Introduction

Unlike the real spectrum of Hermitian Hamiltonians, the complex non-Hermitian spectrum can be fragile against both the imposed boundary condition and the size of the matrix system1. Specifically, the Bloch bands of certain one-dimensional (1D) ring-shaped non-Hermitian systems form spectral loops on the complex energy plane under periodic boundary conditions (PBCs). Upon adding a perturbation to the boundary coupling of the ring, the energy spectrum, which is called the non-Bloch band2, shrinks and eventually collapses into a line segment when approaching the open boundary condition (OBC)3,4,5. Especially at the intermediate states between PBCs and OBCs, termed the generalized boundary conditions (GBCs)1, the non-Bloch bands are highly sensitive against boundary perturbations. Besides, the sensitivity exhibits a typical exponential enhancement when increasing the system size6, implying that even an infinitesimal perturbation can reshape the non-Bloch bands in a sufficiently large system. Such a size-dependent spectral sensitivity has a topological origin4 and underlies various advanced technologies, including topological sensors7,8, optomechanically induced transparency9, and scale-tailored localizations10. Consequently, ultrahigh-precision control of both the boundary condition and size is important for non-Hermitian topological systems to fully exploit their spectral sensitivity, even if it is still a challenge.

Additionally, non-Hermitian spectra feature another topological structure known as the braid11,12,13,14. Under PBCs, the multiple Bloch bands of 1D non-Hermitian systems are intertwined and form braids in the R3 space spanned by the complex energy and quasi-momentum. Owing to the periodicity of the Brillouin zone (BZ), the start and end of the energy strings can be joined together, thereby closing the braid into a knot or link15. The braid topology of N separable bands is classified by the conjugacy class of the braid group \({{{{\rm{B}}}}}_{N}\)16,17, enriching the classification of non-Hermitian systems based on the homotopy theory. Recently, the braid interpretation has been restored for non-Bloch bands under OBCs18,19, which, in the thermodynamic limit, can be parameterized by the non-Bloch wavenumber in generalized Brillouin zones (GBZs)2,20,21,22. Associating the two non-Hermitian spectral topologies under distinct boundary conditions, another challenge is whether the braid operation persists for GBC non-Bloch bands and if so, how the spectral sensitivity affects the non-Bloch braid and phase transition.

In this study, we tackle these challenges by realizing the ultrahigh-precision control of boundary condition and size in a non-Hermitian static mechanical lattice, based on which we demonstrate and measure the fragile non-Bloch braids and associated phase transitions. The achieved minimal boundary coupling in our experiments outperforms tens of orders of magnitude than current wave-based platforms, while the largest size can be hundreds or more. We further establish the framework of non-Bloch braid topologies under GBCs. Two unique phase transition mechanisms stimulated by either the boundary coupling or size, along with the size-dependent boundary sensitivity of non-Bloch braids, are uncovered. The topologically distinct non-Bloch braids, such as the unknot and Hopf link, and the braid phase transition at the extreme boundary coupling ~10−18 relative to bulk in a system having 160 sites, are observed. Our study not only maneuvers the intricate non-Hermitian boundary and size with an ultrahigh precision, but also unifies two non-Hermitian spectral topologies, i.e., spectral sensitivity and braid topology, that are still uncorrelated.

Results

Non-Hermitian Bloch braids in static mechanical lattices

A sketch of the considered two-dimensional (2D) static mechanical lattice is shown in Fig. 1a, where the unit cell contains two nodes and is tessellated orthogonally along the m and n axes. The nodes are perfect hinges and are connected by linear elastic struts (i.e., with only axial deformations) with bulk intracellular (intercellular) stiffnesses \({k}_{1}\) and \({k}_{2}\) (\({k}_{3}\) and \({k}_{4}\)). These stiffnesses are normalized by the relative value, as the nodal equilibrium equations are homogeneous. There are M (infinite) cells along the m (n) axis, and the intercellular couplings between the sublattice-1 in boundary cell \((1,n)\) and sublattice-2 in \((M,n\pm 1)\) are \({k}_{3}^{{{{\rm{b}}}}}\) and \({k}_{4}^{{{{\rm{b}}}}}\). We enforce the nodes to translate vertically. When \({k}_{3}^{{{{\rm{b}}}}}={k}_{3}\) and \({k}_{4}^{{{{\rm{b}}}}}={k}_{4}\), the lattice is under the m-directional PBC and the nodal displacement field respects the static Rayleigh (SR) form23,24,25,26, \({u}_{m,n}^{(j)}={p}^{(j)}{e}^{iqm}{\lambda }^{n}\) with j = 1, 2. Here, q is the real Bloch wavenumber and \({{{\boldsymbol{p}}}}=({p}^{(1)},{p}^{(2)})\) is the polarization vector. The decay factor \(\lambda\) quantifies the spatial attenuation of displacements along the n axis. Applying the SR solution to the equilibrium equation yields the eigen-equation for an effective Bloch Hamiltonian, given by

$$H(q)={\sum }_{j=0}^{3}{\sigma }_{j}\otimes {d}_{j}(q)$$
(1)

where \({\sigma }_{0}\) is the identity matrix and \({\sigma }_{j}\) (j = 1–3) are Pauli matrices, with the 2 × 2 matrices \({d}_{j}\) given in Supplementary Note 1. The eigenvalue and eigenstate of \(H(q)\) are respectively \(\lambda\) and \({{{\boldsymbol{\psi }}}}={({{{\boldsymbol{p}}}},\lambda {{{\boldsymbol{p}}}})}^{T}\). Upon regarding the horizontal m axis of the 2D lattice as the physical space while the n axis as a pseudo-time that supports state evolution, the lattice is topologically equivalent to a 1D Su-Schrieffer-Heeger (SSH) model27. The eigen-decay factor \(\lambda\) serves as an effective energy and is physically observable in static systems. Under PBCs, \(\lambda\) governs the evolution of the eigenstate, corresponding to the sinusoidal displacement field distributed along the m axis yet with different amplitudes and phases for the two sublattices. These differences are encoded in \({{{\boldsymbol{p}}}}\). Non-Hermiticity is introduced by tuning the strut stiffness to invoke geometric asymmetry24,26,28 (Supplementary Note 1). Moreover, the four bands \({\lambda }_{j}(q)\) (j = 1–4) are partitioned into two sets: two with \(|{\lambda }_{1,2}|\le 1\) govern the forward state evolution, namely the spatial attenuation of displacement fields along the \(+n\) direction, while the other two with \(|{\lambda }_{3,4}|\ge 1\) correspond to the backward evolution with the attenuation occurring along the \(-n\) direction. As these evolution processes are uncoupled, we focus on the forward evolution and \({\lambda }_{1,2}\). Under this circumstance, the lattice is explicitly mapped to the SSH model.

