Quantum phase transition
Encyclopedia
In physics
, a quantum phase transition (QPT) is a phase transition
between different quantum phases
(phases of matter
at zero temperature
). Contrary to classical phase transitions, quantum phase transitions can only be accessed by varying a physical parameter - such as magnetic field
or pressure - at absolute zero
temperature. The transition describes an abrupt change in the ground state
of a many-body system due to its quantum fluctuations. Such quantum phase transitions can be second-order phase transition.
To understand quantum phase transitions, it is useful to contrast them to classical phase transitions
(CPT) (also called thermal phase transitions). A CPT describes a cusp in the thermodynamic properties of a system. It signals a reorganization of the particles; A typical example is the freezing
transition of water describing the transition between liquid and ice. The classical phase transitions are driven by a competition between the energy
of a system and the entropy
of its thermal fluctuations. A classical system does not have entropy at zero temperature and therefore no phase transition can occur. Their order is determined by the first discontinuous derivative of a thermodynamic potential.
A phase transition from water to ice, for example, involves latent heat (a discontinuity of the heat capacity
) and is of first order.
A phase transition from a ferromagnet to a paramagnet is continuous and is of second order.
These continuous transitions from an ordered to a disordered phase are described by an order parameter, which is zero in the
disordered and non-zero in the ordered phase. For the aforementioned ferromagnetic transition, the order parameter would represent the total magnetization of
the system. Although the thermodynamic average of the order parameter is zero in the disordered state, its fluctuations can be non-zero and become
long-ranged in the vicinity of the critical point, where their typical length scale
(correlation length) and typical fluctuation decay time
scale
(correlation time) diverge:
where ε is defined as the relative deviation from the citical temperature, ε=(T-Tc)/Tc.
We call ν the (correlation length) critical exponent and z the dynamical critical exponent. Critical behavior of finite temperature phase
transitions is fully described by classical thermodynamics, quantum mechanics does not play any role even if the actual phases require a quantum
mechanical description (e.g. superconductivity).
Talking about quantum phase transitions means talking about transitions at
: by tuning a non-temperature parameter like pressure,
chemical composition or magnetic field, one could suppress e.g. some transition temperature like the Curie- or Néel- temperature to
K. As
such a QPT cannot be driven by thermal fluctuations, quantum fluctuations have to be the underlying source. The terminology used for the
description of classical phase transitions is nevertheless applied. As for a classical second order transition,
a quantum second order transition has a quantum critical point (QCP) where the quantum fluctuations driving the transition diverge and
become scale invariant in space and time. Also at finite temperatures, quantum fluctuations with an energy scale of
and classical
fluctuations with an energy scale of k
T compete. For 
, quantum fluctuations will dominate the system's
properties.
is the characteristic frequency of a quantum oscillation and inversely proportional to the correlation time. As a
consequence it should be possible to detect remnants or traces of the transition also at low enough finite temperatures: a quantum critical
region in proximity of the actual transition or the quantum critical point. These traces become manifest in unconventional and unexpected
physical behavior like novel non Fermi liquid phases. From a theoretical point of view, a phase diagram like the one shown on the right is
expected: the QPT separates an ordered from a disordered phase (often, the low temperature disordered phase is referred to as 'quantum'
disordered). At high enough temperatures, the system is disordered and purely classical. Around the classical phase transition, the system is
governed by classical thermal fluctuations (light blue area). This region becomes narrower with decreasing energies and converges towards the
quantum critical point (QCP). Experimentally, the 'quantum critical' phase, which is still governed by quantum fluctuations, is the most interesting
one.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a quantum phase transition (QPT) is a phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
between different quantum phases
Quantum phases
Quantum phases are quantum states of matter at zero temperature. Even at zero temperature a quantum-mechanical system has quantum fluctuations and therefore can still support phase transitions. As a physical parameter is varied, quantum fluctuations can drive a phase transition into a different...
(phases of matter
Phase (matter)
In the physical sciences, a phase is a region of space , throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, and chemical composition...
at zero temperature
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....
). Contrary to classical phase transitions, quantum phase transitions can only be accessed by varying a physical parameter - such as magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
or pressure - at absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....
temperature. The transition describes an abrupt change in the ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...
of a many-body system due to its quantum fluctuations. Such quantum phase transitions can be second-order phase transition.
