Uhlmann curvature in dissipative phase transitions

A novel approach based on the Uhlmann curvature is introduced for the investigation of non-equilibrium steady-state quantum phase transitions (NESS-QPTs). Equilibrium phase transitions fall invariably into two markedly non-overlapping categories: classical phase transitions and quantum phase transitions. NESS-QPTs offer a unique arena where such a distinction fades off. We propose a method to reveal and quantitatively assess the quantum character of such critical phenomena. We apply this tool to a paradigmatic class of lattice fermion systems with local reservoirs, characterised by Gaussian non-equilibrium steady states. The relations between the behaviour of the Uhlmann curvature, the divergence of the correlation length, the character of the criticality and the dissipative gap are demonstrated. We argue that this tool can shade light upon the nature of non equilibrium steady state criticality in particular with regard to the role played by quantum vs classical fluctuations.

This definition leaves a U(n) gauge freedom in the choice of w, because w′ = wU, for any U ∈ U(n), generates the same ρ.
Let ρ λ(t) be a family of density matrices, with γ λ = ∈ ∈  t t T : { ( ) , [0, ]} a smooth closed path in a parameter manifold , and w λ(t) is a corresponding path of amplitudes. To lift the U(n) gauge freedom, Uhlmann introduced a parallel transport condition on w λ(t) 13 . For a closed trajectory ρ λ(t) , initial and final amplitudes are related by a unitary transformation w λ(T) = w λ(0) V γ . If the Uhlmann parallel transport condition is fullfilled, V γ is a holonomy, i.e. a non-Abelian generalisation of the Berry phase 13 , and reads  1 2 , where : / λ ∂ = ∂ ∂ μ μ , and L μ 's are Hermitian operators known as symmetric logarithmic derivatives, implicitly defined as the operator solutions of ρ ρ ρ ∂ = + μ μ μ L L : ( ) 1 2 . Unless otherwise stated, we will assume that ρ is full-rank. If ρ is singular, L μ and A μ are not unique. However, we will show that any quantity of interest to us can be extended by continuity to singular ρ's 57 . It follows also that A μ are Hermitian operators obeying the transformation rule of a non-abelian gauge potential, A U AU iU dU † † → + under w → wU, while L μ are gauge invariant. The Uhlmann curvature, defined as , is equal to the Uhlmann holonomy per unit area associated to an infin- , where γ μν is the infinitesimal parallelogram spanned by two independent directions ê ε μ and ε ν eˆ in . We focus on the Uhlmann GP per unit area for an infinitesimal loop, i.e. It can be shown that (see Methods) Tr [ , ] (1)  The above expression bears a striking resemblance with a pivotal quantity of quantum metrology, the quantum Fisher information matrix, defined as ρ = μν μ ν J L L Tr { , } 1 2 . The quantum Fisher information matrix determines a figure of merit of the estimation precision of parameters labelling a quantum state, known as the quantum Cramér-Rao bound 58,59 . Given a set of locally unbiased estimators λ of the parameters  λ ∈ , the covariance matrix Cov( ) is lower bounded (in a matrix sense) as follows 1 For single parameter estimation, the Cramér-Rao bound can always be saturated by the projective measurement on the eigenbasis of the symmetric logarithmic derivative. However, in a multi-parameter scenario this is not always the case, due to the non-commutativity of measurements associated to independent parameters. Within the framework of quantum local asymptotic normality [60][61][62] , one can prove that the multi-parameter quantum Cramér-Rao bound is attainable iff = μν 0  for all λ μ , λ ν 53 . In this sense,  μν marks the incompatibility between λ μ and λ ν , and such incompatibility arises from the inherent quantum nature of the underlying physical system. For a