Fig. 1: Comparison of the three phase transition mechanisms for non-Hermitian energy braids.
figure 1

a Schematic of the static mechanical lattice. The stiffnesses of bulk elastic struts are \({k}_{1}\) to \({k}_{4}\) (solid lines), while the boundary stiffnesses between cells \(m=1\) and M are \({k}_{3}^{{{{\rm{b}}}}}\) and \({k}_{4}^{{{{\rm{b}}}}}\) (dashed lines). The leftmost (rightmost) column of nodes are equivalent to the sublattice-2 (sublattice-1) in cells \(m=M\) (\(m=1\)). b Summary of the three braid phase transition mechanisms: Bloch braid phase transition driven by the bulk modulation, and non-Bloch braid phase transition driven by the boundary or size modulation. Each 3D graph shows schematically a braid structure in the \(({{{\mathrm{Re}}}}\lambda ,{{{\rm{Im}}}}\lambda ,q)\) space for Bloch bands or the \(({{{\mathrm{Re}}}}\lambda ,{{\mbox{Im}}}\lambda ,\bar{q})\) space for non-Bloch bands, with its projection on the complex \(\lambda\) plane also presented. The 2D graph adjacent to the braid structure depicts the distribution of EPs for Riemann surfaces and BZ/GBZ on the complex z plane, see the notations in (c).

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Figure 2a–c present three paradigmatic band diagrams of \({\lambda }_{1}\) and \({\lambda }_{2}\) in the \(({{{\mathrm{Re}}}}\lambda ,{{{\rm{Im}}}}\lambda ,q)\) space, where q encircles the first BZ from 0 to \(2\pi\). Clearly, the two bands are isolated in Fig. 2a and braid once in Fig. 2c, corresponding to the unlink and unknot29, respectively. Since chiral symmetry is preserved for \(H(q)\), \(\varGamma H{\varGamma }^{-1}=-H\) with \(\varGamma ={\sigma }_{3}\otimes {\sigma }_{3}\), the braiding degree for the two strands of braid can be characterized by the spectral winding number pertaining to \(\lambda =0\)14,30

$$\nu =\frac{1}{2\pi i}{\int }_{0}^{2\pi }{d}_{q}\,{{\mathrm{ln}}}({\lambda }_{1}{\lambda }_{2})dq$$
(2)
Fig. 2: Non-Hermitian Bloch braids under the bulk modulation.
figure 2

a–c Bloch braids in the R3 space. Their projections on the complex \(\lambda\) plane are marked by the colored dots that encode the phase of the Bloch wavenumber, q. The dashed line refers to the line gap, \({{{\mathrm{Re}}}}(\lambda )=0\), while the black dot marks the EP at \(\lambda =0\). d–f Bloch vectors on the Bloch sphere. The insets depict the corresponding braid diagrams. g Sketch of the Bloch braid phase transition under the modulation of bulk stiffness \({k}_{2}\), which is 8 (a, d), 2 (b, e), and 0.1 (c, f), respectively. Other stiffnesses are common, \({k}_{1}=10\), \({k}_{3}=0.5\), and \({k}_{4}=2\).

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We have a trivial \(\nu =0\) and a topological \(\nu =1\) for Fig. 2a, c, respectively. These phases are bridged by Fig. 2b, where the two bands merge at the exceptional point (EP)31 \(\lambda =0\) and become inseparable. Mechanically, \(\lambda =0\) corresponds to a state of self-equilibrium at the lattice boundary, where the internal forces exerted on \(n=1\) generated by the tension/compression of elastic struts linking \(n=0\) and 1 suffice to assure equilibrium, leaving all interior nodes at \(n\ge 1\) motionless and giving rise to a vanishing decay factor, \(\lambda =0\), see details in Supplementary Note 1.

In parallel with the Bloch bands, the eigenstates also exhibit nontrivial braid topology. We calculate the Bloch vector, \({{{{\boldsymbol{S}}}}}_{j}(q)=\langle {\psi }_{j}^{{{{\rm{R}}}}}|{{{\boldsymbol{\sigma }}}}|{\psi }_{j}^{{{{\rm{R}}}}}\rangle\) with \(|{\psi }_{j}^{{{{\rm{R}}}}}(q)\rangle ={{{{\boldsymbol{p}}}}}_{j}(q)\) (j = 1, 2), \({{{\boldsymbol{\sigma }}}}=({\sigma }_{1},{\sigma }_{2},{\sigma }_{3})\), and project it onto the Bloch sphere32,33, see Fig. 2d–f. Evidently, the Bloch vectors display accordant braid topologies with their spectra, either forming two separate loops (Fig. 2d) or one single loop (Fig. 2f). The phase transition at the EP \(\lambda =0\) are also captured, where \({{{{\boldsymbol{S}}}}}_{1}\) and \({{{{\boldsymbol{S}}}}}_{2}\) coalesce at the north pole when \(q=\pi\), see Fig. 2e. The eigenstate evolution on the Bloch sphere is characterized by the global biorthogonal Zak phase34, Q. For unlink, we have \(Q=0\), indicating an even permutation parity (i.e., the eigenstate returns to itself) when q encircles the BZ once, see the calculation in Supplementary Note 3. By contrast, the unknot in Fig. 2f has an odd permutation, \(Q=\pi\), where the eigenstates interchange after one-round trip of q. The above Bloch braid phase transition driven by the bulk stiffness is sketched in Fig. 2g, as widely studied in existing works. More sophisticated Bloch braids (e.g., the Hopf link with \(\nu =2\)) can be constructed by exploiting all four bands \({\lambda }_{1}\) to \({\lambda }_{4}\), as demonstrated in Supplementary Note 2.

