Curving the Space by NonHermiticity
Abstract
Quantum systems are often classified into Hermitian and nonHermitian ones. Extraordinary nonHermitian phenomena, ranging from the nonHermitian skin effect to the supersensitivity to boundary conditions, have been widely explored. Whereas these intriguing phenomena have been considered peculiar to nonHermitian systems, we show that they can be naturally explained by a duality between nonHermitian models in flat spaces and their counterparts, which could be Hermitian, in curved spaces. For instance, prototypical onedimensional (1D) chains with uniform chiral tunnelings are equivalent to their duals in twodimensional (2D) hyperbolic spaces with or without magnetic fields, and nonuniform tunnelings could further tailor local curvatures. Such a duality unfolds deep geometric roots of nonHermitian phenomena, delivers an unprecedented routine connecting Hermitian and nonHermitian physics, and gives rise to a theoretical perspective reformulating our understandings of curvatures and distance. In practice, it provides experimentalists with a powerful twofold application, using nonHermiticity to engineer curvatures or implementing synthetic curved spaces to explore nonHermitian quantum physics.
Introduction
Systemenvironment couplings lead to a plethora of intriguing nonHermitian phenomena^{1,2,3,4,5,6,7,8,9}, such as nonorthogonal eigenstates, the nonHermitian skin effect^{10,11,12,13,14,15,16}, real energy spectra in certain parameter regimes^{17,18}, collapsed energy spectra, and coalesced eigenstates at an exceptional point (EP)^{12,19,20}, and drastic responses to boundary conditions^{21,22}. While these phenomena have been extensively explored in quantum sciences and technologies^{2,3,4,5,8,23,24,25,26,27}, peculiar theoretical tools are often required to study nonHermitian physics. Though biorthogonal vectors and metric operators are introduced to restore orthogonality^{11,19,28,29,30}, the underlying physics of these mathematical tools is not clear yet. Moreover, it remains challenging to prove the real energy spectra of certain nonHermitian systems, as the existence of the ({{{{{{{mathcal{PT}}}}}}}}) symmetry does not guarantee a real energy spectrum and sophisticated mathematical techniques are required^{17,18,31}.
In this work, we show a duality between nonHermitian 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 nonHermitian 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 nonHermitian systems. Moreover, in sharp contrast to existing schemes of studying curved spaces^{32,33,34,35}, which were built on the conventional wisdom that a flat space needs to be physically distorted to become curved, our results show that nonHermiticity 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 twofold implication. On the one hand, it establishes nonHermiticity as a unique tool to simulate intriguing quantum systems in curved spaces. For instance, it offers an approach of using nonHermitian systems in flat spaces to solve the grand challenge of accessing gravitational responses of quantum Hall states (QHS) in curved spaces^{34,36,37,38}. On the other hand, the duality allows experimentalists to use curved spaces to explore nonHermitian physics. Whereas a variety of nonHermitian phenomena have been addressed in experiments, delicate designs of dissipations are often required^{2,4,7,8,9,39,40,41,42}. Our results show that curved spaces can serve as an unprecedented means to explore nonHermitian Hamiltonians without resorting to dissipations.
