Signature of a continuous quantum phase transition in non-equilibrium energy absorption: Footprints of criticality on higher excited states

Understanding phase transitions in quantum matters constitutes a significant part of present day condensed matter physics. Quantum phase transitions concern ground state properties of many-body systems, and hence their signatures are expected to be pronounced in low-energy states. Here we report signature of a quantum critical point manifested in strongly out-of-equilibrium states with finite energy density with respect to the ground state and extensive (subsystem) entanglement entropy, generated by an external pulse. These non-equilibrium states are evidently completely disordered (e.g., paramagnetic in case of a magnetic ordering transition). The pulse is applied by switching a coupling of the Hamiltonian from an initial value (λI) to a final value (λF) for sufficiently long time and back again. The signature appears as non-analyticities (kinks) in the energy absorbed by the system from the pulse as a function of λF at critical-points (i.e., at values of λF corresponding to static critical-points of the system). As one excites higher and higher eigenstates of the final Hamiltonian H(λF) by increasing the pulse height , the non-analyticity grows stronger monotonically with it. This implies adding contributions from higher eigenstates help magnifying the non-analyticity, indicating strong imprint of the critical-point on them. Our findings are grounded on exact analytical results derived for Ising and XY chains in transverse field.

A continuous quantum phase transition (henceforth QPT) entails non-analytic behaviour of the ground-state properties of a system at the quantum critical point [1][2][3][4] . QPT is observed ubiquitously in diverse areas of condensed matter physics and has extensively been studied through last few decades 3 . Interest in different aspects of it, with many important open issues is nevertheless growing rapidly with time.
One important and long-standing question concerns the existence of the fingerprints of a QPT on higher excited states (i.e., states with finite energy density with respect to the ground state). It is known that QPT has definitive signatures in the ground-state (and low-lying states close to it), as reflected in certain ground-state properties like the derivative of ground-state entanglement (concurrence) 5 , the ground-state fidelity susceptibility (see, e.g., [6][7][8][9][10][11] or scaling of ground-state entanglement entropy at the critical point(see, e.g., [12][13][14]. For a physical (local) Hamiltonian, it is not unexpected for the states moderately close to the ground state (with vanishing energy density) to retain such direct mark of the singularity on them. But as one goes up in energy scale (and hence down in length-scale) the non-universal microscopic details of the system tend to dominate most of the physical properties of the system, and the signature of the ground-state singularity gets obscured. This however does not rule out existence of such a signature lying hidden in those excited states in some form.
From the point of view of equilibrium physics, a faint positive indication follows from the existence of the so-called quantum critical region in the finite temperature phase diagram of a quantum system, where the behaviors of the system is strongly affected by the existence of an underlying quantum critical point even at T ≠ 0 3 . Similar indication is obtained from non-equilibrium response mechanism of a system driven at a finite rate across a quantum critical point, largely known as the Kibble-Zurek mechanism 15,16 , which predicts universal scaling behaviour of defect density with respect to the drive rate. It has been observed that similar to the classical case 17,18 , the said universality holds even for rapid drives that produces finite density of non-adiabatic excitation also in the quantum case [19][20][21][22][23][24] .
However, a clear signature of quantum criticality on high-lying states has remained elusive so far. Here we bridge this gap by demonstrating a case where the fingerprints of the QPT on the higher-excited states play an essential role in the occurrence of a non-analyticity in the non-equilibrium response of the system. The signature of the critical point is observed in the energy absorbed by the system from a pulse of large duration. Stronger signature is observed as contribution from higher excited states are incurred by applying pulse of larger amplitude. Though a non-analyticity does not necessarily imply a phase transition (see, e.g. 25 , where non-analytic behaviour of correlation length is observed at a point with no phase transition), in our case the signature always and exclusively occurs at the critical points.

