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Resources tagged with: Symmetry

There are 71 NRICH Mathematical resources connected to Symmetry , you may find related items under Transformations and constructions .

Always, Sometimes or Never? Shape

Are these statements always true, sometimes true or never true?

Use the information on these cards to draw the shape that is being described.

National Flags

This problem explores the shapes and symmetries in some national flags.

Poly Plug Pattern

Create a pattern on the small grid. How could you extend your pattern on the larger grid?

Exploded Squares

This practical activity challenges you to create symmetrical designs by cutting a square into strips.

Attractive Rotations

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Counters in the Middle

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

Reflector ! Rotcelfer

Can you place the blocks so that you see the reflection in the picture?

Coordinate Challenge

Use the clues about the symmetrical properties of these letters to place them on the grid.

Stringy Quads

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

Octa-flower

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?

Symmetry Challenge

How many symmetric designs can you make on this grid? Can you find them all?

Shady Symmetry

How many different symmetrical shapes can you make by shading triangles or squares?

Reflecting Squarely

In how many ways can you fit all three pieces together to make shapes with line symmetry?

A Cartesian Puzzle

Find the missing coordinates which will form these eight quadrilaterals. These coordinates themselves will then form a shape with rotational and line symmetry.

Attractive Tablecloths

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

Witch of Agnesi

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

Folium of Descartes

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

Logosquares

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

Mean Geometrically

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

Colouring Triangles

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

Symmetric Trace

Points off a rolling wheel make traces. What makes those traces have symmetry?

Building Patterns

Can you deduce the pattern that has been used to lay out these bottle tops?

Emmy Noether

Find out about Emmy Noether, whose ideas linked physics and algebra, and whom Einstein described as a 'creative mathematical genius'.

Watch Those Wheels

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!

What mathematical words can be used to describe this floor covering? How many different shapes can you see inside this photograph?

Plot the graph of x^y = y^x in the first quadrant and explain its properties.

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

Symmetrical Semaphore

Someone at the top of a hill sends a message in semaphore to a friend in the valley. A person in the valley behind also sees the same message. What is it?

Hidden Meaning

What is the missing symbol? Can you decode this in a similar way?

Pattern Power

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

Two Triangles in a Square

Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.

Some local pupils lost a geometric opportunity recently as they surveyed the cars in the car park. Did you know that car tyres, and the wheels that they on, are a rich source of geometry?

Tournament Scheduling

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

Classifying Solids Using Angle Deficiency

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

The Frieze Tree

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Frieze Patterns in Cast Iron

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.

A red square and a blue square overlap. Is the area of the overlap always the same?

Square Pizza

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Prime Magic

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

Rhombicubocts

Each of these solids is made up with 3 squares and a triangle around each vertex. Each has a total of 18 square faces and 8 faces that are equilateral triangles. How many faces, edges and vertices does each solid have?

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.

Eight Dominoes

Using the 8 dominoes make a square where each of the columns and rows adds up to 8

A Problem of Time

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

Maltese Cross

Sketch the graph of $xy(x^2 - y^2) = x^2 + y^2$ consisting of four curves and a single point at the origin. Convert to polar form. Describe the symmetries of the graph.

Sketch the graphs for this implicitly defined family of functions.

Find the shape and symmetries of the two pieces of this cut cube.

This activity investigates how you might make squares and pentominoes from Polydron.

Reflection and Symmetry

Two lessons on reflection:

An introduction which focuses on symmetry and reflecting simple shapes with a fully differentiated main task.

The second lesson is more advanced and features linear graphs as lines of symmetry.   Extension task is credit of TES user TristanJones.

Line Symmetry Practice Questions

Click here for questions, click here for answers, gcse revision cards.

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Symmetry Worksheets Line Symmetry Easier

Welcome to the Math Salamanders Line Symmetry Worksheets page. Here you will find a range of free printable symmetry worksheets, which will help your child to practice their reflecting and flipping skills.

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Symmetry Help

The Math Salamanders have a large bank of free printable symmetry worksheets. Each symmetry sheet comes complete with answers for support.

Handy Hints

Each point or block that has been reflected must remain the same distance from the mirror line as the original point. So if point A is 3 squares away from the mirror line, then the reflection of point A must also be 3 squares away.