Results
HatanoNelson model and hyperbolic surfaces
Our duality can be demonstrated using the celebrated HatanoNelson (HN) model^{43}, which reads,
$${t}_{R}{psi }_{n1}{t}_{L}{psi }_{n+1}=E{psi }_{n},$$
(1)
where n = 0, 1, . . . N − 1 is the lattice index of a onedimensional (1D) chain, ψ_{n} is the eigenstate, E is the corresponding eigenenergy, and t_{L} and t_{R} are the nearestneighbor tunneling amplitudes towards the left and the right, respectively. Under the open boundary condition (OBC), ψ_{0} = ψ_{N−1} = 0, ({psi }_{n}={e}^{nln (gamma )}sin ({k}_{m}nd)/sqrt{(N1)/2}), where (gamma =sqrt{{t}_{R}/{t}_{L}}) characterizes the strength of nonHermiticity, k_{m} = mπ/((N − 1)d), m = 1, 2, . . . N − 2, and d is the lattice constant. The eigenenergy reads ({E}_{m}=2sqrt{{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 nonrelativistic (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
$$frac{{hslash }^{2}}{2M}kappa left({y}^{2}{partial }_{y}^{2}+frac{1}{4}right)psi (y)=Epsi (y),$$
(2)
where (M={hslash }^{2}/(2sqrt{{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}_{N1}/{y}_{0})}{(y/{y}_{0})}^{frac{1}{2}}sin left[{k}_{y}ln (y/{y}_{0})/sqrt{kappa }right]) with ({k}_{y}=mpi sqrt{kappa }/ln ({y}_{N1}/{y}_{0})) and ψ(y_{0}) = ψ(y_{N−1}) = 0. y_{0} and y_{N−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 halfplane,
$$frac{{hslash }^{2}}{2M}kappa left({y}^{2}{nabla }^{2}+frac{1}{4}right){{Psi }}(x,y)=E{{Psi }}(x,y),$$
(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 k_{x} is a good quantum number, Eq. (3) reduces to Eq. (2) when k_{x} = 0. A finite k_{x} adds an onsite potential to the HN model,
$$V_{n}{psi }_{n}{t}_{R}{psi }_{n1}{t}_{L}{psi }_{n+1}=E{psi }_{n},$$
(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 V_{n}.
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 s_{n} = nd, such that the eigenstate of the HN model, ψ_{n}, defined on discrete lattice sites is extended to ψ(s) as a function of a continuous variable s. Since ψ_{n}, under OBC, includes a part that changes exponentially, i.e., ({e}^{nln (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 ψ_{n} into Eq. (1) and using the Taylor expansion for ϕ(s), (phi ({s}_{npm 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 =Ephi). Consequently, ψ(s) satisfies
$$sqrt{{t}_{L}{t}_{R}}{d}^{2}left({partial }_{s}^{2}2q{partial }_{s}+{q}^{2}+2/{d}^{2}right)psi (s)=Epsi (s).$$
(5)
It describes a nonrelativistic particle subject to an imaginary vector potential, A ~ iq^{39,43}. 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/y_{0} = e^{2qs} and applying (M={hslash }^{2}/(2sqrt{{t}_{L}{t}_{R}}{d}^{2})), (kappa =4{ln }^{2}( gamma  )/{d}^{2}), we obtain Eq. (2) up to a constant energy shift (2sqrt{{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, ψ_{n}, the wavefunction at the nth lattice site of the NH model is identical to ψ(y_{n}), the wavefunction on the Poincaré halfplane evaluated at ({y}_{n}={y}_{0}{e}^{nsqrt{kappa }d}). The lowenergy limit of the eigenenergy of Eq. (1) is also identical to the eigenenergy of Eq. (2) as shown by Table 1.
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 = y_{0}e^{2qs}. We obtain the effective theory near K_{0}d ≠ 0, ±π,
$$big[E({K}_{0})pm isqrt{kappa }hslash {v}_{F}y({partial }_{y}1/(2y))big]psi (y)=Epsi (y),$$
(6)
where (E({K}_{0})=2sqrt{{t}_{L}{t}_{R}}(cos ({K}_{0}d)+{K}_{0}dsin ({K}_{0}d))), ({v}_{F}=2sqrt{{t}_{L}{t}_{R}}dsin ({K}_{0}d)/hslash), and ± corresponds to the left and right moving waves centered near ± K_{0}, 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}_{N1}/{y}_{0})}{(y/{y}_{0})}^{frac{1}{2}}sin big[{k}_{y}ln (y/{y}_{0})/sqrt{kappa }big]) with eigenenergy of (2sqrt{{t}_{L}{t}_{R}}[cos ({K}_{0}d)+({K}_{0}{k}_{y})dsin ({K}_{0}d)]), which recovers the results of the HN model near a finite K_{0}.
