The competition between order and disorder is a fundamental problem in condensed-matter physics, which directly impacts many different systems, such as crystalline solids7,11, electronic or magnetic arrangements12, localization in metals and superconductors, or vortex lattices in superconductors and condensates4,13. In 2D systems, long wavelength fluctuations induce deviations in the atomic positions from the perfect lattice, with the mean-squared displacement diverging logarithmically at large distances14. One major consequence is the so-called Mermin–Wagner–Hohenberg (MWH) theorem14,15, which states that no true order exists in 2D systems at any finite temperature. Usually, we can distinguish between static quenched disorder and fluctuations. In the absence of quenched disorder, thermal fluctuations drive the 2D melting transition which is described by Berezinskii–Kosterlitz–Thouless–Halperin–Nelson–Young (BKTHNY) theory through the two-stage proliferation and unbinding of topological defects9,10,16,17. Quenched disorder, on the other hand, is expected to suppress long-range correlations more effectively than temperature18. It can be classified as pinning with identifiable length scales, such as impurities or defects in 2D crystals, or as scale-invariant (random) disorder, as for example in an amorphous film. Pinning destroys long-range 2D correlations at any strength19,20. Scale-invariant disorder produces power-law decaying correlations and a transition to a disordered lattice with exponentially falling correlations above a critical disorder strength6,7. The order–disorder transition induced by scale-invariant disorder has been investigated in a wide range of physical systems, such as 2D disordered XY models6, 2D solids7, Josephson junction arrays21, colloids or Lennard-Jones systems22. However, the disorder mechanism—the way in which disorder proliferates at zero temperature—has not been observed directly. Disorder-induced order has been recently proposed when quenched disorder breaks the continuous 2D symmetry, for example, by introducing a 1D periodic disorder potential3. Within this scenario, true long-range order may be favoured by the 1D disorder, breaking the MWH theorem. Calculations show the stabilization of the quantum Hall ferromagnetic state in graphene monolayers due to strain-induced easy-plane anisotropy23 or improved control of the relative phase in randomly coupled condensates24. The experimental realization of such a disorder-induced order in the absence of thermal fluctuations has not yet been reported. The effect of symmetry breaking on microscopic properties and the critical exponents of the order–disorder transition are unknown.

Here we address these questions by directly imaging the order–disorder transition in a 2D vortex lattice induced by a 1D periodic potential using scanning tunnelling microscopy at 0.1 K. By changing the magnetic field, we modify the coupling strength between the 1D periodic potential—produced by a surface corrugation with period w—and the vortex lattice, as well as the intervortex distance a0 = (3/4)1/4(ϕ0/B)1/2 (see Methods and Supplementary Information for a detailed description of sample preparation and the experimental procedure). This allows us to go from a locked 2D solid, where the lattice is commensurate with the 1D potential (Fig. 1a, left), to a floating 2D solid at larger densities, where it becomes incommensurate with the 1D modulation (Fig. 1a, right).

Figure 1: Unlocking of the 2D vortex lattice on a linear potential.
figure 1

a, Sketch of a commensurate arrangement locked to the 1D potential for small p (left) and an unlocked lattice producing a floating solid incommensurate with the 1D modulation for large p (right). b, Scanning electron microscopy image of the sample (left). STM topography of a 1 × 1.2 μm2 area (right) with the profile along the blue dashed line given below. Red dashed lines indicate the 1D modulation. c, Sequence of vortex lattice images taken at 0.1 K with increasing magnetic field up to 0.5 T (see Supplementary Movie 1 for the entire sequence). Red dashed lines indicate the 1D modulation and vortices are shown as black points. Blue lines are Delaunay triangulation. d, Dependence on magnetic field of the angle between the 1D modulation and the vortex lattice. Lines joining points are guides to the eye. When p < 5 (H < 0.4 T; blue background), the vortex lattice oscillates between the two primary commensurate arrangements θ = 0° (green dotted line) and θ = 30° (yellow dotted line). This satisfies, respectively, and w = ma0, with n and m integers (the n = 1 and m = 1 cases are highlighted in the upper (yellow) and lower (green) right panels of d, whereas the n = 1 and m = 4 commensurate arrangements are identified in the vortex images of c at respectively 0.05 T and 0.25 T). At fields above 0.4 T (w > p; khaki coloured in c and d), the lattice unlocks and adopts an orientation independent of the 1D potential.

The coupling strength between the vortex lattice and the 1D modulation depends on the commensurability ratio p, defined as p = w/a0, and θ the relative orientation between them (ref. 25). Figure 1c shows a sequence of vortex lattice images obtained at lower magnetic fields. Below 0.4 T, with p 5, the lattice suffers a series of commensurate to incommensurate transitions that produces a rotation of its overall orientation between θ = 0° and 30°, while maintaining a perfect hexagonal arrangement26. Figure 1d shows θ as a function of p for vortex lattice images taken with increasing magnetic field. On increasing p above 0.4 T (p 5), the rotation of the vortex lattice ceases and its orientation becomes independent of the 1D potential. The lattice is incommensurate with the 1D potential and forms a floating 2D solid.