To understand quantum phase transitions, it is useful to contrast them to classical phase transitions
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
(CPT) (also called thermal phase transitions). A CPT describes a cusp in the thermodynamic properties of a system. It signals a reorganization of the particles; A typical example is the freezing
Freezing
Freezing or solidification is a phase change in which a liquid turns into a solid when its temperature is lowered below its freezing point. The reverse process is melting....
transition of water describing the transition between liquid and ice. The classical phase transitions are driven by a competition between the energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
of a system and the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
of its thermal fluctuations. A classical system does not have entropy at zero temperature and therefore no phase transition can occur. Their order is determined by the first discontinuous derivative of a thermodynamic potential.
A phase transition from water to ice, for example, involves latent heat (a discontinuity of the heat capacity
) and is of first order.A phase transition from a ferromagnet to a paramagnet is continuous and is of second order.
These continuous transitions from an ordered to a disordered phase are described by an order parameter, which is zero in the
disordered and non-zero in the ordered phase. For the aforementioned ferromagnetic transition, the order parameter would represent the total magnetization of
the system. Although the thermodynamic average of the order parameter is zero in the disordered state, its fluctuations can be non-zero and become
long-ranged in the vicinity of the critical point, where their typical length scale
(correlation length) and typical fluctuation decay timescale
(correlation time) diverge:where ε is defined as the relative deviation from the citical temperature, ε=(T-Tc)/Tc.
We call ν the (correlation length) critical exponent and z the dynamical critical exponent. Critical behavior of finite temperature phase
transitions is fully described by classical thermodynamics, quantum mechanics does not play any role even if the actual phases require a quantum
mechanical description (e.g. superconductivity).
Talking about quantum phase transitions means talking about transitions at
: by tuning a non-temperature parameter like pressure,chemical composition or magnetic field, one could suppress e.g. some transition temperature like the Curie- or Néel- temperature to
K. Assuch a QPT cannot be driven by thermal fluctuations, quantum fluctuations have to be the underlying source. The terminology used for the
description of classical phase transitions is nevertheless applied. As for a classical second order transition,
a quantum second order transition has a quantum critical point (QCP) where the quantum fluctuations driving the transition diverge and
become scale invariant in space and time. Also at finite temperatures, quantum fluctuations with an energy scale of
and classicalfluctuations with an energy scale of k
T compete. For , quantum fluctuations will dominate the system'sproperties.
is the characteristic frequency of a quantum oscillation and inversely proportional to the correlation time. As aconsequence it should be possible to detect remnants or traces of the transition also at low enough finite temperatures: a quantum critical
region in proximity of the actual transition or the quantum critical point. These traces become manifest in unconventional and unexpected
physical behavior like novel non Fermi liquid phases. From a theoretical point of view, a phase diagram like the one shown on the right is
expected: the QPT separates an ordered from a disordered phase (often, the low temperature disordered phase is referred to as 'quantum'
disordered). At high enough temperatures, the system is disordered and purely classical. Around the classical phase transition, the system is
governed by classical thermal fluctuations (light blue area). This region becomes narrower with decreasing energies and converges towards the
quantum critical point (QCP). Experimentally, the 'quantum critical' phase, which is still governed by quantum fluctuations, is the most interesting
one.
See also
- Superconductor Insulator TransitionSuperconductor Insulator TransitionThe Superconductor Insulator Transition is an example of a quantum phase transition, whereupon tuning some parameter in the Hamiltonian, a dramatic change in the behavior of the electrons occurs. The nature of how this transition occurs is disputed, and many studies seek to understand how the order...
- Quantum phasesQuantum phasesQuantum phases are quantum states of matter at zero temperature. Even at zero temperature a quantum-mechanical system has quantum fluctuations and therefore can still support phase transitions. As a physical parameter is varied, quantum fluctuations can drive a phase transition into a different...
- Quantum critical pointQuantum Critical PointA quantum critical point is a special class of continuous phase transition that takes place at the absolute zero of temperature, typically in a material where the phase transition temperature has been driven to zero by the application of a pressure, field or through doping.Conventional phase...