Introduction

System-environment couplings lead to a plethora of intriguing non-Hermitian phenomena1,2,3,4,5,6,7,8,9, such as non-orthogonal eigenstates, the non-Hermitian skin effect10,11,12,13,14,15,16, real energy spectra in certain parameter regimes17,18, collapsed energy spectra, and coalesced eigenstates at an exceptional point (EP)12,19,20, and drastic responses to boundary conditions21,22. While these phenomena have been extensively explored in quantum sciences and technologies2,3,4,5,8,23,24,25,26,27, peculiar theoretical tools are often required to study non-Hermitian physics. Though bi-orthogonal vectors and metric operators are introduced to restore orthogonality11,19,28,29,30, the underlying physics of these mathematical tools is not clear yet. Moreover, it remains challenging to prove the real energy spectra of certain non-Hermitian systems, as the existence of the \({{{{{{{\mathcal{PT}}}}}}}}\) symmetry does not guarantee a real energy spectrum and sophisticated mathematical techniques are required17,18,31.

In this work, we show a duality between non-Hermitian Hamiltonians in flat spaces and their counterparts in curved spaces. On the theoretical side, this duality leads to a geometric framework providing a unified explanation of several non-Hermitian phenomena. For instance, it is the finite curvature that requires an orthonormal condition distinct from that in flat spaces, enforces eigenstates to localize at edges, and gives rise to the supersensitivity to boundary conditions. Dual models in curved spaces could be Hermitian, providing a simple proof of the existence of real energy spectra in certain non-Hermitian systems. Moreover, in sharp contrast to existing schemes of studying curved spaces32,33,34,35, which were built on the conventional wisdom that a flat space needs to be physically distorted to become curved, our results show that non-Hermiticity is a controllable knob for tuning curvatures even when the space appears to be flat, for instance, in lattices with fixed lattice spacing. This duality therefore may reform our understanding of distance and curvatures.

In practice, our duality has a two-fold implication. On the one hand, it establishes non-Hermiticity as a unique tool to simulate intriguing quantum systems in curved spaces. For instance, it offers an approach of using non-Hermitian systems in flat spaces to solve the grand challenge of accessing gravitational responses of quantum Hall states (QHS) in curved spaces34,36,37,38. On the other hand, the duality allows experimentalists to use curved spaces to explore non-Hermitian physics. Whereas a variety of non-Hermitian phenomena have been addressed in experiments, delicate designs of dissipations are often required2,4,7,8,9,39,40,41,42. Our results show that curved spaces can serve as an unprecedented means to explore non-Hermitian Hamiltonians without resorting to dissipations.

Results

Hatano-Nelson model and hyperbolic surfaces

Our duality can be demonstrated using the celebrated Hatano-Nelson (HN) model43, which reads,

$$-{t}_{R}{\psi }_{n-1}-{t}_{L}{\psi }_{n+1}=E{\psi }_{n},$$
(1)

where n = 0, 1, . . . N − 1 is the lattice index of a one-dimensional (1D) chain, ψn is the eigenstate, E is the corresponding eigenenergy, and tL and tR are the nearest-neighbor tunneling amplitudes towards the left and the right, respectively. Under the open boundary condition (OBC), ψ0 = ψN−1 = 0, \({\psi }_{n}={e}^{n\ln (\gamma )}\sin ({k}_{m}nd)/\sqrt{(N-1)/2}\), where \(\gamma =\sqrt{{t}_{R}/{t}_{L}}\) characterizes the strength of non-Hermiticity, km = mπ/((N − 1)d), m = 1, 2, . . . N − 2, and d is the lattice constant. The eigenenergy reads \({E}_{m}=-2\sqrt{{t}_{L}{t}_{R}}\cos ({k}_{m}d)\).

