The active iTBS sessions will be delivered using the Magstim Rapid2 stimulator. The iTBS protocol consists of 3-pulse 50-Hz bursts given every 200 ms (at 5 Hz) for 2 s at 8-s intervals for 60 cycles. A 2-s train of iTBS will be repeated every 10 s for a total of 1800 pulses per session. The intensity of stimulation will be set at 80% resting motor threshold (RMT). The target is the left DLPFC with the coil centered at the MNI coordinate [-38, 44, 26] calculated from T1-weighted MRI. In the sham HD-tDCS stimulation, short continuous currents without neuromodulatory effects will be applied to mimic real-stimulation sensations. Specifically, sham stimulation will deliver 40-sec, 2 mA normal-like stimulation, followed by a tiny current pulse (110 μA over 15 ms) for impedance control taking place every 550 ms for the remaining time. The other procedure is the same as the active HD-tDCS stimulation.

In the sham iTBS condition, the patients receive the same iTBS regimen and exact positioning of the coil but stimulations will be delivered using a commercial identical looking figure 8 sham coil (Magstim D70 Air film sham coil) that can produce a similar sound and sensation. In the active HD-tDCS condition, stimulation will be applied by a battery-operated device (NeuroConn DC Stimulator Plus) via 5 carbon rubber electrodes (1 cm radius, high-definition 4 × 1 rings configuration). To target the left DLPFC, the central electrode (anode) will be placed over International 10-20 electrode position F3, with return peripheral electrodes at Fp1, Fz, C3 and F7. Stimulation will be applied at an intensity of 2 milliamp (mA), 8-sec fade in and 5-sec fade out, for 20 min. During each session, the subject has to perform a computerized working memory task (i.e., 2-back task). The two times daily sessions will be separated by at least 2 hours.

In the sham iTBS condition, patients receive the same iTBS regimen and exact positioning of the coil but stimulations will be delivered using a commercial identical looking figure 8 sham coil (Magstim D70 Air film sham coil) that can produce a similar sound and sensation. In the sham HD-tDCS condition, short continuous currents without neuromodulatory effects will be applied to mimic real-stimulation sensations. Specifically, sham stimulation will deliver 40-sec, 2 mA normal-like stimulation, followed by a tiny current pulse (110 μA over 15 ms) for impedance control taking place every 550 ms for the remaining time. The other procedure is the same as the active HD-tDCS stimulation.

The participants' depressive symptoms will be measured by Calgary Depression Rating Scale for Schizophrenia (CDSS). The changes in the CDSS score immediately after the 3-week intervention, as well as at one-month and three-month follow-ups, will be collected as the secondary outcome.

PSP is a clinician-administered rating scale for measuring the psychosocial functioning of the patients with schizophrenia spectrum disorder. The PSP scale measures psychosocial functioning within four domains: socially useful activities, personal and social relationships, self-care, and disturbing and aggressive behavior.The patient is rated from 1 to 6 on each item of the four domains. A higher score indicates greater psychosocial functioning in any of the four domains.

The final global score is defined according to a summary instruction table. This scale provides a single, overall rating from 1 to 100, where a higher score represents better personal and social function. The changes in the Personal and Social Performance scale (PSP) score immediately after the 3-week intervention, as well as at one-month and three-month follow-ups, will be collected as the secondary outcome.

Cognitive insight was measured by the Taiwanese version of the Beck Cognitive Insight Scale (BCIS), a self-reported instrument comprising 15 items.The Taiwanese BCIS is composed of 2 subscales including reflective attitude (9 items) and certain attitude (6 items). We obtained a R-C (reflective attitude minus certain attitude) index of the Taiwanese BCIS, representing the measurement of cognitive insight by subtracting the score of the certain attitude subscale from that of the reflective attitude subscale. Lower R-C index scores indicate poorer cognitive insight. The changes in the score of the Beck Cognitive Insight Scale (BCIS) immediately after the 3-week intervention, as well as at one-month and three-month follow-ups, will be collected as the secondary outcome.

The Chinese version of the Schizophrenia Quality of Life Scale Revision Four (SQOLR4) is a self-administered questionnaire of 33 items in two domains: psychosocial and vitality. All but four items are coded on a scale of 0-4 according to the frequency of occurrence during the previous 7 days (0 = always, 4 = never; the four exceptions are coded 0 = never, 4

= always). A higher score indicates higher health-related QoL. The changes in the score of Schizophrenia Quality of Life Scale Revision Four (SQOLR4) immediately after the 3-week intervention, as well as at one-month and three-month follow-ups, will be collected as the secondary outcome.

The Tower of London Drexel University Test 2nd Edition is a neuropsychological test for the assessment of executive functioning specifically to detect deficits in planning, which may occur due to a variety of medical and neuropsychiatric conditions. The changes in the results of the Tower of London Drexel University Test 2nd Edition immediately after the 3-week intervention will be collected as the secondary outcome.

The CTT, a culture-neutral version of the Trail Making Test, was selected to measure sustained visual attention. The CTT consists of two parts (CTT-1 and CTT-2). The CTT-1 requires participants to connect a series of numbered circles that are randomly printed on a sheet of paper. In the CTT-2, numbered circles of 1 to 25 are shown twice (printed in pink and in yellow) randomly on a sheet of paper. Participants are asked to connect the numbers from 1 to 25 alternating between the two colors. The changes in the results of the Color Trails Test (CTT) immediately after the 3-week intervention will be collected as the secondary outcome.

Stroop Color Word Test (SCWT) will be administered to measure selective attention and cognitive flexibility. SCWT is composed of three parts, each lasting for 45 seconds. The changes in the results of the Stroop Color Word Test (SCWT) immediately after the 3-week intervention will be collected as the secondary outcome.

HRV indices represent autonomic functioning. ECG electrodes will be placed on bilateral arms just below the elbows, with a ground electrode placed just above the right wrist bone. Lead I electrocardiogram of each patient will be taken for 5 min after sitting and having a rest for 20 min in a soundproof, dim-lighted room with thermostatic control.

The ECG signals will be acquired, stored, pre-processed according to the recommended procedures and processed by an HRV analyser. The time domain of HRV is obtained: Standard deviation of NN intervals (SDNN). Power spectrum of HRV is quantified into the standard frequency-domain measurements including low-frequency power (LF, 0.04-0.15 Hz), high-frequency power (HF, 0.15- 0.40 Hz). The changes in indices of heart rate variability (HRV) derived from R-R intervals of ECG signals immediately after the 3-week intervention will be collected as the secondary outcome.

In a dimly lit, electrically shielded room, patients' electroencephalogram (EEG) will be recorded. Resting-state EEG will be collected with eye opened (5min) and closed (5min). Signals will be amplified by using EEG amplifiers and stored for offline analyses. The changes in resting-state EEG features (power, coherence, and microstates within different frequency bands including delta, theta, alpha and beta) immediately after the 3-week intervention will be collected as the secondary outcome.

Functional near infrared spectroscopy (fNIRS) system is used to measure the relative changes in oxyhemoglobin (Hb) of the frontal regions. Resting-state fNIRS fo 180s and verbal fluency task (VFT)-based fNIRS for 160s will be collected in a quiet room. The changes in fNIRS features immediately after the 3-week intervention will be collected as the secondary outcome.

Magnetic Resonance Imaging (MRI) scans will be performed on the hybrid 3.0T PET-MR imaging unit (GE Healthcare, Waukesha, WI). MRI scans will be carried out using the same imaging protocol, including T1-weighted images and resting-state functional MRI (rs-fMRIs).The changes in resting-state fMRI features immediately after the 3-week intervention will be collected as the secondary outcome.