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Lukasz Fidkowski

Publications and source records attributed to Lukasz Fidkowski.

At least 19 recordsLinked to original sources
arXiv2026

1+1d Lattice Dirac Fermions from Non-Onsite Vector and Axial Symmetries

We construct an exactly solvable Hamiltonian lattice model realizing a 1+1d Dirac fermion, with exact microscopic vector and axial vector $U(1)$ symmetries. The mixed anomaly between these is accommodated by the not-on-site action of the symmetries. Our Hilbert space is a $Z_2$-graded tensor product of local $Z_2$-graded Hilbert spaces which include infinite dimensional rotor degrees of freedom. The Hamiltonian becomes manifestly exactly solvable after a locality-preserving unitary mapping to an equivalent fermionic Villain Hilbert space. Our construction also allows an exactly solvable realization of interacting fermionic Luttinger liquids. At the free Dirac fixed point, our Hamiltonian contains irrelevant interactions, which we compute to leading order.

cond-mat.str-el↗
arXiv2026

Non-Invertible Symmetries Mixing with Witt Non-Trivial Quantum Cellular Automata

Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a $\mathbb{Z}_p$ one-form symmetry in 3+1d these correspond to the Kramers-Wannier-Wegner duality $S$, which is the gauging operation underlying non-invertible duality symmetries, and the stacking of a 1-form symmetry SPT $T$. In the continuum, they form a central extension of $PSL(2,\mathbb{Z}_4)$ for $p=2$, and of $SL(2,\mathbb{Z}_p)$ for odd primes $p$, whose central elements are invertible theories with purely gravitational response. These central extensions are governed by a twisted, graded generalization of the Witt group of abelian anyon theories, which we determine. For $p=2$ the resulting group is the single-qubit Clifford group, with duality and entangler acting as the Hadamard and phase gates. We realize this entire structure microscopically as quantum cellular automata (QCA) acting on a certain local operator algebra associated with a spin lattice Hilbert space on a cubic lattice. Specifically, our local operator algebra is built by starting with all local operators commuting with a $\mathbb{Z}_p$ 1-form symmetry, and taking the quotient by all the (local) 1-form symmetry generators. The central elements can always be extended to the full tensor product algebra with a uniquely defined QCA class. For $p=2$ they are generated by the non-trivial semion QCA, and for odd prime $p$ they are generated by the non-trivial $\mathbb{Z}_p$ Clifford QCA. Consequently the lattice fusion rules reproduce the continuum ones only up to these QCAs and lattice translations, giving rise to fusion rules refined by QCAs.

quant-ph↗
arXiv2026

Chiral Lattice Gauge Theories from Symmetry Disentanglers

We propose a Hamiltonian framework for constructing chiral gauge theories on the lattice based on symmetry disentanglers: constant-depth circuits of local unitaries that transform not-on-site symmetries into on-site ones. When chiral symmetry can be realized not-on-site and such a disentangler exists, the symmetry can be implemented in a strictly local Hamiltonian and gauged by standard lattice methods. Using lattice rotor models, we realize this idea in 1+1 and 3+1 spacetime dimensions for $U(1)$ symmetries with mixed 't Hooft anomalies, and show that symmetry disentanglers can be constructed when anomalies cancel. As an example, we present an exactly solvable Hamiltonian lattice model of the (1+1)-dimensional "3450" chiral gauge theory, and we argue that a related construction applies to the $U(1)$ hypercharge symmetry of the Standard Model fermions in 3+1 dimensions. Our results open a new route toward fully local, nonperturbative formulations of chiral gauge theories.

hep-th↗
arXiv2025

Non-invertible bosonic chiral symmetry on the lattice

In this work we realize the 3 + 1 dimensional non-invertible ${\mathbb{Z}}_N$ chiral symmetry generator as an operator in a many body lattice Hilbert space. A crucial ingredient in our construction is the use of infinite dimensional $U(1)$ rotor site Hilbert spaces. Specifically, our Hilbert space is that of a $U(1)$ lattice gauge theory coupled to a charge $1$ scalar in the Villain formulation, which allows for direct access to monopoles and for a simple definition of a magnetic ${\mathbb{Z}}_N$ one-form symmetry $Z^{(1)}_m$ , at the lattice Hamiltonian level. We construct the generator of the ${\mathbb{Z}}_N$ chiral symmetry as as a unitary operator in the subspace of $Z^{(1)}_m$-invariant states, and show that it cannot be extended to the entire Hilbert space while preserving locality and unitarity. Using a lattice-level duality based on gauging $Z^{(1)}_m$, we find a dual description of this subspace, as the subspace of a charge $1/N$ gauge theory invariant under an electric one-form symmetry $Z^{(1)}_e$. We show that in this dual formulation, the chiral symmetry generator does extend unitarily to the entire Hilbert space, but has a mixed anomaly with the $Z^{(1)}_e$ symmetry.