Similar to Hermitian lattice models, the effective theory of Eq. (1) in the continuum limit describes the motion of a non-relativistic (relativistic) particle at (away from) the band bottom and top, with a quadratic (linear) dispersion relation, as shown in Fig. 1a. At the band bottom, the effective theory is written as

$$-\frac{{\hslash }^{2}}{2M}\kappa \left({y}^{2}{\partial }_{y}^{2}+\frac{1}{4}\right)\psi (y)=E\psi (y),$$
(2)

where \(M={\hslash }^{2}/(2\sqrt{{t}_{L}{t}_{R}}{d}^{2})\) is the effective mass, and \(\kappa =4{\ln }^{2}(| \gamma | )/{d}^{2}\). Solutions to Eq. (2), \({y}^{\frac{1}{2}}{y}^{\pm i{k}_{y}/\sqrt{\kappa }}\), have the same energy, \({\hslash }^{2}{k}_{y}^{2}/(2M)\). An eigenstate under OBC is their superposition, \(\psi (y)=\sqrt{2/\ln ({y}_{N-1}/{y}_{0})}{(y/{y}_{0})}^{\frac{1}{2}}\sin \left[{k}_{y}\ln (y/{y}_{0})/\sqrt{\kappa }\right]\) with \({k}_{y}=m\pi \sqrt{\kappa }/\ln ({y}_{N-1}/{y}_{0})\) and ψ(y0) = ψ(yN−1) = 0. y0 and yN−1 specify the positions of the two edges. At the band top, we have M → − M. Eq. (2) is a dimension reduction of the Schrödinger equation on a Poincaré half-plane,

$$-\frac{{\hslash }^{2}}{2M}\kappa \left({y}^{2}{\nabla }^{2}+\frac{1}{4}\right){{\Psi }}(x,y)=E{{\Psi }}(x,y),$$
(3)

where \({\nabla }^{2}\equiv \left({\partial }_{x}^{2}+{\partial }_{y}^{2}\right)\), \({{\Psi }}(x,y)={e}^{i{k}_{x}x}\psi (y)\), and − κ is the curvature (Supplementary Material). The metric tensor is \({{{{{{{\bf{g}}}}}}}}=\frac{1}{\kappa {y}^{2}}({{{{{{{\rm{d}}}}}}}}{x}^{2}+{{{{{{{\rm{d}}}}}}}}{y}^{2})\), \(g=\det ({{{{{{{\bf{g}}}}}}}})=1/({\kappa }^{2}{y}^{4})\). Since kx is a good quantum number, Eq. (3) reduces to Eq. (2) when kx = 0. A finite kx adds an onsite potential to the HN model,

$$V_{n}{\psi }_{n}-{t}_{R}{\psi }_{n-1}-{t}_{L}{\psi }_{n+1}=E{\psi }_{n},$$
(4)

where \(V_{n}={a}^{2}\sqrt{{t}_{R}{t}_{L}}{\gamma }^{4n}\). The dimensionless quantity \({a}^{2}=4({\ln }^{2}| \gamma | ){y}_{0}^{2}{k}_{x}^{2}\) characterizes the strength of Vn.

Fig. 1: The duality between the Hatano-Nelson (HN) model and a hyperbolic surface.
figure 1

a A HN chain and its energy spectrum as a function of k. Near a vanishing (finite) K0, the effective theory in curved space is non-relativistic (relativistic). Eigenstates on the HN chain are localized at the edge, ∣ψn∣2 ∝ ∣γ∣2n. b A HN chain is mapped to the shaded strip on the Poincaré half-plane, in which an eigenstate with kx = 0 satisfies ∣ψ∣2 ∝ y. This shaded strip on the Poincaré half-plane with PBC in the x-direction is equivalent to a pseudosphere embedded in 3D Euclidean space. c The curvature and the inverse of the effective mass, as functions of tL for a fixed tR. The unites of κ and M−1 are 1/d2 and 2tRd2/(ℏ2), respectively. (i–v) show the dual pseudospheres of the HN model at various tL > 0. A pseudosphere for tL < 0 is the same as that for −tL.

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To derive the duality between the continuum limit of Eq. (1) at the band bottom and Eq. (2), we define \({\psi }_{n}\equiv \sqrt{d}\psi ({s}_{n})\) with sn = nd, such that the eigenstate of the HN model, ψn, defined on discrete lattice sites is extended to ψ(s) as a function of a continuous variable s. Since ψn, under OBC, includes a part that changes exponentially, i.e., \({e}^{n\ln (\gamma )}\), so does ψ. We thus define ϕ(s) ≡ ψ(s)e−qs with \(q=\ln (\gamma )/d=\frac{1}{2d}\ln ({t}_{R}/{t}_{L})\) determining the inverse of the localization length, and ϕ(s) varies slowly with changing s. Then we have \({\psi }_{n}=\sqrt{d}\phi ({s}_{n}){e}^{q{s}_{n}}\). Substituting ψn into Eq. (1) and using the Taylor expansion for ϕ(s), \(\phi ({s}_{n\pm 1})=\phi ({s}_{n})\pm d{\partial }_{s}\phi +\frac{1}{2}{d}^{2}{\partial }_{s}^{2}\phi\), we obtain \(-\sqrt{{t}_{L}{t}_{R}}(2+{\partial }_{s}^{2})\phi =E\phi\). Consequently, ψ(s) satisfies

