KMS state

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In the statistical mechanics of quantum mechanical systems and quantum field theory, the properties of a system in thermal equilibrium can be described by a mathematical object called a Kubo–Martin–Schwinger (KMS) state: a state satisfying the KMS condition.

Overview

The simplest case to study is that of a finite-dimensional Hilbert space, in which one does not encounter complications like phase transitions or spontaneous symmetry breaking. The density matrix of a thermal state is given by

${\displaystyle \rho _{\beta ,\mu }={\frac {e^{-\beta \left(H-\mu N\right)}}{\mathrm {Tr} \left[e^{-\beta \left(H-\mu N\right)}\right]}}={\frac {e^{-\beta \left(H-\mu N\right)}}{Z(\beta ,\mu )}}}$

where H is the Hamiltonian operator and N is the particle number operator (or charge operator, if we wish to be more general) and

${\displaystyle Z(\beta ,\mu )\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {Tr} \left[e^{-\beta \left(H-\mu N\right)}\right]}$

is the

.

In the Heisenberg picture, the density matrix does not change with time, but the operators are time-dependent. In particular, translating an operator A by τ into the future gives the operator

${\displaystyle \alpha _{\tau }(A)\ {\stackrel {\mathrm {def} }{=}}\ e^{iH\tau }Ae^{-iH\tau }}$.

A combination of

"rotation" gives the more general

${\displaystyle \alpha _{\tau }^{\mu }(A)\ {\stackrel {\mathrm {def} }{=}}\ e^{i\left(H-\mu N\right)\tau }Ae^{-i\left(H-\mu N\right)\tau }}$

A bit of algebraic manipulation shows that the expected values

${\displaystyle \left\langle \alpha _{\tau }^{\mu }(A)B\right\rangle _{\beta ,\mu }=\mathrm {Tr} \left[\rho \alpha _{\tau }^{\mu }(A)B\right]=\mathrm {Tr} \left[\rho B\alpha _{\tau +i\beta }^{\mu }(A)\right]=\left\langle B\alpha _{\tau +i\beta }^{\mu }(A)\right\rangle _{\beta ,\mu }}$

for any two operators A and B and any real τ (we are working with finite-dimensional Hilbert spaces after all). We used the fact that the density matrix commutes with any function of (H − μN) and that the trace is cyclic.

As hinted at earlier, with infinite dimensional Hilbert spaces, we run into a lot of problems like phase transitions, spontaneous symmetry breaking, operators that are not trace class, divergent partition functions, etc..

The

of z, ${\displaystyle \left\langle \alpha _{z}^{\mu }(A)B\right\rangle }$ converges in the complex strip ${\displaystyle -\beta <\Im {z}<0}$ whereas ${\displaystyle \left\langle B\alpha _{z}^{\mu }(A)\right\rangle }$ converges in the complex strip ${\displaystyle 0<\Im {z}<\beta }$ if we make certain technical assumptions like the within the strip they are defined over as their derivatives,

${\displaystyle {\frac {d}{dz}}\left\langle \alpha _{z}^{\mu }(A)B\right\rangle =i\left\langle \alpha _{z}^{\mu }\left(\left[H-\mu N,A\right]\right)B\right\rangle }$

and

${\displaystyle {\frac {d}{dz}}\left\langle B\alpha _{z}^{\mu }(A)\right\rangle =i\left\langle B\alpha _{z}^{\mu }\left(\left[H-\mu N,A\right]\right)\right\rangle }$

exist.

However, we can still define a KMS state as any state satisfying

${\displaystyle \left\langle \alpha _{\tau }^{\mu }(A)B\right\rangle =\left\langle B\alpha _{\tau +i\beta }^{\mu }(A)\right\rangle }$

with ${\displaystyle \left\langle \alpha _{z}^{\mu }(A)B\right\rangle }$ and ${\displaystyle \left\langle B\alpha _{z}^{\mu }(A)\right\rangle }$ being analytic functions of z within their domain strips.

${\displaystyle \left\langle \alpha _{\tau }^{\mu }(A)B\right\rangle }$ and ${\displaystyle \left\langle B\alpha _{\tau +i\beta }^{\mu }(A)\right\rangle }$ are the boundary distribution values of the analytic functions in question.

This gives the right large volume, large particle number thermodynamic limit. If there is a phase transition or spontaneous symmetry breaking, the KMS state is not unique.

The density matrix of a KMS state is related to

.

See also

• Gibbs state

References