Geometric interpretations of nonHermitian phenomena
Our duality provides a natural explanation of several peculiar nonHermitian 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 x_{0 }≤ x ≤ x_{0} + L, its width in the xdirection depends on y, ({L}_{x}(y)=intnolimits_{x = {x}_{0}}^{{x}_{0}+L}{{{{{{{rm{d}}}}}}}}x/(sqrt{kappa }y)=L/(sqrt{kappa }y)). Thus, a wave packet traveling in the ydirection must include an extra factor ({y}^{frac{1}{2}}) to guarantee the conservation of particle numbers. In the scoordinate, 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 operator^{30}. The mapping to a curved space thus establishes an explicit physical interpretation of orthonormal conditions in nonHermitian systems.
Secondly, the duality allows us to equate the nonHermitian skin effect to its counterpart on the Poincaré halfplane we found recently^{44}. This can be best visualized using the embedding of a hyperbolic surface in threedimensional (3D) Euclidean space. We define (y={r}_{0}cosh (eta )), x = r_{0}φ, where r_{0} 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 = y_{0}e^{2qs} 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 t_{L} = t_{R}, the pseudosphere reduces to a cylinder with a vanishing κ. For a given t_{R}( > t_{L}), κ increases with decreasing t_{L}. Increasing the nonHermiticity thus makes the space more curved, as shown by Fig. 1 redc. Approaching EP, t_{L} → 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, t_{L}t_{R} < 0, and the effective mass becomes imaginary, all previous results of positive t_{L}t_{R} 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}_{N1}^{1})}^{frac{1}{2}}{(y/{y}_{0})}^{i{k}_{y}/sqrt{kappa }}), where ({k}_{y}=2mpi sqrt{kappa }/ln ({y}_{N1}/{y}_{0}))) so that ψ(y_{0}) = ψ(y_{N−1}). Correspondingly, eigenenergies become complex. This can be explicitly shown from the timedependent Schrödinger equations. For instance, at the band bottom(top), we multiply ψ^{*}(y) to both sides of (ihslash {partial }_{t}psi =frac{{hslash }^{2}kappa }{2M}left({y}^{2}{partial }_{y}^{2}+1/4right)psi), subtract from the resultant expression its complex conjugate, and integrate over y from y_{0} to y_{N−1}. We find that the total particle number ({{{{{{{{mathcal{N}}}}}}}}}_{p}=intnolimits_{{y}_{0}}^{{y}_{N1}}{{{{{{{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), (ihslash {partial }_{t}psi =left[E({K}_{0})+isqrt{kappa }hslash {v}_{F}yleft({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 y_{0} ≠ y_{N−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, V_{L }≥ 0, in one of the lattice sites of the HN model continuously changes PBC to OBC once V_{L} 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 V_{L}. Similarly, an external potential can be added to the Poincaré halfplane,
$${V}_{delta }=d{V}_{L}sqrt{kappa }ymathop{sum}limits_{l}delta (y{Y}_{l}),$$
(11)
where ({Y}_{l}={y}_{0}{e}^{Nlsqrt{kappa }d}) is the lattice site of the superlattice. The ydependent amplitude of the deltafunctions guarantees the scale invariance and the equivalence between each section between Y_{l} and Y_{l+1}. With V_{L} increasing from zero to infinity, eigenstates evolve from those under PBC to the ones under OBC. (intnolimits_{{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 nonHermiticity is, the more sensitive of the system is to the boundary condition.
