmetry, bi-lateral symmetry, translational symmetry, planar symmetry, parity or inversion symmetry, rotational symmetry, gauge freedom, time-reversal symmetry, etc are usually the common symmetry we talk about in physics. In solid state physics, phase transition relates to discrete symmetry. The best example of discrete symmetry is any crystal. A classic example of continuous symmetry is our universe. Wherever we go from any point in the universe will be symmetrical. It will remain unchanged in rotation about any particular point in the universe. For the universe rotational symmetry will be maintained.

In several branches of physics physicists try to understand different principles by using the varieties rules of symmetry. Many complex equations can be simplified to some extent by applying selections of symmetry rules. To simply put, in general, some complex problems in physics are greatly simplified by the correct use of symmetry arguments. In the following we will try to understand with specific examples how symmetry can be used to understand the phase transitions in condensed matter systems. There comes the concept of symmetry breaking. The next section deals with this.

Symmetry Breaking

Application of symmetry arguments is widespread in various fields of science today. Chemistry, physics, biology and mathematics are essentially the main areas of applications. An important connection, perhaps, can even be made with the theory of evolution. When naturalist Charles Darwin published his theory of evolution, resistance to his theory came from various affiliates, either directly or indirectly. A major obstacle to the wide acceptance of Darwin's revolutionary doctrines was the timely concepts of science. Then the effect of Newton’s principles of determinism was very profound in physics. It means that in physics all events happen according to causality rules. There is no chance of anything happening suddenly. There is no possibility of any sudden change. As a result, Darwinism was not easily accepted by a group of physicists and philosophers. But the question is can someone connect this with the phase transitions of matter. The answer is, perhaps, yes. In condensed matter systems phase transition can take place rather suddenly. The properties of matter can change suddenly. The symmetry can break rather spontaneously. The change of states of matter is a result of spontaneous broken symmetry. The phase transition means the emergence of a completely new state. In such a scenario do we need to bother the principles of causality? If someone gives you a glass of water at room temperature, do we have any clue by looking at this water that with lowering the temperature below 0 degree Celsius the water can become ice! Do we have any clue by investigating a paramagnet; the same paramagnetic state by lowering temperature can be ferromagnetic, antiferromagnetic or any other magnetic phase? Do we have any clue by investigating a superconductor at normal condition it can be a superconductor below a certain temperature? In all three cases the answer is no, actually we don’t know the phase transition can take place in these systems. The change of states occurs suddenly with small changes of temperature, pressure or other external parameters.

Russian (Soviet) physicist Lev Landau was, most likely, the first to realize that changing a state of matter would also modify its symmetry. Contextually, the more correct statement would be the change in the symmetry in a matter leads to the changes of the phases or this leads to phase transition. Lev Landau was an outstanding theoretical physicist. He made significant contributions to various branches of physics in particular for theoretical condensed matter physics research. By applying his theories, it is very useful to know and understand basic ideas of phase transition and its related phenomenon. He made significant contributions in the phenomenological theories in superfluid (a state with no friction), ferromagnetism and superconductor. Landau was the first to introduce the concept of order parameters. In technical words the order parameter is an observable that was not present above the ordering or the phase transition. This suddenly appears along with the phase transition, and develops as one approaching deep in the order state. For example in magnetic phase transition magnetic moment is an order parameter, which is not present or zero above the ordering temperature. But in the ordered state the value of the ordered moment increases and deep in the ordered state the value of the magnetic moment becomes saturated. In different phase transition, such as gas to liquid to solid, ordinary conductor to superconductor, paramagnetic to ferromagnetic or antiferromagnet, or nematic liquid crystal to liquid (liquid), or ferroelectric to non-polar crystal order parameter is also different. In general, order parameters depend on the type of phase transitions. The value of the order parameter changes with modifying the temperature. As the system goes to a new state, so to say ground state of the system, the value of the order parameter becomes constant. As the system reaches the ground state by changing external parameter order, the parameter hardly modifies. The development of the order parameter along the phase transitions, and how this changes with the temperature, helps us to know how that state evolves in addition to the microscopic properties of the system. Landau most likely realized that if the various states of matter were to be accurately described, microscopic theory, perhaps, would not do much. Instead, if the order parameter approach could be taken into action along with the symmetry arguments associated with that. The new states of the matter might be possible to explain. Now the main important point is that how the symmetry of one state of matter is changed as the system reach to a new state. This helps to extract much complex information about the system. The proper utilization of the symmetry arguments is in general very useful to know the properties of the matter. Therefore, symmetry breaking aspects become very important in physics to understand the mechanism of phase transitions and exotic low temperature phases of condensed matter systems.

