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Dynamical phase transitions at finite temperature from fidelity and interferometric Loschmidt echo induced metrics Bruno Mera 1,2,4 , Chrysoula Vlachou 3,4 , Nikola Paunkovi´ c 3,4 , V´ ıtor R. Vieira 1,2 and Oscar Viyuela 5,6 1 CeFEMA, Instituto Superior T´ ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal 2 Departamento de F´ ısica, Instituto Superior T´ ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal 3 Departamento de Matem´ atica, Instituto Superior T´ ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal 4 Instituto de Telecomunica¸ oes, 1049-001 Lisbon, Portugal 5 Department of Physics, Harvard University, Cambridge, MA 02318, USA and 6 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA We study finite-temperature Dynamical Quantum Phase Transitions (DQPTs) by means of the fidelity and the interferometric Loschmidt Echo (LE) induced metrics. We analyse the associated dynamical susceptibilities (Riemannian metrics), and derive analytic expressions for the case of two-band Hamiltonians. At zero temperature the two quantities are identical, nevertheless, at finite temperatures they behave very differently. Using the fidelity LE, the zero temperature DQPTs are gradually washed away with temperature, while the interferometric counterpart exhibits finite- temperature Phase Transitions (PTs). We analyse the physical differences between the two finite- temperature LE generalisations, and argue that, while the interferometric one is more sensitive and can therefore provide more information when applied to genuine quantum (microscopic) systems, when analysing many-body macroscopic systems, the fidelity-based counterpart is a more suitable quantity to study. Finally, we apply the previous results to two representative models of topological insulators in 1D and 2D. I. INTRODUCTION Equilibrium Phase Transitions (PTs) are characterised by a non-analytic behaviour of relevant thermodynamic observables with respect to the change of temperature. Quantum Phase Transitions (QPTs) [1], traditionally described by Landau theory [2], occur when we adiabatically change a physical parameter of the system at zero temperature, i.e., the transition is driven by purely quantum fluctuations. A particularly interesting case arises when one studies QPTs in the context of topological phases of matter [3–7]. They are quite different from the standard QPTs, since they do not involve symmetry breaking and are characterised by global order parameters [8]. These novel phases of matter, which include topological insulators and superconductors [9–11], have potentially many applications in emerging fields such as spintronics, photonics or quantum computing. Although many of their remarkable properties have traditionally been studied at zero temperature, there has been a great effort to generalise these phases from pure to mixed states, and finite temperatures [12–24]. The real time evolution of closed quantum systems out of equilibrium has some surprising similarities with thermal phase transitions, as noticed by Heyl, Polkovnikov and Kehrein [25]. They coined the term Dynamical Quantum Phase Transitions (DQPTs) to describe the non-analytic behaviour of certain dynamical observables after a sudden quench in one of the parameters of the Hamiltonian. Since then, the study of DQPTs became an active field of research and a lot of progress has been achieved in comparing and connecting them to the equilibrium PTs [26–33]. Along those lines, there exist several studies of DQPTs for systems featuring non-trivial topological properties [34–39]. DQPTs have been experimentally observed in systems of trapped ions [40] and cold atoms in optical lattices [? ]. The figure of merit in the study of DQPTs is the Loschmidt Echo (LE) and its derivatives, which have been extensively used in the analysis of quantum criticality [41–45] and quantum quenches [46]. At finite temperature, generalisations of the zero temperature LE were proposed, based on the mixed-state Uhlmann fidelity [45, 47], and the interferometric mixed-state geometric phase [48, 49]. For an alternative approach to finite-temperature DPTs, see [50, 51]. Fidelity is a measure of state distinguishability, which has been employed numerous times in the study of PTs [42, 52–55], while the interferometric mixed-state geometric phase was introduced in [56]. The two quantities are in general different and it comes as no surprise that they give different predictions for the finite temperature behaviour [49]: the fidelity LE does not show DPTs at finite temperatures, while the interferometric LE indicates their persistence at finite temperature. Thus, it remains unclear what the fate of DQPT at finite temperature truly is, and which of the two opposite predictions better captures the many-body nature of these PTs. In this paper, we discuss the existence of finite temperature Dynamical Phase Transitions (DPTs) for the broad class of two-band Hamiltonians in terms of both the fidelity and the interferometric LEs. We derive analytic expressions for the metrics (susceptibilities) induced by the fidelity and the interferometric LEs, respectively, showing explicitly that the two approaches give different behaviours: the fidelity susceptibility shows a gradual disappearance arXiv:1712.01314v2 [cond-mat.stat-mech] 13 Apr 2018

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Dynamical phase transitions at finite temperature from fidelity and interferometricLoschmidt echo induced metrics

Bruno Mera1,2,4, Chrysoula Vlachou3,4, Nikola Paunkovic3,4, Vıtor R. Vieira1,2 and Oscar Viyuela5,61 CeFEMA, Instituto Superior Tecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

2 Departamento de Fısica, Instituto Superior Tecnico,Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

3 Departamento de Matematica, Instituto Superior Tecnico,Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

4 Instituto de Telecomunicacoes, 1049-001 Lisbon, Portugal5 Department of Physics, Harvard University, Cambridge, MA 02318, USA and

6 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

We study finite-temperature Dynamical Quantum Phase Transitions (DQPTs) by means of thefidelity and the interferometric Loschmidt Echo (LE) induced metrics. We analyse the associateddynamical susceptibilities (Riemannian metrics), and derive analytic expressions for the case oftwo-band Hamiltonians. At zero temperature the two quantities are identical, nevertheless, at finitetemperatures they behave very differently. Using the fidelity LE, the zero temperature DQPTsare gradually washed away with temperature, while the interferometric counterpart exhibits finite-temperature Phase Transitions (PTs). We analyse the physical differences between the two finite-temperature LE generalisations, and argue that, while the interferometric one is more sensitive andcan therefore provide more information when applied to genuine quantum (microscopic) systems,when analysing many-body macroscopic systems, the fidelity-based counterpart is a more suitablequantity to study. Finally, we apply the previous results to two representative models of topologicalinsulators in 1D and 2D.

I. INTRODUCTION

Equilibrium Phase Transitions (PTs) are characterised by a non-analytic behaviour of relevant thermodynamicobservables with respect to the change of temperature. Quantum Phase Transitions (QPTs) [1], traditionally describedby Landau theory [2], occur when we adiabatically change a physical parameter of the system at zero temperature,i.e., the transition is driven by purely quantum fluctuations. A particularly interesting case arises when one studiesQPTs in the context of topological phases of matter [3–7]. They are quite different from the standard QPTs, sincethey do not involve symmetry breaking and are characterised by global order parameters [8]. These novel phasesof matter, which include topological insulators and superconductors [9–11], have potentially many applications inemerging fields such as spintronics, photonics or quantum computing. Although many of their remarkable propertieshave traditionally been studied at zero temperature, there has been a great effort to generalise these phases from pureto mixed states, and finite temperatures [12–24].