two-parameter model, the discrepancy between the attainable multi-parameter bound and the quantum Cramér-Rao bound can be estimated by the ratio | | μν J 2 /Det  , and the MUC is upper bounded by (see Methods) When saturated, bound (3) marks the condition of maximal incompatibility, in which the quantum indeterminacy in the estimation problem reaches the order of Det(J) −1/2 , the same of the quantum Cramér-Rao bound (2). Dissipative quadratic models. We now investigate the scaling law of the MUC, in dissipative Markovian models whose dynamics are generated by a master equation of Lindblad type 63 SCIeNTIFIC REPORTS | (2018) 8:9852 | DOI:10.1038/s41598-018-27362-9 The Hamiltonian is assumed quadratic in the fermion operators, i.e. ω ω = H : jk j k . Let λ ∈  be the set of parameters on which H and l α 's depend. Due to uniqueness,  parametrises the admissible NESS ρ(λ). The correlation matrix of the NESS is the solution of the (continuous time) Lyapunov equation XΓ In Methods, we show that for a generic Gaussian Fermionic state the MUC can be expressed in a parameter-independent way, as where K is the operator solution of the (discrete time) Lyapunov equation dΓ = ΓKΓ − K, which can be formally † is the adjoint action. According to 64 , the condition of NESS uniqueness is , where x j is an eigenvalue of X, and Δ is the Liouvillian spectral gap. When this condition is met, any state will eventually decay into the NESS in a time scale τ Δ 1/  . In the thermodynamical limit n → ∞ a vanishing gap Δ(n) → 0 may be accompained, though not-necessarily, by non-differentiable properties of the NESS 2,65 . For this reason, the scaling of Δ(n) has been used as an indication of NESS criticality [65][66][67][68][69] . NESS-QPT has been investigated through the scaling of the Bures metrics 19 , whose super-extensivity has been connected to a vanishing Δ 41 .
A similar relation between the super-extensivity of the MUC and Δ is implied by the inequality (1 ) 1 and g is the Bures metric tensor, which, except in pathological cases 57 , is equal to g = J/4. This bound shows that if  P (1)  Γ , a scaling of n 1  | | ∝ α+ entails a dissipative gap that vanishes at least as Δ ∝ n −α/2 , providing a relation between the dynamical properties of the NESS-QPT and the MUC.
However, as stated above, the scaling of MUC does indeed signal the presence of a NESS-QPT, but provide also a way of revealing the quantum character of the fluctuations that drive the criticality. On the one hand, the relation between MUC and quantum nature of the underlying physical system is apparent from the expression (1). The MUC arises from the commutator of two SLD, and, as such, its super-extensive properties cannot arise from classical fluctuations, as in equilibrium thermal phase transitions, but can only arise as a consequence of non-commutativity of close-by density matrices ρ(λ) and ρ(λ + dλ). In this sense,  is a signature of criticality associated to quantum fluctuations, as it cannot be sensitive to criticality induced by classical fluctuations, i.e. those associated only to changes in eigenvalues and not eigenstates of the density matrices.
Moreover, the comparison between the scaling laws of the MUC and quantum Fisher Information provides a means to estimate the quantum vs classical contributions to the fluctuations driving the criticality. This comparison is quantified by the ratio  Q J : Det2 /Det = | | , which according to the inequality (3) is upper bounded by Q ≤ 1, hence its scaling law is at most ∼ Q n 0 . When the above scaling law is saturated, the condition of maximal incompatibility of the associated quantum estimation problem is asymptotically satisfied. This implies that, in the thermodynamic limit n → ∞, the quantum character of the fluctuations driving the criticality cannot be neglected.