In Fig. 2a we show a sequence of vortex lattice images between 0.5 T and 5.5 T. We identify three different phases. In phase I, between 0.5 T and 2 T, there are no topological defects, and all vortices are surrounded by six nearest neighbours. However, the vortex positions show small deviations from those expected for a perfect hexagonal lattice, which become gradually more pronounced on increasing the magnetic field. Between 2.5 T and 4.5 T, in phase II, we observe the appearance of dislocations, that is pairs of five-fold and seven-fold coordinated vortices. We identify bound dislocation pairs as well as isolated dislocations. Above 4.5 T, in phase III, the density of dislocations increases strongly and we identify the appearance of free disclinations in the form of isolated five-fold or seven-fold coordinated vortices. The images at 5 T and 5.5 T show that defects exist over the whole sample, producing a disordered vortex lattice.

Figure 2: Order–disorder transition in the 2D vortex lattice at 0.1 K.
figure 2

a, Vortex lattice images taken at 0.1 K on increasing the magnetic field from 0.5 T to 5.5 T (see Supplementary Movie 2). Vortices are shown by black points and the Delaunay triangulated lattice as blue lines. Vortices with five and seven nearest neighbours are identified by green and orange points. Dislocations formed by five and seven nearest neighbour pairs of vortices are identified by black triangles, pairs of dislocations by black rectangles and isolated disclinations by black circles. Above 4 T, the image size is decreased to better resolve individual vortices. b, Positional fluctuations of the vortex density during the disorder process calculated from the vortex images shown in a. Colour scale is given at the top right (see Section III.3 in Supplementary Information). Deviations in the local vortex density become stronger with increasing magnetic field. c, Histograms of the vortex density obtained from the images in b using the same colour scale.

One important observation—the appearance of fluctuations in the local vortex density ρ—is shown in Fig. 2b, c. The standard deviation in ρ increases with the field-induced proliferation of defects—from less than 5% in phase I to up to 20% at 5.5 T in phase III (Fig. 2c). Density fluctuations are characteristic of a long wavelength or fully uncorrelated quenched disorder potential27.

To further quantify the spatial dependence of vortex disorder, we calculate the translational and orientational correlation functions, GK(r) and G6(r) (Methods and Supplementary Information). Figure 3a shows the evolution of GK(r) and G6(r) with the magnetic field. In phase I, between 0.5 T and 2 T, we observe that G6(r) remains close to 1 and is independent of the distance, whereas GK(r) decays following a power-law dependence, GK(r) r η K , with ηK increasing with field. Above 2 T, in phase II, GK(r) decays exponentially at large distances when ηK = 1/3, when a finite amount of defects starts to be observed in the images. Here, G6(r) shows a power-law decay, G6(r) r η 6 , with η6 increasing continuously from 2 T up to 4.5 T. In phase III, above 5 T, G6(r) decays exponentially when η6 = 1/4, and the defect density diverges—reaching 0.4—indicating that nearly half of all vortices have five or seven nearest neighbours at 5.5 T (Fig. 3b). The observed behaviour follows the microscopic two-step sequence for the proliferation of disorder described by BKTHNY theory, with critical values for the exponents ηKc = 1/3 and η6c = 1/4 (ref. 28).

Figure 3: Correlation functions and critical exponents.
figure 3

a, Field-induced changes in the correlation functions for translational order GK(r) (left) and orientational order G6(r) (right) between 0.5 T and 5.5 T. Blue lines in the graphs are the fits for the envelopes of the correlation functions (black lines). Green (left) and magenta (right) lines stand for power decay laws with, respectively, the critical exponents ηKc = 1/3 and η6c = 1/4. The background colour (yellow for GK(r) and green for G6(r)) represents the range of the order and turns from dark (long-range) to light (short-range) when the exponents ηK and η6 become higher than the critical values and the correlation functions change from a power law to exponential decay with the distance r. b, (Top) Field dependence of the power-law decay exponents ηK (green points) and η6 (magenta points). (Bottom) Density of five-fold and seven-fold coordinated vortices as a function of the magnetic field. c, Correlation function B(r)/a02 at 1.5 T (black dots, see text and Supplementary Information for details). Red line is ≈ (3ηK/8π2)ln(r/a0), obtained within the Gaussian approximation for a scale-invariant random potential (see text and Supplementary Section VI). Blue line gives the ln2(r) dependence expected within correlated pinning disorder. d, Spatial dependence of the mean-squared correlation of the 1D surface potential 〈[V1D(r) − V1D(r′)]2〉 at 1.2 T, where V1D(r) is as defined in the text and r, r′ are the actual vortex positions obtained from the vortex image at 1.2 T (see Section III.5 in Supplementary Information for details of the calculations). The red line was obtained by simulating the topography with a periodic modulation and extrapolating the vortex position in an area 100 times larger than in the experiment (see Supplementary Information). The blue line is the fit to equation (1) in the main text. e, Magnetic field dependence of Eel (black line) and Edis (blue and green circles). Edis (blue dashed line) is estimated independently from power-law exponents ηK in positional correlation (green circles) and long-range logarithmic correlations of V1D(r) (blue circles).