When reflecting a shape, look at the corners of the shape and reflect each corner first as a dot in the mirror line. The dots can then be joined up (in the correct order!)

For lines of symmetry at angles of 45°, it is often better to rotate your paper so that the line of symmetry is vertical or horizontal, and the rest of the paper is at an angle.

The basis and understanding of symmetry starts at about Grade 2, and then develops further in Grades 3,4 and 5.

Line Symmetry Worksheets

On this webpage you will find our range of line symmetry sheets for kids.

The sheets have been carefully graded with the easier sheets coming first. The first 3 worksheets involve only horizontal and vertical lines only. The next 3 worksheets involve reflecting diagonal lines as well.

There are also some templates at the end of this section for you to create your own shapes for your child to reflect, or, even better, for your child to create their own symmetric patterns!

Using these sheets will help your child to:

• learn to reflect a shape in a vertical or horizontal mirror line;
• learn to reflect a shape in both a vertical and horizontal mirror line;

Reflecting in 1 mirror line

Horizontal and vertical lines only.

• Line Symmetry 1
• PDF version
• Line Symmetry 2
• Line Symmetry 3
• Line Symmetry 4
• Line Symmetry 5
• Line Symmetry 6

Reflecting in Diagonal mirror lines

• Line Symmetry 9
• Line Symmetry 10

Reflecting in vertical, horizontal and diagonal lines

• Line Symmetry 11

Reflecting in 2 mirror lines

• Line Symmetry 7
• Sheet 7 Answers
• Line Symmetry 8
• Sheet 8 Answers

Reflecting in vertical and horizontal or 2 diagonal lines

• Line Symmetry 12

Line Symmetry templates

• Line Symmetry Template 1
• Line Symmetry Template 2
• Line Symmetry Template 3
• Line Symmetry Template 4
• Line Symmetry Template 5
• Line Symmetry Template 6

Looking for something harder?

Here you will find a range of line symmetry activity sheets with one or two mirror lines.

The sheets in this section are similar to those on this page, but are more complicated and at a harder level.

• Harder Symmetry Activities

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Block Symmetry Worksheets

Here you will find a range of symmetry worksheets reflecting blocks instead of lines.

These sheets are at an easier level than the ones on this page.

• Symmetry Worksheets - Block Symmetry
• Explore 2d Shapes Worksheets

Looking for some geometry worksheets to get children thinking and reasoning about 2d shapes?

The shapes on this page are all about children really understanding what 2d shapes are all about, and using their reasoning skills to justify their thinking.

• know the properties of a range of 2d shapes;
• recognise that some shapes can also be described as being other shapes; e.g. a square is also a rhombus;
• recognise and understand right angles, parallel lines, lines of symmetry;
• develop their geometric reasoning skills.

Coordinate Sheets

Here is our collection of printable coordinate plane grids and coordinate worksheets.

Using these fun coordinate sheets is a great way to learn math in an enjoyable way.

• plot and write coordinates.
• Coordinate Plane Grid templates
• Coordinate Worksheets (1st Quadrant)
• Coordinate Plane Worksheets (All 4 Quadrants)

Captain Recommends

Have a look at these online symmetry games - a great way to learn symmetry and get instant feedback!

• Softschools Symmetry Game
• Sheppard Software Symmetry Activities

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Lines of Symmetry Worksheet

FREE DOWNLOAD

Help your students prepare for their Maths GCSE with this free lines of symmetry worksheet of 34 questions and answers

• Section 1 of the lines of symmetry worksheet contains 27 skills-based lines of symmetry questions, in 3 groups to support differentiation
• Section 2 contains 4 applied lines of symmetry questions with a mix of worded symmetry activities and deeper problem solving questions
• Section 3 contains 3 foundation and higher level GCSE exam style lines of symmetry questions 
• Answers and a mark scheme for all lines of symmetry questions are provided
• Questions follow variation theory with plenty of opportunities for students to work independently at their own level
• All questions created by fully qualified expert secondary maths teachers
• Suitable for GCSE maths revision for AQA, OCR and Edexcel exam boards

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Raise maths attainment across your school with hundreds of flexible and easy to use GCSE maths worksheets and lessons designed by teachers for teachers.

Lines of symmetry at a glance

A shape has a line of symmetry if a line can be drawn through the shape so that each half of the shape is an exact mirror image of the other half (identical halves). Some 2D shapes have multiple lines of symmetry, some have one and others don’t have any.