The real time evolution of closed quantum systems out of equilibrium has some surprising similarities with thermalphase transitions, as noticed by Heyl, Polkovnikov and Kehrein [25]. They coined the term Dynamical Quantum PhaseTransitions (DQPTs) to describe the non-analytic behaviour of certain dynamical observables after a sudden quenchin one of the parameters of the Hamiltonian. Since then, the study of DQPTs became an active field of research anda lot of progress has been achieved in comparing and connecting them to the equilibrium PTs [26–33]. Along thoselines, there exist several studies of DQPTs for systems featuring non-trivial topological properties [34–39]. DQPTshave been experimentally observed in systems of trapped ions [40] and cold atoms in optical lattices [? ]. The figureof merit in the study of DQPTs is the Loschmidt Echo (LE) and its derivatives, which have been extensively usedin the analysis of quantum criticality [41–45] and quantum quenches [46]. At finite temperature, generalisations ofthe zero temperature LE were proposed, based on the mixed-state Uhlmann fidelity [45, 47], and the interferometricmixed-state geometric phase [48, 49]. For an alternative approach to finite-temperature DPTs, see [50, 51]. Fidelity isa measure of state distinguishability, which has been employed numerous times in the study of PTs [42, 52–55], whilethe interferometric mixed-state geometric phase was introduced in [56]. The two quantities are in general differentand it comes as no surprise that they give different predictions for the finite temperature behaviour [49]: the fidelityLE does not show DPTs at finite temperatures, while the interferometric LE indicates their persistence at finitetemperature. Thus, it remains unclear what the fate of DQPT at finite temperature truly is, and which of the twoopposite predictions better captures the many-body nature of these PTs.

In this paper, we discuss the existence of finite temperature Dynamical Phase Transitions (DPTs) for the broadclass of two-band Hamiltonians in terms of both the fidelity and the interferometric LEs. We derive analyticexpressions for the metrics (susceptibilities) induced by the fidelity and the interferometric LEs, respectively, showingexplicitly that the two approaches give different behaviours: the fidelity susceptibility shows a gradual disappearance

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of DPTs as the temperature increases, while the interferometric susceptibility indicates their persistence at finitetemperature (consistent with recent studies on interferometric LE [48, 49]). We analyze the reasons for such differentbehaviours. The fidelity LE quantifies state distinguishability in terms of measurements of physical properties,inducing a metric over the space of quantum states, while the interferometric LE quantifies the effects of quantumchannels acting upon a state, inducing a pullback metric over the space of unitaries. Thus, we argue that the fidelityLE and its associated dynamical susceptibility are more suitable for the study of many-body systems, while themore sensitive interferometric counterparts are optimal when considering genuine microscopic quantum systems. Inaddition, interferometric experiments that are suitable for genuine (microscopic) quantum systems involve coherentsuperpositions of two states, which could be, in the case of many-body macroscopic systems, experimentally infeasiblewith current technology. To confirm our analysis for the fidelity LE, we also present quantitative results for thefidelity-induced first time derivative of the rate function in the case of the 1D Su-Schrieffer-Heeger (SSH) topologicalinsulator and the 2D Massive Dirac (MD) model of a Chern insulator [57].

II. DYNAMICAL (QUANTUM) PHASE TRANSITIONS AND ASSOCIATED SUSCEPTIBILITIES

The authors in [25] introduce the concept of DQPTs and illustrate their properties on the case of the transverse-fieldIsing model. They observe a similarity between the partition function of a quantum system in equilibrium, Z(β) =Tr(e−βH), and the overlap amplitude of some time-evolved initial quantum state |ψi〉 with itself, G(t) = 〈ψi|e−iHt|ψi〉.During a temperature-driven PT, the abrupt change of the properties of the system is indicated by the non-analyticityof the free energy density f(β) = − limN→∞

1N lnZ(β) at the critical temperature (N being the number of degrees

of freedom). It is then possible to establish an analogy with the case of the real time evolution of a quantum systemout of equilibrium, by considering the rate function

g(t) = − 1

Nlog |G(t)|2, (1)

where |G(t)|2 is a mixed-state LE, as we detail below. The rate function g(t) may exhibit non-analyticities at somecritical times tc, after a quantum quench. This phenomenon is termed DPT.

We study DPTs for mixed states using the fidelity and the interferometric LEs. We first investigate the relationbetween the two approaches for DQPTs at zero temperature. More concretely, we perform analytical derivationsof the corresponding susceptibilities in the general case of a family of static Hamiltonians, parametrised by somesmooth manifold M , {H(λ) : λ ∈M}.

A. DQPTs for pure states

At zero temperature, the LE G(t) from (1) between the ground state for λ = λi ∈ M and the evolved state withrespect to the Hamiltonian for λ = λf ∈M is given by the fidelity between the two states

F(t;λf , λi) ≡ |〈ψ(λi)|e−itH(λf )|ψ(λi)〉|. (2)

For λi = λf , the fidelity is trivial, since the system remains in the same state. Fixing λi ≡ λ and λf = λ+ δλ, withδλ << 1, in the t → ∞ limit Eq. (2) is nothing but the familiar S-matrix with an unperturbed Hamiltonian H(λ)and an interaction Hamiltonian V (λ), which is approximated by

V (λ) ≡ H(λf )−H(λ) ≈ ∂H

∂λa(λ)δλa. (3)

After applying standard perturbation theory techniques (for details, see Appendix), we obtain

F(t;λf , λ) ≈ 1− χab(t;λ)δλaδλb, (4)

where the dynamical susceptibility χab(t;λ) is given by

χab(t;λ) =

∫ t

0

∫ t

0

dt2dt1

(1

2〈{Va(t2), Vb(t1)}〉 − 〈Va(t2)〉〈Vb(t1)〉

), (5)

with Va(t, λ) = eitH(λ)∂H/∂λa(λ)e−itH(λ) and 〈∗〉 = 〈ψ(λ)| ∗ |ψ(λ)〉. The family of symmetric tensors {ds2(t) =χab(t, λ)dλadλb}t∈R defines a family of metrics in the manifold M , which can be seen as pullback metrics of the Buresmetric (Fubini-Study metric) in the manifold of pure states [58]. Specifically, at time t, the pullback is given by themap Φt : λf 7→ e−itH(λf )|ψ(λ)〉〈ψ(λ)|eitH(λf ), evaluated at λf = λ.

3

B. Generalizations at finite temperatures

The generalization of DQPTs to mixed states is not unique. There are several ways to construct a LE for a generaldensity matrix. In what follows, we derive two finite temperature generalizations, such that they have the same zerotemperature limit.

1. Fidelity Loschmidt Echo at T > 0

First, we introduce the fidelity LE between the state ρ(β;λi) = e−βH(λi)/Tr{e−βH(λi)} and the one evolved by theunitary operator e−itH(λf ) as

F(t, β;λi, λf )=F (ρ(β;λi), e−itH(λf )ρ(β;λi)e

itH(λf )), (6)

where F (ρ, σ) = Tr{√√

ρσ√ρ} is the quantum fidelity between arbitrary mixed states ρ and σ. For λf close to

λi = λ, we can write

F(t, β;λf , λ) ≈ 1− χab(t, β;λ)δλaδλb, (7)

with χab(t, β, λ) being the Dynamical Fidelity Susceptibility (DFS). Notice that limβ→∞ χab(t, β;λ) = χab(t;λ), where

χab(t;λ) is given by Eq. (5). At time t and inverse temperature β, we have a map Φ(t,β) : λf 7→ e−itH(λf )ρ(β;λ)eitH(λf ).

The 2-parameter family of metrics defined by ds2(β, t) = χab(t, β;λ)dλadλb is the pullback by Φ(t,β) of the Buresmetric on the manifold of full-rank density operators, evaluated at λf = λ (see Appendix).

The fidelity LE is closely related to the Uhlmann connection: F (ρ1, ρ2) equals the overlap between purifications

W1 and W2, 〈W1,W2〉 = Tr{W †1W2

}, satisfying discrete parallel transport condition (see, for instance [59]).

2. Interferometric Loschmidt Echo at T > 0

Here, we consider an alternative definition of the LE for mixed states [G(t) from Eq. (1)]. In particular, we definethe interferometric LE as

L(t, β;λf , λi) =

∣∣∣∣∣Tr{e−βH(λi)eitH(λi)e−itH(λf )

}Tr{e−βH(λi)

} ∣∣∣∣∣ . (8)

The eitH(λi) factor does not appear at zero temperature, since it just gives a phase which is canceled by takingthe absolute value. This differs from previous treatments in the literature [36] (see Section 5.5.4 of [58], where thevariation of the interferometric phase, Tr{ρ0e−itH}, exposes this structure). However, it is convenient to introduce itin order to have the usual form of the perturbation expansion, as will become clear later.

