# P2: Effective spin models for deconfined quantum criticality in strongly interacting Ising-gauged Fermi systems

## Project Description

The study of unconventional quantum phase transitions, for which the critical properties differ fundamentally from well-known universality classes of thermal phase transitions such as in the O(n) models, is a central theme in modern quantum many-body physics. A specific scenario of such unconventional criticality refers to quantum phase transitions that separate two ground state phases with different symmetry breaking patterns, such as magnetic order and valence bond crystals in quantum spin systems. Within conventional Ginzburg-Landau theory, such a transition is expected to be generically discontinuous. However, within the so-called deconfined quantum criticality scenario [1], it has been proposed that such quantum phase transitions may be generically continuous with, e.g., an unconventionally large anomalous exponent η that results from the emergence of deconfined fractionalized excitations at the quantum critical point. In the past, various studies were performed on specifically designed quantum spin models, for which the stability of such deconfined quantum critical points was, indeed, suggested, e.g. in Ref. [2]. More recently, however, it has been proposed, that such unconventional quantum critical points may also be realized in specific correlated fermion models with competing interactions [3]. These recent findings result mainly from quantum Monte Carlo studies of correlated fermion models that interact with Ising-gauge fields residing on the links of the fermion lattice model. This construction allows for sign problem-free quantum Monte Carlo simulations [4,5]. It has been concluded, that these fermionic systems exhibit regimes of unconventional continuous quantum criticality with large anomalous exponents, as well as an enlarged emerging symmetry, which provides another hallmark for deconfined criticality [6].

Since the quantum critical points reside within the region of strong correlations in the Mott-insulating regime of the fermion systems, we propose to derive effective quantum spin models that describe these fermionic systems in the strong-coupling regime, thus allowing us to study the proposed unconventional quantum phase transitions using highly-efficient quantum Monte Carlo schemes that are available for quantum spin models [7]. However, in con-trast to conventional quantum spin models, these effective quantum spin systems will also include the quantum Ising degrees of freedom [8]. Separately efficient quantum Monte Carlo schemes that allow the simulation of system sizes two orders-of-magnitude larger than in the original fermionic description exist for both systems. Moreover, for the spin systems, there are also very efficient cluster update schemes available which reduce the critical slowing down of the simulations at quantum critical points which typically plagues local update schemes. Such cluster updates are not available for the algorithms for the original fermionic models. In this project, we will combine the cluster efficient update schemes for quantum spin models and quantum Ising models in order to design algorithms for these effective models of the proposed fermionic deconfined critical points. This will allow us to assess and evaluate the quantum critical properties of these unconventional quantum critical points in the relevant strong-coupling regime on much larger system sizes than accessible in the fermionic description. In a first step, we will use perturbation theory to derive the effective spin models, and then appropriately combine the algorithmic advances of the quantum Monte Carlo approaches to study the critical regime of these spin systems.

Furthermore, we will use tensor-network-based methods to examine the ground states and the quantum phase transitions of these effective spin models, profiting from the remarkable recent progress in using tensor-network-based methods to study quantum spin systems as well as gauge theories in two dimensions [9]. Such a complementary analysis will extend beyond the quantum Monte Carlo approach and allows us to obtain an understanding of the entanglement structure of the wave function that accounts for the expected fractionalization in the quantum critical parameter regime.

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[2] A. W. Sandvik, Phys. Rev. Lett. **98**, 227202 (2007)

[3] T. Sato, M. Hohenadler, and F. F. Assaad, Phys. Rev. Lett. **119**, 197203 (2017)

[4] F. F. Assaad and T. Grover, Phys. Rev. X **6**, 041049 (2016)

[5] S. Gazit, M. Randeria, and A. Vishwanath, Nature Phys. **13**, 484 (2017)

[6] A. Nahum, P. Serna, J. T. Chalker, M. Ortuno, and A. M. Somoza, Phys. Rev. Lett. **115**, 267203 (2015)

[7] F. Alet, S. Wessel, and M. Troyer, Phys. Rev. E **71**, 036706 (2005)

[8] A. W. Sandvik, Phys. Rev. E **68**, 056701 (2003)

[9] P. Corboz and F. Mila, Phys. Rev. Lett. **112**, 147203 (2014)