diff --git a/cppcore/include/numeric/dense.hpp b/cppcore/include/numeric/dense.hpp
index 7251c5d4..6bf02571 100644
--- a/cppcore/include/numeric/dense.hpp
+++ b/cppcore/include/numeric/dense.hpp
@@ -318,4 +318,19 @@ ArrayX<T> make_integer_range(idx_t size) {
     return result;
 }
 
+template<class Vector, class Bools>
+Vector slice(Vector const& v, Bools const& keep) {
+    using std::begin; using std::end;
+    auto const original_size = v.size();
+    auto const result_size = std::accumulate(begin(keep), end(keep), idx_t{0});
+
+    auto result = Vector(result_size);
+    auto count = 0;
+    for (auto i = idx_t{0}; i < original_size; ++i) {
+        if (keep[i])
+            result[count++] = v[i];
+    }
+    return result;
+};
+
 } // namespace cpb
diff --git a/cppcore/include/system/CompressedSublattices.hpp b/cppcore/include/system/CompressedSublattices.hpp
index da39f561..6a577353 100644
--- a/cppcore/include/system/CompressedSublattices.hpp
+++ b/cppcore/include/system/CompressedSublattices.hpp
@@ -71,6 +71,10 @@ class CompressedSublattices {
 
     /// Start a new sublattice block or increment the site count for the existing block
     void add(SubAliasID id, idx_t norb);
+
+    /// Remove sites for which `keep == false`
+    void filter(VectorX<bool> const& keep);
+
     /// Verify that the stored data is correct: `sum(site_counts) == num_sites`
     void verify(idx_t num_sites) const;
 
diff --git a/cppcore/include/system/HoppingBlocks.hpp b/cppcore/include/system/HoppingBlocks.hpp
index 8d941346..22b0a4bc 100644
--- a/cppcore/include/system/HoppingBlocks.hpp
+++ b/cppcore/include/system/HoppingBlocks.hpp
@@ -121,6 +121,9 @@ class HoppingBlocks {
     /// Append a range of coordinates to the given family block
     void append(HopID family_id, ArrayXi&& rows, ArrayXi&& cols);
 
+    /// Remove sites for which `keep == false`
+    void filter(VectorX<bool> const& keep);
+
     /// Return the matrix in the CSR sparse matrix format
     SparseMatrixX<storage_idx_t> tocsr() const;
 
diff --git a/cppcore/include/system/StructureModifiers.hpp b/cppcore/include/system/StructureModifiers.hpp
index c67eeb97..4fc16ecb 100644
--- a/cppcore/include/system/StructureModifiers.hpp
+++ b/cppcore/include/system/StructureModifiers.hpp
@@ -75,6 +75,8 @@ void apply(SiteStateModifier const& m, Foundation& f);
 void apply(PositionModifier const& m, Foundation& f);
 
 template<class M> void apply(M const&, System&) {}
+void apply(SiteStateModifier const& m, System& s);
+void apply(PositionModifier const& m, System& s);
 void apply(HoppingGenerator const& g, System& s);
 
 template<class M> constexpr bool requires_system(M const&) { return false; }
diff --git a/cppcore/include/system/System.hpp b/cppcore/include/system/System.hpp
index a69975a1..05e7d4ff 100644
--- a/cppcore/include/system/System.hpp
+++ b/cppcore/include/system/System.hpp
@@ -28,6 +28,7 @@ struct System {
     CompressedSublattices compressed_sublattices;
     HoppingBlocks hopping_blocks;
     std::vector<Boundary> boundaries;
+    ArrayX<bool> is_valid;
 
     System(Lattice const& lattice) : lattice(lattice) {}
 
@@ -69,6 +70,7 @@ namespace detail {
     void populate_system(System& system, Foundation const& foundation);
     void populate_boundaries(System& system, Foundation const& foundation,
                              TranslationalSymmetry const& symmetry);
+    void remove_invalid(System& system);
 } // namespace detail
 