For λf close to λi = λ, we get

L(t, β;λf , λ) ≈

∣∣∣∣∣∣Tr{e−βH(λ)Te−i

∫ t0dt′Va(t,λ)δλ

a}

Tr{e−βH(λ)

}∣∣∣∣∣∣ , (9)

so that the perturbation expansion goes as in Eq. (5), yielding

L(t, β;λf , λ) ≈ 1− χab(t, β;λ)δλaδλb, (10)

with the dynamical susceptibility given by

χab(t, β;λ) =

∫ t

0

∫ t

0

dt2dt1

(1

2〈{Va(t2), Vb(t1)}〉 − 〈Va(t2)〉〈Vb(t1)〉

), (11)

where 〈∗〉 = Tr{e−βH(λ)∗}/Tr{e−βH(λ)}. Notice that Eqs (11) and (5) are formally the same with the averageover the ground state replaced by the thermal average. This justifies the extra eitH(λi) factor. Since this suscep-tibility comes from the interferometric LE, we call it Dynamical Interferometric Susceptibility (DIS). The quantity

4

ds2(β, t) = χab(t, β;λ)dλadλb defines a 2-parameter family of metrics over the manifold M , except that they cannotbe seen as pullbacks of metrics on the manifold of density operators with full rank. However, it can be interpreted asthe pullback by a map from M to the unitary group associated with the Hilbert space of a particular Riemannianmetric. For a detailed analysis, see Appendix. Additionally, we point out that this version of LE is related to theinterferometric geometric phase introduced by Sjoqvist et. al [56, 60].

C. Two-band systems

Many representative examples of topological insulators and superconductors can be described by effective two-bandHamiltonians. Therefore, we derive closed expressions of the previously introduced dynamical susceptibilities fortopological systems within this class.

The general form of such Hamiltonians is {H(λ) = ~x(λ) · ~σ : λ ∈M}, where ~σ is the Pauli vector. The interactionHamiltonian V (λ), introduced in Eq. (2), casts the form

V (λ) ≈(∂~x

∂λa· ~σ)δλa.

It is convenient to decompose ∂~x/∂λa into one component perpendicular to ~x and one parallel to it:

∂~x

∂λa=

(∂~x

∂λa

)⊥+

(∂~x

∂λa

)‖= ~ta + ~na.

The first term is tangent, in R3, at ~x(λ), to a sphere of constant radius r = |~x(λ)|. Hence, this kind of perturbationsdoes not change the spectrum of H, only its eigenbasis. The second term is a variation of the length of ~x and hence,it changes the spectrum of H, while keeping the eigenbasis fixed. The DFS and the DIS are given by (for the detailsof the derivation, see Appendix)

χab = tanh2(β|~x(λ)|) sin2(|~x(λ)|t)|~x(λ)|2

~ta·~tb (12)

χab =sin2(|~x(λ)|t)|~x(λ)|2

~ta·~tb+t2(1−tanh2(β|~x(λ)|))~na·~nb. (13)

While the DIS (13) depends on the variation of both the spectrum and the eigenbasis of the Hamiltonian, theDFS (12) depends only on the variations which preserve the spectrum, i.e., changes in the eigenbasis. This is veryremarkable, in general the fidelity between two quantum states, being their distinguishability measure, does depend onboth the variations of the spectrum and the eigenbasis. In our particular case of a quenched system, the eigenvaluesare preserved (see Eq. (6)), as the system is subject to a unitary evolution. The tangential components of bothsusceptibilities are modulated by the function sin2(Et)/E2, where E is the gap. This captures the Fisher zeros, i.e.,the zeros of the (dynamical) partition function which here is given by the fidelity F from (6) (see [61–63]). Observethat whenever t = (2n+1)π/2E, n ∈ Z, this factor is maximal and hence, both LEs decrease abruptly. The differencebetween the two susceptibilities is given by

χab − χab = (1− tanh2(β|~x(λ)|))(

sin2(|~x(λ)|t)|~x(λ)|2

~ta · ~tb + t2~na · ~nb).

The quantity (1− tanh2(βE)) is nothing but the static susceptibility, see [64]. Therefore, the difference between DISand DFS is modulated by the static susceptibility at finite temperature.

To illustrate the relationship between the two susceptibilities, in FIG. 1 we plotted the modulating functionfor the tangential components of both. We observe that at zero temperature they coincide. As the temperatureincreases, in the case of the fidelity LE, the gap-vanishing points become less important. On the contrary, for theinterferometric LE, the associated tangential part of the susceptibility does not depend on temperature, thus thegap-vanishing points remain prominent. The DFS from Eq. (12) thus predicts gradual smearing of critical behaviour,consistent with previous findings that showed the absence of phase transitions at finite temperatures in the staticcase [14, 15]. The DIS from Eq.(13) has a tangential term that is not coupled to the temperature, persisting at highertemperatures and giving rise to abrupt changes in the finite-temperature system’s behaviour. This is also consistentwith previous studies in the literature, where DPTs were found even at finite temperatures [48, 49]. Additionally, theinterferometric LE depends on the normal components of the variation of ~x. In other words, the finite-temperature

5

phase transitions inferred by the behavior of the interferometric LE occur due to the change of the parameters of theHamiltonian and not due to temperature.

-10 -5 5 10

0.2

0.4

0.6

0.8

1.0

FIG. 1: The susceptibility modulating function for the tangential components at t = 1.

D. Comparing the two approaches

The above analysis of the two dynamical susceptibilities (metrics) reflects the essential difference between the twodistinguishability measures, one based on the fidelity, the other on interferometric experiments. From the quantuminformation theoretical point of view, the two quantities can be interpreted as distances between states, or betweenprocesses, respectively. The Hamiltonian evaluated at a certain point of parameter space M defines the macroscopicphase. Associated to it we have thermal states and unitary processes. The fidelity LE is obtained from the Bures

distance between a thermal state ρ1 in phase 1 and the one obtained by unitarily evolving this state, ρ2 = U2ρ1U†2 ,

with U2 associated to phase 2. Given a thermal state ρ1 prepared in phase 1, the interferometric LE is obtained fromthe distance between two unitary processes U1 and U2 (defined modulo a phase factor), associated to phases 1 and 2.

The quantum fidelity between two states is in fact the classical fidelity between the probability distributions obtainedby performing an optimal measurement on them. Measuring an observable M on the two states ρ1 and ρ2, oneobtains the probability distributions {p1(i)} and {p2(i)}, respectively. The quantum fidelity FQ between the twostates ρ1 and ρ2 is bounded by the classical fidelity FC between the probability distributions {p1(i)} and {p2(i)},FQ(ρ1, ρ2) = Tr

√√ρ1ρ2√ρ1 ≤

∑i

√p1(i)p2(i) = Fc(p1(i), p2(i)), such that the equality is obtained by measuring an

optimal observable, given by Mop = ρ−1/21

√√ρ1ρ2√ρ1ρ−1/21 (note that optimal observable is not unique). For that

reason, one can argue that the fidelity is capturing all order parameters (i.e., measurements) through its optimal

observables Mop. Fidelity-induced distances, the Bures distance DB(ρ1, ρ2) =√

2(1− FQ(ρ1, ρ2)), the sine distance

DS(ρ1, ρ2) =√

1− F 2Q(ρ1, ρ2) and the F-distance DF (ρ1, ρ2) = 1− FQ(ρ1, ρ2) satisfy the following set of inequalities

DF (ρ1, ρ2) ≤ DT (ρ1, ρ2) ≤ DS(ρ1, ρ2) ≤ DB(ρ1, ρ2),

where the trace distance is given by DT (ρ1, ρ2) = 12Tr|ρ1 − ρ2|. In other words, the fidelity-induced distances and the

trace distance establish the same order on the space of quantum states. This is important, as the trace distance is givingthe optimal value for the success probability in ambiguously discriminating in a single-shot measurement between twoa priori equally probable states ρ1 and ρ2, given by the so-called Helstrom bound PH(ρ1, ρ2) = (1+DT (ρ1, ρ2))/2 [65].

