Noncommuting Charges and Generalized Gibbs States

Specified expectation values of noncommuting conserved charges still license a maximum-entropy state of exponential form, but not simultaneous sharp charge assignments. The multipliers combine into one modular generator, select a direction in charge space, and preserve only its centralizer. The inferred state can depend on the representation, reference frame, preparation constraints, and thermodynamic limit; it must not be conflated with an integrable GGE built from a complete commuting charge family.

A maximum-entropy and resource-theoretic derivation for noncommuting conserved quantities is given by Yunger Halpern et al. 2016, §§ Results–Discussion.

Required background. Conserved Charges and Grand-Canonical States gives compatible chemical potentials. Helpful background. Generalized Gibbs Ensembles for Integrable Charges supplies the distinct commuting integrable construction.

Maximum entropy does not require mutual commutation

Let Hermitian charges $Q_a$ be exactly conserved,

$$[H,Q_a]=0,$$

but satisfy a Lie algebra

$$[Q_a,Q_b]=if_{ab}{}^cQ_c.$$

Maximizing

$$S=-\operatorname{Tr}(\rho\log\rho)$$

subject to $\operatorname{Tr}\rho=1$ and $\operatorname{Tr}(\rho Q_a)=q_a$ gives

$$\rho_\lambda= \frac{e^{-\sum_a\lambda_aQ_a}}{Z(\lambda)}, \qquad Z=\operatorname{Tr}e^{-\sum_a\lambda_aQ_a}.$$

No step in the variational derivation requires $[Q_a,Q_b]=0$. What fails is the classical interpretation of one joint probability distribution over simultaneous eigenvalues.

The response of constraints to multipliers is not generally the equal-time covariance. Duhamel differentiation gives the Kubo–Mori matrix

$$-\frac{\partial\langle Q_a\rangle}{\partial\lambda_b} =\int_0^1\mathrm ds\, \operatorname{Tr} \left[ \rho^{1-s}\delta Q_b\rho^s\delta Q_a \right].$$

This positive semidefinite information matrix controls local identifiability of $\lambda_a$ modulo redundant directions. Replacing it by $\langle\delta Q_a\delta Q_b\rangle$ silently assumes commutation.

Residual symmetry and a spin example

For $SU(2)$ charges $\mathbf S$,

$$\rho_{\boldsymbol\lambda} =Z^{-1}e^{-\boldsymbol\lambda\cdot\mathbf S}.$$

A global rotation sends $\boldsymbol\lambda$ and $\langle\mathbf S\rangle$ together. Once a direction $\hat{\lambda}$ is fixed by a reference frame or preparation, the state commutes only with rotations about that axis. The residual algebra is the centralizer of $\boldsymbol\lambda\cdot\mathbf S$.

For one spin $1/2$,

$$Z=2\cosh(\lambda/2), \qquad \langle\mathbf S\rangle =-\frac12\tanh(\lambda/2)\hat{\lambda}.$$

The state has a definite mean spin vector but no simultaneous sharp values of $S_x,S_y,S_z$. Rotating the external reference frame rotates the multiplier; it does not produce three independent commuting chemical potentials.

Preparation and thermodynamic qualifications

Expectation constraints may arise from a maximum-entropy inference, weak coupling to non-Abelian reservoirs, or a particular preparation protocol. These routes need not be equivalent. A resource-theoretic equilibrium construction can require a reference frame or a work-storage system carrying noncommuting charges; a finite isolated system may not operationally conserve each reservoir charge in the same way.

In the thermodynamic limit, symmetry breaking can make the map between $\lambda_a$ and $q_a$ nonunique or discontinuous. Boundary conditions and order of limits then select phases. Maximum entropy supplies the least-biased state relative to the declared constraints; it does not prove equilibration, uniqueness across representations, or completeness of the constraint set.

Contrast with an integrable GGE

Feature	Commuting integrable GGE	Noncommuting constrained state
Charge algebra	mutually commuting local/quasilocal family	genuinely non-Abelian or noncommuting
Primary issue	completeness for a thermodynamic macrostate	joint incompatibility and reference-frame dependence
Response matrix	ordinary covariance in common eigenbasis	Kubo–Mori/Duhamel covariance
Sharp values	simultaneous in principle	generally impossible
Typical application	post-quench integrable stationary data	non-Abelian charge constraints or reservoirs

Both are exponential maximum-entropy states, but that shared syntax does not erase their different hypotheses.

The schematic below organizes the relationships used on this page. Inspect it with this question in mind: Which charge structures license chemical potentials or generalized equilibrium states?

A grand-canonical state requires conserved charges and a declared algebra; commuting integrable charges, noncommuting constraints, temporal holonomies, and density thresholds demand distinct constructions and checks. Connections classify the charge structure and required construction; their direction is organizational, not a causal-time ordering. Dashed marks show qualifications and failure boundaries. The diagram is schematic and not to scale.

The surrounding discussion supplies the relevant equations and checks in text form; the figure is a navigational summary.

Exercise

For $\rho\propto e^{-\lambda S_z}$, show that constraining $\langle S_x\rangle$ as well requires rotating the single multiplier vector rather than multiplying by an independent commuting factor $e^{-\lambda_xS_x}$.

Solution

Because $[S_x,S_z]\ne0$, $e^{-\lambda_zS_z}e^{-\lambda_xS_x}$ depends on ordering and is not the maximum-entropy exponential of the two constraints. The variational solution is $\rho\propto e^{-(\lambda_xS_x+\lambda_zS_z)}$, equal to an exponential along one resultant spin direction. Its mean aligns oppositely with that vector.

References

• Jaynes, E. T. “Information Theory and Statistical Mechanics.” Physical Review 106, no. 4 (1957): 620–630. doi:10.1103/PhysRev.106.620.
• Yunger Halpern, Nicole, Philippe Faist, Jonathan Oppenheim, and Andreas Winter. “Microcanonical and Resource-Theoretic Derivations of the Thermal State of a Quantum System with Noncommuting Charges.” Nature Communications 7 (2016): 12051. doi:10.1038/ncomms12051; Open PDF.