 } // namespace cpb
diff --git a/cppcore/src/Model.cpp b/cppcore/src/Model.cpp
index f0ec59fd..b4d2e534 100644
--- a/cppcore/src/Model.cpp
+++ b/cppcore/src/Model.cpp
@@ -153,6 +153,8 @@ std::shared_ptr<System> Model::make_system() const {
         apply(modifier, *sys);
     }
 
+    detail::remove_invalid(*sys);
+
     if (sys->num_sites() == 0) { throw std::runtime_error{"Impossible system: 0 sites"}; }
 
     return sys;
diff --git a/cppcore/src/system/CompressedSublattices.cpp b/cppcore/src/system/CompressedSublattices.cpp
index 2ff8a912..fa8d02b8 100644
--- a/cppcore/src/system/CompressedSublattices.cpp
+++ b/cppcore/src/system/CompressedSublattices.cpp
@@ -20,6 +20,22 @@ void CompressedSublattices::add(SubAliasID id, idx_t norb) {
     }
 }
 
+void CompressedSublattices::filter(VectorX<bool> const& keep) {
+    using std::begin;
+
+    auto new_counts = std::vector<storage_idx_t>();
+    new_counts.reserve(data.size());
+
+    for (auto const& sub : *this) {
+        new_counts.push_back(std::accumulate(begin(keep) + sub.sys_start(),
+                                             begin(keep) + sub.sys_end(), storage_idx_t{0}));
+    }
+
+    for (auto i = size_t{0}; i < data.size(); ++i) {
+        data[i].num_sites = new_counts[i];
+    }
+}
+
 void CompressedSublattices::verify(idx_t num_sites) const {
     using std::begin; using std::end;
 
diff --git a/cppcore/src/system/HoppingBlocks.cpp b/cppcore/src/system/HoppingBlocks.cpp
index d321da41..618f36ad 100644
--- a/cppcore/src/system/HoppingBlocks.cpp
+++ b/cppcore/src/system/HoppingBlocks.cpp
@@ -62,6 +62,17 @@ void HoppingBlocks::append(HopID family_id, ArrayXi&& rows, ArrayXi&& cols) {
     block.erase(std::unique(block.begin(), block.end()), block.end());
 }
 
+void HoppingBlocks::filter(VectorX<bool> const& keep) {
+    using std::begin; using std::end;
+
+    num_sites = std::accumulate(begin(keep), end(keep), idx_t{0});
+    for (auto& block : blocks) {
+        block.erase(std::remove_if(block.begin(), block.end(), [&](COO coo) {
+            return !keep[coo.row] || !keep[coo.col];
+        }), block.end());
+    }
+}
+
 SparseMatrixX<storage_idx_t> HoppingBlocks::tocsr() const {
     auto csr = SparseMatrixX<storage_idx_t>(num_sites, num_sites);
     csr.reserve(nnz());
diff --git a/cppcore/src/system/StructureModifiers.cpp b/cppcore/src/system/StructureModifiers.cpp
index 013b7633..7b8205b5 100644
--- a/cppcore/src/system/StructureModifiers.cpp
+++ b/cppcore/src/system/StructureModifiers.cpp
@@ -16,6 +16,23 @@ void apply(SiteStateModifier const& m, Foundation& f) {
     }
 }
 
+void apply(SiteStateModifier const& m, System& s) {
+    if (s.is_valid.size() == 0) {
+        s.is_valid = ArrayX<bool>::Constant(s.num_sites(), true);
+    }
+
+    for (auto const& sub : s.compressed_sublattices) {
+        m.apply(s.is_valid.segment(sub.sys_start(), sub.num_sites()),
+                s.positions.segment(sub.sys_start(), sub.num_sites()),
+                s.lattice.sublattice_name(SubID(sub.alias_id())));
+    }
+
+    if (m.min_neighbors > 0) {
+        throw std::runtime_error("Eliminating dangling bonds after a generator "
+                                 "has not been implemented yet");
+    }
+}
+
 void apply(PositionModifier const& m, Foundation& f) {
     for (auto const& pair : f.get_lattice().get_sublattices()) {
         auto slice = f[pair.second.unique_id];
@@ -23,7 +40,16 @@ void apply(PositionModifier const& m, Foundation& f) {
     }
 }
 