On the other hand, the interferometric phase is based on some interferometric experiment to distinguish two states,

ρ1 =∑i ri|i〉〈i| and ρ2 = U2ρ1U

†2 : it measures how the intensities at the outputs of the interferometer are affected by

applying U2 to only one of its arms [56]. Therefore, to set up such an experiment, one does not need to know the stateρ1 that enters the interferometer, as only the knowledge of U2 is required. Note that this does not mean that the outputintensities do not depend on the interferometric LE: indeed, the inner product 〈U1, U2〉ρ1 is defined with respect to thestate ρ1. This is a different type of experiment, not based on the observation of any physical property of a system. Itis analogous to comparing two masses with weighing scales, which would show the same difference of ∆m = m1−m2,regardless of how large the two masses m1 and m2 are. For that reason, interferometric distinguishability is moresensitive than the fidelity (fidelity depends on more information, not only how much the two states are different, butin what aspects this difference is observable). Indeed, the interferometric LE between ρ1 and ρ2 can be written as

6

the overlap L(ρ1, ρ2) = |〈ρ1|ρ2〉| between the purifications |ρ1〉 =∑i

√ri|i〉|i〉 and |ρ2〉 = (U ⊗ I)|ρ1〉. On the other

hand, the fidelity satisfies F (ρ1, ρ2) = max|ψ〉,|ϕ〉 |〈ψ|ϕ〉|, where |ψ〉 and |ϕ〉 are purifications of ρ1 and ρ2, respectively,i.e., L(ρ1, ρ2) ≤ F (ρ1, ρ2). Moreover, what one does observe in interferometric experiments are the mentioned outputintensities, i.e., one needs a number of identical systems prepared in the same state to obtain results in interferometricmeasurements. This additionally explains why interferometric LE is more sensible than the fidelity one, as the latteris based on the observations performed on single systems. The fact that interferometric LE is more sensitive than thefidelity LE is consistent with the result that the former is able to capture the changes of some of the system’s featuresat finite temperatures (thus they predict DPTs), while the latter cannot.

In terms of experimental feasibility, the fidelity is more suitable for the study of many-body macroscopic systemsand phenomena, while the interferometric measurements provide a more detailed information on genuinely quantum(microscopic) systems. Finally, interferometric experiments involve coherent superpositions of two states. Therefore,when applied to many-body systems, one would need to create genuine Schrodinger cat-like states, which goes beyondthe current, and any foreseeable, technology (and could possibly be forbidden by more fundamental laws of physics;see for example objective collapse theories [66]).

III. DPTS OF TOPOLOGICAL INSULATORS AT FINITE TEMPERATURES

Our general study of two-band Hamiltonians showed that the fidelity-induced LE predicts a gradual smearing ofDPTs with temperature. In order to test this result, we study the fidelity LE on concrete examples of two topologicalinsulators (as noted in the main text, the analogous study for the interferometric LE on concrete examples has alreadybeen performed, and is consistent with our findings [48, 49]). In particular, we present quantitative results obtainedfor the first derivative of the rate function, dg/dt, where g(t) = − 1

N logF , and

F(t, β;λi, λf )=F (ρ(β;λi), e−itH(λf )ρ(β;λi)e

itH(λf )).

The fidelity F is obtained by taking the product of the single-mode fidelities, each of which has the form

F (ρ(β;λi), e−itH(λf )ρ(β;λi)e

itH(λf )) =

√1 + cosh2(βEi) + sinh2(βEi) [cos(2Ef t) + (1− cos(2Ef t))(~ni · ~nf )2)]

2 cosh2(βEi),

with Ha = Ea~na · ~σ and a = i, f . This expression can be obtained by using Eq. (17) and the result found in theSupplemental Material of [14]. The quantity dg/dt is the figure of merit in the study of the DQPTs, therefore wepresent the respective results that confirm the previous study: the generalisation of the LE with respect to thefidelity shows the absence of finite temperature dynamical PTs. We consider two paradigmatic models of topologicalinsulators, namely the SSH [67] and the MD [57] models.

A. SSH model (1D)

The SSH model was introduced in [67] to describe polyacetilene, and it was later found to describe diatomicpolymers [68]. In momentum space, the Hamiltonian for this model is of the form H(k,m) = ~x(k,m) · ~σ, with mbeing the parameter that drives the static PT. The vector ~x(k,m) is given by:

~x(k,m) = (m+ cos(k), sin(k), 0).

By varying m we find two distinct topological regimes. For m < mc = 1 the system is in a non-trivial phase withwinding number 1, while for m > mc = 1 the system is in a topologically trivial phase with winding number 0.

We consider both cases in which we go from a trivial to a topological phase and vice versa (FIGs 2 and 3, respec-tively). We notice that non-analyticities of the first derivative appear at zero temperature, and they are smeared outfor higher temperatures.

B. MD model (2D)

The Massive Dirac model (MDM) captures the physics of a 2D Chern insulator [57], and shows different topologically

distinct phases. In momentum space, the Hamiltonian for the MDM is of the form H(~k,m) = ~x(~k,m) · ~σ, with m

7

0.5 1.0 1.5 2.0 2.5

0.1

0.2

0.3

0.4

0.5

FIG. 2: We plot the time derivative of the rate function, dg/dt, as a function of time for different values of the inversetemperature β = 1/T . We consider a quantum quench from a trivial phase (m = 1.2) to a topological phase (m = 0.8).

0.5 1.0 1.5 2.0 2.5

0.1

0.2

0.3

0.4

FIG. 3: The time derivative of the rate function, dg/dt, as a function of time for different values of the inverse temperature.The quench is from a topological (m = 0.8) to a trivial phase (m = 1.2).

being the parameter that drives the static PT. The vector ~x(~k,m) is given by

~x(~k,m) = (sin(kx), sin(ky),m− cos(kx)− cos(ky)).

By varying m we find four different topological regimes:

• For −∞ < m < mc1 = −2 it is trivial (the Chern number is zero) – Regime I

• For −2 = mc1 < m < mc2 = 0 it is topological (the Chern number is −1) – Regime II

• For 0 = mc2 < m < mc3 = 2 it is topological (the Chern number is +1) – Regime III

• For 2 = mc3 < m <∞ it is trivial (the Chern number is zero) – Regime IV

In FIGs 4, 5 and 6 we plot the first derivative of the rate function g(t), as a function of time for different temperatures.We only consider quenches that traverse a single phase transition point.

We observe that at zero temperature there exist non-analyticities at the critical times – the signatures of DQPTs. Aswe increase the temperature, these non-analyticities are gradually smeared out, resulting in smooth curves for higherfinite temperatures. We note that the peak of the derivative dg(t)/dt is drifted when increasing the temperature, inanalogy to the usual drift of non-dynamical quantum phase transitions at finite temperature[69].

Next, we proceed by considering the cases in which we cross two phase transition points, as shown in FIGs 7 and 8.At zero temperature we obtain a non-analytic behaviour, which gradually disappears for higher temperatures.