Let's apply the above analysis to a specific model, the boundary-driven spin-1/2 XY chain 2 . In this model, an open chain of spin-1/2 particles interacts via the XY-Hamiltonian, where the σ j x y z , , are Pauli operators acting on the spin on the j-th site. At each boundary, the chain is in contact with two different reservoirs, described by Lindblad operators . A Jordan-Wigner transform converts the system into a quadratic fermionic dissipative model with Gaussian In Table 1, the MUC scaling law is compared with the scaling of ||J|| ∞ , DetJ and Δ in each region of the phase diagram. Figure 1 clearly shows that  h | | δ maps faithfully the phase diagram. A super-extensive behaviour of the MUC characterises the LRMC phase with a scaling U O | | = δ n ( ) h 2 , while in the short range phase the MUC is size independent. Thus, differently from Δ, the MUC discriminates these phases, with no need of crossing the critical line h = h c . Figure 2 shows that in the LRMC phase, the scaling law of the MUC saturates the upper bound (3), in contrast to the short range phase. This shows the striking different nature of the two phases. In the LRMC region, the system behaves as an inherently two-parameter quantum estimation model, where the parameter incompatibility cannot be neglected even in the thermodynamical limit. On the short-range phase, instead, the system is of the MUC, and the ratio  = | | Q J : Det2 /Det for each phase of the boundary driven XY model 2 . The ratio Q ≤ 1 when Q ~ n 0 marks the condition of maximal asymptotic incompatibility.  asymptotically quasi-classical. The critical line δ = 0 (with |h| ≤ h c ) and the critical line h = 0, which mark regions of short range correlations embedded in a LRMC phase, show a MUC which grows super-extensively, with scaling n ( ) 3

 , and a nearly saturated inequality (3). In the critical line
, the scaling law of h | | δ  drops to a constant, revealing an asymptotic quasi-classical behaviour of the model at the phase transition.

Translationally invariant systems. An important subclass of quadratic Liouvillian Fermionic models
are those enjoying the translational invariance symmetry. In such systems one can employ the whole wealth of powerful tools stemming out of the Fourier transform and work directly in the thermodynamical limit. This enables one to quantitatively define criticality in terms of singularities in the quasi-momentum space, thereby secluding the kinematics of the NESS-QPT from the dynamical properties of the model. The most natural notion of many-body criticality is in terms of diverging correlation length, which in a translationally invariant system is relatively straightforward to handle. This way of defining criticality enables one to bypass the difficulties arising from the ambiguous relation between NESS-QPTs and the vanishing dissipative gap.
The object of investigation is the covariance matrix, which in a translationally invariant system can be conveniently studied through its Fourier components. It is the non-analytical behaviour in the Fourier basis which conveys information on the long-wavelength limit, i.e. on the divergence of the correlation length.
Consider an explicit translationally invariant d-dimensional lattice of Fermions located at sites r L d ∈  , and assume finite (or quasi-finite) range interaction. The system size is n = L d , and subsequently, one takes the thermodynamical limit L → ∞. One can define the covariance matrix over a discrete quasi-momentum space. However the considerations on the long-wavelength limit that will follow truly apply only at the thermodynamical limit: hence divergences of correlation lengths manifest genuine quantum many-body effects.
In a translationally invariant chain, the Fermions can be labelled as ω r = (ω r,1 , ω r,2 ) T , where ω r,β with β = 1, 2 are the two types of Majorana fermions on each site ∈ r . The Hamiltonian can be written as and similarly the Lindbladians In the limit of infinite large system, both Hamiltonian and bath matrix are circulant. And the correlation matrix of the unique steady state solution is circulant, too: , . The Fourier component of the covariance matrix, called the covariance symbol, reads In terms of the symbol functions, the continuous Lyapunov equation reduces to a set of 2 × 2 matrix equations is a positive semidefinite matrix. The spatial correlation between Majorana Fermions are then recovered from the inverse Fourier transform of the symbol function 54,70 , here we will define criticality by the divergence of correlation length, which is defined as In the thermodynamical limt, the divergence may only arise as a consequence of the non-analytical dependence of γ(r) on the system parameters. Let's confine ourselves to the case of a one-dimensional Fermionic chain. In order to derive informations on the large distance behaviour of the correlations, it is convenient to express the integral of the inverse Fourier transform in the complex plane, though the analytical continuation e iφ → z. This results in the following expression for the correlation function where Res z indicates the residues of the poles inside the unit circle = || | ≤ S z z : { 1} 1 . Since γ z ( )  is the solution of a finite dimensional matrix equation (7), it may only possess simple poles. Thus, the above expression may become singular only when an isolated pole of z ( ) γ  approaches the unit circle from the inside 54,70 . This may happen for some specific critical values λ λ = ∈ 0 . As λ approaches λ 0 the correlation length ξ diverges. One can show that the long wave-length behaviour is governed by the closest pole to unit circle | | z 0 , and indeed the correlation length is given by ξ = | | z ln 0 .

