Determination of Dynamical Quantum Phase Transitions in Strongly Correlated Many-Body Systems Using Loschmidt Cumulants

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Determination of Dynamical Quantum Phase Transitions in Strongly Correlated Many-Body Systems Using Loschmidt Cumulants

Sebastiano Peotta, Fredrik Brange, Aydin Deger, Teemu Ojanen, and Christian Flindt

Phys. Rev. X 11, 041018 – Published 26 October 2021

Abstract

Dynamical phase transitions extend the notion of criticality to nonstationary settings and are characterized by sudden changes in the macroscopic properties of time-evolving quantum systems. Investigations of dynamical phase transitions combine aspects of symmetry, topology, and nonequilibrium physics; however, progress has been hindered by the notorious difficulties of predicting the time evolution of large, interacting quantum systems. Here, we tackle this outstanding problem by determining the critical times of interacting many-body systems after a quench using Loschmidt cumulants. Specifically, we investigate dynamical topological phase transitions in the interacting Kitaev chain and in the spin-1 Heisenberg chain. To this end, we map out the thermodynamic lines of complex times, where the Loschmidt amplitude vanishes, and identify the intersections with the imaginary axis, which yield the real critical times after a quench. For the Kitaev chain, we can accurately predict how the critical behavior is affected by strong interactions, which gradually shift the time at which a dynamical phase transition occurs. We also discuss the experimental perspectives of predicting the first critical time of a quantum many-body system by measuring the energy fluctuations in the initial state, and we describe the prospects of implementing our method on a near-term quantum computer with a limited number of qubits. Our work demonstrates that Loschmidt cumulants are a powerful tool to unravel the far-from-equilibrium dynamics of strongly correlated many-body systems, and our approach can immediately be applied in higher dimensions.

Received 11 January 2021
Revised 12 August 2021
Accepted 30 August 2021

DOI:https://doi.org/10.1103/PhysRevX.11.041018

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Dynamical phase transitions Magnetic phase transitions Magnetization dynamics Topological phase transition Topological phases of matter Topological superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Sebastiano Peotta ^1,2,3, Fredrik Brange ², Aydin Deger ², Teemu Ojanen ^1,3,*, and Christian Flindt ²

¹Computational Physics Laboratory, Physics Unit, Faculty of Engineering and Natural Sciences, Tampere University, FI-33014 Tampere, Finland
²Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland
³Helsinki Institute of Physics, FI-00014 University of Helsinki, Finland

^*teemu.ojanen@tuni.fi

Popular Summary

Whether or not quantum many-body systems out of equilibrium can be understood in terms of well-defined phases of matter is a central question in condensed-matter physics. While the well-known principles of equilibrium statistical physics are not applicable, the concepts of criticality and far-from-equilibrium dynamics have recently been elegantly unified through the discovery of dynamical phase transitions. Here, we formulate a new method of studying the dynamical phase transitions in strongly correlated quantum systems.

The dynamics of strongly correlated quantum systems poses one of the most difficult challenges in contemporary physics. This fact has substantially complicated the theoretical studies of dynamical phase transitions in these systems. Our novel method of treating dynamical phase transitions can predict the occurrence of the phase transitions by accurately extrapolating properties of the small systems. The power of the procedure is illustrated by solving, for the first time, dynamical phase transitions in paradigmatic strongly correlated topological systems.

Our work will pave the way for systematic studies of dynamical quantum phase transitions in correlated systems that have been out of reach of previous methods, and it will shed new light on the far-from-equilibrium dynamics of strongly correlated quantum systems.

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Vol. 11, Iss. 4 — October - December 2021

