Understanding quantum phase transitions i.e phase transitions at absolute zero, are the key to understanding high-temperature superconductors. Depending on the geometry of the underlying lattice structure, antiferromagnetically interacting spins can find themselves in a multiply degenerate ground-state in which the orientation of certain electron spins does not change the energy of the ground state.

In isolation, spins will naturally be in a superposition of two possible states.

Consider an equilateral triangle, with a spin living on each vertex. Since spins can only be defined in one axis, in this case, consider the vertical axis, one of the three spins will not be able to align antiferromagnetically with another. Thus these spins are frustrated meaning they cannot achieve the lowest possible energy state. Since nature’s guiding principle is energy minimization, a superposition of all ground-states, each having an equal probability of existing, is created. This is analogous to the superposition of the spin of an electron but for a configuration of spins. Due to the dependence of a spins state on the state of other spins, the superposition of the degenerate ground-states is entangled. The entanglement depends on the geometry of the lattice and thus there is a deep relationship between lattice geometry and entanglement. Such a spin structure forms a quantum spin liquid whose main characteristic is its inherent finite entropy even at absolute zero.

Such frustrated quantum spin liquids can undergo a phase transition when parameters such as the externally applied pressure or magnetic field strength are varied. To see this imagine a horizontally applied magnetic field defined by its field strength g, acting on a quantum spin liquid whose spins are defined in the vertical axis. At zero-g, the superposition of the degenerate ground-states is symmetric under spin inversion symmetry and does not possess a magnetic moment while each configuration of spins does. This would not be the case in a conventional lattice in which spins are not frustrated. As the field strength is increased spins find it energetically more favorable to align with the field and eventually all spins will point in the direction of the field thus no longer being in an antiferromagnetic state. The frustration due to the individual spin interactions will be trumped by the strength of the field and go to zero as the field strength goes to infinity. As the field forces the spins into the horizontal direction, all individual spins will be in a superposition of up and down and thus the structure is again spin inversion symmetric. However, now the degeneracy of the ground-state is singular while the overall configuration remains entangled. Thus the degeneracy of a spin quantum liquids ground-state evolves as the magnetic field strength is varied. The maximum value for the degeneracy of the system is defined to occur at a critical value of g which lies between 0 and infinity. Current theoretical calculations imply that a larger difference in energy of each spin configuration results in the broader region of minimum entropy around zero kelvin, which increases more slowly with temperature, therefore predicting a higher but less abrupt transition temperature to a low entropy state. The entropy calculated oscillates around its minimum value very subtly as the temperature is varied. As the number of total spins increases however, this oscillation becomes ever more present as the temperature is lowered to zero kelvin.

This contrasts the evolution of a non-frustrated system in which the spin inversion symmetry is restored as the field strength is taken to infinity implying the initial antiferromagnetic state is not spin inversion symmetric which it is.

Quantum Superposition: Example of Quantum slit experiment, the electron must decide to go through a slit but it has equal chances of going through both so it goes through each slit half the time. Each electron thus passes through both slits and its state is the linear superposition of the state corresponding through the left state and state corresponding through the right state. Similarly frustrated electrons are in a state corresponding to a linear superposition of the up and downstate. These electrons are perfectly anti-correlated meaning that the state of one electron completely determines the state of the other(s). Such a collective state is non-local and is shared by both electrons.

By removing some electrons in an antiferromagnet, some spins no longer find themselves in a preferred direction as the energy of the electron in both configurations is equal. Nevertheless, this depends on the orientation of a neighboring spin and when this spin finds itself in a similar situation, the spins pair up in a linear superposition and become entangled. When this process happens for all electrons in a system, the entangled pairs can move around and collectively acquire a state representing a superposition of all possible configurations. In this way, each electron pair is no longer locally defined but has an equal probability of being everywhere in the system simultaneously. Such a state is described as a Bose-Einstein condensate.

Entangled pairs of electrons can exchange their partners and due to the non-locality of the state, these exchanges must not occur with adjacent electrons but can occur with electrons at the opposite end of the system thus leading to long-range entanglement. At the quantum critical point, the long-range entanglement is believed to be maximized and the critical temperature is maximized.

Combined pairs of electrons result in a combined degree of freedom. Combining this with a similar degree of freedom results in another degree of freedom and so on. In this way, the depth of entanglement can be increased. String theory allows calculation of what happens to particles under conditions of long-range quantum entanglement.

When entangled electrons are separated by a black holes event horizon they are still able to exchange information. This seems to imply that entanglement is a process that occurs in an extra dimension surpassing any three-dimensional boundary and therefore being non-local.

Long-range quantum entanglement enhances the probability of pair production which then results in higher critical temperature for superconductivity.

If ER=EPR is considered valid, the reason for superconductivity is explained by the fact that entangled electrons can travel through an extra dimension thus avoiding the atoms in their path and therefore never colliding with them. The path through this extra dimension is analogous to the path of a wormhole through space-time.

Entangled electrons can travel through this extra dimension and thereby change their spin. In this way, electrons can travel without interacting with any atoms leading to high temperature.