$$-\sqrt{{t}_{L}{t}_{R}}{d}^{2}\left({\partial }_{s}^{2}-2q{\partial }_{s}+{q}^{2}+2/{d}^{2}\right)\psi (s)=E\psi (s).$$
(5)

It describes a nonrelativistic particle subject to an imaginary vector potential, A ~ iq39,43. Unlike a real vector potential that amounts to a U(1) gauge field, here, an imaginary vector potential curves the space. Performing a coordinate transformation y/y0 = e2qs and applying \(M={\hslash }^{2}/(2\sqrt{{t}_{L}{t}_{R}}{d}^{2})\), \(\kappa =4{\ln }^{2}(| \gamma | )/{d}^{2}\), we obtain Eq. (2) up to a constant energy shift \(-2\sqrt{{t}_{L}{t}_{R}}\). The mapping between these two models is summarized in Table 1, which provides a dictionary translating microscopic parameters between them. For instance, ψn, the wavefunction at the n-th lattice site of the NH model is identical to ψ(yn), the wavefunction on the Poincaré half-plane evaluated at \({y}_{n}={y}_{0}{e}^{n\sqrt{\kappa }d}\). The low-energy limit of the eigenenergy of Eq. (1) is also identical to the eigenenergy of Eq. (2) as shown by Table 1.

Table 1 The mapping between the continuum limit of the HN model near the band bottom under OBC and the Poincaré half-plane.
Full size table

Away from the band bottom(top), similar calculations can be performed by defining \(\psi (s)={e}^{\pm i{K}_{0}s}{e}^{qs}\phi (s)\) using Taylor expansions of the slowly varying ϕ(s) and the same coordinate transformation y = y0e2qs. We obtain the effective theory near K0d ≠ 0, ±π,

$$\big[E({K}_{0})\pm i\sqrt{\kappa }\hslash {v}_{F}y({\partial }_{y}-1/(2y))\big]\psi (y)=E\psi (y),$$
(6)

where \(E({K}_{0})=-2\sqrt{{t}_{L}{t}_{R}}(\cos ({K}_{0}d)+{K}_{0}d\sin ({K}_{0}d))\), \({v}_{F}=-2\sqrt{{t}_{L}{t}_{R}}d\sin ({K}_{0}d)/\hslash\), and ± corresponds to the left and right moving waves centered near ± K0, respectively, The previously defined \(\kappa =4{\ln }^{2}(| \gamma | )/{d}^{2}\) has been used. The eigenstate under OBC includes both the left and right moving waves and is written as \(\sqrt{2/\ln ({y}_{N-1}/{y}_{0})}{(y/{y}_{0})}^{\frac{1}{2}}\sin \big[{k}_{y}\ln (y/{y}_{0})/\sqrt{\kappa }\big]\) with eigenenergy of \(-2\sqrt{{t}_{L}{t}_{R}}[\cos ({K}_{0}d)+({K}_{0}-{k}_{y})d\sin ({K}_{0}d)]\), which recovers the results of the HN model near a finite K0.

Geometric interpretations of non-Hermitian phenomena

Our duality provides a natural explanation of several peculiar non-Hermitian phenomena. Firstly, the orthonormal condition of effective theories in Eq. (2) and Eq. (6) reads

$$\int \frac{{{{{{{{\rm{d}}}}}}}}y}{\kappa {y}^{2}}{\psi }^{* }(y;{k}_{y})\psi (y;{k}_{y}^{\prime})={{{{{{{\mathcal{N}}}}}}}}{\delta }_{{k}_{y},{k}_{y}^{\prime}},$$
(7)