We might be asked to draw lines of symmetry on different shapes or to identify the number of lines of symmetry a polygon may have. We could also be asked to create symmetrical shapes by adding sections or shading parts of a diagram. 

Regular polygons have a number of lines of symmetry equal to their number of sides. For example, an equilateral triangle has three lines of symmetry and a regular hexagon has six lines of symmetry. Drawing a symmetric figure in different orientations will not change its lines of symmetry. 

Usually diagrams are placed on grids or a set of xy axes. 

Vertical lines of symmetry can be labelled as x=a where a is an integer or a decimal.

Horizontal lines of symmetry can be labelled as y=b where b is an integer or a decimal.

Two types of shape symmetry include line symmetry and rotational symmetry.

Looking forward, students can then progress to additional symmetry worksheets and other geometry worksheets , for example a simplifying expressions worksheet or simultaneous equations worksheet .

For more teaching and learning support on Geometry our GCSE maths lessons provide step by step support for all GCSE maths concepts.

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Year 4 Lines of Symmetry Maths Challenge

This Year 4 Lines of Symmetry Challenge checks children’s understanding of finding lines of symmetry. Children will help Matt to select the correct shapes to find a given number of lines of symmetry.

If you would like to access  additional resources which link to this maths challenge, you can purchase a subscription for only £5.31 per month on our sister site, Classroom Secrets .

Teacher Specific Information

This Year 4 Lines of Symmetry Challenge checks pupils’ understanding of finding lines of symmetry. Pupils will help Matt to select the correct shapes to find a given number of lines of symmetry.

National Curriculum Objectives

Properties of Shapes

Mathematics Year 4: (4G2b) Identify lines of symmetry in 2-D shapes presented in different orientations

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Outstanding Maths Lesson for Interview/Lesson Observation Problem Solving - Years 5 and 6

Subject: Mathematics

Age range: 7-11

Resource type: Lesson (complete)

Last updated

10 May 2024

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An exceptional maths problem solving lesson, complete with written lesson plan and interactive slides. Crafted to engage and challenge high-ability Year 5/6 students, this lesson offers a rich tapestry of activities and tasks designed to ignite mathematical thinking.

Perfect for a lesson observation or job interview, these comprehensive materials will impress all observers and demonstrate your expertise in delivering high-quality maths education.

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• Published: 15 May 2024

Wavefunction matching for solving quantum many-body problems

• Serdar Elhatisari   ORCID: orcid.org/0000-0002-7951-1991 1 , 2 ,
• Lukas Bovermann   ORCID: orcid.org/0000-0001-7765-1643 3 ,
• Yuan-Zhuo Ma   ORCID: orcid.org/0000-0002-0892-4457 4 , 5 ,
• Evgeny Epelbaum   ORCID: orcid.org/0000-0002-7613-0210 3 ,
• Dillon Frame 6 , 7 ,
• Fabian Hildenbrand 6 , 7 ,
• Myungkuk Kim 8 ,
• Youngman Kim 8 ,
• Hermann Krebs 3 ,
• Timo A. Lähde   ORCID: orcid.org/0000-0003-3251-1035 6 , 7 ,
• Dean Lee   ORCID: orcid.org/0000-0002-3630-567X 4 ,
• Ning Li 9 ,
• Bing-Nan Lu 10 ,
• Ulf-G. Meißner   ORCID: orcid.org/0000-0003-1254-442X 2 , 6 , 7 , 11 ,
• Gautam Rupak 12 ,
• Shihang Shen   ORCID: orcid.org/0000-0002-8051-6466 6 , 7 ,
• Young-Ho Song   ORCID: orcid.org/0000-0003-0361-3251 13 &
• Gianluca Stellin 14  

Nature ( 2024 ) Cite this article

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• Quantum simulation
• Theoretical nuclear physics
• Theoretical physics
• Ultracold gases