Let us try to understand this with a simple everyday life example. Water has three phases, gas, liquid and solid (ice). The above schematic shows three phases of gas, liquid and solid. At a first glance, our conventional outlook suggests that gaseous state is less symmetric with respect to the liquid and/or solid phase. The gas particles are randomly distributed. Apparently, this phase reflects highly low symmetry. But when the phase’s changes from gas to liquid, its symmetry seems to higher, and when it becomes solid, since each atom in the lattice is arranged in a highly aligned fashion, the symmetry of the object actually increases from the gaseous state to the solid state. That's what we all might think at first sight. However, the reality is something else. Gas particles in a closed container move randomly within the container depending on the temperature of the container. Higher the temperature the average velocity will increase. From outside as we look at the gas particles in the enclosed container, at any given moment all particles look the same. It does not matter how we look at it, it does not matter which direction we look at it. To simply put, one gas particle will look the same with respect to another gas particle. Now if we place this as symmetry arguments, when we look at a gas, it will be the same no matter how we look at a particular point in the gas. Similarly, if we rotate the gas about a fixed point, individual gas particles cannot be simply distinguished between its initial and final states. The gas as a whole will retain its symmetry after all these operations. In symmetry language, it will be said that both translational and rotational symmetries are preserved here. Let us move one step further.

When the gas (steam) turns into a liquid (water), there is no change of symmetry. Both translational and rotational symmetries are unchanged. But now if water becomes ice, that is liquid to solid phase transition, things are not the same any more. Solid atoms are uniformly placed at unit cells. Atoms are located in such a way that individual atoms are separated at a certain equal distance. All atoms maintain a fixed distance. This is how solid is formed. In a real crystal to identify a distance between two random atoms one just has to multiply with certain numbers. Unlike the case of gas where all gas particles look the same, solids represent something else. Some degree of freedom has been lost. However, a discrete symmetry remains preserved.

What happens now if the lattice is rotated about any arbitrary axis? For any arbitrary rotation the initial and the final state will not appear the same with respect to the observer. Only with some specific angle of rotation the initial and final configuration appear the same, which is different than the case of gas or liquid. Therefore for solids one degree of freedom has been reduced. So the symmetry has been broken and/or reduced.

In technical terms rotational symmetry is maintained for all solids, but some limitations are imposed there. So because of the phase transition from gas/liquid to solid the symmetry of the system has changed. The gas/liquid system possesses higher symmetry, but its symmetry is reduced in the solid (ice) state with lowering the temperature. This is called symmetry breaking in physics. It turns out that in a condensed matter system with varying external parameters such as temperature, pressure (hydrostatic and uniaxial) broken symmetry associates phase transition. So far we have discussed a simple example for gas, liquid and solid phase. But this idea of symmetry breaking concepts is not restricted there. Exactly in a similar way one can demonstrate the case of magnetic phase transition.

*Left shows a higher symmetric object and the right shows a lower symmetric object. The right figure shows the ferromagnetic state of the spins. Spins are aligned in the same direction. Left figure shows disordered phase of spins at high temperatures. *

At first glance, the left picture appears to be less symmetrical. The spins are randomly oriented, there is no apparent correlation between the spins and the system is in the paramagnetic (disordered) state. However, this does not mean that it is a lower symmetric object. On the contrary, if we change our perspective, this displays a different scenario. In fact, no matter how we look at the system, no matter how we rotate the object, this turns out to be symmetric at any given time with respect to the observer as compared one spin to the other. This is in similar analogy to gas particles in a given confined container. We have already discussed the symmetric nature of gas. In the case of a magnet, when magnetic phase transition occurs, say paramagnetic (disordered) to ferromagnetic (ordered) state, and then evidently symmetry breaks. In the ferromagnetic state all the spins are oriented in a specific direction, and spins are correlated. Now we notice immediately things are different. This lowers the symmetry. In ferromagnetic transition spontaneous symmetry is broken.

Similarly, if we talk about the antiferromagnetic transition which is another type of magnetic phase transition shown on the right side. In this case, in the ordered state all the spins are not oriented in a specific direction; rather two neighboring spins are aligned exactly opposite to each other. Evidently in the ordered state the whole object is no more symmetric. In this case also we see that if the whole system is turned a bit, the system is no longer the same. From different perspectives the spins are no longer symmetric as compared to the paramagnetic state. This tells in case of antiferromagnetic phase transition as well there is a symmetry breaking. So far we have discussed how phase transition and symmetry breaking aspects are connected in case of liquid to solid or magnetic phase transition. But these general concepts are also being used to understand other kinds of phase transition: normal conductor to superconductor or normal fluid to superfluid. However, there we talk about different kinds of symmetry.

In conclusion, we try to give a simple physical picture of symmetry and symmetry breaking in condensed matter systems. We have seen that breaking symmetry breaking or lowering symmetry leads to phase transition in condensed matter systems. Breaking symmetry not only leads to phase transition of a matter, but also these lead to other consequences in a substance. Basically four things can be observed in a condensed matter system as a result of broken-symmetry, and everything else can be described based on these. These four things are 1) phase transition, 2) rigidity, 3) excitation, and 4) defects. In our follow up writing we will discuss all these aspects in detail.

References:

P. W. Anderson, More Is Different, Science177, 393 (1972)

Steve Blundell, Magnetism in Condensed Matter, Oxford (2000)

Juan Maldacena, The symmetry and simplicity of the laws of physics and the Higgs boson arXiv:1410.6753 (2014)

Author:

Rajib Sarkar

Scientist

Technical University of Dresden, Germany

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