where the normalization constant \({{{{{{{\mathcal{N}}}}}}}}\) can be chosen freely. As a common feature of curved spaces, a finite curvature appears in the above equation. Considering a strip in the domain x0 ≤ x ≤ x0 + L, its width in the x-direction depends on y, \({L}_{x}(y)=\int\nolimits_{x = {x}_{0}}^{{x}_{0}+L}{{{{{{{\rm{d}}}}}}}}x/(\sqrt{\kappa }y)=L/(\sqrt{\kappa }y)\). Thus, a wave packet traveling in the y-direction must include an extra factor \({y}^{\frac{1}{2}}\) to guarantee the conservation of particle numbers. In the s-coordinate, Eq. (7) is written as \(\int \frac{{{{{{{{\rm{d}}}}}}}}s}{\sqrt{\kappa }{y}_{0}}{e}^{-2qs}{\psi }_{{k}_{y}}^{* }(s){\psi }_{{k}_{y}^{\prime}}(s)={{{{{{{\mathcal{N}}}}}}}}{\delta }_{{k}_{y},{k}_{y}^{\prime}}\). Discretizing this equation with \({{{{{{{\mathcal{N}}}}}}}}={(\sqrt{\kappa }{y}_{0})}^{-1}\), and transforming it to the HN model, we obtain,

$${\sum }_{n}| \gamma {| }^{-2n}{\psi }_{n}^{* }({k}_{m}){\psi }_{n}({k}_{{m}^{\prime}})={\delta }_{{k}_{m},{k}_{{m}^{\prime}}},$$
(8)

where ∣γ∣−2n is precisely the difference between the left and right eigenvectors, or the metric operator30. The mapping to a curved space thus establishes an explicit physical interpretation of orthonormal conditions in non-Hermitian systems.

Secondly, the duality allows us to equate the non-Hermitian skin effect to its counterpart on the Poincaré half-plane we found recently44. This can be best visualized using the embedding of a hyperbolic surface in three-dimensional (3D) Euclidean space. We define \(y={r}_{0}\cosh (\eta )\), x = r0φ, where r0 is an arbitrary constant and η > 0, φ ∈ (−π, π). The embedding can then be written as

$$(u,v,w)=\frac{1}{\sqrt{\kappa }}\left((\eta -\tanh (\eta )),\frac{\cos (\varphi )}{\cosh (\eta )},\frac{\sin (\varphi )}{\cosh (\eta )}\right).$$
(9)

This is a parameterization of a pseudosphere with a constant negative curvature and a radius of \(1/\sqrt{\kappa }\), which satisfies \({(u-{{{{{{{\rm{arcsech}}}}}}}}(\sqrt{({v}^{2}+{w}^{2})\kappa })/\sqrt{\kappa })}^{2}+{v}^{2}+{w}^{2}={\kappa }^{-1}\). As shown in Fig. 1b, a pseudosphere features a funnel shape, since the circumference of the circle with a fixed y(η) changes with changing y(η). As previously explained, a coordinate transformation y = y0e2qs maps eigenstates on the hyperbolic surface, \({y}^{\frac{1}{2}}{y}^{i{k}_{y}/\sqrt{\kappa }}\), to \({e}^{qs}{e}^{i(2q/\sqrt{\kappa }){k}_{y}s}\), which exponentially localizes near the funneling mouth, the smaller end.

Thirdly, the collapsed energy spectrum at EP of the HN models has a natural geometric interpretation. When tL = tR, the pseudosphere reduces to a cylinder with a vanishing κ. For a given tR( > tL), κ increases with decreasing tL. Increasing the non-Hermiticity thus makes the space more curved, as shown by Fig. 1 redc. Approaching EP, tL → 0, κ diverges, and the localization length, \(1/\ln (| \gamma | )\), vanishes, forcing all eigenstates to coalesce. As eigenenergies read \(E={\hslash }^{2}{k}_{y}^{2}/(2M)\) with divergent M, eigenenergies collapse to zero with a massive degeneracy. Across EP, tLtR < 0, and the effective mass becomes imaginary, all previous results of positive tLtR still apply provided that M → ±iM. Particles moving in hyperbolic spaces are thus dissipative, and stationary states no longer exist.