Ab initio calculations have an essential role in our fundamental understanding of quantum many-body systems across many subfields, from strongly correlated fermions 1 , 2 , 3 to quantum chemistry 4 , 5 , 6 and from atomic and molecular systems 7 , 8 , 9 to nuclear physics 10 , 11 , 12 , 13 , 14 . One of the primary challenges is to perform accurate calculations for systems where the interactions may be complicated and difficult for the chosen computational method to handle. Here we address the problem by introducing an approach called wavefunction matching. Wavefunction matching transforms the interaction between particles so that the wavefunctions up to some finite range match that of an easily computable interaction. This allows for calculations of systems that would otherwise be impossible owing to problems such as Monte Carlo sign cancellations. We apply the method to lattice Monte Carlo simulations 15 , 16 of light nuclei, medium-mass nuclei, neutron matter and nuclear matter. We use high-fidelity chiral effective field theory interactions 17 , 18 and find good agreement with empirical data. These results are accompanied by insights on the nuclear interactions that may help to resolve long-standing challenges in accurately reproducing nuclear binding energies, charge radii and nuclear-matter saturation in ab initio calculations 19 , 20 .

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Quantum Monte Carlo simulations are a powerful and efficient ab initio method for describing quantum many-body systems using stochastic processes 1 , 9 , 15 , 16 , 21 , 22 , 23 . If the Monte Carlo amplitudes are positive, then the computational effort grows only as a low power of the number of particles. For many problems of interest, a simple Hamiltonian H S can be found that is easily computable using Monte Carlo methods and describes the energies and other observable properties of the many-body system in fair agreement with empirical data 24 , 25 , 26 , 27 . However, realistic high-fidelity Hamiltonians usually suffer from severe sign problems with positive and negative contributions cancelling each other so that Monte Carlo calculations become impractical. Here we solve the problem using an approach called wavefunction matching. While keeping the observable physics unchanged, wavefunction matching creates a new high-fidelity Hamiltonian H ′ such that the two-body wavefunctions up to some finite range match that of a simple Hamiltonian H S , which is easily computed. This allows for a rapidly converging expansion in powers of the difference H ′ −  H S . Although wavefunction matching can be used with any computational scheme, we focus here on quantum Monte Carlo simulations where the method presents a practical strategy for evading sign oscillations in high-fidelity calculations. While H S and H ′ act on many-body systems, the wavefunction-matching process is done at the two-body level only. For the sake of clarity, we define H S and H ′ as containing only two-body interactions. Later we also consider the inclusion of three-body interactions. However, that analysis is separate from wavefunction matching.

A unitary transformation U is a linear transformation that maps normalized orthogonal states to other normalized orthogonal states. Starting from a high-fidelity Hamiltonian H with only two-body interactions, wavefunction matching defines a new Hamiltonian H ′ =  U † HU , where U † is the Hermitian conjugate of U . The unitary transformation is performed at the two-body level. In each two-body angular momentum channel, the unitary transformation U is active only when the separation distance between two particles is less than some chosen distance R . For the calculations presented here, the value R  = 3.72 fm is used. The dependence on R is extensively discussed in  Supplementary Information .

Let us write ψ 0 ( r ), \({\psi }_{0}^{{\prime} }(r)\) and \({\psi }_{0}^{{\rm{S}}}(r)\) for the two-body ground-state wavefunctions of H , H ′ and the simple Hamiltonian H S , respectively. Here r is the distance between the two particles. The transformation U is defined such that \({\psi }_{0}^{{\prime} }(r)\) is proportional to \({\psi }_{0}^{{\rm{S}}}(r)\) for r  <  R . The simple Hamiltonian is chosen so that the constant of proportionality is close to 1. For r  >  R , however, U is not active and so \({\psi }_{0}^{{\prime} }(r)\) remains equal to ψ 0 ( r ). The key point to notice here is that \({\psi }_{0}^{{\prime} }(r)\) and \({\psi }_{0}^{{\rm{S}}}(r)\) are numerically close to each other for all values of r . This can be seen visually in Fig. 1a and is the reason why perturbation theory in powers of H ′ −  H S converges quickly when starting from low-energy states of H S .

a , Pictorial representation of wavefunction matching. The simple Hamiltonian H S is an easily computable Hamiltonian whereas the high-fidelity Hamiltonian H is not. A unitary transformation on the two-nucleon interaction with finite range R is used to produce a new Hamiltonian H ′ that is close to H S . In each two-body channel, the ground-state wavefunction of H ′ matches the ground-state wavefunction of H for r  >  R and is proportional to the ground-state wavefunction of H S for r  <  R . b , The Tjon band correlation between the binding energies of 3 H ( B 3 ) and 4 He ( B 4 ). The grey band is the predicted result from ref. 31 . The black open box shows the empirical point. The green diamond, blue circle and red square points show the results at LO, NLO and N3LO in chiral effective field theory, respectively. The open points show the results from the first-order perturbative calculations using the Hamiltonian H and the filled points are the results of the first-order perturbative calculations using the Hamiltonian H ′. The error bars show standard deviations.