Mean Uhlmann Curvature and Criticality in Translationally Invariant Models.
We will show that the MUC is sensitive to the criticality, but only in the sense of a truly diverging correlation length. Indeed one can show that the Uhlmann curvature is insensitive to the vanishing of the dissipative gap, if the latter, as it may happen, is not accompanied by a diverging correlation length. In this sense, the Uhlmann curvature confirms its role as a witness of the purely kinematic aspects of the criticality, and it is only indirectly affected by the dynamical features of the NESS-QPT. Thanks to the translational symmetry, one can exploit the formalism of Fourier transform and derive a quite compact expression of the MUC. By applying the convolution theorem on the equation (5), one obtains the following expression for the MUC per site In the above expression, κ μ (φ) is the operator solution of the 2 × 2 discrete time Lyapunov equation In the eigenbasis of ( ) γ φ  , with eigenvalues  j γ , the explicit solution of (12) reads Notice that the diagonal terms (κ μ (φ)) jj provide vanishing contributions to eq. (11) (they commute with ( ) γ φ  ). Hence, eq. (11) can be cast in the following basis independent form 2~~Ñ otice that the condition Det ( ) 1  γ φ = is equivalent to having two eigenvalues of correlation matrix equal to (γ i , γ k ) = ±(1, 1). Such extremal values cause no singularity in MUC, but result in a vanishing contribution to the MUC.
In Methods, we will demonstrate that a singularity of  signals the occurrence of a criticality. Specifically, employing the analytical extension in the complex plane of u μν (φ) leads to z S z 1 1  Notice that u μν (z) has at most isolated poles, due to its rational dependence on z. Assume that as λ λ → ∈ 0 , a pole z 0 of u μν (z) approaches the unit circle from inside, which is the only condition under which  is singular in λ 0 . One can show that, whenever a pole z 0 of u μν (z) approaches the unit circle, also a pole z of γ z ( )  approaches the same value, causing the correlation length to diverge. Therefore the singular behaviour of the Uhlmann phase necessarily represents a sufficient criterion for a NESS-QPT. Notice also (see Methods) that such criticalities are necessarily accompanied by the closure of the dissipative gap, however, the converse is in general not true. Indeed, a vanishing dissipative gap is not a sufficient condition for criticality, but only necessary. This fact can be readily checked with the model discussed in the next subsection, which shows a closing dissipative gap without the occurrence of a diverging correlation length.
Moreover, a singularity in the MUC may only arise as the result of criticality and are otherwise insensitive to a vanishing dissipative gaps. These features are exemplified in the following translational invariant dissipative fermionic chain: the rotated XY model with periodic boundary conditions 25,26 where each site j is coupled to two local reservoirs with Lindblad operators εμσ Λ = ± ± j j . The spin-system is converted into a quadratic fermionic model via Jordan-Wigner transformations. The Liouvillian spectrum can be solved exactly 2,64,68 and it is independent of θ. In the weak coupling limit ε → 0, the symbol function of the NESS correlation matrix reads The system shows criticality in the same critical regions of the XY hamiltonian model 68 . By using expression (14) we can calculate the exact values of the mean Uhlmann curvature. We find that  h δ vanishes, while δθ  and θ  h are plotted in Fig. 3. As predicted, the Uhlmann curvature shows a singular behaviour only across criticality. In particular,  θ h is discontinuous in the XY critical points |h| = 1, while  δθ is discontinuous in the XX type criticalities δ = 0, h < 1.
