Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems (2024)

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Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems

Chao Chen Ye, W. L. Vleeshouwers, S. Heatley, V. Gritsev, and C. Morais Smith

Phys. Rev. Research 6, 023202 – Published 23 May 2024

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Abstract

Topological insulators have been studied intensively over the last decades. Earlier research focused on Hermitian Hamiltonians, but recently, peculiar and interesting properties were found by introducing non-Hermiticity. In this work, we apply a quantum geometric approach to various Hermitian and non-Hermitian versions of the Su-Schrieffer-Heeger (SSH) model. We find that this method allows one to correctly identify different topological phases and topological phase transitions for all SSH models, but only when using the metric tensor containing both left and right eigenvectors. Whereas the quantum geometry of Hermitian systems is Riemannian, introducing non-Hermiticity leads to pseudo-Riemannian and complex geometries, thus significantly generalizing from the quantum geometries studied thus far. One remarkable example of this is the mathematical agreement between topological phase transition curves and lightlike paths in general relativity, suggesting a possibility of simulating space-time in non-Hermitian systems. We find that the metric in non-Hermitian phases degenerates in such a way that it effectively reduces the dimensionality of the quantum geometry by one. This implies that within linear response theory, one can perturb the system by a particular change of parameters while maintaining a zero excitation rate.

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Received 27 June 2023
Revised 4 March 2024
Accepted 8 March 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.023202

Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems (8)

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)

Research Areas

Topological insulatorsTopological phase transitionTopological phases of matter

Physical Systems

Authors & Affiliations

Chao Chen Ye^1,2,*, W. L. Vleeshouwers^1,3,†, S. Heatley^3,‡, V. Gritsev^3,§, and C. Morais Smith^1,∥

¹Institute for Theoretical Physics, Utrecht University, Princetonplein 5, Utrecht, 3584 CC, Netherlands
²Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
³Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Postbus 94485, 1098 XH Amsterdam, The Netherlands

^*c.chen.ye@rug.nl
^†w.l.vleeshouwers@uva.nl
^‡sarahmheatley@gmail.com
^§v.gritsev@uva.nl
^∥C.deMoraisSmith@uu.nl

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Vol. 6, Iss. 2 — May - July 2024

Subject Areas

Condensed Matter Physics
Quantum Physics
Topological Insulators

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Quantum metric of non-Hermitian Su-Schrieffer-Heeger systems (12)

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Figure 1
A general schematic representation of an SSH chain with sublattices $A$ and $B$ . The hopping parameters are the right-handed intracell $t_{R}$ , the left-handed intracell $t_{L}$ , and the intercell $t_{2}$ .
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Depth of Field (DoF), Angle of View, and Equivalent Lens Calculator • Points in Focus Photography
Figure 2
Metric tensor $g_{y y}$ in the parameter space $y$ for $N = {50, 200, 800}$ and $N \to \infty$ .
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Figure 3
The QMT components $g_{y z}^{α β}$ with $α, β \in {L, R}$ in the parameter space $y, z \in R$ for $N = 200$ . Their imaginary parts are nonzero (order $10^{- 19}$ ) due to finite-size effects: $Im [g_{μ ν}^{α β}] \approx 0 \forall λ^{μ}, λ^{ν} \in {y, z}$ . Note: keep in mind that $g_{μ ν}^{α α}$ with $α \in {L, R}$ are not legitimate QMT components.
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Figure 4
QMT components $Re [g_{μ ν}^{L R}]$ with $x^{μ}, x^{ν} \in {y, z}$ in the parameter space $y, z \in R$ for $N ⟶ \infty$ . Top viewpoint. Their imaginary parts are zero: $Im [g_{μ ν}^{L R}] = 0 \forall λ^{μ}, λ^{ν} \in {y, z}$ . Therefore, $g_{μ ν}^{L R} = Re [g_{μ ν}^{L R}]$ .
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Figure 5
The phase diagram with associated metric components $g_{y y}^{L R}$ , $g_{z z}^{L R}$ , and $g_{y z}^{L R}$ in the parameter space $y, z \in R$ . The symmetry by exchanging $x^{μ} \to - x^{μ}$ relies on the metric equation $d s^{2} = g_{μ ν}^{L R} d λ^{μ} d λ^{ν}$ , where the line element also has the negative sign, instead of the QMT itself. This domain matches perfectly with the topological phase diagram found by using a more conventional method in Refs.[12, 41]. Different topological phases are distinguished by colours and the two-component winding number $(ν_{1}, ν_{2})$ . Hermitian topological phases have $ν_{1} = ν_{2}$ , while $ν_{1} \neq ν_{2}$ only happens for NH regimes. Specifically, (0,0) represents a Hermitian and NH trivial phase (I and II, respectively), while $(1 / 2, 1 / 2)$ is a Hermitian topological phase, and $(0, 1 / 2)$ and $(1 / 2, 0)$ represent NH topological phases. Notice that the Hermitian phase diagram is recovered when $z = 0$ .
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Figure 6
$g_{y y}$ and $g_{θ θ}$ in the parameter space ${y \equiv t / t_{2}, θ}$ for $N \to \infty$ (i.e., thermodynamic limit). Note that both QMT components are independent of the parameter $θ$ .
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Figure 7
Real and imaginary parts of the QMT component $g_{u u}^{L R}$ in the parameter space ${u_{R}, u_{I}} \in R$ for $N \to \infty$ .
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Figure 8
$Re [g_{μ ν}^{α β}]$ with $α, β \in {L, R}$ in the parameter space $λ^{μ}, λ^{ν} \in {y, z} \in R$ for $N = 200$ . Top viewpoint.
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Figure 9
The winding vector for different values of $t$ , $t_{2}$ , and $δ$ , $(t, t_{2}, δ)$ : (a) $(1.45, 1, 0.3)$ , (b) $(0.5, 1, 1)$ , (c) $(0.5, 1, 1.8)$ , (d) $(0.5, 1, 0.3)$ , (e) $(0.5, 1, 0)$ , (f) $(- 1.45, 1, 0.3)$ . Their corresponding topological phase and winding number $W = (W_{1}, W_{2})$ are also depicted.
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Figure 10
$Re [g_{u u}^{L R}]$ , $Im [g_{u u}^{L R}]$ , $Re [g_{u u^{*}}^{L L}]$ , and $Re [g_{u u^{*}}^{R R}]$ in the parameter space $u_{R}, u_{I} \in R$ for $N = 200$ . $Im [g_{u u^{*}}^{L L}] \approx 0$ and $Im [g_{u u^{*}}^{R R}] \approx 0$ due to finite-size effects.
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Figure 11
Comparison between $Re [g_{u u^{*}}^{L L}]$ and $Re [g_{u u^{*}}^{R R}]$ in the parameter space $u_{R}, u_{I} \in R$ for $N = 200$ .
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