+void apply(PositionModifier const& m, System& s) {
+    for (auto const& sub : s.compressed_sublattices) {
+        m.apply(s.positions.segment(sub.sys_start(), sub.num_sites()),
+                s.lattice.sublattice_name(SubID(sub.alias_id())));
+    }
+}
+
 void apply(HoppingGenerator const& g, System& s) {
+    detail::remove_invalid(s);
+
     auto const& lattice = s.lattice;
     auto const sublattices = s.compressed_sublattices.decompressed();
     auto const family_id = lattice.hopping_family(g.name).family_id;
diff --git a/cppcore/src/system/System.cpp b/cppcore/src/system/System.cpp
index a0a62a70..b4a6ce6d 100644
--- a/cppcore/src/system/System.cpp
+++ b/cppcore/src/system/System.cpp
@@ -157,5 +157,21 @@ void populate_boundaries(System& system, Foundation const& foundation,
     }
 }
 
+void remove_invalid(System& s) {
+    if (s.is_valid.size() == 0) { return; }
+
+    s.positions.x = slice(s.positions.x, s.is_valid);
+    s.positions.y = slice(s.positions.y, s.is_valid);
+    s.positions.z = slice(s.positions.z, s.is_valid);
+    s.compressed_sublattices.filter(s.is_valid);
+    s.hopping_blocks.filter(s.is_valid);
+
+    for (auto& b : s.boundaries) {
+        b.hopping_blocks.filter(s.is_valid);
+    }
+
+    s.is_valid.resize(0);
+}
+
 } // namespace detail
 } // namespace cpb
diff --git a/cppcore/tests/fixtures.cpp b/cppcore/tests/fixtures.cpp
index ae669d7a..95c3f0fe 100644
--- a/cppcore/tests/fixtures.cpp
+++ b/cppcore/tests/fixtures.cpp
@@ -185,3 +185,14 @@ cpb::HoppingModifier force_complex_numbers() {
 }
 
 } // namespace field
+
+
+namespace generator {
+
+cpb::HoppingGenerator do_nothing_hopping(std::string const& name) {
+    return {name, 0, [](cpb::CartesianArray const&, SubIdRef) {
+        return HoppingGenerator::Result{ArrayXi{}, ArrayXi{}};
+    }};
+}
+
+} // namespace generator
diff --git a/cppcore/tests/fixtures.hpp b/cppcore/tests/fixtures.hpp
index ed1acebf..f19bbfc3 100644
--- a/cppcore/tests/fixtures.hpp
+++ b/cppcore/tests/fixtures.hpp
@@ -37,3 +37,9 @@ cpb::HoppingModifier force_double_precision();
 cpb::HoppingModifier force_complex_numbers();
 
 } // namespace field
+
+namespace generator {
+
+cpb::HoppingGenerator do_nothing_hopping(std::string const& name = "_t");
+
+} // namespace generator
diff --git a/cppcore/tests/test_modifiers.cpp b/cppcore/tests/test_modifiers.cpp
index 199019e8..0cc85acd 100644
--- a/cppcore/tests/test_modifiers.cpp
+++ b/cppcore/tests/test_modifiers.cpp
@@ -7,29 +7,87 @@ TEST_CASE("SiteStateModifier") {
     auto model = Model(lattice::square_2atom(), Primitive(2));
     REQUIRE(model.system()->num_sites() == 4);
 