Non-Hermitian non-Bloch braids under GBCs

Next, we deviate from PBCs and establish the framework of braid topology for GBC non-Bloch bands. In particular, we unveil two phase transition mechanisms of non-Bloch braids and demonstrate their size-dependent boundary sensitivity. Specifically, the system is under an OBC when boundary couplings \({k}_{3}^{{{{\rm{b}}}}}={k}_{4}^{{{{\rm{b}}}}}=0\), while it is under a GBC when \(0 < {k}_{3}^{{{{\rm{b}}}}} < {k}_{3}\) and \(0 < {k}_{4}^{{{{\rm{b}}}}} < {k}_{4}\). Due to the lack of translation symmetry, the eigen-problem can only be solved in real space. There are 2 M discrete eigenvalues that constitute the two strands of non-Bloch bands. The eigenstate concomitant with each eigenvalue \(\lambda\) is a displacement vector \(\{{u}_{0}\}\), such that when employing it as the quasistatic boundary load at \(n=0\), its spatial evolution along the \(+n\) direction is \(\{{u}_{n}\}={\lambda }^{n}\{{u}_{0}\}\) (Methods). According to the non-Bloch band theory2,35,36, we first solve the characteristic polynomial of the bulk lattice under the substitution \(z=\exp (iq)\) (Supplementary Note 4)

$$\begin{array}{c}f(z,\lambda )=({k}_{2}{k}_{3}{\lambda }^{2}+{k}_{1}{k}_{3}+{k}_{2}{k}_{4}+{k}_{1}{k}_{4}{\lambda }^{-2}){z}^{2}+\\ ({k}_{1}{k}_{2}{\lambda }^{2}+{k}_{3}{k}_{4}{\lambda }^{2}+{k}_{1}{k}_{2}{\lambda }^{-2}+{k}_{3}{k}_{4}{\lambda }^{-2}-{k}_{{t}}^{2}+{\sum }_{j=1}^{4}{k}_{j}^{2})z+\\ {k}_{1}{k}_{4}{\lambda }^{2}+{k}_{1}{k}_{3}+{k}_{2}{k}_{4}+{k}_{2}{k}_{3}{\lambda }^{-2}\end{array}$$
(3)

with \({k}_{{{{\rm{t}}}}}={\sum }_{j=1}^{4}{k}_{j}\). For given decay factor \(\lambda\), the roots of Eq. (3) are sorted as \(|{z}_{1}(\lambda )|\le |{z}_{2}(\lambda )|\). Then, the general solution of the bulk displacement field is a linear superposition of the two, given by

$${{{{\boldsymbol{u}}}}}_{m,n}=[{c}_{1}{z}_{1}^{m}{{{{\boldsymbol{p}}}}}_{1}+{c}_{2}{z}_{2}^{m}{{{{\boldsymbol{p}}}}}_{2}]{\lambda }^{n}$$
(4)

where \({{{{\boldsymbol{p}}}}}_{1}\) and \({{{{\boldsymbol{p}}}}}_{2}\) are normalized polarizations associated with \({z}_{1}\) and \({z}_{2}\), which are extracted from bulk equations (Supplementary Note 4). The coefficients \({c}_{1}\) and \({c}_{2}\) are determined by the equilibrium of boundary nodes, \({u}_{1,n}^{(1)}\) and \({u}_{M,n}^{(2)}\). Upon inserting the non-Bloch bands into Eq. (3), we obtain two individual sets of solutions for \({z}_{1}\) and \({z}_{2}\), which form two closed manifolds on the complex z plane and constitute the two contours of the GBZ defined uniquely for a finite-sized system1,37. GBZ is the preimage of non-Bloch bands, such that it deviates from a unit circle (i.e., BZ) when non-Hermitian skin effect exists4,5,20.

For PBC Bloch bands, we have \({c}_{1}(\lambda )=0\) and \(|{z}_{2}(\lambda )|=1\). Here, the outer contour of GBZ coincides with the BZ with its argument \({{\rm{arg}}}({z}_{2})=q\) parameterizing the Bloch braids. Upon departing from PBCs, the outer contour of GBZ shrinks while the inner contour expands. Both \({c}_{1}\) and \({c}_{2}\) are nonzero and contribute to the formation of non-Bloch wavefunction in Eq. (4). As a continuous extension, we employ the argument of the outer contour of GBZ, i.e., \({{\rm{arg}}}({z}_{2})\), as a parameterization of the non-Bloch bands under any chosen boundary condition. Such a one-to-one mapping eliminates the ambiguity of the preimage and respects the continuity criterion19, and it degrades to the conventional Bloch case when back to the PBC. It also preserves the handedness between the Bloch and non-Bloch braids (Supplementary Note 5). Because GBZ is a closed manifold homeomorphic to the 1-sphere S1, the homotopic interpretation of knots and links can be restored for GBC non-Bloch bands, provided that they are separable38. In this sense, each of the M eigenvalues form a strand of the two-band braids in the \(({{{\mathrm{Re}}}}\lambda ,{{{\rm{Im}}}}\lambda ,\bar{q})\) space, with \(\bar{q}={{\rm{arg}}}({z}_{2})\in [0,2\pi )\). It is noted that this mapping is not injective, since for \(\lambda \in {{{\rm{R}}}}\) we have \({z}_{1}(\lambda )={z}_{2}^{\ast }(\lambda )\) and \(|{c}_{1}(\lambda )|=|{c}_{2}(\lambda )|\) (Supplementary Note 5), suggesting that the two contours of GBZ overlap and contribute equally to the wavefunction. Hence, each real eigenvalue is a two-bifurcation point on the GBZ39 and is thus assigned with two arguments, \(\pm {{\rm{arg}}}({z}_{2})\). In addition, the right eigenstate in the “non-Bloch vector”, defined as \({{{{\boldsymbol{S}}}}}_{j}=\langle {\psi }_{j}^{{{{\rm{R}}}}}|{{{\boldsymbol{\sigma }}}}|{\psi }_{j}^{{{{\rm{R}}}}}\rangle\) with j = 1, 2…2 M the mode index, is \(|{\psi }_{j}^{{{{\rm{R}}}}}\rangle ={{{{\boldsymbol{p}}}}}_{j,2}\), i.e., the polarization associated with the non-Bloch wavenumber \({z}_{2}\) in terms of the eigenvalue \({\lambda }_{j}\).