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Figure 1
Dynamical phase transitions. (a) A sudden quench of the system parameters causes a dynamical phase transition in a quantum spin chain with $L$ sites. (b) In the thermodynamic limit, such phase transitions give rise to singularities in the rate function at the critical times, $t_{c, 1}, t_{c, 2} \dots$ [see Eqs. (1) and (2) for definitions]; however, in finite-size systems, they are smeared out. (c) The singularities in the rate function are associated with the zeros (circles) of the Loschmidt amplitude in the complex-time plane. In the thermodynamic limit, they form continuous lines, and the real critical times are given by the crossing points with the imaginary axis. We determine the zeros of the Loschmidt amplitude from the Loschmidt cumulants evaluated at the basepoint $τ$ .
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Figure 2
Interacting Kitaev chain. We quench the chemical potential from the trivial phase $μ = - 1.4$ to the topological phase $μ = - 0.2$ for $t > 0$ with fixed $Δ = 0.3$ (all parameters and the inverse time $τ^{- 1}$ are expressed in units of $J = 1$ ). (a)–(c) Complex zeros for different system sizes ( $L = 7 - 20$ ) and boundary conditions ( $Φ = 0, π$ ) in the noninteracting case, compared with the exact thermodynamic lines of zeros. Only the zeros within a finite range from the basepoint can be accurately obtained from the Loschmidt cumulants. This fact is illustrated by moving the basepoint $τ$ along two different paths [paths A and B in panels (a) and (b)]. Panel (c) combines the results from panels (a) and (b). (d)–(f) Loschmidt zeros and critical times obtained with repulsive interactions ( $V > 0$ ) along the two paths. The lines of zeros for $V = 0$ (dashed line) are shown for comparison. The critical time $t_{c}$ , shown in each panel as a red cross, is obtained as the intersection between the imaginary axis and the line drawn from the zero $τ_{-}$ with the smallest negative real part (in absolute value) to the zero $τ_{+}$ with the smallest positive real part. The error on the critical time is estimated as $Δ t_{c} = \max (| t_{c} - Im τ_{-} |, | t_{c} - Im τ_{+} |)$ . (g)–(l) Evolution of zeros and critical times with increasing attractive interactions ( $V < 0$ ). For very strong interactions [ $V = - 1$ , panel (l)], the zeros do not cross the imaginary axis, signaling the absence of a dynamical quantum phase transition. As discussed in Appendix pp2, the numerical error in the zeros is of the order of $10^{- 3}$ .
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Figure 3
Heisenberg chain. We quench the system from the Haldane phase to the large- $D$ phase. The initial state is the ground state of the model (11) with $J = J_{z} = 3 B$ and $D = 0$ (the AKLT state [58]). For the postquench Hamiltonian, we set $B = 0$ , while $D$ can take the values 2, 3, or 4. Here, $J = J_{z} = 1$ is the unit of energy and inverse time. (a) Loschmidt zeros for $D = 2$ . Panel (a1) is a magnified view of the area within the black rectangle in panel (a). From panels (a) and (a1), one can clearly see how the position of a Loschmidt zero for fixed $L$ depends on the twist angle $Φ$ , which is a finite-size effect. It is also useful to consider a fixed twist angle and vary the system size as in the case of the zeros connected by the dash-dotted line in panel (a1) ( $Φ = 0$ , $L = 13$ , 14, 15, 16). The finite-size dependence is suppressed for the zeros corresponding to the twist angle $Φ = π / 2$ , defining the effective thermodynamic line of zeros (solid line, see Appendix pp4). The critical time, determined by the crossing of the effective line with the imaginary axis, is in excellent agreement with the result of Ref. [27] (red bar) obtained using matrix product states (MPS). (b,c) Same as in panel (a) but with $D = 3$ , 4. Finite-size effects are suppressed with increasing $D$ . In panels (b1), (b2), (c1), and (c2), the crossings of the effective thermodynamic lines of zeros with the imaginary axis are shown and compared again with the critical times obtained in Ref. [27].
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Figure 4
Quench from the Haldane phase to the Néel phase. The initial state is the ground state of the model (11) with $J_{z} / J = 1 / 2$ and $D = B = 0$ . The quench is performed by changing the parameter $J_{z}$ to the values $J_{z} = 1$ , 2, 3, 4 (in units of $J$ ). (a) Loschmidt zeros for the quench to $J_{z} = 1$ , which is not sufficiently large to reach the Néel phase. In this case, the zeros do not cross the imaginary axis, and no dynamical phase transition occurs. (b)–(d) Similar to panel (a), but with $J_{z} = 2$ , 3, 4. In this case, several dynamical phase transitions occur as shown, for example, in panels (e1)–(e3), (f), and (g). The critical times are shown as red crosses and are estimated as done in Fig. 3 using the zeros for $Φ = π / 2$ only (see Appendix pp4 for details).
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Figure 5
Determination of the critical time from the initial energy fluctuations. Loschmidt zeros for the Heisenberg chain (11) are obtained from the energy fluctuations in the ground state of the model for $J = J_{z} = 3 B$ and $D = 0$ at the initial time $τ = 0$ , while the energy is determined by the postquench Hamiltonian with $B = 0$ and $D = 4$ . Here, $J = J_{z} = 1$ is the unit of energy and inverse time. The zeros correspond to chains of lengths $L = 5, \dots, 9$ , and in the upper (lower) panel, we have extracted the zeros using energy cumulants of orders $n = 4, \dots, 14$ ( $n = 8, \dots, 19$ ). Importantly, the zeros converge to their exact positions with increasing cumulant orders as can be seen by comparing the panels. The gray line corresponds to the zeros in panel (c1) of Fig. 3, and the estimate of the critical time is indicated with a red cross.
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