-    auto remove_site = [](Eigen::Ref<ArrayX<bool>> state, CartesianArrayConstRef, string_view s) {
-        if (s == "A") {
+    auto count = std::unordered_map<std::string, idx_t>();
+
+    auto remove_site = [&](Eigen::Ref<ArrayX<bool>> state, CartesianArrayConstRef, string_view s) {
+        count[s] = state.size();
+        if (s == "A" && state.size() != 0) {
             state[0] = false;
         }
     };
-    model.add(SiteStateModifier(remove_site));
-    REQUIRE(model.system()->num_sites() == 3);
-    model.add(SiteStateModifier(remove_site, 1));
-    REQUIRE(model.system()->num_sites() == 2);
-    model.add(SiteStateModifier(remove_site, 2));
-    REQUIRE_THROWS(model.system());
+
+    SECTION("Apply to foundation") {
+        model.add(SiteStateModifier(remove_site));
+        REQUIRE(model.system()->num_sites() == 3);
+        REQUIRE(count["A"] == 2);
+        REQUIRE(count["B"] == 2);
+
+        model.add(SiteStateModifier(remove_site, 1));
+        REQUIRE(model.system()->num_sites() == 2);
+        REQUIRE(count["A"] == 2);
+        REQUIRE(count["B"] == 2);
+
+        model.add(SiteStateModifier(remove_site, 2));
+        REQUIRE_THROWS(model.system());
+        REQUIRE(count["A"] == 2);
+        REQUIRE(count["B"] == 2);
+    }
+
+    SECTION("Apply to system") {
+        model.add(SiteStateModifier(remove_site));
+        model.add(generator::do_nothing_hopping());
+
+        model.add(SiteStateModifier(remove_site));
+        REQUIRE(model.system()->num_sites() == 2);
+        REQUIRE(count["A"] == 1);
+        REQUIRE(count["B"] == 2);
+
+        model.add(generator::do_nothing_hopping("_t2"));
+        model.add(SiteStateModifier(remove_site));
+        REQUIRE(model.system()->num_sites() == 2);
+        REQUIRE(count["A"] == 0);
+        REQUIRE(count["B"] == 2);
+
+        model.add(SiteStateModifier(remove_site, 1));
+        REQUIRE_THROWS_WITH(model.system(), Catch::Contains("has not been implemented yet"));
+        REQUIRE(count["A"] == 0);
+        REQUIRE(count["B"] == 2);
+    }
 }
 
 TEST_CASE("SitePositionModifier") {
-    auto model = Model(lattice::square_2atom());
-    REQUIRE(model.system()->positions.y[1] == Approx(0.5));
+    auto model = Model(lattice::square_2atom(), shape::rectangle(2, 2));
+    REQUIRE(model.system()->num_sites() == 6);
+    REQUIRE(model.system()->positions.y[1] == Approx(-1));
+
+    auto count = std::unordered_map<std::string, idx_t>();
+    constexpr auto moved_pos = 10.0f;
 
-    model.add(PositionModifier([](CartesianArrayRef position, string_view sublattice) {
+    auto move_site = PositionModifier([&](CartesianArrayRef position, string_view sublattice) {
+        count[sublattice] = position.size();
         if (sublattice == "B") {
-            position.y()[0] = 1;
+            position.y().setConstant(moved_pos);
         }
-    }));
-    REQUIRE(model.system()->positions.y[1] == Approx(1));
+    });
+
+    SECTION("Apply to foundation") {
+        model.add(move_site);
+        model.eval();
+        REQUIRE(count["A"] == 25);
+        REQUIRE(count["B"] == 25);
+        REQUIRE(model.system()->num_sites() == 6);
+        REQUIRE(model.system()->positions.y.segment(4, 2).isApproxToConstant(moved_pos));
+    }
+
+    SECTION("Apply to system") {
+        model.add(generator::do_nothing_hopping());
+        model.add(move_site);
+        model.eval();
+        REQUIRE(count["A"] == 4);
+        REQUIRE(count["B"] == 2);
+        REQUIRE(model.system()->num_sites() == 6);
+        REQUIRE(model.system()->positions.y.segment(4, 2).isApproxToConstant(moved_pos));
+    }
 }
 
 TEST_CASE("State and position modifier ordering") {