For fixed size M with reducing boundary couplings, the non-Bloch bands collapse from spectral loops to line segments on the complex \(\lambda\) plane due to the spectral flow1,3,40. In our systems, all OBC eigenvalues are real, implying that the two bands can only trivially unlink. In this context, if the PBC Bloch braid is nontrivial, e.g., the unknot in Fig. 2c, there must be a braid phase transition at the intermediate GBC, where the EP degeneracy occurs. As another paradigm, for given states between PBCs and OBCs, increasing the site number can also drive two non-Bloch braids from isolated (i.e., unlink when \(M\to 0\)) to tangled (i.e., unknot when \(M\to \infty\)), accompanied with the scale-free localization of bulk eigenstates37,41,42. This ensures the EP at a specific size, \(M={M}_{0}\). The above formal analysis underpins the two mechanisms of non-Bloch braid phase transition: the first is driven by the local boundary coupling while the second is driven by the system size. These mechanisms are fundamentally different from the Bloch braid phase transition, as stimulated by the synchronous modulation of all bulk parameters12,13,14,43,44,45,46. When \({k}_{1} > {k}_{3}\) and \({k}_{2} < {k}_{4}\) (cases with a nontrivial Bloch braid), the critical boundary couplings separating distinct non-Bloch braid phases can be explicitly derived (Methods)

$${k}_{4}^{{{{\rm{b}}}}}={\bar{k}}_{4}^{{{{\rm{b}}}}}={k}_{2}{\left(\frac{{k}_{2}}{{k}_{4}}\right)}^{M-1}\,0 < {k}_{3}^{{{{\rm{b}}}}} < {k}_{3}$$
(5)

Hence, when \({k}_{4}^{{{{\rm{b}}}}} < {\bar{k}}_{4}^{{{{\rm{b}}}}}\), the non-Bloch bands are OBC-like and are trivially unlinked, while for \({k}_{4}^{{{{\rm{b}}}}} > {\bar{k}}_{4}^{{{{\rm{b}}}}}\) the non-Bloch bands, approaching PBCs, braid as an unknot, with the braid phase transition taking exactly at \({k}_{4}^{{{{\rm{b}}}}}={\bar{k}}_{4}^{{{{\rm{b}}}}}\). Markedly, Eq. (5) manifests an exponentially decreased phase transition threshold when increasing the size M, which is the hallmark of GBC non-Bloch bands1,6. It suggests that in a large system, even a sufficiently small boundary perturbation can drastically affect the non-Bloch braid topology, highlighting its extreme sensitivity. Our theory thus unifies the two non-Hermitian spectral topologies, by demonstrating the braiding and phase transition of GBC non-Bloch bands and showing their size-dependent boundary sensitivity. This unification underpins the topological classification of non-Bloch bands based on their homotopic characterizations15.

As an illustration, consider the bulk parameters \({k}_{1}=8\), \({k}_{2}=1\), and \({k}_{3}={k}_{4}=2\). We first set \(M=60\) and \({k}_{3}^{{{{\rm{b}}}}}={k}_{4}^{{{{\rm{b}}}}}\), such that \({\bar{k}}_{4}^{{{{\rm{b}}}}}=1.73\times {10}^{-18}\). Figure 3a–c depict the GBZs when \({k}_{4}^{{{{\rm{b}}}}} < {\bar{k}}_{4}^{{{{\rm{b}}}}}\) (Fig. 3a), \({k}_{4}^{{{{\rm{b}}}}}={\bar{k}}_{4}^{{{{\rm{b}}}}}\) (Fig. 3b), and \({k}_{4}^{{{{\rm{b}}}}} > {\bar{k}}_{4}^{{{{\rm{b}}}}}\) (Fig. 3c), respectively. The bordered dots mark the EPs of two Riemann sheets \(\lambda (z)\), as solved from Eq. (3). The blue lines connecting adjacent EPs are branch cuts. Figure 3d–f show the non-Bloch braids. We see the two strands are isolated in Fig. 3d while braid once in Fig. 3f, with the phase transitions reached in Fig. 3e. These distinct braid topologies are intimately related to the enclosure of EPs by GBZ (Supplementary Note 6). For instance, GBZ encloses two EPs in Fig. 3a while not crossing the branch cut, for which the non-Bloch bands do not braid. Upon increasing \({k}_{4}^{{{{\rm{b}}}}}\), GBZ expands and touches the EPs at \({\bar{k}}_{4}^{{{{\rm{b}}}}}\) (Fig. 3b), where phase transition occurs. The GBZ further intersects the branch cut when \({k}_{4}^{{{{\rm{b}}}}} > {\bar{k}}_{4}^{{{{\rm{b}}}}}\) (Fig. 3c), giving rise to a nontrivial unknot, owing to the permutation of eigenstates on Riemann surfaces12. The braiding degrees of non-Bloch bands, calculated from a discretization of Eq. (2), are 0 for Figs. 1 and 3d for Fig. 3f respectively. Figure 3g–i present the corresponding non-Bloch vectors, exhibiting the same braiding patterns as their bands. The non-Bloch braid phase transition driven by boundary coupling is sketched in Fig. 3j.