cond-mat.str-el↗
arXiv2024

Stabilizer Scars

Quantum many-body scars are eigenstates in non-integrable isolated quantum systems that defy typical thermalization paradigms, violating the eigenstate thermalization hypothesis and quantum ergodicity. We identify exact analytic scar solutions in a 2+1 dimensional lattice gauge theory in a quasi-1d limit as zero-magic resource stabilizer states.

quant-ph↗
arXiv2024

A QCA for every SPT

In three dimensions, there is a nontrivial quantum cellular automaton (QCA) which disentangles the three-fermion Walker--Wang model, a model whose action depends on Stiefel--Whitney classes of the spacetime manifold. Here we present a conjectured generalization to higher dimensions. For an arbitrary symmetry protected topological phase of time reversal whose action depends on Stiefel--Whitney classes, we construct a corresponding QCA that we conjecture disentangles that phase. Some of our QCA are Clifford, and we relate these to a classification theorem of Clifford QCA. We identify Clifford QCA in $4m+1$ dimensions, for which we find a low-depth circuit description using non-Clifford gates but not with Clifford gates.

cond-mat.str-el↗
arXiv2023

Pumping Chirality in Three Dimensions

Using bosonization, which maps fermions coupled to a ${\mathbb{Z}}_2$ gauge field to a qubit system, we give a simple form for the non-trivial 3-fermion quantum cellular automaton (QCA) as a unitary operator realizing a phase depending on the framing of flux loops, building off work by Shirley et al. We relate this framing dependent phase to a pump of $8$ copies of a $p+ip$ state through the system. We give a resolution of an apparent paradox, namely that the pump is a shallow depth circuit (albeit with tails), while the QCA is nontrivial. We discuss also the pump of fewer copies of a $p+ip$ state, and describe its action on topologically degenerate ground states. One consequence of our results is that a pump of $n$ $p+ip$ states generated by a free Fermi evolution is a free fermion unitary characterized by a non-trivial winding number $n$ as a map from the third homotopy group of the Brilliouin Zone $3$-torus to that of $SU(N_ b)$, where $N_b$ is the number of bands. Using our simplified form of the QCA, we give higher dimensional generalizations that we conjecture are also nontrivial QCAs, and we discuss the relation to Chern-Simons theory.

cond-mat.str-el↗
arXiv2023

A no-go result for implementing chiral symmetries by locality-preserving unitaries in a 3 dimensional Hamiltonian lattice model of fermions

We argue that the chiral $U(1)_A$ symmetry of a Weyl fermion cannot be implemented by a shallow depth quantum circuit operation in a fermionic lattice Hamiltonian model with finite dimensional onsite Hilbert spaces. We also extend this result to discrete ${\mathbb{Z}}_{2N}$ subgroups of $U(1)_A$, in which case we show that for $N_f$ Weyl fermions of the same helicity, this group action cannot be implemented with shallow depth circuits when $N_f$ is not an integer multiple of $2N$.

cond-mat.str-el↗
arXiv2022

Entanglement transitions with free fermions

We use Majorana operators to study entanglement dynamics under random free fermion unitary evolution and projective measurements in one dimension. For certain choices of unitary evolution, namely those which swap neighboring Majorana operators, and measurements of neighboring Majorana bilinears, one can map the evolution to the statistical model of completely packed loops with crossings (CPLC) and study the corresponding phase diagram. We generalize this model using the language of fermionic Gaussian states to a general free fermion unitary evolution acting on neighboring Majorana operators, and numerically compute its phase diagram. We find that both the Goldstone and area law phases persist in this new phase diagram, but with a shifted phase boundary. One important qualitative aspect of the new phase boundary is that even for the case of commuting measurements, the Goldstone phase persists up to a finite non-zero measurement rate. This is in contrast with the CPLC, in which non-commuting measurements are necessary for realizing the Goldstone phase. We also numerically compute the correlation length critical exponent at the transition, which we find to be near to that of the CPLC, and give a tentative symmetry based explanation for some differences in the phase transition line between the CPLC and generalized models.