Lastly, similar to the HN model, changing OBC to PBC leads to drastic changes in the curved space. Eigenstates of Eq. (2) and Eq. (6) normalized to \({{{{{{{\mathcal{N}}}}}}}}\) become \(\sqrt{\kappa {{{{{{{\mathcal{N}}}}}}}}}{({y}_{0}^{-1}-{y}_{N-1}^{-1})}^{-\frac{1}{2}}{(y/{y}_{0})}^{i{k}_{y}/\sqrt{\kappa }}\), where \({k}_{y}=2m\pi \sqrt{\kappa }/\ln ({y}_{N-1}/{y}_{0})\)) so that ψ(y0) = ψ(yN−1). Correspondingly, eigenenergies become complex. This can be explicitly shown from the time-dependent Schrödinger equations. For instance, at the band bottom(top), we multiply ψ*(y) to both sides of \(i\hslash {\partial }_{t}\psi =-\frac{{\hslash }^{2}\kappa }{2M}\left({y}^{2}{\partial }_{y}^{2}+1/4\right)\psi\), subtract from the resultant expression its complex conjugate, and integrate over y from y0 to yN−1. We find that the total particle number \({{{{{{{{\mathcal{N}}}}}}}}}_{p}=\int\nolimits_{{y}_{0}}^{{y}_{N-1}}{{{{{{{\rm{d}}}}}}}}y| \psi (y){| }^{2}/(\kappa {y}^{2})\) satisfies,

$${\partial }_{t}{{{{{{{{\mathcal{N}}}}}}}}}_{p}=\hslash \sqrt{\kappa }{{{{{{{\mathcal{N}}}}}}}}{k}_{y}/M,$$
(10)

which signifies the absence of a stationary state and explains complex eigenenergies under PBC. Using \({\partial }_{t}{{{{{{{{\mathcal{N}}}}}}}}}_{p}=\frac{2}{\hslash }{{{{{{{\rm{Im}}}}}}}}(E){{{{{{{{\mathcal{N}}}}}}}}}_{p}\), we find \({{{{{{{\rm{Im}}}}}}}}(E)={\hslash }^{2}\sqrt{\kappa }{k}_{y}/(2M)\). This is distinct from the result for OBC, where \(\psi (y) \sim {y}^{1/2}{y}^{\pm i{k}_{y}/\sqrt{\kappa }}\) such that \({\partial }_{t}{{{{{{{{\mathcal{N}}}}}}}}}_{p}=0\). Similar calculations can be performed for effective theories away from the band top (bottom), \(i\hslash {\partial }_{t}\psi =\left[E({K}_{0})+i\sqrt{\kappa }\hslash {v}_{F}y\left({\partial }_{y}-1/(2y)\right)\right]\psi\). Straightforward calculations show that \({\partial }_{t}{{{{{{{{\mathcal{N}}}}}}}}}_{p}=-\sqrt{\kappa }{v}_{F}{{{{{{{\mathcal{N}}}}}}}}\), which explains the imaginary part of the eigenenergy, \({{{{{{{\rm{Im}}}}}}}}(E)=-\sqrt{\kappa }\hslash {v}_{F}/2\).

Despite that y0 ≠ yN−1, these two edges of a hyperbolic surface can be identified in mathematics, since the solutions under PBC exist, as we previously discussed. In physical systems, such PBC can also be realized. In fact, the boundary condition can be continuously tuned. An onsite energy offset, VL ≥ 0, in one of the lattice sites of the HN model continuously changes PBC to OBC once VL increase from 0 to ∞. We consider a superlattice of a lattice spacing of Nd, whose unit cell is a HN chain, as shown in Fig. 2a. Figure 2b shows eigenenergies as functions of VL. Similarly, an external potential can be added to the Poincaré half-plane,

$${V}_{\delta }=d{V}_{L}\sqrt{\kappa }y\mathop{\sum}\limits_{l}\delta (y-{Y}_{l}),$$
(11)

where \({Y}_{l}={y}_{0}{e}^{Nl\sqrt{\kappa }d}\) is the lattice site of the superlattice. The y-dependent amplitude of the delta-functions guarantees the scale invariance and the equivalence between each section between Yl and Yl+1. With VL increasing from zero to infinity, eigenstates evolve from those under PBC to the ones under OBC. \(\int\nolimits_{{Y}_{l}^{-}}^{{Y}_{l}^{+}}\sqrt{g}{{{{{{{\rm{d}}}}}}}}y{V}_{\delta } \sim 1/\sqrt{\kappa }\) sets the energy scale of the potential, such that the larger the non-Hermiticity is, the more sensitive of the system is to the boundary condition.