Fig. 3: Non-Hermitian non-Bloch braids under the boundary modulation.
figure 3

a–c Outer contours of GBZ (colored dots) lying inside the BZ (gray circles). The color encodes the phase of non-Bloch wavenumber, \(\bar{q}\). The bordered dots mark the EPs within the axis range, and the blue lines linking two EPs are branch cuts. d–f Non-Bloch braids in the R3 space and their projections on the complex \(\lambda\) plane. g–i Non-Bloch vectors on the Bloch sphere. The insets depict the corresponding braid diagrams. j Sketch of the non-Bloch braid phase transition under the modulation of boundary stiffness \({k}_{4}^{{{{\rm{b}}}}}\), which is 0 (a, d, g), \({\bar{k}}_{4}^{{{{\rm{b}}}}}\) (b, e, h), and 10-10 (c, f, i), respectively. The bulk stiffnesses are \({k}_{1}=8\), \({k}_{2}=1\), and \({k}_{3}={k}_{4}=2\), with the size \(M=60\).

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Next, we fix \({k}_{3}^{{{{\rm{b}}}}}={k}_{4}^{{{{\rm{b}}}}}=1.73\times {10}^{-18}\) while varying the system size. The GBZs and associated non-Bloch braids with \(M=40\), 60 (which is equivalent to Fig. 3b and e), and 80 are shown in Fig. 4a–f. In this circumstance, enlarging the size expands the GBZ and enforces its intersection with the branch cut above the threshold \({M}_{0}=60\). Therefore, the non-Bloch braids are trivially unlinked (Fig. 4d) for small size, while being a topological unknot (Fig. 4f) when exceeding the transition point Fig. 4e. The braid topologies of non-Bloch vectors are accordant with their bands, see Fig. 4g–i. Figure 4j sketches the non-Bloch braid phase transition driven by the size modulation.

Fig. 4: Non-Hermitian non-Bloch braids under the size modulation.
figure 4

a–c Outer contours of GBZ (colored dots) lying inside the BZ (gray circles). The color encodes the phase of non-Bloch wavenumber, \(\bar{q}\). The bordered dots mark the EPs within the axis range, and the blue lines linking two EPs are branch cuts. d–f Non-Bloch braids in the R3 space and their projections on the complex \(\lambda\) plane. g–i Non-Bloch vectors on the Bloch sphere. The insets depict the corresponding braid diagrams. j Sketch of the non-Bloch braid phase transition under the modulation of size M, which is 40 (a, d, g), 60 (b, e, h), and 80 (c, f, i), respectively. The stiffnesses are common, \({k}_{1}=8\), \({k}_{2}=1\), \({k}_{3}={k}_{4}=2\), and \({k}_{3}^{{{{\rm{b}}}}}={k}_{4}^{{{{\rm{b}}}}}={\bar{k}}_{4}^{{{{\rm{b}}}}}\).

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Mathematically, the preimage (i.e., BZ) of Bloch bands under bulk modulation is invariant while the EPs on Riemann surfaces move, inducing the phase transition of Bloch braids. This is in contrast with the non-Bloch braids, where the EPs are fixed while the GBZs deform when modifying the boundary coupling or size, underlying the topological origin of non-Bloch braid phase transition. These distinct mechanisms are summarized in Fig. 1b. Similar results apply to non-Hermitian wave systems, as all our theories rely only on a tight-binding model. Nevertheless, despite Bloch braids have been widely measured in, for example, optical ring resonators12, acoustic cavities14,43, optomechanical systems13, and electric circuits47, experimentally detecting the non-Bloch braids and associated phase transitions resorts to the careful arrangement of boundary condition and size. These morphologies can be precisely manipulated in our static mechanical system, as outlined below.

Observation of the ultra-sensitive non-Bloch braids and phase transitions

Figure 5a sketches an assembled sample lattice having 5 (3) unit cells along the m (n) axis, with four additional nodes (purple shaded) pinned at the corners (34 nodes in total). The elastic struts connecting adjacent nodes are linear springs, where the stiffnesses \({k}_{j}\) are 1.6, 0.2, 0.4, and 0.4 N mm−1 for j = 1–4, respectively, in accordance with Figs. 3 and 4. The nodal displacements at \(n=0\) and 2 are preset while those at \(n=1\) are to be measured, which contain all information about a complex eigenvalue to be tested. The details of the sample construction and displacement measurement are provided in Methods. Therein, we also demonstrate the capability of such a finite-sized sample, with finite-valued spring constants, to measure the complete spectrum of a target lattice under arbitrarily small boundary couplings \({k}_{3,4}^{{{{\rm{b}}}}}\) with an arbitrarily large size M. The rationale is outlined in Fig. 5b, which relies on two aspects (Supplementary Note 8). The first is intrinsic to non-Hermitian non-Bloch bands1, given that a small boundary perturbation to a large-sized target lattice induces a large change in eigenstates. The second aspect is unique to static systems, where any GBC eigenvalue can be measured from the bulk displacement field upon applying a set of quasistatic boundary loads corresponding to the eigenstate of that eigenvalue. These two aspects together amplify the microscopic boundary perturbation on a large target system (marked by the hollow dots in Fig. 5c) to the macroscopic change of bulk displacements in a finite-sized subdomain (solid dots in Fig. 5c), eventually enabling the eigenvalue after perturbation being experimentally measurable.