Wavefunction matching will now be applied to ab initio Monte Carlo nuclear lattice simulations 15 , 16 , 25 , 26 , 28 using the framework of chiral effective field theory (χEFT) 17 , 29 . For our realistic Hamiltonian H , we use χEFT two-nucleon interactions at next-to-next-to-next-to-leading order (N3LO) with lattice spacing a  = 1.32 fm using a low-energy scheme described in  Supplementary Information . For our simple Hamiltonian H S , we use a χEFT interaction at leading order. Details of the interactions can be found in  Supplementary Information . In the following, we use the term ‘local’ for interactions that do not change the positions of particles and ‘non-local’ refers to interactions that do change the relative positions of particles. The ‘range’ of the interaction refers to the separation distance beyond which the interaction between particles becomes negligible.

We calculate all quantities up to first order in perturbation theory, which corresponds to one power in the difference H ′ −  H S . As a first test, we consider the energy of the deuteron, 2 H. The wavefunction-matching calculation gives a binding energy of 2.02 MeV, compared with 2.21 MeV for the true binding energy of H and 2.22 MeV for the experimentally observed value. The residual error of 0.1 MeV per nucleon is due to corrections beyond first order in powers of H ′ −  H S . If one does not use wavefunction matching and instead performs the analogous calculation to first order in H  −  H S , the result is a much less accurate binding energy of 0.68 MeV.

As a second test of wavefunction matching, we calculate the binding energies of 3 H and 4 He. The Tjon band describes the universal correlations between the 3 H and 4 He binding energies 30 , 31 . Provided that there are no long-range non-local interactions, any realistic two-nucleon interaction produces binding energies that lie on the Tjon band. The inclusion of any short-range three-nucleon interaction also preserves this universal relation. In Fig. 1 , we show wavefunction-matching calculations using two-nucleon interactions only. At leading order (LO) the calculated point falls outside the Tjon band as the Coulomb interaction is not included, whereas the next-to-leading order (NLO) and N3LO results lie squarely in the middle of the band. We are using a low-energy scheme where the two-nucleon interaction is the same at NLO and next-to-next-to-leading order (NNLO) 32 . The empirical point is also shown in Fig. 1 . The good agreement with the Tjon band suggests a residual error of 0.1 MeV per nucleon or less for 3 H and 4 He. In  Supplementary Information , we present numerical evidence that the estimate of 0.1 MeV error per nucleon is also valid for light and medium-mass nuclei. This can be compared with the substantial deviation from the Tjon line if one does not use wavefunction matching and performs the analogous calculation to first order in H  −  H S . Before proceeding to larger nuclei and many-body systems, we first comment on the current status of ab initio calculations of nuclear structure using χEFT. The following analysis is not directly connected to wavefunction matching. Instead, it is a separate theoretical framework designed to help push beyond the current limitations of ab initio nuclear structure theory.

There has been tremendous progress in the past few years towards producing accurate results for nuclear structure across much of the nuclear chart using a variety of different computational approaches 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 . But there is also ample evidence that the calculations are sensitive to the manner in which the short-distance features of the interactions are regulated 20 , 45 , 46 , 47 , 48 , a warning sign that systematic errors are not fully under control. Current ab initio calculations have difficulty simultaneously maintaining high-fidelity two-nucleon phase shifts and mixing angles and describing the saturation energy and density of symmetric nuclear matter as well as the binding energies and charge radii of light and medium-mass nuclei. Previous ab initio nuclear structure calculations have either not addressed some of the relevant observables or require further improvement in one or more of these areas. We aim to identify the problem and point to a viable solution.