cond-mat.str-el↗
arXiv2022

Gravitational anomaly of 3+1 dimensional Z_2 toric code with fermionic charges and fermionic loop self-statistics

Quasiparticle excitations in $3+1$ dimensions can be either bosons or fermions. In this work, we introduce the notion of fermionic loop excitations in $3+1$ dimensional topological phases. Specifically, we construct a new many-body lattice invariant of gapped Hamiltonians, the loop self-statistics, that distinguishes two bosonic topological orders that both superficially resemble $3+1$ d ${\mathbb{Z}}_2$ gauge theory coupled to fermionic charged matter. The first has fermionic charges and bosonic ${\mathbb{Z}}_2$ gauge flux loops (FcBl) and is just the ordinary fermionic toric code. The second has fermionic charges and fermionic loops (FcFl), and, as we argue, can only exist at the boundary of a non-trivial 4+1d invertible bosonic phase, stable without any symmetries, i.e. it possesses a gravitational anomaly. We substantiate these claims by constructing an explicit exactly solvable $4+1$ d Walker-Wang model and computing the loop self-statistics in the fermionic ${\mathbb{Z}}_2$ gauge theory hosted at its boundary. We also show that the FcFl phase has the same gravitational anomaly as all-fermion quantum electrodynamics. Our results are in agreement with the recent classification of nondegenerate braided fusion 2-categories by Johnson-Freyd, and with the cobordism prediction of a non-trivial ${\mathbb{Z}}_2$ classified $4+1$ d invertible phase with action $S=\frac{1}{2} \int w_2 w_3$.

cond-mat.str-el↗
arXiv2022

How Dynamical Quantum Memories Forget

Motivated by recent work showing that a quantum error correcting code can be generated by hybrid dynamics of unitaries and measurements, we study the long time behavior of such systems. We demonstrate that even in the "mixed" phase, a maximally mixed initial density matrix is purified on a time scale equal to the Hilbert space dimension (i.e., exponential in system size), albeit with noisy dynamics at intermediate times which we connect to Dyson Brownian motion. In contrast, we show that free fermion systems -- i.e., ones where the unitaries are generated by quadratic Hamiltonians and the measurements are of fermion bilinears -- purify in a time quadratic in the system size. In particular, a volume law phase for the entanglement entropy cannot be sustained in a free fermion system.

quant-ph↗
arXiv2022

Bootstrapping Lieb-Schultz-Mattis anomalies

We incorporate the microscopic assumptions that lead to a certain generalization of the Lieb-Schultz-Mattis (LSM) theorem for one-dimensional spin chains into the conformal bootstrap. Our approach accounts for the "LSM anomaly" possessed by these spin chains through a combination of modular bootstrap and correlator bootstrap of symmetry defect operators. We thus obtain universal bounds on the local operator content of (1+1)$d$ conformal field theories (CFTs) that could describe translationally invariant lattice Hamiltonians with a $\mathbb Z_N\times \mathbb Z_N$ symmetry realized projectively at each site. We present bounds on local operators both with and without refinement by their global symmetry representations. Interestingly, we can obtain non-trivial bounds on charged operators when $N$ is odd, which turns out to be impossible with modular bootstrap alone. Our bounds exhibit distinctive kinks, some of which are approximately saturated by known theories and others that are unexplained. We discuss additional scenarios with the properties necessary for our bounds to apply, including certain multicritical points between (1+1)$d$ symmetry protected topological phases, where we argue that the anomaly studied in our bootstrap calculations should emerge.