To investigate the microscopic disorder mechanism, we further analyse the first entrance of disorder in the ordered phase I. Deviations in the vortex positions with respect to a perfect hexagonal lattice can be quantified by the relative displacement correlator B(r) (given by B(r) = 〈[u(r) − u(0)]2〉/2, where u(r) = rrp is the displacement of each vortex at r from its position in the perfect lattice rp; see ref. 27). We find that B(r) grows as ln(r/a0) in the dislocation-free Phase I. In Fig. 3c we plot the result at 1.5 T. In 2D systems, this is the expected behaviour in response to a scale-invariant disorder19. Next, we fit the data using the Gaussian approximation GK(r) = e - K 2 B ( r ) / 2 (valid for Gaussian disorder potentials) and the translational correlation function GK(r) (shown in Fig. 3a). A comparison reveals a very good agreement (Fig. 3c). Therefore, all three independent observations (local vortex density fluctuations, logarithmic growth of B(r), and a Gaussian distribution of displacements) show that a random potential drives the transition.

We next focus on the source of scale-invariant disorder driving the transition. No thermally induced or quantum-induced fluctuations are available to effectively disorder the vortex lattice here. At 0.1 K, the transition induced by either thermal or quantum fluctuations is expected to occur at a magnetic field extremely close to Hc2 (see Supplementary Information). Our sample is amorphous and compositionally homogeneous, both laterally and across its thickness, thus, there is no quenched disorder from compositional or structural changes. Instead, thickness variations given by the 1D modulation emerge as the only source for quenched disorder. The fundamental property of a scale-invariant potential V (r) is that it has long-range logarithmic correlations6

where J = μda02/2π is the elastic interaction strength (the magnetic field dependence of the shear modulus μ is discussed in the Supplementary Information) and σ is the disorder strength. In Fig. 3d we calculate the spatial correlations of V (r) (the first term in equation (1)) by taking V1D(r) = z(r)ɛL, where z(r) is the topography and ɛL = (ϕ02/4πμ0λ2)ln(λ/ξ) is the vortex energy per unit length (see Supplementary Information for details). We find that V1D(r) has long-range logarithmic correlations and short-range smooth periodic correlations at integer multiples of w, which are strongly damped at large distances. Thus, incommensurate 1D correlation behaves as a quasi-random disorder potential.

We can write the free energy, F, following available renormalization group (RG) theory for random disorder as6,7:

where the thermal energy Eth, elastic energy Eel, and disorder energy Edis (first to third terms, respectively) diverge logarithmically with the system size L (ref. 6, 7). Here, σc is the critical value for the disorder strength. In the ordered phase at low temperatures, the relative strength between Eel and Edis determines the algebraical decay of the translational correlations, with the exponent ηK in a hexagonal solid given by19,21,

Following the transition from power law to exponential decay in GK(r), we found ηKc = 1/3 (Fig. 3a), which corresponds to σc = 1/2. We now calculate the value of Edis produced by V1D(r) (using equations (1) and (2)) and, independently, from the power-law decay of the positional correlation functions (using equations (2) and (3) with the ηK values from Fig. 3b). The results obtained by the two methods are plotted in Fig. 3e (as blue and green circles, respectively) together with Eel (black line) versus the magnetic field. Note that Eth at 0.1 K is three orders of magnitude smaller than both Eel and Edis, so it is negligible here. The agreement between Edis determined from the exponents in the correlation functions ηK (green circles) and from logarithmic correlations in the 1D disorder potential (blue circles) is almost perfect. This demonstrates that the 1D surface corrugation produces the disorder through incommensuration and provides the energy scale for the random field driving the transition in the 2D lattice. Figure 3e shows that Edis begins to exceed Eel at the magnetic field where we start to observe dislocations in the images.