A model with closing dissipative gap without criticality. In this section we will show an example of a 1D fermionic dissipative system in which the closure of the dissipative gap does not necessarily lead to a diverging correlation length. Consider a chain of fermions on a ring geometry, with no Hamiltonian and a reservoir defined by the following set of Lindblad operators where r = 1, …, n, l 0 = (cos θ, −sin θ) T , l 1 = l 2 = i(sin θ, cos θ) T , and n(λ) = 4(λ 2 + λ + 1), with  λ ∈ , θ = [0, 2π). This is a simple extension of a model introduced in 71 , which, under open boundary conditions, shows a dissipative topological phase transition for λ = ±1. In the thermodynamical limit n → ∞, the eigenvalues of  φ x( ) are x 1 = 4(1 + λ) 2 /n(λ) 2 , and x 2 = 4(1 + 2λ cos φ + λ 2 )/n(λ) 2 , showing a closure of the dissipative gap at λ = ±1. For |λ| ≠ 1 the unique NESS is found by solving the continuous Lyapunov equation (7). The symbol function, in a Pauli matrix representation, results ( ) γ σ γ φ = ⋅ , with g( ) (sin sin 2 ) cos 2 (cos cos 2 ) (sin sin 2 ) sin 2 2 cos 2 . This shows that γ  is critical in the sense of diverging correlation, only for λ = −1 and not for λ = 1, even if the dissipative gap closes in both cases. Figure 4 shows the dependence of the inverse correlation length of the bulk, the dissipative gap and the mean Uhlmann curvature λφ  on the parameter λ. Notice a discontinuity of the Uhlmann phase corresponding to the critical point λ 0 = −1, while it does not show any singularity for λ = 1 where the gap closes.

Discussion
We have introduced an entirely novel approach to quantitatively assess the "quantum-ness" of critical phenomena. To this end, we resorted to ideas borrowed from quantum estimation theory, which endow the geometric phase approach with an operationally well defined character. The geometrical interpretation offers an intuitive explanation as to why singularities of MUC emerge in criticalities, and leads to a unified interpretation for equilibrium and out-of-equilibrium QPTs. In quantum metrology, the MUC accounts for the discrepancy between an inherently quantum and a quasi-classical multi-parameter estimation problem, shading a new light onto the nature of correlations in NESS-QPTs. We have explored the properties of the MUC in the physically relevant class of dissipative NESS-QPTs exhibited by quadratic fermionic Liouvillian models. A relation between the singular behaviour of the MUC and the criticality has been analytically demonstrated. We have employed specific prototypical models, showing that the scaling laws and the singularities of  map faithfully the phase diagrams. This approach goes well beyond the application to the important class of quadratic dissipative models analysed here, and  introduces a tool suitable for the systematic investigation of out-of-equilibrium quantum critical phenomena. It immediately extends to phase transitions with and without order parameters, quenched dynamics in open and closed systems, topological dissipative phase transitions, dynamical critical phenomena. Moreover, this idea is also a promising tool which may glean insight on the interplay between competing orders both in equilibrium and non-equilibrium QPTs.

Methods
Uhlmann geometric phase and mean Uhlmann curvature. Here we will briefly review the idea of the Uhlmann geometric phase, and derive the expression of the mean Uhlmann curvature as a function of the symmetric logarithmic derivatives (SLDs). Given a density operator ρ acting on a Hilbert space  of dimension n, an exteded Hilbert space is defined by attaching an ancilla a: A purification is defined as any pure state  ψ ∈ such that ρ = Tr a |ψ〉〈ψ|, where Tr a is the partial trace over the ancilla. A standard choice for  a is the dual of , then  ext becomes the space of operator w over , with Hilbert-Schmidt scalar product , then the phase difference ϕ B between initial and final state ψ is