Fig. 2: Changing boundary conditions.
figure 2

a A constant potential, VL, is added to the first site in each unit cell of the superlattice. The corresponding potential in the curved space depends on the position. b Eigenenergies for the ground state \({E}_{0}\in {\mathbb{R}}\) and the first two excited states \({E}_{1,2}\in {\mathbb{C}}\) as functions of VL. \(\sqrt{{t}_{R}/{t}_{L}}=1.5\) and N = 12 are used.

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Generalizations to long-range and non-uniform tunnelings

Whereas the HN model provides an illuminating example of the duality, applications of our approach to generic non-Hermitian models are straightforward. We consider

$$-\mathop{\sum }\limits_{m=1}^{M}{t}_{Rm}{\psi }_{n-m}-\mathop{\sum }\limits_{m=1}^{M}{t}_{Lm}{\psi }_{n+m}=E{\psi }_{n},$$
(12)

where tRm and tLm are tunneling amplitudes from the (n ∓ m)th to nth sites. An eigenstate under OBC in the bulk is written as eiknd+qnd, where kd ∈ [0, 2π] and q is real. Unlike the HN model, where \(q=\ln ({t}_{R}/{t}_{L})/(2d)\) is a constant, once beyond the nearest neighbor tunnelings exist, q becomes a function of k and defines the so-called generalized Brillouin zone (BZ) in the complex plane10,15,27,45,46. Near any point in the generalized BZ specified by K0d ∈ [0, 2π], we define \(\psi (s)={e}^{i{K}_{0}s}{e}^{q({K}_{0})s}\phi (s)\), where ϕ(s) changes slowly as a function of s, corresponding to small deviations of the momentum in the continuum limit. Similar to discussions about the HN model, the effective theory can be formulated straightforwardly using \(\phi ({s}_{n\pm 1})=\phi ({s}_{n})\pm d{\partial }_{s}\phi \,+\frac{1}{2}{d}^{2}{\partial }_{s}^{2}\phi\). The Schrödinger equation satisfied by ψ(s) is written as

$$-{{{{{{{\mathcal{B}}}}}}}}({K}_{0})[{\partial }_{s}^{2}-2{{{{{{{\mathcal{A}}}}}}}}({K}_{0}){\partial }_{s}+{{{{{{{\mathcal{C}}}}}}}}({K}_{0})]\psi (s)=E\psi (s),$$
(13)

where \({{{{{{{\mathcal{A}}}}}}}}({K}_{0})\), \({{{{{{{\mathcal{B}}}}}}}}({K}_{0})\) and \({{{{{{{\mathcal{C}}}}}}}}({K}_{0})\) depend on K0, as well as tRm and tLm. When only the nearest neighbor tunnelings exist, the above equation recovers Eq. (5) at K0 = 0 and \({{{{{{{\mathcal{A}}}}}}}}({K}_{0})\) becomes real and reduces to a constant imaginary vector potential \(\sim \ln ({t}_{R}/{t}_{L})/(2d)\) that we have discussed in the HN model. In the most generic case, \({{{{{{{\mathcal{A}}}}}}}}({K}_{0})\) provides a complex vector potential, whose real part curves the space. Using a coordinate transformation \(y={y}_{0}{e}^{2{{{{{{{{\mathcal{A}}}}}}}}}_{R}({K}_{0})s}\), where \({{{{{{{{\mathcal{A}}}}}}}}}_{R}({K}_{0})\) is the real part of \({{{{{{{\mathcal{A}}}}}}}}({K}_{0})\), a hyperbolic surface is thus obtained in the same manner as the HN model. The only difference is that κ now is written as \(\kappa =4{{{{{{{{\mathcal{A}}}}}}}}}_{R}^{2}\) and depends on K0. Such K0-dependent curvature provides a geometric interpretation for the generalized BZ. Explicit calculations for a model including the next-nearest-neighbor interaction are given in Supplementary Materials (Fig. S1).