Fig. 5: Observation of the non-Bloch braids and associated phase transitions.
figure 5

a The assembled lattice. Inset is a zoom-in view of the unit cell. b Rationale of the experiment. c Indexing the assembled finite-sized lattice (solid dots) in an arbitrarily large target lattice (hollow dots). d, e Photograph (d) and measured nodal displacement intensities (e) when operating at the EP \(\lambda =0\). f Phase diagram of the non-Bloch braids. Solid dots and lines are measured and analytically predicted eigenvalues. The parameters are followed from Figs. 3 and 4, where \({k}_{4}^{{{{\rm{b}}}}}\) varies from 0, \(1.73\times {10}^{-18}\), to 10−10 along the horizontal axis, while M varies from 40, 60 to 80 along the vertical axis. The non-Bloch braid evolves from the trivial unlink to the nontrivial unknot when increasing \({k}_{4}^{{{{\rm{b}}}}}\) or M, with the phase transition reached at the dashed line. The negative stiffness regime \({k}_{4}^{{{{\rm{b}}}}} < 0\) corresponds to mechanical instability and is not considered, as highlighted by the dashed horizontal axis. g, h Simulated (solid lines) and measured (dots) gap widths of two non-Bloch bands under the fixed \(M=60\) (g) and \({k}_{3}^{{{{\rm{b}}}}}={k}_{4}^{{{{\rm{b}}}}}=1.73\times {10}^{-18}\) (h). The error bar refers to the maximum deviation of ten measurements from the average, see “Methods”.

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We measure the topologically distinct non-Bloch braids and associated phase transitions in Figs. 3 and 4. Figure 5f shows the phase diagram of non-Bloch braids on the \({k}_{4}^{{{{\rm{b}}}}}-M\) plane. Solid dots are measured non-Bloch bands, agreeing well with the analytical solutions (solid lines). The isolation of the two strands of braids for small (i) boundary couplings or (ii) size and the nontrivial braids for large (iv) couplings or (v) size are observed. We note the results in (i) are indeed the observation of energy braid for OBC bands proposed in recent theoretical studies18,19, further corroborating the versatility of our platform. The non-Bloch braid phase transitions at the EP \(\lambda =0\) are also captured, i.e., (iii). For \(\lambda =0\), the loads applied at the boundary \(n=0\) are obstructed and all nodes at \(n=1\) are nearly unmoved, as visualized in the snapshot Fig. 5d and measured displacement intensities Fig. 5e. Figure 5g, h further report the measured gap widths between two non-Bloch bands, under the varying \({k}_{3}^{{{{\rm{b}}}}}={k}_{4}^{{{{\rm{b}}}}}\) (Fig. 5g) and M (Fig. 5h). We see the gap closes and reopens when crossing \({\bar{k}}_{4}^{{{{\rm{b}}}}}=1.73\times {10}^{-18}\) and \({M}_{0}=60\) (dashed lines), signifying the braid phase transition. Additional measurements for both the Bloch and non-Bloch braids, along with a higher-order non-Bloch braid phase transition from unlink to Hopf link, are given in Supplementary Note 7. We also demonstrate the repeatability and stability of experiments in Supplementary Note 8. Overall, the achieved minimum boundary coupling in experiments is ~10−22, far away from that for PBCs (i.e., ~100), while the maximum size approaches 80 cells (160 nodes). These extreme values highlight the ultra-sensitivity of non-Bloch braids and the superiority of our static platforms.

Conclusions

In conclusion, we have fulfilled the ultrahigh-precision manipulation of both the boundary condition and size in a non-Hermitian static mechanical lattice. The lower bound of the boundary couplings accessible in our experiments can be tens of orders of magnitude improved than current studies7,48, while the largest size surpasses 102. Based on the static lattice, we further demonstrate that the spectral sensitivity under GBCs has a nontrivial impact on the braid topology of fragile non-Bloch bands, where the phase transition threshold decreases exponentially with the size. Both the topologically distinct non-Bloch braids and associated phase transitions at extremely low boundary couplings in a large system are observed. With the capacity of precisely engineering the boundary condition and size, static mechanical systems can serve as a valuable platform for exploring highly susceptible non-Hermitian physics, including non-Abelian braiding of non-Bloch bands13,43,46, graph topology of non-Hermitian spectra49, and enhanced nonnormalities50.

Methods

Critical boundary couplings of non-Bloch braid phase transitions

Thanks to the chiral symmetry, the two non-Bloch bands merge and the non-Bloch braid phase transition occurs at \(\lambda =0\) even under GBCs. For \(\lambda =0\), all nodes at \(n=1\) are under the self-equilibrium states and are thus motionless. In this circumstance, the governing equations of bulk nodes at unit cell \((m,1)\) are given by

$$\left\{\begin{array}{c}{k}_{3}{u}_{m-1,0}^{(2)}+{k}_{1}{u}_{m,0}^{(2)}=0\\ {k}_{2}{u}_{m,0}^{(1)}+{k}_{4}{u}_{m+1,0}^{(1)}=0\end{array}\right.$$
(6)

from which we can set \({u}_{m,0}^{(1)}={a}_{1}{(-{k}_{2}/{k}_{4})}^{m}\) and \({u}_{m,0}^{(2)}={a}_{2}{(-{k}_{3}/{k}_{1})}^{m}\), with \({a}_{1}\) and \({a}_{2}\) the amplitudes. The equilibrium equations of boundary nodes, \({u}_{1,1}^{(1)}\) and \({u}_{M,1}^{(2)}\), are written as

$$\left\{\begin{array}{c}{k}_{3}^{{{{\rm{b}}}}}{u}_{M,0}^{(2)}+{k}_{1}{u}_{1,0}^{(2)}=0\\ {k}_{4}^{{{{\rm{b}}}}}{u}_{1,0}^{(1)}+{k}_{2}{u}_{M,0}^{(1)}=0\end{array}\right.$$
(7)

Substituting the expressions of \({u}_{m,0}^{(1)}\) and \({u}_{m,0}^{(2)}\) into Eq. (7), we have

$$\left\{\begin{array}{c}{k}_{3}^{{{{\rm{b}}}}}={(-1)}^{M}{k}_{1}{({k}_{1}/{k}_{3})}^{M-1}\\ {k}_{4}^{{{{\rm{b}}}}}={(-1)}^{M}{k}_{2}{({k}_{2}/{k}_{4})}^{M-1}\end{array}\right.$$
(8)