purely geometric in nature, i.e. it solely depends on the path γ, regardless of parameterisation and re-gauging. This phase is called Berry phase and its value reads is the Berry connection one-form, whose components are is the Berry curvature two-form, whose components are The parallel transport condition is equivalent to choose the representative path ψ λ(t) that minimizes the length of the path on the Hilbert space measure by Similarly, we can have a smooth closed trajectory of density matrices, ρ λ(t) , parametrised by a path γ: , t ∈ [0, T], and, correspondingly, a path of Hilbert-Schmidt operators w λ(t) in  ext . The choice of amplitudes is quite redundant due to the local U(n) gauge freedom. Similarly to the pure state case, this redundancy can be mitigated by imposing the so called Uhlmann parallel transport condition, which prescribes that, given any two ρ 1 and ρ 2 , their respective amplitudes w 1 and w 2 are parallel whenever This equivalently means that the chosen w 1 and w 2 are those that maximise their Hilbert Schmidt scalar product = w w w w ( , ) : Tr( ) where the maximum is taken over all w 2 ′ purifying ρ 2 . The above maximal value depends on ρ 1 and ρ 2 only, and it is equal to ( , ) Tr , the so called Uhlmann fidelity of ρ 1 and ρ 2 . Through the fidelity one can define a geometric measure of statistical indistinguishability between states ρ 1 and ρ 2 16 , the Bures distance which, for infinitesimally closed states, defines a Riemannian metrics on the manifold of density operators, the Bures metrics μν μν μ μ λ λ λ + Applied to any two neighbouring points w λ(t) and w λ(t+dt) of a smooth path of amplitudes, the parallel transport condition (17) becomes where dots denote derivatives with respect to t. The maximisation of the overlap (w λ(t) , w λ(t+dt) ) is equivalent to the minimisation of the "velocity"   v w w : ( , ) = , which in turns means that the path of amplitudes fullfilling the Uhlmann condition are those with the shortest length, measured by ) . According to 56 , the parallel transport condition (18) t t t L t can be determined by differentiating ww ρ = † and inserting (19), which yields t  where {., .} is the anticommutator. L t , known as the symmetric logarithmic derivative (SLD), is implicitly defined as the (unique) operator solution of (20) with the auxiliary requirement that 〈ψ|L t |ψ〉 = 0, whenever ρ|ψ〉 = 0. As already mentioned, as far as the definition of the SLD is concerned, we will actually confine ourselves to full-rank density matrices. In the case of singular density matrices, quantities of interest to us can be calculated consistently by a limiting procedure from the set of full rank matrices. In terms of L t , the "velocity" can be cast as , which in turn means that the Bures metrics can be expressed in the following form where L μ is the restriction of L t along the coordinate λ μ , and it is determined by the analog of equation (20), . We can also define the operator-valued differential one-form L Ld : λ = ∑ μ μ μ . In the closed path ρ λ(t) , initial and final amplitudes are related by a unitary transformation, i.e. w λ(T) = w λ(0) V γ . If the path of amplitudes w λ(t) fullfills the Uhlmann condition, V γ is a holonomy, the non-Abelian generalisation of Berry phase 13 . The holonomy is expressed as where  is the path ordering operator and A is the Uhlmann connection one-form. The Uhlmann connection can be derived from the following ansatz 56 which is the generalisation of (19) when the parallel transport condition is lifted. By differentiating † ww ρ = and using the defining property of the SLD (see eq. (20)), it follows that A is Hermitian and it is implicitly defined by the equation with the auxiliary constraint that 〈ψ′|A|ψ′〉 = 0, for w|ψ′〉 = 0. From eq. (22), it can be checked that A obeys the expected transformation rule of non-Abelian gauge potentials, A U AU iU dU under w t → w t U t , and that L is gauge invariant.