Consider a quantum many-body Hamiltonian
with a set of d independent coupling parameters defining a d-dimensional parameter space, where a position vector λ  represents a set of values for the couplings. We start with the initial state ψ ( ) 0 to be the ground-state ε I 0 of At t = 0 we apply a pulse of strength r in a given direction r by switching the value of λ , and again at t = τ, switch it back to λ  I (see Fig. 1). The initial energy of the system at t = 0 is the ground-state energy ε I 0 . The final energy of the system (after the pulse is withdrawn) is We show (our main result), that if and only if for a particular value of r (r = r c ), λ λ → = → F C corresponds to quantum phase transition point, then the first derivative of E abs with respect to r is discontinuous at r = r c (a general mathematical argument is given under the heading"General Outline of our Calculation"). Henceforth we drop the arrows and hats when unambiguous. Note that it is actually sufficient to study the final energy of the system instead of E abs but energy transfer is usually easier to measure than absolute energy.
For the sake of illustration we consider the XY-chain in a transverse field: where σ x,y,z represents the three components of a Pauli spin, γ and h denote the anisotropy and the transverse field respectively (we take ħ = J = 1). The T = 0 phase diagram of the system has three critical lines as shown in (Fig. 1). Pulses of varying height r along a chosen direction r in the h-γ plane are applied by changing parameters from λ γ ≡ ( , ) , δ being the angle between r and the positive h-axis (see Fig. 1).
The general expression for E abs can be obtained as follows. Hamiltonian (2) can be written as direct product of / N 2 mutually commuting operators  k , each acting on a separate 2-D Hilbert space spanned by occupation number states , − 0 0 k k and , − 1 1 k k , in the momentum (k)-space (see, e.g., 3,4 ).
corresponding respectively to eigenvectors . We start at t = 0 with the ground-state of γ ( , ) H h I I , given by ψ ε 26 ). The absorbed energy as defined in Eq.
limit. Now we consider the so called dephasing limit τ ,  J h, in which all phases that include τ can be considered rapidly oscillating and are dropped out from the calculation of any physical quantity following Riemann-Lebesgue Lemma (see, e.g., 27 ). In this limit µ µ = ( − ) / sin 1 cos 2 2 2 is hence replaced by 1 2 . Using this, and expressions for ϵ k and φ (θ), we get is that the derivative changes discontinuously as a function of r at r = r c . The analytic expression for the measure of discontinuity (size of the jump in the derivative) is given by (where the subscripts  refer respectively to the values η =  r r c , with η → 0 + ) which is the quantitative measure of the non-analyticity.
General Outline of our Calculation. The occurrence of the non-analyticity can be qualitatively seen from the general outline of our calculation given in the following. We switch to the complex plane by substituting z = e ik in the above expression and convert the integral in Eq. (3) to one over a circular contour  with unit radius centering the origin in the z-plane: Note that W is analytic within , Q brings in branch-points into the problem which can be avoided by drawing proper branch-cuts and indenting the contour so that the only singularities enclosed within  are the poles, coming from the zeros of z 2 Q. The non-analytic behaviour of E abs with respect to r occurs because the pole structure within  are different for r > r c and r < r c -poles move in/out of the contour as r is varied across r c . Thus the functional dependence of E abs on r (which depends on the residues of the poles within ) are in general different on two sides of r c , giving rise to the non-analyticity at (and only at) the critical point. In this context it is interesting to note that no such non-analyticity is observed in E abs (see Suppl. Mat.) for a single quench λ λ ( ) → ( ) H H I F as a function of λ F in an XY chain (see e.g. 28-30 ). In the following we outline our approach for calculating Δ exactly analytically for pulses in arbitrary directions in the parameter plane.
Crossing the Ising (h = 1) Critical Line. Here we consider pulses from a general initial point (h I , γ I ) towards a direction δ such that the Ising transition line is crossed (e.g., in Fig. 2c). Evaluating the integral for E abs (Eq. 4) following the general outline given above (see the Method section for details) we get, for h F = 1, γ F ≠ 0, where m = tan(δ). Δ = 0 for any other point on the pulse-line that is not a QPT point. This is of course under the assumption that we are considering the nearest point of intersection between the pulse-line and the critical line from the initial point. If the line intersects more than one critical lines, E abs shows discontinuity (Δ ≠ 0) at each of those (see Fig. 3). A further simplification occurs if we consider the Ising for h F ≥ 1 (Fig. 2d).
Crossing the Anisotropic (γ = 0) Critical Line. In this case (as in Fig. 2b), when we cross γ F = 0 critical line in the region − 1 < h < 1, the size of discontinuity is given by Note that for > h 1, the γ = 0 line is not critical since there is no restructuring of the locations of poles (see Methods section).
The Multi-Critical Points (MCP) are characterized by a discontinuity in Δ as a function of δ. Suppose we keep (h I , γ I ) fixed and vary the pulse direction δ, then Δ (δ) is discontinuous at the values of δ for which the pulse-direction passes through the MCPs at (h = ± 1, γ = 0). For example, choosing the initial point to be (h I , γ I ) = (0, 1), we get the following values of Δ depending on δ within the infinitesimal neighborhood of the MCP.  (Fig. 3a) so that the E abs − λ F curve has two kinks (Fig. 3b) due to (independent) changes in the pole structure twice corresponding to the crossing of two critical points (see Methods Sec. for details).  tribution χ ε I 00 4 0 vanishes as N → ∞ (note that χ 00 is nothing but the fidelity between ε I 0 and ε F 0 .) Hence all the terms contributing to E abs (and hence its non-analyticity) involves higher excited states.

Signature on the Higher Excited
The question that naturally arises is how high can we go up in the energy and still find eigenstates (of the critical Hamiltonian H(λ C )) contributing significantly to the signature? One quantitative way of probing this is to monitor the magnitude of the discontinuity Δ as the function of the pulse height r, as one increases it by moving λ  I away from the critical point deeper into a phase. An increase in r will result in excitation of higher energy eigenstates of H(λ C ), and hence more weight of such states in E abs . Hence, if the signature weakens on the eigenstates of H c as one goes up in energy, then increase in r is expected to decrease Δ . But here Δ increases monotonically with such increase in r without bound. For example, for crossing the Ising line (Eq. 6), we get ∆ ∝ − h 1 I , and for crossing the anisotropy line (Eq. 7), γ ∆ ∝ I . This indicates that eigenstates with higher and higher energy (even those with non-vanishing energy density) excited by abitrary increase in r, actually carry the imprint of the critical point, which is filtered out and added up to enhance Δ .
Finally, the state ψ τ ( ) itself, from which we read our signature off, has extensive subsystem entanglement entropy (proportional to the linear dimension of the sub-system) 32,33 . Our signature is thus visible on a non-equilibrium state not only with extensive energy density, but also with extensive entanglement entropy -a state truly far from the ground state manifold.