Whereas uniform chiral tunnelings lead to a hyperbolic surface with a constant curvature, we could also consider non-Hermitian models with non-uniform tunnelings,

$$-{t}_{R,n-1}{\psi }_{n-1}-{t}_{L,n}{\psi }_{n+1}=E{\psi }_{n},$$
(14)

which gives rise to inhomogeneous local curvatures. For slowly varying tR,n and tL,n, we define \(\bar{t}(s)\) and \(\bar{\gamma }(s)\) such that \(\bar{t}(nd)=\frac{2M{d}^{2}}{{\hslash }^{2}}\sqrt{{t}_{R,n}{t}_{L,n}}\) and \(\bar{\gamma }(nd)=\sqrt{{t}_{R,n}/{t}_{L,n}}\). We introduce a slowly changing function ϕ(s) = eν(s)/2ψ(s) with \(\nu (s)=\frac{2}{d}\int\nolimits_{0}^{s}\ln (\bar{\gamma }(s^{\prime} )){{{{{{{\rm{d}}}}}}}}s^{\prime}\). This is a generalization of the uniform case, where ν(nd) reduces to a linear function of n, i.e., the previously discussed \(n\ln (\gamma )\) in the HN model. Using the same procedure, we obtain the effective theory of Eq. (14). For instance, the non-relativistic theory is written as

$$\frac{{\hslash }^{2}}{2M}\left(-\frac{1}{\sqrt{g}}{\partial }_{i}{g}^{ij}\sqrt{g}{\partial }_{j}-\frac{\kappa }{4}+{V}_{c}\right){{\Psi }}(x,y)=E{{\Psi }}(x,y),$$
(15)

where \({g}_{xx}={g}_{yy}=\sqrt{g}=\bar{t}({s}_{y}){e}^{-\frac{4}{d}\int\nolimits_{0}^{{s}_{y}}\ln (\bar{\gamma }(s^{\prime} )){{{{{{{\rm{d}}}}}}}}s^{\prime} }\), gxy = gyx = 0, \({V}_{c}=\frac{{\hslash }^{2}}{2M{d}^{2}}\left(\frac{d}{2}{\partial }_{s}\ln \bar{\gamma }{| }_{{s}_{y}}-2\right)\bar{t}({s}_{y})\), and the position-dependent curvature is written as \(\kappa (y)=\bar{t}({s}_{y})\left(4\ln {\bar{\gamma }}^{2}({s}_{y})-2d{\partial }_{s}\ln \bar{\gamma }{| }_{{s}_{y}}\right)/{d}^{2}\). In these expressions, sy is obtained from \(y-{y}_{0}=\int\nolimits_{0}^{{s}_{y}}{{{{{{{\rm{d}}}}}}}}s^{\prime} {e}^{\frac{2}{d}\int\nolimits_{0}^{s^{\prime} }\ln (\bar{\gamma }(s^{\prime\prime} )){{{{{{{\rm{d}}}}}}}}s^{\prime\prime} }/\bar{t}(s^{\prime} )\). The constant κ of a hyperbolic surface is recovered when tR,n and tL,n are constants. Changing tR,n and tL,n then tunes local curvatures. For instance, when \({t}_{R,n}=\frac{{\hslash }^{2}}{2M{d}^{2}}{e}^{-{{\Theta }}(n-{n}^{* })/(2n)}\), \({t}_{L,n}=\frac{{\hslash }^{2}}{2M{d}^{2}}{e}^{{{\Theta }}(n-{n}^{* })/(2n)}\), where Θ(x) is the Heaviside step function, the curvature vanishes everywhere except at a particular location, i.e., κ ~ δ(y − y*), where y* = y0 + n*d.

In addition to one dimension, many non-Hermitian models in higher dimensions can be constructed based on the HN model. For instance, 1D HN chains can be assembled to access higher dimensional curved spaces. Whereas curved spaces in higher dimensions are, in general, more complex than those in two dimensions, inter-chain couplings can be engineered to access different higher dimensional curved spaces (see Fig. S2 of Supplementary Materials).

Non-Hermitian realization of QHS in curved spaces

The duality we established has a wide range of profound applications. For instance, Fig. 3a shows a non-Hermitian realization of QHS in curved spaces. When a particle with a charge − e is subjected to a uniform magnetic field, y2∇2 in Eq. (3) is replaced by \({y}^{2}[{({\partial }_{x}-i\frac{eB}{\hslash \kappa }\frac{1}{y})}^{2}+{\partial }_{y}^{2}]\), where we have chosen the gauge with the vector potential A = (−B/(κy), 0)47 such that kx is still a good quantum number. Wavefunctions of the lowest Landau level (LLL) are written as,

$${\psi }_{{{{{{{{\rm{LLL}}}}}}}}}={(2{k}_{x})}^{\frac{eB}{\hslash \kappa }-\frac{1}{2}}\sqrt{\frac{{{{{{{{\mathcal{N}}}}}}}}\kappa }{{{\Gamma }}(2\frac{eB}{\hslash \kappa }-1)L}}{e}^{-{k}_{x}y+i{k}_{x}x}{y}^{\frac{eB}{\hslash \kappa }},$$
(16)