Finally, we have also studied the case in which we move inside the same topological regime from left to right andvice versa. We obtained smooth curves without non – analyticities, which we omit for the sake of briefness.

8

1 2 3 4 5

-1.0

-0.5

0.5

FIG. 4: The time derivative of the rate function, dg/dt, as a function of time for different values of the inverse temperature.We quench the system from a trivial to a topological regime (Regimes from I to II and from IV to III).

1 2 3 4 5

-0.5

0.5

1.0

FIG. 5: The time derivative of the rate function, dg/dt, as a function of time for different values of the inverse temperature.The quench is from a topological to a trivial regime (Regimes from II to I and from III to IV).

IV. CONCLUSIONS

We analysed the fidelity and the interferometric generalisations of the LE for general mixed states, and applied themto the study of finite temperature DPTs in topological systems. We showed that the dynamical fidelity susceptibilityis the pullback of the Bures metric in the space of density matrices, i.e., states. On the other hand, the dynamicalinterferometric susceptibility is the pullback of a metric in the space of unitaries (i.e., quantum channels).

The difference between the two metrics reflects the fact that the fidelity is a measure of the state distinguishabilitybetween two given states ρ and σ in terms of observations, while the “interferometric distinguishability” quantifieshow a quantum channel (a unitary U) changes an arbitrary state ρ to UρU†.

Therefore, while the “interferometric distinguishability” is in general more sensitive, and thus appropriate for thestudy of genuine (microscopic) systems, it is the fidelity that is the most suitable for the study of many-body systemphases.

Moreover, interferometric experiments involve coherent superpositions of two states, which for many-body systemswould require creating and manipulating genuine Schrodinger cat-like states. This seems to be experimentally beyondcurrent technology.

We presented analytic expressions for the dynamical susceptibilities in the case of two-band Hamiltonians. At finitetemperature, the fidelity LE indicates gradual disappearance of the zero-temperature DQPTs, while the interfero-metric LE predicts finite-temperature DPTs. We have performed finite temperature study on two representativesof topological insulators: the 1D SSH and the 2D MD models. In perfect agreement with the general result, thefidelity-induced first derivatives gradually smear down with temperature, not exhibiting any critical behaviour atfinite temperatures. This is consistent with recent studies of 1D symmetry protected topological phases at finitetemperatures [14, 15]. On the contrary, the interferometric LE exhibits critical behaviour even at finite temperatures(confirming previous studies on DPTs [48, 49]).

Note added. During the preparation of this manuscript, we became aware of a recent related work [39].

9

1 2 3 4 5

-1

1

2

3

FIG. 6: The time derivative of the rate function, dg/dt, as a function of time for different inverse temperatures. The quantumquench is from a topological to a topological regime (Regimes from II to III and vice versa).

1 2 3 4 5

-20

-10

10

20

30

FIG. 7: The time derivative of the rate function, dg/dt, as a function of time for different values of the inverse temperature, inthe case that we quench the system from a trivial to a topological regime (Regimes from I to III and from IV to II).

Acknowledgments

B. M. and C. V. acknowledge the support from DP-PMI and FCT (Portugal) through the grantsSFRH/BD/52244/2013 and PD/BD/52652/2014, respectively. N. P. acknowledges the IT project QbigD funded byFCT PEst-OE/EEI/LA0008/2013, UID/EEA/50008/2013 and the Confident project PTDC/EEI-CTP/4503/2014.Support from FCT (Portugal) through Grant UID/CTM/04540/2013 is also acknowledged. O.V. thanks FundacionRafael del Pino, Fundacion Ramon Areces and RCC Harvard.

V. APPENDIX: ANALYTICAL DERIVATION OF THE DYNAMICAL SUSCEPTIBILITIES

A. Zero Temperature case

Let H be a Hilbert space. Suppose we have a family of Hamiltonians {H(λ) : λ ∈M} where M is a smooth compactmanifold of the Hamiltonian’s parameters. We assume that aside from a closed finite subset of M , C = {λi}ni=1 ⊂M ,the Hamiltonian is gapped and the ground state subspace is one-dimensional. Locally, on M − C, we can find aground state (with unit norm) described by |ψ(λ)〉. Take λi ∈ C, and let U be an open neighbourhood containing λi.Of course, for sufficiently small U , on the open set U − {λi} one can find a smooth assignment λ 7→ |ψ(λ)〉. Considera curve [0, 1] 3 s 7→ λ(s) ∈ U , with initial condition λ(0) = λ0, such that λ(s0) = λi for some s0 ∈ [0, 1]. Thefamily of Hamiltonians H(s) := H(λ(s)) is well-defined for every s ∈ [0, 1]. The family of states |ψ(s)〉 ≡ |ψ(λ(s))〉 iswell-defined for s 6= s0 and so is the ground state energy

E(s) := 〈ψ(s)|H(s)|ψ(s)〉.

10

1 2 3 4 5

-40

-20

20

40

60

FIG. 8: The time derivative of the rate function, dg/dt, as a function of time for different values of the inverse temperature.The system is quenched from a topological to a trivial regime (Regimes from III to I and from II to IV).

The overlap,

A(s) := 〈ψ(0)| exp(−itH(s))|ψ(0)〉,

is well-defined. We can write

exp(−itH(s)) = exp(−itH(0))T exp

{−i∫ t

0

dτV (s, τ)

}.

If we take a derivative with respect to t of the equation, we find,

H(s) = H(0) + exp(−itH(0))V (s, t) exp(itH(0))

so,

V (s, t) = exp(itH(0))(H(s)−H(0)) exp(−itH(0)).

We can now write, since |ψ(0)〉 is an eigenvector of H(0),

A(s) = e−itE(0)〈ψ(0)|T exp

{−i∫ t

0

dτV (s, τ)

}|ψ(0)〉.

We now perform an expansion of the overlap

〈ψ(0)|T exp

{−i∫ t

0

dτV (s, τ)

}|ψ(0)〉

in powers of s. Notice that

T exp

{−i∫ t

0

dτV (s, τ)

}= I − i

∫ t

0

dτV (s, τ)− 1

2

∫ t

0

∫ t

0

dτ2dτ1T{V (s, τ2)V (s, τ1)}+ ...

and hence

d

ds

(T exp

{−i∫ t

0

dτV (s, τ)

}) ∣∣∣∣s=0

= −i∫ t

0

dτ∂V

∂s(0, τ)

and

d2

ds2

(T exp

{−i∫ t

0

dτV (s, τ)

}) ∣∣∣∣s=0

= −i∫ t

0

dτ∂2V

∂s2(0, τ)−

∫ t

0

∫ t

0

dτ2dτ1T

{∂V

∂s(0, τ2)

∂V

∂s(0, τ1)

}.

Therefore,

11

〈ψ(0)|T exp

{−i∫ t

0

dτV (s, τ)

}|ψ(0)〉 = 1− is〈ψ(0)|

∫ t

0

dτ∂V

∂s(0, τ)|ψ(0)〉

+s2

2

[−i〈ψ(0)|

∫ t

0

dτ∂2V

∂s2(0, τ)|ψ(0)〉−〈ψ(0)|

∫ t

0

∫ t

0

dτ2dτ1T

{∂V

∂s(0, τ2)

∂V

∂s(0, τ1)

}|ψ(0)〉

]+ O(s3).

Thus, by using the identity θ(τ) + θ(−τ) = 1 of the Heaviside theta function, we obtain

|A(s)|2 = 1− s2[∫ t

0

∫ t

0

dτ2dτ1〈ψ(0)|12

{∂V

∂s(0, τ2),

∂V

∂s(0, τ1)

}|ψ(0)〉 − 〈ψ(0)|∂V

∂s(0, τ2)|ψ(0)〉〈ψ(0)|∂V

∂s(0, τ1)|ψ(0)〉

]+ O(s3).