cond-mat.str-el↗
arXiv2022

Nontrivial Quantum Cellular Automata in Higher Dimensions

We construct a three-dimensional quantum cellular automaton (QCA), an automorphism of the local operator algebra on a lattice of qubits, which disentangles the ground state of the Walker-Wang three fermion model. We show that if this QCA can be realized by a quantum circuit of constant depth, then there exists a two-dimensional commuting projector Hamiltonian which realizes the three fermion topological order which is widely believed not to be possible. We conjecture in accordance with this belief that this QCA is not a quantum circuit of constant depth, and we provide two further pieces of evidence to support the conjecture. We show that this QCA maps every local Pauli operator to a local Pauli operator, but is not a Clifford circuit of constant depth. Further, we show that if the three-dimensional QCA can be realized by a quantum circuit of constant depth, then there exists a two-dimensional QCA acting on fermionic degrees of freedom which cannot be realized by a quantum circuit of constant depth; i.e., we prove the existence of a nontrivial QCA in either three or two dimensions. The square of our three-dimensional QCA can be realized by a quantum circuit of constant depth, and this suggests the existence of a $\mathbb{Z}_2$ invariant of a QCA in higher dimensions, totally distinct from the classification by positive rationals (i.e., by one integer index for each prime) in one dimension. In an appendix, unrelated to the main body of this paper, we give a fermionic generalization of a result of Bravyi and Vyalyi on ground states of 2-local commuting Hamiltonians.

quant-ph↗
arXiv2020

Disentangling the Generalized Double Semion Model

We analyze the class of Generalized Double Semion (GDS) models in arbitrary dimensions from the point of view of lattice Hamiltonians. We show that on a $d$-dimensional spatial manifold $M$ the dual of the GDS is equivalent, up to constant depth local quantum circuits, to a group cohomology theory tensored with lower dimensional cohomology models that depend on the manifold $M$. We comment on the space-time topological quantum field theory (TQFT) interpretation of this result. We also investigate the GDS in the presence of time reversal symmetry, showing that it forms a non-trivial symmetry enriched toric code phase in odd spatial dimensions.

math-ph↗
arXiv2020

An Exactly Solvable Model for a $4+1D$ Beyond-Cohomology Symmetry Protected Topological Phase

We construct an exactly solvable commuting projector model for a $4+1$ dimensional ${\mathbb Z}_2$ symmetry-protected topological phase (SPT) which is outside the cohomology classification of SPTs. The model is described by a decorated domain wall construction, with "three-fermion" Walker-Wang phases on the domain walls. We describe the anomalous nature of the phase in several ways. One interesting feature is that, in contrast to in-cohomology phases, the effective ${\mathbb Z}_2$ symmetry on a $3+1$ dimensional boundary cannot be described by a quantum circuit and instead is a nontrivial quantum cellular automaton (QCA). A related property is that a codimension-two defect (for example, the termination of a ${\mathbb Z}_2$ domain wall at a trivial boundary) will carry nontrivial chiral central charge $4$ mod $8$. We also construct a gapped symmetric topologically-ordered boundary state for our model, which constitutes an anomalous symmetry enriched topological phase outside of the classification of arXiv:1602.00187, and define a corresponding anomaly indicator.

cond-mat.str-el↗
arXiv2019

A tensor network approach to 2D bosonization

We present a 2D bosonization duality using the language of tensor networks. Specifically, we construct a tensor network operator (TNO) that implements an exact 2D bosonization duality. The primary benefit of the TNO is that it allows for bosonization at the level of quantum states. Thus, we use the TNO to provide an explicit algorithm for bosonizing fermionic projected entangled pair states (fPEPs). A key step in the algorithm is to account for a choice of spin-structure, encoded in a set of bonds of the bosonized fPEPS. This enables our tensor network approach to bosonization to be applied to systems on arbitrary triangulations of orientable 2D manifolds.

cond-mat.str-el↗
arXiv2019

Disentangling interacting symmetry protected phases of fermions in two dimensions

We construct fixed point lattice models for group supercohomology symmetry protected topological (SPT) phases of fermions in 2+1D. A key feature of our approach is to construct finite depth circuits of local unitaries that explicitly build the ground states from a tensor product state. We then recover the classification of fermionic SPT phases, including the group structure under stacking, from the algebraic composition rules of these circuits. Furthermore, we show that the circuits are symmetric, implying that the group supercohomology phases can be many body localized. Our strategy involves first building an auxiliary bosonic model, and then fermionizing it using the duality of Chen, Kapustin, and Radicevic. One benefit of this approach is that it clearly disentangles the role of the algebraic group supercohomology data, which is used to build the auxiliary bosonic model, from that of the spin structure, which is combinatorially encoded in the lattice and enters only in the fermionization step. In particular this allows us to study our models on 2d spatial manifolds of any topology and to define a lattice-level procedure for ungauging fermion parity.

cond-mat.str-el↗