The results in refs. 49 , 50 showed that the range and locality of the nuclear interactions have a strong influence on nuclear binding and that the α–α interaction is highly sensitive to the range and locality of the nucleonic interactions as well as omitted higher-order interactions. These same arguments apply to other interactions involving α particles and nucleons. In  Supplementary Information , we use the formalism of cluster effective field theory 51 , 52 , 53 , 54 for α-particles and nucleons to provide a simple counting argument for the number of parameters that require tuning to reduce unwanted errors. Our strategy is to tune the short-distance features of the three-nucleon interactions to achieve this error cancellation. We should emphasize that our calculations are full A -body calculations, and cluster effective field theory is only used to diagnose sensitivities to short-distance physics.

In χEFT, three-nucleon forces first appear at order NNLO. These include terms associated with the exchange of two pions and whose coefficients are determined from pion–nucleon scattering. There are also two interactions with singular short-distance properties that must be regulated and the corresponding couplings fitted to empirical data. As shown in Fig. 2a , c D corresponds to the short-range interaction of two nucleons linked to a third nucleon through the exchange of a pion, and c E corresponds to the short-range interaction of all three nucleons. At N3LO, there are additional terms associated with the exchange of two pions as well as readjustments of the c D and c E coefficients 55 , 56 , 57 . Four-nucleon interactions also appear at N3LO but are not considered in this work.

a , Short-range three-nucleon forces at NNLO. The first is the one-pion exchange term c D shown on the left. The other is the purely short-range term c E shown on the right. At order N3LO, there are additional three-nucleon interactions associated with the exchange of two pions, as well as the corrections from the renormalization of the c D and c E terms. b , Results for nuclear binding energies ( B A ) using wavefunction matching. Calculated ground-state and excited-state energies of some selected nuclei with up to A  = 58 at N3LO in χ EFT and comparison with experimental data. The symbols with a black border indicate nuclei with unequal numbers of protons and neutrons. The nuclei used in the fit of the higher-order three-nucleon interactions are labelled with open squares and the other nuclei are predictions denoted with filled diamonds. The error bars show standard deviations.

We tune the short-distance features of the c D and c E three-nucleon interactions to minimize errors in the binding energies of selected light and medium-mass nuclei. A total of six additional three-nucleon parameters are adjusted, and in  Supplementary Information we present the details of these parameters along with a detailed description of the fitting procedure and the resulting uncertainty. We find that with just one parameter, the root-mean-square-deviation (RMSD) for the energy per nucleon drops from 1.2 MeV down to 0.4 MeV. With the addition of a few additional parameters, the RMSD per nucleon drops further to about 0.1 MeV. These results are consistent with the hypothesis that the α–α interaction has a key role in nuclear binding and that there are several additional cluster interactions that are sensitive to short-distance physics.

In Fig. 2b , we present the results for the nuclear binding energies using wavefunction matching. We show ground-state and excited-state energies of selected nuclei with up to A  = 58 nucleons and comparison with experimental data. The symbols with a black border indicate nuclei with unequal numbers of protons and neutrons. The nuclei used in the fit of the three-nucleon interactions are labelled with open squares, and the other nuclei are predictions denoted with filled diamonds. The one-standard-deviation error bars shown in Fig. 2 represent uncertainties due to Monte Carlo errors, infinite-volume extrapolations and infinite projection time extrapolations. As described in  Supplementary Information , we estimate the additional systematic errors due to truncation of the expansion in powers of H ′ −  H S to be approximately 0.1 MeV per nucleon. However, this source of systematic error can be significantly reduced by allowing for variational optimization of the Hamiltonian used to prepare the nuclear many-body wavefunction. We perform this variational optimization so that the remaining systematic error is smaller than the estimated computational error due to other sources. In  Supplementary Information , we also compute the additional systematic errors due to uncertainties in the chiral interactions.

In Fig. 3a , we present the results for the charge radii of nuclei with up to A  = 58 nucleons. No charge radii data were used to fit any interaction parameters. The one-standard-deviation point estimate error bars shown in Fig. 3 represent computational uncertainties due to Monte Carlo errors, infinite-volume extrapolation and infinite-time extrapolation. The agreement with empirical results is quite good, with an RMSD of about 0.03 fm. An extended analysis for selected nuclei that also includes uncertainties from the interactions are presented in  Supplementary Information . We note that the larger errors for the heaviest nuclei are statistical and can be decreased by utilizing greater computational resources. The specific terms included in the calculations of the charge radii are detailed in  Supplementary Information .