Finally, let us discuss the critical behaviour of the observed transitions. Our experiments closely reproduce the expected features for the zero-temperature phases as induced by random disorder, with, in particular, positional fluctuations which increase as ln(r) below the critical disorder and correlations that decay exponentially for high disorder strengths. But here we find critical values, σc = 1/2 and ηKc = 1/3, which are far above those proposed in theory on the basis of RG calculations and models for random quenched disorder (σc = 1/8 and ηKc = 1/12; refs 6, 7, 18, 29). The difference between random field theories and our experiments is the presence of symmetry-breaking 1D modulation. A recent proposal shows that the XY model with 1D symmetry-breaking disorder has an increased order parameter at all temperatures3. An earlier work also points out that correlations in the disorder potential enhance the critical value of σ (ref. 21). This strongly suggests that the 1D modulation, by breaking symmetry, modifies the screening of the interactions among dislocations to enhance the critical point and exponents with respect to random field theories.

Our experiments show that, in the presence of the 1D symmetry-breaking disorder, the critical exponents increase to the values expected by BKTHNY theory and the microscopic disordering behaviour follows the sequence defined by the two-step thermal melting transition, suggesting that this route for creating disorder possibly describes more phenomena than just 2D thermal melting. Inherent to this is the presence both of an intermediate hexatic phase and bound dislocations in the ordered phase which are not expected within random field models29,30. The question is: why does our experiment follow BKTHNY theory? To answer this question, one needs to calculate new critical points of the order–disorder transition at zero temperature by taking into account symmetry-breaking correlations within randomness and their influence on the renormalization of the parameters involved in the transition.

Overall, our data represent the first evidence that incommensurate 1D modulation widens the stability range of the ordered phase in 2D systems at zero temperature and raise questions that will motivate a detailed examination of the effect of correlations in the critical behaviour of disordered systems. 2D random environments are usually unavoidable in different fields, such as colloids, optical lattices, quantum condensates, 2D crystals or graphene. The experimental approach presented here reveals an exciting new opportunity to produce coherence in the presence of 1D symmetry-breaking fields, as for example nematicity.



Our sample is an ultraflat amorphous thin film with a thickness, d, of 200 nm and a lateral size of 100 μm, fabricated by focused-ion-beam-induced deposition (microscopy and nanofabrication methods are described in detail in sections I and II of the Supplementary Information). d is far below Lc, the characteristic length for vortex bending along the field direction; thus the vortex lattice forms a 2D solid. The surface roughness is less than 1% of d and consists of a smooth 1D modulation with period w = 400 nm (Fig. 1b).

Low-temperature STM/S vortex imaging.

We use scanning tunnelling microscopy/spectroscopy (STM/S) to directly determine modifications in the spatial correlations induced by disorder in the vortex lattice and visualize the microscopic details of the ordered and disordered phases. The STM is carried out in a dilution refrigerator and we work at temperatures low enough (0.1 K) to neglect any temperature-induced effects. The sample is biased through W-based contact pads, as shown in Fig. 1b (see Supplementary Information). The order–disorder transition was followed by imaging up to thousands of individual vortices from fields of 0.01 T up to just below Hc2, with the vortex density increasing by a factor of 500. The average intervortex distance a0 decreases with field, following the expected dependence in a triangular vortex lattice (shown in Section III.3 of Supplementary Information).

Locked and floating vortex lattice.

We modify the interaction strength between the vortex lattice and the underlying disorder potential by increasing the magnetic field (see Section V in Supplementary Information). At low magnetic fields, when the lattice constant and disorder wavelength are similar (small p), commensurate vortex configurations are observed, as expected, at matching conditions p = n or , with n an integer, and θ = 30° or 0° degrees, respectively26. Generally, commensurate lattices locked to the 1D potential are favoured, because they lower the elastic energy of the lattice (Fig. 1). At higher fields, when the lattice constant is much smaller than the 1D modulation, the increase in elastic energy obtained by adjusting to the potential decreases and the lattice can show incommensurate configurations forming a floating solid (Figs 1 and 2).

Calculation of the correlation functions of the vortex images.

The order–disorder transition has been characterized in real space through the calculation of translational and orientational correlation functions, GK(r) and G6(r), vortex density fluctuations ρ(r), and the relative displacement correlator B(r) (see details for the calculation in Sections III.2, III.3 and VI of Supplementary Information). GK(r) and G6(r) are directly obtained from the individual vortex positions in the images. Peaks in the correlation functions appear at the distances to nth nearest neighbours. Deviations with respect to the perfect lattice produce a decay with r of the envelopes of GK(r) and G6(r) that describes, respectively, weakening of translational and orientational correlations. The crossovers between phases I (yellow), II (green) and III (magenta) are determined as the fields at which the translational and orientational order become short range and the exponents ηK and η6 reach the critical values (dotted lines in Fig. 3b, see text for further details). The relative displacement correlator B(r) is calculated by minimizing the averaged relative deviation between the experimental vortex positions and the simulated perfect hexagonal lattice (see Section VI in Supplementary Information for details of the calculations and B(r) at different fields in Phase I). The evolution of the disordering process in reciprocal space is shown in Section IV of the Supplementary Information through the gradual changes of the height and width of the vortex lattice Bragg peaks.