Abstract
Given a quasi-lattice ordered group (G, P) and a compactly aligned product system X of essential $$\hbox {C}^*$$
C
∗
-correspondences over the monoid P, we show that there is a bijection between the gauge-invariant $$\hbox {KMS}_\beta $$
KMS
β
-states on the Nica-Toeplitz algebra $$\mathcal {NT}(X)$$
NT
(
X
)
of X with respect to a gauge-type dynamics, on one side, and the tracial states on the coefficient algebra A satisfying a system (in general infinite) of inequalities, on the other. This strengthens and generalizes a number of results in the literature in several directions: we do not make any extra assumptions on P and X, and our result can, in principle, be used to study KMS-states at any finite inverse temperature $$\beta $$
β
. Under fairly general additional assumptions we show that there is a critical inverse temperature $$\beta _c$$
β
c
such that for $$\beta >\beta _c$$
β
>
β
c
all $$\hbox {KMS}_\beta $$
KMS
β
-states are of Gibbs type, hence gauge-invariant, in which case we have a complete classification of $$\hbox {KMS}_\beta $$
KMS
β
-states in terms of tracial states on A, while at $$\beta =\beta _c$$
β
=
β
c
we have a phase transition manifesting itself in the appearance of $$\hbox {KMS}_\beta $$
KMS
β
-states that are not of Gibbs type. In the case of right-angled Artin monoids we show also that our system of inequalities for traces on A can be reduced to a much smaller system, a finite one when the monoid is finitely generated. Most of our results generalize to arbitrary quasi-free dynamics on $$\mathcal {NT}(X)$$
NT
(
X
)
.
Spinons as Composite Fermions