If we denote the expectation value 〈ψ(0)| ∗ |ψ(0)〉 ≡ 〈∗〉 we can write,

|A(s)|2 = 1− s2∫ t

0

∫ t

0

dτ2dτ1

[〈12

{∂V

∂s(0, τ2),

∂V

∂s(0, τ1)

}〉 − 〈∂V

∂s(0, τ2)〉〈∂V

∂s(0, τ1)〉

]+ O(s3) = 1− χs2 + O(s3),

where

χ ≡∫ t

0

∫ t

0

dτ2dτ1

[〈12

{∂V

∂s(0, τ2),

∂V

∂s(0, τ1)

}〉 − 〈∂V

∂s(0, τ2)〉〈∂V

∂s(0, τ1)〉

]is the dynamical susceptibility and is naturally nonnegative. In fact, defining Va(τ) = eiτH(0)∂H/∂λa(λ0)e−iτH(0)

such that, by the chain rule,

∂V

∂s(0, τ) = Va(τ)

∂λa

∂s(0),

we can write,

χ = gab(λ0)∂λa

∂s(0)

∂λb

∂s(0),

with the metric tensor given by

gab(λ0) =

∫ t

0

∫ t

0

dτ2dτ1

[〈12{Va(τ2), Vb(τ1)}〉 − 〈Va(τ2)〉〈Vb(τ1)〉

]. (14)

B. Dynamical fidelity susceptibility χ at finite temperature

A possible generalisation of the zero-temperature LE to finite temperatures is through the Uhlmann fidelity, sincethe zero temperature |A(s)| is precisely the fidelity between the states |ψ(0)〉 and exp(−itH(s))|ψ(0)〉. Since theUhlmann fidelity between two close mixed states is determined by the Bures metric, we begin by revisiting thederivation of the latter for the case of interest, i.e., two-level systems.

C. Bures metric for a two-level system

Take a curve of full-rank density operators t 7→ ρ(t) and an horizontal lift t 7→ W (t), with W (0) =√ρ(0). Then

the Bures metric is given by

gρ(t)(dρ

dt,dρ

dt) = Tr

{dW †

dt

dW

dt

}.

The horizontality condition is given by

W †dW

dt=dW

dt

†W,

12

for each t. In the full-rank case, we can find a unique Hermitian matrix G(t), such that

dW

dt= G(t)W

solves the horizontality condition

W †dW

dt= W †GW =

dW

dt

†W.

Also, G is such that

dt=

d

dt(WW †) = Gρ+ ρG.

If Lρ (Rρ) is left (right) multiplication by ρ, we have, formally,

G = (Lρ +Rρ)−1 dρ

dt.

Therefore,

gρ(t)(dρ

dt,dρ

dt) = Tr

{dW †

dt

dW

dt

}= Tr

{G2ρ

}=

1

2Tr{G(ρG+Gρ)} =

1

2Tr{Gdρ

dt} =

1

2Tr

{(Lρ +Rρ)

−1 dρ

dt

dt

}.

If we write ρ(t) in the diagonal basis,

ρ(t) =∑i

pi(t)|i(t)〉〈i(t)|

we find

gρ(t)(dρ

dt,dρ

dt) =

1

2Tr

{(Lρ +Rρ)

−1 dρ

dt

dt

}=

1

2

∑i,j

1

pi(t) + pj(t)〈i(t)|dρ

dt|j(t)〉〈j(t)|dρ

dt|i(t)〉.

Hence, using the diagonal basis of ρ, we can read off the metric tensor at ρ as

gρ =1

2

∑i,j

1

pi + pj〈i|dρ|j〉〈j|dρ|i〉.

This is the result for general full rank density operators. For two-level systems, writing

ρ =1

2(1−Xµσµ),

and defining variables |X| = r and nµ = Xµ/|X|, we can express gρ as

gρ =

[1

1 + r+

1

1− r

]dρ211 + dρ12dρ21 =

1

1− r2dρ211 + dρ12dρ21,

where we used dρ11 = −dρ22. Notice that

dρ11 =1

2Tr{dρUσ3U−1} =

1

2Tr{dρnµσµ},

where U is a unitary matrix diagonalizing ρ, Uσ3U−1 = nµσµ. Now,

dρ = −1

2dXµσµ,

and hence

dρ11 = −1

2dXµnµ = −1

2dr.

On the other hand,

13

dρ12dρ21 =1

4Tr{dρU(σ1 − iσ2)U−1}Tr{dρU(σ1 − iσ2)U−1}

=1

4

[Tr{dρUσ1U−1}Tr{dρUσ1U−1}+ Tr{dρUσ2U−1}Tr{dρUσ2U−1}

]=

1

4δµν(dXµ − nµnλdXλ)(dXν − nνnσdXσ) =

1

4r2dnµdnµ,

where we used the fact that the vectors (u, v) defined by the equations Uσ1U−1 = uµσµ and Uσ2U

−1 = vµσµ forman orthonormal basis for the orthogonal complement in R3 of the line generated by nµ (which corresponds to thetangent space to the unit sphere S2 at nµ). Thus, we obtain the final expression for the squared line element

ds2 =1

4

(dr2

1− r2+ r2δµνdn

µdnν). (15)

The above expression is ill-defined for the pure state case of r = 1. Nevertheless, the limiting case of r → 1 as themetric is smooth as we will now show by introducing another coordinate patch. The set of pure states is defined byr = 1, i.e., they correspond to the boundary of the 3-dimensional ball B = {X : |X| = r ≤ 1} which, topologically,is the set of all density matrices in dimension 2. Introducing the change of variable r = cosu, with u ∈ [0, π/2), themetric becomes

ds2 =1

4

(du2 + (cosu)2δµνdn

µdnν),

which is well defined also for the pure-state case of r = cos(0) = 1. Restricting it to the unit sphere, the metriccoincides with the Fubini-Study metric, also known as the quantum metric, on the space of pure states CP 1 ∼= S2,i.e., the Bloch sphere. Therefore, there is no problem on taking the pure-state limit of this metric on the space ofstates, since it reproduces the correct result.

D. Pullback of the Bures metric

We have a map

M 3 λ 7→ ρ(λ) = U(λ)ρ0U(λ)−1 =1

2U(λ) (I −Xµσµ)U(λ)−1,

with

U(λ) = exp(−itH(λ)),

and we take

ρ0 =exp(−βH(λ0))

Tr {exp(−βH(λ0)}, for some λ0 ∈M.

We use the curve [0, 1] 3 s 7→ λ(s), with λ(0) = λ0, to obtain a curve of density operators

s 7→ ρ(s) := ρ(λ(s)).

Notice that ρ(0) = ρ0. The fidelity we consider is then

F (ρ(0), ρ(s)) = Tr

(√√ρ(0)ρ(s)

√ρ(0)

).

Recall that for 2× 2 density operators of full rank the Bures line element reads

ds2 =1

4

(dr2

1− r2+ r2δµνdn

µdnν),

where

nµ = Xµ/|X| and r = |X|,

14

with

ρ =1

2(I −Xµσµ) .

Now,

ρ(λ) =1

2(I −Rµν(λ)Xνσµ) ,

with Rµν(λ) being the unique SO(3) element satisfying

U(λ)σµU(λ)−1 = Rνµ(λ)σν .

We then have, pulling back the coordinates,

r(λ) = |X| = constant and nµ(λ) = Rµν(λ)nν .