According to Eq. (8), we first set M as even to avoid a negative stiffness which, despite can be realized by using the bistable elements51,52,53, may induce mechanical instability and is thus not considered. For bulk parameters \({k}_{1} > {k}_{3}\) and \({k}_{2} < {k}_{4}\) (with nontrivial Bloch braid topologies), \({k}_{3}^{{{{\rm{b}}}}}\) increases exponentially with M and is physically inadmissible. This can only be resolved by imposing \({a}_{2}=0\), such that all sublattices-2 at \(n=0\) are motionless and the equilibrium of \({u}_{1,1}^{(1)}\) naturally holds. Therefore, we obtain the critical boundary couplings in Eq. (5), which impose no constraint on \({k}_{3}^{{{{\rm{b}}}}}\). Furthermore, a switch of bulk stiffnesses \({k}_{1}\leftrightarrow {k}_{2}\) and \({k}_{3}\leftrightarrow {k}_{4}\) reverses the handedness of Bloch braids, such that the critical boundary couplings of non-Bloch braid phase transitions are modified as \({k}_{3}^{{{{\rm{b}}}}}={\bar{k}}_{3}^{{{{\rm{b}}}}}={k}_{1}{({k}_{1}/{k}_{3})}^{M-1}\) with arbitrary \(0 < {k}_{4}^{{{{\rm{b}}}}} < {k}_{4}\).

Non-Hermitian spectral measurement based on static mechanical lattices

Below, we outline the procedure for spectral measurement based on the sample lattice in Fig. 5a. Crucially, we show that this finite-sized sample with finite-valued spring constants suffices to measure the complete spectrum of a target non-Hermitian lattice under the arbitrary boundary condition with an arbitrary size. We begin with a target lattice under a GBC, the equilibrium equation for the nth row of nodes (\(n > 0\)) is given by

$$[{K}_{1}]\{{u}_{n-1}\}+[{K}_{2}]\{{u}_{n+1}\}=[{K}_{0}]\{{u}_{n}\}$$
(9)

where \(\{{u}_{n}\}\) refers to the displacement vector of the 2 M nodes in the nth row. The stiffness matrices \([{K}_{j}]\) (j = 0–2) are expressed as

$$[{K}_{1}]={[{K}_{2}]}^{T}={\left[\begin{array}{ccccc} & {k}_{1} & & & {k}_{3}^{{{{\rm{b}}}}}\\ {k}_{2} & & {k}_{4} & & \\ & {k}_{3} & & \ddots & \\ & & \ddots & & {k}_{1}\\ {k}_{4}^{{{{\rm{b}}}}} & & & {k}_{2} & \end{array}\right]}_{2M\times 2M} [{K}_{0}]={\left[\begin{array}{ccccc}{k}_{{{{\rm{t}}}}}^{(1)} & & & & \\ & {k}_{{{{\rm{t}}}}} & & & \\ & & \ddots & & \\ & & & {k}_{{{{\rm{t}}}}} & \\ & & & & {k}_{{{{\rm{t}}}}}^{(2)}\end{array}\right]}_{2M\times 2M}$$
(10)

where \({k}_{{{{\rm{t}}}}}^{(1)}={k}_{1}+{k}_{2}+{k}_{3}+{k}_{4}^{{{{\rm{b}}}}}\) and \({k}_{{{{\rm{t}}}}}^{(2)}={k}_{1}+{k}_{2}+{k}_{4}+{k}_{3}^{{{{\rm{b}}}}}\). Applying the ansatz \(\{{u}_{n}\}=\lambda \{{u}_{n-1}\}\) to Eq. (9) yields \(([{K}_{1}]{\lambda }^{-1}+[{K}_{2}]\lambda -[{K}_{0}])\{{u}_{n}\}=0\), from which the eigenvalue \(\lambda\) and associated eigenstate \(\{{u}_{n}\}\) can be solved by imposing a vanishing determinant and finding the null space of the coefficient matrix. Under GBCs, the eigenvalue and eigenstate are generally complex, which are mechanically inadmissible since the nodal displacements should be real. To facilitate experimental measurement, we expand \(\{{u}_{n}\}\) as \({{{\mathrm{Re}}}}\{{u}_{n}\}+i{{{\rm{Im}}}}\{{u}_{n}\}\), and get two real equations from Eq. (9)

$$\left\{\begin{array}{c}[{K}_{1}]{{{\mathrm{Re}}}}\{{u}_{n-1}\}+[{K}_{2}]{{{\mathrm{Re}}}}\{{u}_{n+1}\}=[K]{{{\mathrm{Re}}}}\{{u}_{n}\}\\ \left[{K}_{1}\right]{{{\rm{Im}}}}\{{u}_{n-1}\}+\left[{K}_{2}\right]{{{\rm{Im}}}}\{{u}_{n+1}\}=[K]{{{\rm{Im}}}}\{{u}_{n}\}\end{array}\right.$$
(11)