Generalizations to longrange and nonuniform tunnelings
Whereas the HN model provides an illuminating example of the duality, applications of our approach to generic nonHermitian models are straightforward. We consider
$$mathop{sum }limits_{m=1}^{M}{t}_{Rm}{psi }_{nm}mathop{sum }limits_{m=1}^{M}{t}_{Lm}{psi }_{n+m}=E{psi }_{n},$$
(12)
where t_{Rm} and t_{Lm} are tunneling amplitudes from the (n ∓ m)th to nth sites. An eigenstate under OBC in the bulk is written as e^{iknd+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 socalled generalized Brillouin zone (BZ) in the complex plane^{10,15,27,45,46}. Near any point in the generalized BZ specified by K_{0}d ∈ [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}_{npm 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
$${{{{{{{mathcal{B}}}}}}}}({K}_{0})[{partial }_{s}^{2}2{{{{{{{mathcal{A}}}}}}}}({K}_{0}){partial }_{s}+{{{{{{{mathcal{C}}}}}}}}({K}_{0})]psi (s)=Epsi (s),$$
(13)
where ({{{{{{{mathcal{A}}}}}}}}({K}_{0})), ({{{{{{{mathcal{B}}}}}}}}({K}_{0})) and ({{{{{{{mathcal{C}}}}}}}}({K}_{0})) depend on K_{0}, as well as t_{Rm} and t_{Lm}. When only the nearest neighbor tunnelings exist, the above equation recovers Eq. (5) at K_{0} = 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 K_{0}. Such K_{0}dependent curvature provides a geometric interpretation for the generalized BZ. Explicit calculations for a model including the nextnearestneighbor 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 nonHermitian models with nonuniform tunnelings,
$${t}_{R,n1}{psi }_{n1}{t}_{L,n}{psi }_{n+1}=E{psi }_{n},$$
(14)
which gives rise to inhomogeneous local curvatures. For slowly varying t_{R,n} and t_{L,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) = e^{ν(s)/2}ψ(s) with (nu (s)=frac{2}{d}intnolimits_{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 (nln (gamma )) in the HN model. Using the same procedure, we obtain the effective theory of Eq. (14). For instance, the nonrelativistic 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}intnolimits_{0}^{{s}_{y}}ln (bar{gamma }(s^{prime} )){{{{{{{rm{d}}}}}}}}s^{prime} }), g_{xy} = g_{yx} = 0, ({V}_{c}=frac{{hslash }^{2}}{2M{d}^{2}}left(frac{d}{2}{partial }_{s}ln bar{gamma }{ }_{{s}_{y}}2right)bar{t}({s}_{y})), and the positiondependent curvature is written as (kappa (y)=bar{t}({s}_{y})left(4ln {bar{gamma }}^{2}({s}_{y})2d{partial }_{s}ln bar{gamma }{ }_{{s}_{y}}right)/{d}^{2}). In these expressions, s_{y} is obtained from (y{y}_{0}=intnolimits_{0}^{{s}_{y}}{{{{{{{rm{d}}}}}}}}s^{prime} {e}^{frac{2}{d}intnolimits_{0}^{s^{prime} }ln (bar{gamma }(s^{primeprime} )){{{{{{{rm{d}}}}}}}}s^{primeprime} }/bar{t}(s^{prime} )). The constant κ of a hyperbolic surface is recovered when t_{R,n} and t_{L,n} are constants. Changing t_{R,n} and t_{L,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^{*} = y_{0} + n^{*}d.
In addition to one dimension, many nonHermitian 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, interchain couplings can be engineered to access different higher dimensional curved spaces (see Fig. S2 of Supplementary Materials).
NonHermitian realization of QHS in curved spaces
The duality we established has a wide range of profound applications. For instance, Fig. 3a shows a nonHermitian realization of QHS in curved spaces. When a particle with a charge − e is subjected to a uniform magnetic field, y^{2}∇^{2} in Eq. (3) is replaced by ({y}^{2}[{({partial }_{x}ifrac{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 k_{x} is still a good quantum number. Wavefunctions of the lowest Landau level (LLL) are written as,
$${psi }_{{{{{{{{rm{LLL}}}}}}}}}={(2{k}_{x})}^{frac{eB}{hslash kappa }frac{1}{2}}sqrt{frac{{{{{{{{mathcal{N}}}}}}}}kappa }{{{Gamma }}(2frac{eB}{hslash kappa }1)L}}{e}^{{k}_{x}y+i{k}_{x}x}{y}^{frac{eB}{hslash kappa }},$$
(16)
whose eigenenergies, ({E}_{LLL}=frac{{hslash }^{2}kappa }{8M}+frac{hslash eB}{2M}), are independent of k_{x}, manifesting the degeneracy of the Landau levels. In the dual nonHermitian systems, a finite magnetic field corresponds to an extra onsite potential in Eq. (4), V_{n} → V_{n} + V_{B,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 V_{B,n} relative to V_{n} is written as
$$b=eB{d}^{2}/(hslash ln  gamma  ),$$
(17)
where ({y}_{0}{k}_{x}=a{(2ln  gamma  )}^{1}) has been used. The magnetic flux density, ({rho }_{phi }=eB/(2pi hslash )=frac{ln  gamma  }{2pi {d}^{2}}b), is thus determined by the ratio of V_{B,n} to V_{n}. For a given B, a finite k_{x} shifts the position of the minimum of the total onsite potential V_{n} + V_{B,n}, similar to the wellknown result of flat spaces where k_{x} determines the location of the minimum of the potential in the Landau gauge.
