Quantum phase transition

In the physics a means quantum phase transition (or English quantum phase transition , QPT ) a phase transition between different quantum phases , the various " states of aggregation " (analogous to "liquid", "fixed"; magnetic, non-magnetic, etc.) at absolute zero temperature , K , where no thermal fluctuations occur, but only quantum fluctuations . The quantum phase transition is based on an abrupt, qualitatively significant change in the ground state of the present many-particle system due to the quantum fluctuations. ${\displaystyle T=0}$

In contrast to the “classic” (thermal) phase transitions, quantum phase transitions can only occur if a non-temperature- related physical parameter such as pressure , chemical composition or a magnetic field is varied at absolute temperature zero .

Classification

A distinction is made between "first" and "second order" phase transitions , depending on whether one of the first or only one of the second derivatives of the thermodynamic potential is zero (usually the first is the case, but at the critical point it is a second order phase transition) . Quantum phase transitions can also be second order phase transitions; they then resemble the transition from the non-magnetic to the magnetic phase of a ferromagnetic system when the temperature falls below the Curie temperature . (Here, however, one is always with .) ${\displaystyle T=0}$

It is also useful to compare quantum phase transitions and classical phase transitions (also called “thermal phase transitions”). A “classic phase transition” describes a sharp, qualitative, significant change in the thermal system properties. It signals a reorganization of the particles (or their characteristic properties). A typical example of a classic phase transition is freezing , which (not only with water!) Describes the transition from a liquid to a solid state. Classical phase transitions are based on the conflict between the energy of the system and the entropy of its thermal fluctuations. In a classical system, the entropy disappears at absolute zero; therefore classically no phase transition can occur. ${\displaystyle T=0}$

Continuous transitions (this includes the "second order") convert an "ordered phase" into a "disordered" one, whereby the state of order is quantitatively described by an order parameter (it is zero in the disordered phase and increases steadily to positive when the transition parameter is undershot Values ​​on). For the ferromagnetic phase transition mentioned above, the order parameter would correspond to the internal magnetization of the system. But although the order parameter itself (a thermal mean value) is zero in the disordered phase, this does not apply to its fluctuations, which get infinite range near the critical point. This is related to the correlation length, and typical fluctuations decay with a characteristic correlation time : ${\displaystyle \xi }$ ${\displaystyle \tau _{\text{c}}}$

${\displaystyle {\begin{aligned}\xi &\propto |\epsilon |^{-\nu }\\\tau _{\text{c}}\propto \xi ^{z}&\propto |\epsilon |^{-\nu z}\end{aligned}}}$

With

• the critical exponent and ${\displaystyle \nu }$ ${\displaystyle z}$
• the relative deviation of the temperature from the critical value: ${\displaystyle \epsilon }$

${\displaystyle \epsilon ={\frac {|T-T_{\text{c}}|}{T_{\text{c}}}}}$.

The critical behavior of thermal phase transitions is fully described by classical physics , even if it is e.g. B. superconductivity is a macroscopic quantum phenomenon .

The phase diagram of a quantum phase transition

At finite temperature , the quantum fluctuations and the thermal fluctuations are in competition with one another. The respective energy scales are ${\displaystyle T>0}$

• ${\displaystyle E\propto \hbar \cdot \omega }$ for the quantum fluctuations

with Planck's reduced quantum of action ${\displaystyle \hbar }$

• ${\displaystyle E\propto k_{\text{B}}\cdot T}$ for the thermal fluctuations

with the Boltzmann constant . ${\displaystyle k_{\text{B}}}$

For quantum fluctuations dominate the system behavior, but for " scaling " along an axis through the critical point  QCP , the respective vertical distance from this axis is decisive; the scale behavior is only violated if z. As with comparable is. This results in a tapering, increasingly broad, quantum- critical scale range around the axis through  QCP . ${\displaystyle \hbar \cdot \omega \gg k_{\mathrm {B} }\cdot T}$ ${\displaystyle \hbar \cdot \delta \cdot \omega }$ ${\displaystyle k_{\mathrm {B} }\cdot T}$ ${\displaystyle y}$

The amount of can be seen as the characteristic frequency of a quantum oscillation and is inversely proportional to the correlation time: ${\displaystyle \omega }$

${\displaystyle |\omega |={\frac {A}{\tau _{\text{c}}}}}$

As a result, it should be possible to see traces of a quantum transition even at finite temperatures. These traces can show up in unconventional physical behavior, e.g. B. in quantum liquids that deviate from the usual Fermi behavior .

So one expects a phase diagram as in the adjacent sketch. The boundary lines outside the ordered state are only vaguely defined as so-called " crossover lines ". The range of visibility of quantum behavior is definitely quite large. ${\displaystyle T>0}$

Systems

Peculiarities that lead to quantum phase transitions occur primarily in one-dimensional  systems, especially since they allow diverse images. Accordingly, such systems, e.g. B. Spin chains and ladders, but also the so-called. Spin ice , primarily examined.

literature

• Matthias Vojta: Quantum phase transitions . In: Reports on Progress in Physics . tape 66 , 2003, p. 2069–2110 , doi : 10.1088 / 0034-4885 / 66/12 / R01 , arxiv : cond-mat / 0309604 .
• Subir Sachdev : Quantum Phase Transitions . 2nd Edition. Cambridge University Press, 2011, ISBN 978-0-521-51468-2 .
• Lincoln Carr: Understanding Quantum Phase Transitions . CRC Press Inc, 2010, ISBN 978-1-4398-0251-9 .
• Thomas Vojta: Quantum phase transitions in electronic systems . In: Annals of Physics . tape 9 , no. 6 , 2000, pp. 403-440 , doi : 10.1002 / 1521-3889 (200006) 9: 6 <403 :: AID-ANDP403> 3.0.CO; 2-R .