Signature in Transverse Spin
Polarization. E abs is not necessarily the only quantity from which the non-analytic signature of QPT can be read off. Here we illustrate this by discussing the special case of 0 → h F → 0 pulse, keeping γ constant. The quantity we concentrate on is the total transverse spin polar- , starting with a ground state of H(h = 0) as the initial state. Thus the non-analytic behaviour of E abs as a function of h F is directly reflected in the behaviour of limτ → ∞M z (τ) vs h F .

Discussion
Though the models we have studied are integrable, we conjecture that E abs will continue to bear similar non-analytic signature of continuous QPT in generic systems, at least when r c is small. The intuition is based on the observation that Δ is a continuous function of (h I , γ I ) for a given critical point, i.e. if we move our initial point towards the transition point (thereby reduce r c ), the value of Δ reduces in a continuous manner. This means, even if our pulse height is small enough so that the ground state of H(λ I ) is a superposition of only the low-lying eigenstates of H(λ c ), still the discontinuity will persist. But appearance of non-analyticity at criticality due to excitation of only the low energy manifold of H by small perturbations is expected to be universal. Hence existence of such non-analyticities in the tractable models suggests their existence in more generic systems. However, if indefinitely higher excited states will still continue to bear the signature in those cases is an open question.
The entire picture fits in if we assume that λ ( )  E bs F a , for a given λ  I and pulse direction is the property of the equilibrium "phase", i.e., it is a smooth function as long as the pulse-line between λ  I and λ  F lies within a single phase. If it crosses a phase boundary (transition line/point), it in general becomes a different smooth function. Hence E abs shows a non-analyticity at the boundary as a function of λ  F . The above picture allows us to consider E abs as a powerful substitute of order parameter for a quantum phase transition, with Δ serving as a sharp locator of the critical point. It is well known that identifying the correct order parameter for complex phases of a condensed matter may often prove to be a hard task. Locating a quantum critical point accurately can also be quite difficult. Our result indicates an alternative non-equilibrium route to the solution of these long-standing problems.
In conclusion, we demonstrate that quantum phase transition (QPT) can be detected from a truly non-equilibrium signature exhibited in the energy absorbed by the concerned system from an externally applied pulse. The non-analyticity essentially involves signature of the QPT on high-lying excited states. This opens up possibilities of devising protocols for detecting and confirming quantum critical points in systems where the location of the critical point and the nature of the phases across it are not precisely known.
Many open issues spring to mind. Do similar signatures exist for QPTs other than continuous ones (e.g., for 1st order) or those without local order parameters, like the topological QPTs? Since the signature involves higher excited states, does it persist at low but finite temperature? Finally, how high in energy can one find eigenstates bearing discernible signatures of the low-lying equilibrium quantum phase transition for a generic quantum critical system with local interactions? Of course, one can design a non-local Hamiltonian, say, by choosing independent orthogonal states as the eigenstates of the Hamiltonian, in which case higher excited states will, by construction, have no information whatsoever regarding the nature of the ground state. Hence the indication we obtain here delves into the deeper issues of correlation between the low-lying and higher eigenstates of a local Hamiltonian. These general open questions are likely to trigger further research in the field. and five poles: 0 (of order 2), (of order 1). Branch points involving (h I , γ I ) and poles involving (h F , γ F ) will be denoted by z I and z F respectively. The positions of the poles change depending on the value of h F and γ F . The critical lines h F = ± 1 and γ F = 0 divide the h F − γ F plane into four different regions where the poles residing inside the contour are different (elaborated in Fig. 4).
First we consider the case of crossing h = 1 critical line from positive γ I (Region I to Region III in Fig. 4). The pole z = 0 are always inside in all cases. Of the other four, are inside the circle and the contour should be indented to avoid branch lines (Fig. 5, left contour). Since the integral over the indentations are continuous functions of (h F , γ F ), the non-analyticity in the behaviour of E abs arises from the restructuring of poles inside the unit circle and we have   This equation leads to the evaluation of the difference of slope before and after crossing γ F = 0 as in Eq. (7). As mentioned above, the γ = 0 line is not critical for > h 1 because the poles z F 1 , For pulses applied along a line that cross two critical lines/points, two independent kinks are observed at the two critical points. For example (Fig. 3b), when λ F crosses the γ F = 0 line, the pole structure changes from ( ) , , 0 z z inside to ( , , ) z z 0 F F 1 2 inside (see Fig. 4), and Δ is given by Eq. (7). Similarly when the pluse line crosses the h F = 1 line, the pole z F 2 goes outside and z 1 F 2 comes inside and Δ is given by Eq. (6).