Therefore,

ds2 =1

4r2δµν

∂nµ

∂λa∂nν

∂λbdλadλb =

1

4r2δµνn

σnτ∂Rµσ∂λa

∂Rντ∂λb

dλadλb,

which in terms of the Euclidean metric on the tangent bundle of R3, denoted by 〈∗, ∗〉, takes the form

gab(λ) =1

4r2〈R−1 ∂R

∂λan,R−1

∂R

∂λbn〉,

written in terms of the pullback of the Maurer-Cartan form in SO(3), R−1dR. We can further pullback by the curves 7→ λ(s) and evaluate at s = 0

χ := gab(λ0)∂λa

∂s(0)

∂λb

∂s(0) =

1

4r2〈R−1 ∂R

∂λan,R−1

∂R

∂λbn〉∂λ

a

∂s(0)

∂λb

∂s(0),

which gives us the expansion of the fidelity

F (s) ≡ F (ρ(0), ρ(s)) = 1− 1

2χs2 + ...

We now evaluate χ. Note that

dUσµU−1 + UσµdU

−1 = U [U−1dU, σµ]U−1 = dRνµσν .

Now, we can parameterise

U = y0I + iyµσµ, with |y|2 = 1.

Therefore,

U−1dU = (y0 − iyµσµ)(dy0 + idyνσν) = i(y0dyµ − yµdy0)σµ +i

2(yµdyν − yνdyµ)ελµνσλ;

,

[U−1dU, σκ] = −2

[(y0dyµ − yµdy0)ετµκ +

1

2(yµdyν − yνdyµ)ελµνε

τλκ

]στ

= −2

[(y0dyµ − yµdy0)ε τ

µκ +1

2(yµdyν − yνdyµ)(δµκδ

τν − δτµδνκ)

]στ

= −2[(y0dyµ − yµdy0)ε τ

µκ + (yκdyτ − yτdyκ)]στ

= 2[(y0dyµ − yµdy0)ε τµ κ + (yτdyκ − yκdyτ )

]στ ≡ (R−1dR)τκστ .

15

Observe that for H(λ) = xµ(λ)σµ we have,

y0(λ) = cos(|x(λ)|t) and yµ = − sin(|x(λ)|t)xµ(λ)

|x(λ)|.

Therefore,

dy0 = − sin(|x(λ)|t)d|x(λ)|,

dyµ = − cos(|x(λ)|t)xµ(λ)

|x(λ)|d|x(λ)| − sin(|x(λ)|t)d

(xµ(λ)

|x(λ)|

).

After a bit of algebra, we get

y0dyµ − yµdy0 =xµ(λ)

|x(λ)|d|x(λ)| − sin(|x(λ)|) cos(|x(λ)|)d

(xµ(λ)

|x(λ)|

),

yµdyν − yνdyµ = 2 sin2(|x(λ)|t)x[µ(λ)

|x(λ)|d

(xν](λ)

|x(λ)|

).

Thus,

(R−1dR)τκ = 2xµ(λ)

|x(λ)|d|x(λ)|ε τµκ − sin(2|x(λ)|t)d

(xµ(λ)

|x(λ)|

)ε τµκ + 4 sin2(|x(λ)|t)x

[τ (λ)

|x(λ)|d

(xκ](λ)

|x(λ)|

).

At s = 0, λ(0) = λ0 and the coordinate nµ(λ(0)) = xµ(λ0)/|x(λ0)|, so the previous expression reduces to

(R−1dR(λ0))τκnκ = (R−1dR(λ0))τκ

xκ(λ0)

|x(λ0)|= − sin(2|x(λ0)|t) 1

|x(λ0)|2ετ µκx

µ(λ0)dxκ(λ0)− (1− cos(2|x(λ0)|t))d(xµ

|x|

)(λ0).

Notice that the first term is perpendicular to the second. Therefore, we find

χds2 =1

4r2|R−1dRn|2

=1

4r2[sin2(2|x(λ0)|t) 1

|x(λ0)|4(δλµδ

σκ − δσµδλκ

)xµ(λ)dxk(λ)xλ(λ0)dxσ(λ0) +

(1− cos(2|x(λ0|t)2〈Pdx(λ0), Pdx(λ0)〉

)]= r2

sin2(|x(λ0)|t)|x(λ0)|2

〈P ∂x

∂λa(λ0), P

∂x

∂λb(λ0)〉∂λ

a

∂s(0)

∂λb

∂s(0)ds2,

where we have introduced the projector P : TxR3 = TxS2|x|⊕NxS

2|x| → TxS

2|x| onto the tangent space of the sphere of

radius |x| at x. In other words, the pullback metric by ρ of the Bures metric at λ0 is given by

gab(λ0) = r2sin2(|x(λ0)|t)|x(λ0)|2

〈P ∂x

∂λa(λ0), P

∂x

∂λb(λ0)〉 = tanh2(β|x(λ0)|) sin2(|x(λ0)|t)

|x(λ0)|2〈P ∂x

∂λa(λ0), P

∂x

∂λb(λ0)〉.

E. Dynamical interferometric susceptibility χ at finite temperature

We can replace the average 〈e−itH(s)〉 ≡ 〈ψ(0)|e−itH(s)|ψ(0)〉 by the corresponding average of eitH(0)e−itH(s) on themixed state ρ(λ0) = ρ(0) = exp(−βH(0))/Tr{exp(−βH(0))} (note its implicit temperature dependence):

A(s) = Tr

{ρ(0)T exp

{−i∫ t

0

dτV (s, τ)

}}.

It is easy to see that |A(s)|2 has the same expansion as before with the average on |ψ(0)〉 replaced by the average onρ(0).

We now proceed to compute χ, or equivalently gab(λ0), in the case of a two-level system, where we can write

ρ(λ) =e−βH(λ)

Tr{e−βH(λ)}=

1

2(I −Xµ(λ)σµ),

16

and define variables r(λ) = |X(λ)| and nµ(λ) = Xµ(λ)/|X(λ)|. Writing H(λ) = xµ(λ)σµ (and H(s) ≡ H(λ(s))), wehave

Va(τ) =∂xµ

∂λa(λ0)eiτH(0)σµe

−iτH(0).

Hence, its expectation value is

〈Va(τ)〉 =1

Tr{e−βH(λ)}Tr{e−βH(0)σµ

} ∂xµ∂λa

(λ0) = r(λ0)nµ(λ0)∂xµ

∂λa(λ0) = Xµ(λ0)

∂xµ

∂λa(λ0),

which is independent of τ . We then have

〈Va(τ2)〉〈Vb(τ1)〉 = (r(λ0))2nµ(λ0)∂xµ

∂λa(λ0)nν(λ0)

∂xν

∂λb(λ0)

= tanh2(β|x(λ0)|) xµ|x(λ0)|

∂xµ

∂λa(λ0)

xν|x(λ0)|

∂xν

∂λb(λ0),

where we used Xµ(λ0) = tanh(β|x(λ0)|)xµ(λ0)/|x(λ0)|. Now, using the cyclic property of the trace, we get

1

2Tr{e−βH(0)}Tr{e−βH(λ){Va(τ2), Vb(τ1)}

}=

1

2Tr{e−βH(0)}Tr{e−βH(0){σµ, σν}

}Rµλ(τ2)Rνσ(τ1)

∂xλ

∂λa(λ0)

∂xσ

∂λb(λ0)

= δµνRµλ(τ2)Rνσ(τ1)

∂xλ

∂λa(λ0)

∂xσ

∂λb(λ0), (16)

where Rµν(τ) is the rotation matrix defined by the equation

eiτH(0)σνe−iτH(0) = Rµν(τ)σµ. (17)

We can explicitly write Rµν(τ) as

Rµν (τ) = cos(2τ |x(λ0)|)δµν + (1− cos(2τ |x(λ0)|))nµ(λ0)nν(λ0) + sin(2τ |x(λ0)|)nλ(λ0)εµλν .