The analog of the Berry curvature, the Uhlmann curvature two-form, is defined as can be understood in terms of the Uhlmann holonomy per unit area associated to an infinitesimal loop in the parameter space. Indeed, for an infinitesimal parallelogram γ μν , spanned by two independent directions δ μ μ e and eˆδ ν ν in the manifold, it reads where δ → 0 is a shorthand of (δ μ , δ ν ) → (0, 0). As already mentioned, the Uhlmann geometric phase is defined as and the Uhlmann phase per unit area for an infinitesimal loop reads We called the latter mean Uhlmann curvature (MUC), on account of the expression By taking the external derivative of the expression (22) and by using the property d 2 = 0, it can be shown that 56 Tr , (27) called the quantum Fisher tensor (QFT) 72 , such that g μν = Re(I μν )/4 and = − μν μν Fermionic Gaussian states. We will specialize our considerations to the case of systems described by fermionic Gaussian states. The fermionic Gaussian states are defined as density matrices ρ that can be expressed as / , : Here Ω is a 2n × 2n real antisymmetric matrix, and ω ω ω is a vector of 2n Majorana fermion operators, defined as: c c : where c k and c k † are annihilation and creation operators of standard fermions, respectively. The anticommutation relations of the Majorana fermion operators read {ω j , ω k } = 2δ jk . The Gaussian state is completely specified by the two-point correlation matrix ρ ω ω Γ = : 1/2Tr( [ , ]) jk j k , which is an imaginary antisymmetric matrix. One can show that Γ and Ω can be simultaneously cast in a canonical form by an orthogonal matrix Q and their eigenvalues are related by γ j = tanh(Ω j /2), which implies that |γ j | ≤ 1. Correspondingly, the density matrix can be factorised as are the Majorana fermions in the eigenmode representation. Notice that |γ k | = 1 corresponds to the fermionic mode = + being in a pure state. For a Gaussian fermionic state, all odd-order correlation functions are zero, and all even-order correlations, higher than two, can be obtained from Γ by Wick's theorem 73  where a : jk jk jk δ = Γ + . We would like to derive a convenient expression for the QFT for Gaussian fermionic states. In order to do this, we first derive the SLD in terms of correlation matrix Γ. Due to the quadratic dependence of (28) in ω, and following the arguments of 74 , it can be shown that L is a quadratic polynomial in the Majorana fermions Kd , with K μ a 2n × 2n hermitian antisymmetric matrix, ζ = ζ μ dλ μ , with ζ μ a 2n real vector, and η = η μ dλ μ a real valued one-form. From the property that Tr(ρω k ) = 0 for any 1 ≤ k ≤ 2n, it is straightforward to show that the linear term in (30) is identically zero where ζ k is the k-th component of ζ, and in the third equality we took into account that the odd order correlations vanish. The quantity η can be determined from the trace preserving condition Tr(dρ) = Tr(ρL) = 0 In order to determine K, we take the differential of Γ jk = 1/2Tr(ρ[ω j , ω k ]) where the last equality is obtained with the help of eq. (29) and using the antisymmetry of Γ and K. Finally, according to eq. (31), the last term vanishes and we obtain the following (discrete time) Lyapunov equation The above equation can be formally solved by where, in the second equality, we made use of the relation γ k = tanh(Ω k /2), which yields the following diagonal in (34) shows that such a singularity is relatively benign. Thanks to this, we can show that the condition γ j = γ k = ±1 produces, at most, removable singularities in the QFT (cf. 57 ). This allows the QFT to be extended by continuity from the set of full-rank density matrices to the submanifolds with γ j = γ k = ±1. Knowing the expression for the SLDs, we can calculate the QFT by plugging Making use of (29) and exploiting the antisymmetry of both Γ and K leads to where the last equality is obtained by plugging in eq. (34). Let's have a closer look at the QFT in the limit of (γ j , γ k ) → ±(1, 1). The boundness K jk , and the multiplicative factors (1 ± γ j ) in (35) causes each term with |γ j | → 1 to vanish. This means that the QFT has a well defined value in the above limit, and we can safely extend by continuity the QTF to the sub-manifolds (γ j , γ k ) = ±(1, 1). The explicit expression of I μν produces the following results for the Bures metrics Tr( ) (38) Sufficient condition for criticality in translationally invariant dissipative models. In this section we will show that a singular dependence of  on the parameters  λ ∈ necessarily implies a criticality, strictly in the sense of a diverging correlation length.
Let's now prove that, in translationally invariant models, a vanishing dissipative gap is a necessary condition for criticality. + with j, k = 1, 2, where x j are the eigenvalues of x( ) φ  . Since Rex j ≥ 0, X Det 0 = implies that there must exist an eigenvalue x 0 of  x( ) φ with vanishing real part, hence x x Re Re 2min 2 0 In 66 it is shown that when 2Rex 0 = 0, the geometric multiplicity of x 0 is equal to its algebraic multiplicity, hence the 2 × 2 matrix  x( ) φ is diagonalisable. Then, let |j〉 be the set of eigenstates with eigenvalues x j . In the eigenbasis | 〉 ⊗ | 〉 j k, j, k = 0, 1 the adjugate matrix has the following diagonal form,^=