Physically, Eq. (11) means that if we set the nodal displacements at \(n-1\) and \(n+1\) as \({{{\mathrm{Re}}}}\{{u}_{n-1}\}\) (\({{{\rm{Im}}}}\{{u}_{n-1}\}\)) and \({{{\mathrm{Re}}}}\{{u}_{n+1}\}\) (\({{{\rm{Im}}}}\{{u}_{n+1}\}\)), then the excited displacement field at n is exactly \({{{\mathrm{Re}}}}\{{u}_{n}\}\) (\({{{\rm{Im}}}}\{{u}_{n}\}\)). To experimentally test an eigenvalue \(\lambda\) based on Fig. 5a, we first employ \({{{\mathrm{Re}}}}\{{u}_{0}\}\) and \({{{\mathrm{Re}}}}\{{u}_{2}\}\) as the displacement loads. Specifically, we extract the 2nd to 13th elements of \({{{\mathrm{Re}}}}\{{u}_{0}\}\) (\({{{\mathrm{Re}}}}\{{u}_{2}\}\)) as the prescribed displacements on the 12 nodes at \(n=0\) (\(n=2\)), and measure the displacements of the 10 nodes at \(n=1\). These nodes are marked by the solid dots in Fig. 5c, corresponding to a subset of an arbitrarily large target lattice (hollow dots). According to the ansatz, we have \({{{\mathrm{Re}}}}\{{u}_{1}\}={\lambda }_{{{{\rm{R}}}}}{{{\mathrm{Re}}}}\{{u}_{0}\}-{\lambda }_{{{{\rm{I}}}}}{{{\rm{Im}}}}\{{u}_{0}\}\), where \(\lambda ={\lambda }_{{{{\rm{R}}}}}+i{\lambda }_{{{{\rm{I}}}}}\). Similar analysis holds when the applied loads at \(n=0\) and 2 are selected from the 2nd to 13th elements of \({{{\rm{Im}}}}\{{u}_{0}\}\) and \({{{\rm{Im}}}}\{{u}_{2}\}\), for which the excited displacements at \(n=1\) should be \({{{\rm{Im}}}}\{{u}_{1}\}={\lambda }_{{{{\rm{R}}}}}{{{\rm{Im}}}}\{{u}_{0}\}+{\lambda }_{{{{\rm{I}}}}}{{{\mathrm{Re}}}}\{{u}_{0}\}\). Because \({{{\mathrm{Re}}}}\{{u}_{0}\}\) and \({{{\rm{Im}}}}\{{u}_{0}\}\) are known while \({{{\mathrm{Re}}}}\{{u}_{1}\}\) and \({{{\rm{Im}}}}\{{u}_{1}\}\) have been measured, then each (associated with a node at \(n=1\)) out of the ten components in the former vector expression about \({{{\mathrm{Re}}}}\{{u}_{1}\}\) or \({{{\rm{Im}}}}\{{u}_{1}\}\) contributes a linear equation about \({\lambda }_{{{{\rm{R}}}}}\) and \({\lambda }_{{{{\rm{I}}}}}\). Finally, the complex eigenvalue can be extracted from any out of the ten pairs of equations, i.e., \(({\lambda }_{{{{\rm{R}}}}}^{(j)},{\lambda }_{{{{\rm{I}}}}}^{(j)})\) with j = 1,2…10. These solutions should be equal, which can be an evaluation of the experiment.

Experimental setup

The assembled lattice in Fig. 5a is a stretch-dominated truss, with the lattice constants along the m and n axes being 80 mm and 50 mm. The elastic struts linking adjacent nodes are hinged and are stacked along the out-of-plane axis to prevent self-crossing. There are linear springs embedded in the struts and are elaborately designed to avoid buckling. Each column of nodes (i.e., with common cell index m and sublattice index 1 or 2) are suspended on a pair of sliders aligned along the n axis to eliminate redundant motional degrees-of-freedom. The end of sliders threads the holes punched on a pair of wing plates placed adjacent to the baseboard.

For given eigenvalue \(\lambda\) to be measured, the prescribed displacements at \(n=0\) and \(n=2\) of the sample are applied by inserting the cylinder-shaped nodes into the blind holes pre-drilled in the baseboard. Upon choosing a gauge for the complex vector \(\{{u}_{0}\}\), the displacement loads at \(n=2\) can be resolved

$$\left\{\begin{array}{c}{{{\rm{Re}}}}\{{u}_{2}\}=({\lambda }_{{{{\rm{R}}}}}^{2}-{\lambda }_{{{{\rm{I}}}}}^{2}){{{\rm{Re}}}}\{{u}_{0}\}-2{\lambda }_{{{{\rm{R}}}}}{\lambda }_{{{{\rm{I}}}}}{{{\rm{Im}}}}\{{u}_{0}\}\\ {{{\rm{Im}}}}\{{u}_{2}\}=({\lambda }_{{{{\rm{R}}}}}^{2}-{\lambda }_{{{{\rm{I}}}}}^{2}){{{\rm{Im}}}}\{{u}_{0}\}+2{\lambda }_{{{{\rm{R}}}}}{\lambda }_{{{{\rm{I}}}}}{{{\rm{Re}}}}\{{u}_{0}\}\end{array}\right.$$
(12)

The nodal displacements at \(n=1\) are measured via the digital-image-correlation (DIC) method. An isolated node anchored on the upper-left of the sample is employed to calibrate the DIC method. Only a partial of the 2 M eigenvalues are measured. The results reported in Fig. 5f, g, and h are extracted by averaging the ten bulk nodes in a single experiment, i.e., \(\bar{\lambda }={\bar{\lambda }}_{{{{\rm{R}}}}}+i{\bar{\lambda }}_{{{{\rm{I}}}}}\) with \({\bar{\lambda }}_{{{{\rm{R}}}},{{{\rm{I}}}}}={\sum }_{j=1}^{10}{\lambda }_{{{{\rm{R}}}},{{{\rm{I}}}}}^{(j)}/10\). The gap width in Fig. 5g and h is defined as the separation between two non-Bloch bands on the complex \(\lambda\) plane when \(\bar{q}=\pi\), with the error bar indicating the maximum deviation from the average. In this study, we enforce the 10 bulk nodes at \(n=1\) to form 5 complete unit cells, meaning that the left (right) two corner nodes at \(n=0\) and 2 correspond to the sublattice-2 (sublattice-1) in the target lattice, as schematically shown in Fig. 5c. In principle, the displacement loads exerted on the 12 nodes at \(n=0\) (\(n=2\)) can be extracted from any \(2l\) th to \(2l+11\) th elements of \({{{\mathrm{Re}}}}\{{u}_{0}\}\) (\({{{\mathrm{Re}}}}\{{u}_{2}\}\)) or \({{{\rm{Im}}}}\{{u}_{0}\}\) (\({{{\rm{Im}}}}\{{u}_{2}\}\)), where l is a positive integer. Finally, the number of bulk nodes at \(n=1\) necessary for measuring a complex eigenvalue can be reduced, which is beneficial for the miniaturization of the device.