whose eigenenergies, \({E}_{LLL}=-\frac{{\hslash }^{2}\kappa }{8M}+\frac{\hslash eB}{2M}\), are independent of kx, manifesting the degeneracy of the Landau levels. In the dual non-Hermitian systems, a finite magnetic field corresponds to an extra onsite potential in Eq. (4), Vn → Vn + VB,n, where \({V}_{B,n}=-ba{\gamma }^{2n}\sqrt{{t}_{L}{t}_{R}}\), as shown in Fig. 3b. The dimensionless b characterizing the strength of VB,n relative to Vn is written as

$$b=eB{d}^{2}/(\hslash \ln | \gamma | ),$$
(17)

where \({y}_{0}{k}_{x}=a{(2\ln | \gamma | )}^{-1}\) has been used. The magnetic flux density, \({\rho }_{\phi }=eB/(2\pi \hslash )=\frac{\ln | \gamma | }{2\pi {d}^{2}}b\), is thus determined by the ratio of VB,n to Vn. For a given B, a finite kx shifts the position of the minimum of the total onsite potential Vn + VB,n, similar to the well-known result of flat spaces where kx determines the location of the minimum of the potential in the Landau gauge.

Fig. 3: Non-Hermitian realization of QHS on hyperbolic surface.
figure 3

a A hyperbolic surface threaded by uniform magnetic fluxes. b An extra onsite energy in HN chain, VB,n, encapsulates the magnetic field.

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A complete description of QHS requires its gravitational responses in curved spaces. For instance, the particle density, ρ, depends on the local curvature36,

$$\rho =\nu {\rho }_{\phi }-\kappa /(4\pi ),$$
(18)

where ν is the filling factor. For integer QHS, ν = 1 and Eq. (18) for a hyperbolic surface can be straightforwardly proved using Eq. (16) (Supplementary Materials). The counterpart of Eq. (18) in the non-Hermitian lattice is

$$| \gamma {| }^{2n}\int {{{{{{{\rm{d}}}}}}}}a{{{{{{{{\mathcal{N}}}}}}}}}_{n}(a,b)=b\ln (| \gamma | )-2{\ln }^{2}(| \gamma | ),$$
(19)

where \({{{{{{{{\mathcal{N}}}}}}}}}_{n}(a,b)=| \gamma {| }^{-2n}{\psi }_{n}^{* }{\psi }_{n}\) is the particle number at lattice site n in the non-Hermitian system. Mapping the magnetic flux, eB/(2πℏ), to the ratio between onsite potentials, \(b\ln (| \gamma | )/(2\pi {d}^{2})\), Eq. (18) and Eq. (19) are equivalent. The dependence of densities of QHS on curvatures is thus readily detectable using this non-Hermitian realization. An alternative scheme is to implement a 2D non-Hermitian lattice model as illustrated in Fig. S3 of Supplementary Materials, which serves as a non-Hermitian generalization of the Harper-Hofstadter Hamiltonian48.

Discussion

In parallel to accessing curved spaces using non-Hermitian systems, experimentalists could also use curved spaces to study non-Hermitian physics32,33,35. In conventional understandings, non-Hermiticity arises when dissipations exist. While dissipations have been engineered in certain apparatuses to deliver desired non-Hermitian Hamiltonians1,6,7, in other platforms, such engineering might be more difficult and sometimes experimentalists may have to use indirect means such as simulating non-Hermitian quantum walks2,8,27. Our results show that many non-Hermitian Hamiltonians are readily accessible using existing curved spaces. For instance, hyperbolic surfaces that have been created in laboratories could be used to realize the HN model directly. In particular, in contrast to current schemes used in the study of non-Hermitian physics, this method does not require engineering losses or gains. It thus provides a conceptually different protocol to access non-Hermiticity without dissipations.

The duality we have found provides insightful perspectives for both the studies of non-Hermitian physics and curved spaces. As the curvature depends on the non-Hermiticity even when the separation between any two points in the system is not physically distorted, our conventional understandings of distance may need to be reformed. We hope that our work will stimulate more interest in studying deep connections between non-Hermitian physics and curved spaces.