A complete description of QHS requires its gravitational responses in curved spaces. For instance, the particle density, ρ, depends on the local curvature^{36},
$$rho =nu {rho }_{phi }kappa /(4pi ),$$
(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 nonHermitian lattice is
$$ gamma { }^{2n}int {{{{{{{rm{d}}}}}}}}a{{{{{{{{mathcal{N}}}}}}}}}_{n}(a,b)=bln ( gamma  )2{ln }^{2}( gamma  ),$$
(19)
where ({{{{{{{{mathcal{N}}}}}}}}}_{n}(a,b)= gamma { }^{2n}{psi }_{n}^{}{psi }_{n}) is the particle number at lattice site n in the nonHermitian system. Mapping the magnetic flux, eB/(2πℏ), to the ratio between onsite potentials, (bln ( gamma  )/(2pi {d}^{2})), Eq. (18) and Eq. (19) are equivalent. The dependence of densities of QHS on curvatures is thus readily detectable using this nonHermitian realization. An alternative scheme is to implement a 2D nonHermitian lattice model as illustrated in Fig. S3 of Supplementary Materials, which serves as a nonHermitian generalization of the HarperHofstadter Hamiltonian^{48}.
Discussion
In parallel to accessing curved spaces using nonHermitian systems, experimentalists could also use curved spaces to study nonHermitian physics^{32,33,35}. In conventional understandings, nonHermiticity arises when dissipations exist. While dissipations have been engineered in certain apparatuses to deliver desired nonHermitian Hamiltonians^{1,6,7}, in other platforms, such engineering might be more difficult and sometimes experimentalists may have to use indirect means such as simulating nonHermitian quantum walks^{2,8,27}. Our results show that many nonHermitian 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 nonHermitian physics, this method does not require engineering losses or gains. It thus provides a conceptually different protocol to access nonHermiticity without dissipations.
The duality we have found provides insightful perspectives for both the studies of nonHermitian physics and curved spaces. As the curvature depends on the nonHermiticity 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 nonHermitian physics and curved spaces.
Data availability
Numerical data for the presented plots are available from the authors upon request.
Code availability
Computer codes for generating the figures presented are available from the authors upon request.
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Acknowledgements
Q.Z. acknowledges useful discussions with Mahdi Hosseini and Pramey Upadhyaya about realizations of chiral couplings. Q.Z., C.L., and Z.Z. are supported by the Air Force Office of Scientific Research under award number FA95502010221, DOE DESC0019202, DOE QuantISED program of the theory consortium “Intersections of QIS and Theoretical Particle Physics” at Fermilab, W. M. Keck Foundation, and a seed grant from PQSEI. R.Z. is supported by the National Key R&D Program of China (Grant No. 2018YFA0307601), NSFC (Grant Nos.12174300, 11804268).
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Lv, C., Zhang, R., Zhai, Z. et al. Curving the space by nonHermiticity.
Nat Commun 13, 2184 (2022). https://doi.org/10.1038/s41467022297748

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DOI: https://doi.org/10.1038/s41467022297748
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