a , Predictions for charge radii ( R ch ) of nuclei up to A  = 58 at N3LO in χEFT and comparison with experimental data. The symbols with a black border indicate nuclei with unequal numbers of protons and neutrons. b , Predictions for pure neutron-matter energy per neutron and symmetric nuclear-matter energy per nucleon as a function of density at N3LO in χEFT. For pure neutron matter, we use the number of neutrons from 14 to 80 and various box sizes from 6.58 fm to 13.2 fm. For symmetric nuclear matter, we use nucleon numbers from 12 to 160 and a periodic box of length 9.21 fm. For comparison, we show the results from variational calculations (APR) 65 , auxiliary-field diffusion Monte Carlo simulations (GCR) 66 , many-body perturbation theory using N3LO/NNLO (two-nucleon (2NF)/three-nucleon (3NF)) chiral interactions (EM 500 MeV, EGM 450/500 MeV and EGM 450/700 MeV) 67 and coupled cluster theory using NNLO chiral interactions with explicit delta degrees of freedom (ΔNNLO) 68 . The empirical saturation point is labelled with a black rectangular box. E denotes energy, ρ is the nucleon density, and ρ 0 is the saturation density of symmetric nuclear matter. The error bars show standard deviations.

In Fig. 3b , we present lattice results for the energy per nucleon versus density for pure neutron matter and symmetric nuclear matter. None of the neutron-matter and symmetric nuclear-matter data were used to fit any interaction parameters. The density is expressed as a fraction of the saturation density for nuclear matter, ρ 0  = 0.16 fm −3 . For the neutron-matter calculations, we consider 14 to 80 neutrons in periodic box lengths ranging from 6.58 fm to 13.2 fm. For the symmetric nuclear-matter calculations, we use system sizes from 12 to 160 nucleons in a periodic box of length 9.21 fm. The comparisons with several other published works are shown and detailed in the figure caption. We see that the neutron-matter calculations agree well with previous calculations. Within the uncertainties due to finite system size corrections, the symmetric nuclear-matter calculations show saturation at an energy and density consistent with the empirical saturation point labelled with the black rectangular box. The relative uncertainties due to finite system size are at the 10% level for the energy. Additional calculations with larger systems are needed to reduce the thermodynamic extrapolation error further.

The one-standard-deviation point estimate error bars shown represent computational uncertainties due to Monte Carlo errors and infinite projection time extrapolation. These lattice simulations of symmetric nuclear matter are qualitatively different to other theoretical calculations that assume a homogeneous phase. The lattice simulations show phase separation and cluster formation, just as in the real physical system. Owing to the finite number of nucleons in these calculations, some oscillations due to nuclear shell effects can be seen in the energy per nucleon.

Another interesting feature of the lattice results is that symmetric nuclear matter without three-nucleon forces is underbound rather than overbound. This is different from what is found in other calculations using renormalization-group methods 58 , 59 , 60 . As discussed in  Supplementary Information , wavefunction matching is very different from renormalization-group transformations. Wavefunction matching implements a unitary transformation that has finite range, and the process can be viewed as defining a new χEFT two-nucleon Hamiltonian H ′. The interaction in H ′ has a range no larger than that of H and H S for the low-energy interactions. Therefore, one does not need to reconstruct the many-body forces induced by the unitary transformation and can simply treat H ′ as the new χEFT two-nucleon Hamiltonian. Wavefunction matching has some characteristics similar to the unitary correlation operator method (UCOM) 61 , 62 , 63 . However, the unitary transformation in UCOM has properties that are more similar to renormalization-group transformations and, therefore, is also quite different from wavefunction matching. The induced forces generated by wavefunction matching have been investigated in a toy model 64 . A detailed discussion of the theory and applications of wavefunction matching and its implementation in continuous space are presented in  Supplementary Information .

In summary, we have presented an approach for solving quantum many-body systems called wavefunction matching. Wavefunction matching uses a transformation of the particle interactions to allow for calculations of systems that would otherwise be difficult or impossible. We have applied the method to lattice Monte Carlo simulations of light nuclei, medium-mass nuclei, neutron matter and nuclear matter using high-fidelity chiral interactions and found good agreement with empirical data. Judging from the accuracy of the predictions, we have been successful in cancelling systematic errors in nuclear structure calculations by tuning the short-distance features of the three-nucleon interactions. These developments may help resolve long-standing challenges in ab initio nuclear structure theory.