Using the previous equation, and because {R(τ)} forms a one-parameter group, we can write

δµνRµλ(τ2)Rνσ(τ1) = δκλR

κσ(τ2 − τ1).

Since χ (i.e., the metric gab; recall its zero-temperature expression (14)) has to be symmetric under the label exchangea↔ b, the relevant symmetric part of (16) is

cos [2(τ2 − τ1)|x(λ0)|] δµν∂xµ

∂λa(λ0)

∂xν

∂λb(λ0) + (1− cos [2(τ2 − τ1)|x(λ0)|]) xµ

|x(λ0)|∂xµ

∂λa(λ0)

xν|x(λ0)|

∂xν

∂λb(λ0).

Putting everything together gives

〈12{Va(τ2), Vb(τ1)}〉 − 〈Va(τ2)〉〈Vb(τ1)〉 = cos [2(τ2 − τ1)|x(λ0)|] 〈P ∂x

∂λa(λ0), P

∂x

∂λb(λ0)〉

+ (1− tanh2(β|x(λ0)|))〈 x(λ0)

|x(λ0)|,∂x

∂λa(λ0)〉〈 x(λ0)

|x(λ0)|,∂x

∂λb(λ0)〉.

The integral on τ1 and τ2 can now be performed, using∫ t

0

∫ t

0

dτ2dτ1 cos[2(τ2 − τ1)ε] =

∫ t

0

∫ t

0

dτ2dτ1 (cos(2τ2ε) cos(2τ1ε) + sin(2τ2ε) sin(2τ1ε))

=1

4ε2[sin2(2tε) + (cos(2tε)− 1)(cos(2tε)− 1)

]=

1

4ε2[2− 2 cos(2tε)] =

sin2(tε)

ε2.

So, the interferometric metric is

gab(λ0) =sin2(|x(λ0)|t)|x(λ0)|2

[〈P ∂x

∂λa(λ0), P

∂x

∂λb(λ0)〉] + t2(1− tanh2(β|x(λ0|))〈

x(λ0)

|x(λ0)|,∂x

∂λa(λ0)〉〈 x(λ0)

|x(λ0)|,∂x

∂λb(λ0)〉.

The dynamical interferometric susceptibility is then given by

17

χ = gab(λ0)∂λa

∂s(0)

∂λb

∂s(0) =

∫ t

0

∫ t

0

dτ2dτ1

[〈12

{∂V

∂s(0, τ2),

∂V

∂s(0, τ1)

}〉 − 〈∂V

∂s(0, τ2)〉〈∂V

∂s(0, τ1)〉

]= { sin2(|x(λ0)|t)

|x(λ0)|2[〈P ∂x

∂λa(λ0), P

∂x

∂λb(λ0)〉] + t2(1− tanh2(β|x(λ0|))〈

x(λ0)

|x(λ0)|,∂x

∂λa(λ0)〉〈 x(λ0)

|x(λ0)|,∂x

∂λb(λ0)〉}∂λ

a

∂s(0)

∂λb

∂s(0),

with the average taken with respect to the thermal state ρ0 = ρ(0) ≡ ρ(λ0). Also, as mentioned previously, we havethe expansion

|A(s)|2 = |Tr [ρ(0) exp(itH(0)) exp(−itH(s))] |2 =

∣∣∣∣Tr

[ρ(0)T exp(−i

∫ t

0

dτV (s, τ))

]∣∣∣∣2 = 1− χs2 + ...

The difference between the two susceptibilities is given by:

χ− χ = (1− tanh2(β|x(λ0)|)){ sin2(|x(λ0)|t)|x(λ0)|2 [〈P ∂x

∂λa(λ0), P

∂x

∂λb(λ0)〉]

+ t2〈 x(λ0)

|x(λ0)| ,∂x

∂λa(λ0)〉〈 x(λ0)

|x(λ0)| ,∂x

∂λb(λ0)〉}∂λ

a

∂s(0)

∂λb

∂s(0).

As β → +∞, i.e., as the temperature goes to zero, the two susceptibilities are equal. Now, the function

f(t) =sin2(εt)

ε2

is well approximated by t2 for small enough ε. In that case the sum of the two terms appearing in the differencebetween susceptibilities is just proportional the pull-back Euclidean metric on TR3.

F. The pullback of the inteferometric (Riemannian) metric on the space of unitaries

We first observe that each full rank density operator ρ defines a Hermitian inner product in the vector space oflinear maps of a Hilbert space H , i.e., End(H ), given by,

〈A,B〉ρ ≡ Tr{ρA†B}.

This inner product then defines a Riemannian metric on the trivial tangent bundle of the vector space End(H ). Sincethe unitary group U(H ) ⊂ End(H ), by restriction we get a Riemannian metric on U(H ). If we choose ρ to bee−βH(λ)/Tr{e−βH(λ)}, then take the pullback by the map Φt : M 3 λf 7→ e−itH(λf ) ∈ U(H ) and evaluate at λf = λ,to obtain the desired metric.

Next, we show that this version of LE is closely related to the interferometric geometric phase introduced by Sjoqvistet. al [56, 60]. To see this, consider the family of distances in U(H ), dρ, parametrised by a full rank density operatorρ, defined as

d2ρ(U1, U2) = Tr{ρ(U1 − U2)†(U1 − U2)} = 2(1− Re〈U1, U2〉ρ),

where 〈∗, ∗〉ρ is the Hermitian inner product defined previously. In terms of the spectral representation of ρ =∑j pj |j〉〈j|, we have

〈U1, U2〉ρ =∑j

pj〈j|U†1U2|j〉.

The Hermitian inner product is invariant under Ui 7→ Ui ·D, i = 1, 2, where D is a phase matrix

D = eiα∑j

|j〉〈j|.

For the interferometric geometric phase, one enlarges this gauge symmetry to the subgroup of unitaries preservingρ, i.e., the gauge degree of freedom is U(1) ⊗ · · · ⊗ U(1). However, since we are interested in the interferometric LEpreviously defined, we choose not to do that, as we only need the diagonal subgroup, i. e., we only have a global

18

phase. Next, promoting this global U(1)-gauge degree of freedom to a local one, i.e., demanding that we only careabout unitaries modulo a phase, we see that, upon changing Ui 7→ Ui ·Di, i = 1, 2, we have

〈U1, U2〉ρ 7→〈U1 ·D1, U2 ·D2〉ρ =∑j

pj〈j|U†1U2|j〉ei(α2−α1).

We can choose gauges, i.e., D1 and D2, minimizing d2ρ(U1 ·D1, U2 ·D2), obtaining

d2ρ(U1 ·D1, U2 ·D2) = 2(1− |〈U1 ·D1, U2 ·D2〉ρ|) = 2(1− |〈U1, U2〉ρ|).

Now, if {Ui = U(ti)}1≤i≤N were the discretisation of a path of unitaries t 7→ U(t), t ∈ [0, 1], applying the minimisationprocess locally, i.e., between adjacent unitaries Ui+1 and Ui, in the limit N →∞ we get a notion of parallel transporton the principal bundle U(H )→ U(H )/U(1). In particular, the parallel transport condition reads as

Tr

{ρU†(t)

dU

dt(t)

}= 0, for all t ∈ [0, 1].

If we take ρ = exp(−βH(λi))/Tr{e−βH(λi)}, U1 = exp(−itH(λi)) and U2 = exp(−itH(λf )), then the interferometricLE is

L(t, β;λf , λi) = |〈U1, U2〉ρ| = 〈U1, U2〉ρ,

where Ui = Ui · Di (i = 1, 2) correspond to representatives satisfying the discrete version of the parallel transportcondition.

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