Although we have focused on Monte Carlo simulations for nuclear physics here, wavefunction matching can be used with any computational method and applied to any quantum many-body system. This also includes quantum computing algorithms where wavefunction matching can be used to reduce the number of quantum gates required. All that is needed is a simple Hamiltonian H S that produces fair agreement with empirical data for the many-body system of interest and is easily computable using the method of choice. Further details on the implementation and theory of wavefunction matching are given in  Supplementary Information .

Data availability

All of the data produced in association with this work have been stored and are publicly available at https://drive.google.com/drive/folders/1MByuG6NMagcgmURe4py-kwr9vksnHCl4 .

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All of the codes produced in association with this work have been stored and can be obtained upon request from the corresponding author, subject to possible export control constraints.

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Acknowledgements

We thank members and partners of the Nuclear Lattice Effective Field Theory Collaboration (J. Drut, G. Jansen, S. Krieg, Z. Ren, A. Sarkar and Q. Wang) and S. Bogner, A. Ekström, H. Hergert, M. Hjorth-Jensen, D. Phillips, A. Schwenk and W. Nazarewicz for discussions. We acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) and the NSFC through the funds provided to the Sino-German Collaborative Research Center TRR110 ‘Symmetries and the Emergence of Structure in QCD’ (DFG project ID 196253076 - TRR 110, NSFC grant number 12070131001), the Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) (grant number 2018DM0034), Volkswagen Stiftung (grant number 93562), the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC AdG EXOTIC, grant agreement number 101018170, and ERC AdG NuclearTheory, grant agreement number 885150), the Scientific and Technological Research Council of Turkey (TUBITAK project number 120F341), the National Natural Science Foundation of China (grants numbers 12105106 and 12275259), NSAF No.U2330401, US National Science Foundation (PHY-1913620, PHY-2209184, PHY-2310620), US Department of Energy (DE-SC0021152, DE-SC0013365, DE-SC0023658, DE-SC0024586, NUCLEI SciDAC-5 project DE-SC0023175), the Rare Isotope Science Project of the Institute for Basic Science funded by the Ministry of Science and ICT (MSICT), the National Research Foundation of Korea (2013M7A1A1075764, RS-2022-00165168), the Institute for Basic Science (IBS-R031-D1, IBS-I001-D1) and the Espace de Structure et de réactions Nucléaires Théorique (ESNT) of the CEA DSM/DAM. Computational resources provided by the Gauss Centre for Supercomputing e.V. ( www.gauss-centre.eu ) for computing time on the GCS Supercomputer JUWELS at Jülich Supercomputing Centre (JSC) and special GPU time allocated on JURECA-DC as well as the Oak Ridge Leadership Computing Facility through the INCITE award ‘Ab-initio nuclear structure and nuclear reactions’, and partially provided by TUBITAK ULAKBIM High Performance and Grid Computing Center (TRUBA resources). Computational resources were also partly provided by the National Supercomputing Center of Korea with supercomputing resources including technical support (KSC-2021-CRE-0429, KSC-2022-CHA-0003, KSC-2023-CHA-0005), and the Southern Nuclear Science Computing Center in the South China Normal University. We have complied with community standards for authorship and all relevant recommendations with regard to inclusion and ethics.

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Serdar Elhatisari

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Code development, testing, optimization and production runs were led by S.E. with contributions from F.H., M.K., T.A.L., D.L., N.L., B.-N.L., Y.-Z.M., G.R., S.S. and Y.-H.S. Conceptual work and tests were led by L.B. with contributions from S.E., E.E., D.F., H.K. and D.L. S.E., T.L., D.L., U.-G.M. and Y.K. supervised the research effort. Additional code development, testing, optimization and production runs in response to reviewer comments were performed by Y.-Z.M. The literature search was led by G.S. All authors contributed to the writing, editing and review of this work.

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Elhatisari, S., Bovermann, L., Ma, YZ. et al. Wavefunction matching for solving quantum many-body problems. Nature (2024). https://doi.org/10.1038/s41586-024-07422-z

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Received : 23 November 2022

Accepted : 15 April 2024

Published : 15 May 2024

DOI : https://doi.org/10.1038/s41586-024-07422-z

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